Adult stem cells have been widely investigated in bioengineering approaches for tissue repair therapy. We evaluated the clinical value and safety of the application of cultured bone marrow-derived allogenic mesenchymal stem cells (MSCs) for treating skin wounds in a canine model.Topical allogenic MSC transplantation can accelerate the closure of experimental full-thickness cutaneous wounds and attenuate local inflammation.n = 10; 3–6 years old; 7.2–13.1 kg) were studied.Adult healthy beagle dogs  and wound healing-related factors .Topical transplantation of MSCs results in paracrine effects on cellular proliferation and angiogenesis, as well as modulation of local mRNA expression of several factors related to cutaneous wound healing.Les cellules souches adultes ont été largement étudiées dans les approches de bio-ingénierie pour la thérapie de réparation tissulaire. Nous évaluons l'efficacité clinique et la sécurité de l'application de cellules souches mésenchymateuses allogéniques en culture dérivées de moelle osseuse (MSCs) pour le traitement de plaies cutanées dans un modèle canin.La transplantation de MSC allogénique topique peut accélérer la fermeture en toute épaisseur de plaies cutanées expérimentales et atténuer l'inflammation locale.n = 10; 3–6 ans; 7.2–13.1 kg) ont été étudiés.Des chiens beagles adultes sains  et des facteurs liés à la cicatrisation .La transplantation topique de MSCs résulte en des effets paracrines de prolifération cellulaire et d'angiogénèse ainsi qu'en une modulation de l'expression locale d'ARNm de plusieurs facteurs liés à la cicatrisation cutanée.Las células madre de adultos se han estudiado extensivamente en estrategias de bioingeniería para la reparación de tejidos. Hemos evaluado el valor clínico y la seguridad de la aplicación de células madre mesenquimales (MSCs) alogenéicas derivadas de la medula ósea para tratar las heridas de la piel en un modelo canino.El transplante tópico alogenéico de MSCs puede acelerar la cicatrización de heridas cutáneas experimentales de todo el grosor de la piel y disminuir la inflamación local.Se estudiaron perros de raza Beagle sanos adultos .Se crearon heridas de todo el grosor de la piel en el dorso de perros Beagle sanos, y se inyectaron MSCs alogenéicas intradérmicamente. El ritmo de reparación de la herida y el grado de producción de colágeno se analizaron histológicamente utilizando tinciones de hematoxilina-eosina y tricrómico. El grado de proliferación celular y la angiogenesis se evaluaron mediante inmunohistoquímica utilizando anticuerpos específicos para el antígeno de proliferación celular nuclear (PCNA), vimentina y α-actina de músculo liso. La expresión local de mRNA para interleuquina-2 (IL-2), interferón- γ (IFN- γ), factor básico de crecimiento de fibroblastos (BFGF) y metaloproteinasa de matriz-2 se evaluaron mediante RT-PCR.comparado con las heridas tratadas solo con el vehículo, las heridas tratadas con MSCs mostraron una reparación más rápida y un aumento en la producción de colágeno, proliferación celular y angiogenesis. Además, las heridas tratadas con MSCs mostraron una expresión aumentada de citoquinas proinflamatorias (IL-2 e IFN- γ) y factores relacionados con la reparación de la herida .El transplante tópico de MSCs resulta en efectos paracrinos en la proliferación celular y angiogenesis, así como en la modulación de la expresión de mRNA de diversos factores relacionados con la cicatrización de heridas.Adulte Stammzellen sind für ihre biotechnologische Verwendung bei der Wiederherstellung von Geweben bereits weit reichend erforscht. Wir haben die klinische Bedeutung und Sicherheit der Applikation von kultivierten allogenen mesenchymalen Stammzellen (MSCs) aus dem Knochenmark zur Behandlung von Hautwunden in einem caninen Modell evaluiert.Die topische allogene MSC Transplantation kann den Wundverschluss experimenteller Hautwunden (Brandwunden Grad III) beschleunigen und die lokale Entzündung mildern.n=10; 3-6 Jahre alt; 7,2-13,1kg) wurden in der Studie verwendet.Erwachsene gesunde Beagles  und Wundheilungsfaktoren .Die topische Transplantation von MSCs resultiert in parakrinen Effekten auf die zelluläre Proliferation und Angiogenese sowie die Modulierung lokaler mRNA Exprimierung verschiedener Faktoren, die mit der Wundheilung der Haut im Zusammenhang stehen.以成体干细胞的生物工程法用于组织修复治疗已被广泛研究。将骨髓培养得到的外源性间充质干细胞(MSCs)用于犬模型上治疗皮肤创伤,我们来评估其临床价值和安全性。局部外源性MSC移植能使试验的皮肤创口全层愈合加快,并使局部炎症减轻。n = 10;3–6 岁;7.2–13.1 kg)。研究成年健康比格犬和与创口愈合相关因子(碱性成纤维细胞生长因子和基质金属蛋白酶-2)表达减少。局部移植MSCs造成了对细胞增殖和血管生成的旁分泌作用,也同样引起几个与皮肤创伤愈合相关因子局部mRNA表达的调节。成熟した幹細胞は組織修復療法に対して生物工学的なアプローチで広く研究されている。著者らは培養した骨髄由来の同種異系の間葉系肝細胞(MSCs)を犬の皮膚創傷の治療に応用し、臨床的な価値と安全性を評価した。同種異系MSCの局所への移植は実験的な皮膚全層創傷の閉鎖と炎症の軽減を加速する可能性がある。成熟した健常ビーグル犬を用いた。健常ビーグルの背部に皮膚全層潰瘍を作成し、同種異系MSCを皮内に注射した。創傷閉鎖率とコラーゲン産生度をヘマトキシリン染色トリクローム染色を用いて組織学的に解析した。細胞増殖度と血管新生は核内増殖抗原−、ビメンチン−、α−平滑筋アクチン−特異的抗体を用いて免疫組織化学染色で評価した。局所のインターロイキン−2、インターフェロンγ、塩基性線維芽細胞増殖因子とマトリックスメタロプロテアーゼ2のmRNA発現レベルをRT-PCRで評価した。賦形剤のみで治療した創傷と比較し、MSCで治療した創傷はより早期に閉鎖し、コラーゲン合成の増加、細胞増殖や血管新生を示した。さらに、MSCで治療した創傷は炎症後サイトカイン(インターロイキン−2とインターフェロン−γ)と創傷治癒関連因子(塩基性線維芽細胞増殖因子とマトリックスメタロプロテアーゼ−2)の発現が減少していた。MSCの局所への移植により細胞増殖と血管新生に対してはパラクライン効果を示すのと同時に,皮膚創傷治癒に関連する様々な因子の局所mRNA発現を調節した。 Normal skin depends on the pool of adult stem cells, such as epidermal skin stem cells and bone marrow (BM) stem cells, for its renewal and maintenance.18Autologous MSC transplantation is time consuming from cell collection to application and requires the patient to have normal BM function. In contrast, allogenic MSC transplantation has the advantage of prompt preparation and can be applied independently of the health status of the patient. However, allogenic cell transplantation has the risk of graft rejection, and the failure of allogenic cell transplantation has also been reported.This study received ethical approval, and all procedures dealing with animal care, handling and sampling were approved by the Institutional Animal Care and Use Committee of Konkuk University .n = 10; 3–6 years old; 7.2–13.1 kg) were used for this study. After overnight fasting, the experimental dogs were premedicated with subcutaneous atropine sulfate  and intravenous cefazolin sodium . Under general anaesthesia using intramuscular (i.m.) tiletamine–zolazepam , 24 full-thickness circular wounds of 6 mm in diameter were created on the back of each dog using disposable dermal biopsy punches. The wounds were at least 2.5 cm apart . The four remaining wounds on each dog were left untreated to monitor the progress of healing. A total of 10 dogs received a set of these treatments, comprising five animals for histological analysis and the other five for cell tracing and molecular analysis , and wounds were harvested using 8 mm dermal biopsy punches. After skin harvesting, tramadol was administered. The samples were bisected along the widest line of the wound, then fixed and paraffin embedded, sectioned into 4-μm-thick slices and prepared for histology. All stained slides were examined under a light microscope , and digital photographs of the sections were taken using imaging software . The photographs were quantitatively analysed using an image analysis program . All measurements were performed by two investigators blinded to the treatment procedure.For comparison of the rate of re-epithelialization on day 7, tissue sections were stained with haematoxylin and eosin, and the epithelial gap (EG) was measured, which was defined as the distance between the advancing opposite edges of epithelial cell migration.7 MSC-treated wounds and PBS-treated wounds harvested on days 7, 14 and 21. The concentration of 1 × 107 was chosen because the 1 × 107 MSC-treated wounds showed the smallest EG and the greatest collagen deposition. Proliferating cell nuclear antigen (PCNA), vimentin and α-smooth muscle actin (α-SMA) were detected using specific monoclonal antibodies . The tissue sections were deparaffinized and hydrated, and sections for PCNA and vimentin immunostaining were pretreated for antigen retrieval using a pressure cooker in 10 mm citrate buffer. Endogenous peroxidases were quenched with 3% hydrogen peroxide for 15 min. Nonspecific antibody binding was blocked by incubation with a protein blocker (Dako). The sections were incubated with the primary antibodies or bovine serum albumin as a negative control for 30 min at 45°C, then incubated with horseradish peroxidase−conjugated anti-mouse and anti-rabbit immunoglobulin (Dako) for 15 min at 45°C, followed by a 5 min incubation with 3,3′-diaminobenzidine . Finally, the sections were counterstained with Gill's haematoxylin, mounted and examined under a light microscope. High-power images were used to quantify the number of DAB-positive cells or the area occupied by DAB-positive cells.To identify the effects of MSC transplantation on cellular proliferation and angiogenesis, an immunohistochemical evaluation was performed using the 1 × 10PCNA-positive cells in the epidermis was counted, and the number of these cells per 0.1 mm2 of epidermis was calculated. The number of proliferating cells in the granulation tissue was analysed in the same manner. To assess the cellularity of mesenchymal origin, the area occupied by vimentin-positive signals was measured, and the percentage of the vimentin-positive area in the granulation tissue was calculated. To analyse the degree of angiogenesis, the number of blood vessels demonstrating α-SMA-positive lumens in five random hpfs was counted. All measurements were repeated by two investigators, and the average value was used for analysis.The degree of epidermal cell proliferation was assessed by the number of proliferating cells in the newly formed epidermis. Three hpfs of the sections were selected randomly, the number of 7 MSC-treated wounds and PBS-treated wounds were harvested from the other five dogs using 8 mm biopsy punches under sedation. Tramadol was administered for pain relief. Harvested wound tissues were embedded in the optimum cutting temperature (OTC) compound, and slides of the 5-μm-thick cryosections were made. The slides were examined under a fluorescence microscope (BK51). Demonstration of transplanted MSCs was performed by detection of CFDA-SE prelabelled cells in tissue slides. Positive fluorescence signals having cell morphology were regarded as the retained MSCs.On days 3, 7 and 14 after cell transplantation, 1 × 107 MSC-treated wounds and PBS-treated wounds on days 3, 7 and 14. Harvested tissues were immediately frozen and, following homogenization, total RNA was extracted using TRIzol , according to the manufacturer's instructions. Then, RT-PCR was performed  using canine-specific primers to evaluate the expression of basic fibroblast growth factor (bFGF), keratinocyte growth factor-1 (KGF-1), interleukin-2 (IL-2), interferon-γ (IFN-γ) and matrix metalloproteinase-2 (MMP-2). The PCR conditions were as follows: 30 s at 94°C for denaturation, followed by 30 cycles of denaturation for 30 s at 94°C, annealing for 30 s at 58°C, and extension for 30 s at 72°C. The primer sequences are listed in GAPDH) after testing for stable expression in skin samples. The bands obtained from the PCR products were measured by densitometry using an image analysis program . The relative mRNA expression was calculated as the ratio of the density of each mRNA to the density of GAPDH mRNA.Local expression of wound healing-related factors was evaluated in 1 × 10spss version 12.0; SPSS Inc., IBM Corporation, Endicott, NY, USA). Nonparametric Kruskal–Wallis tests were used to compare the EG and collagen deposition among five different wounds. Mann–Whitney U-tests were used to assess the immunohistochemistry and RT-PCR results between the 1 × 107 MSC-treated and PBS-treated wounds. Spearman rank correlations were used to analyse the relationship between the number of transplanted MSCs and the EG or collagen deposition. A value of P < 0.05 was considered statistically significant.Statistical analyses were performed using the Statistical Package for the Social Sciences , although there was no significant difference among the MSC-treated wounds  were significantly smaller than those of the PBS-treated wounds  was significantly higher than that of the PBS-treated wounds  on day 7  on days 7 and 21  on day 7. The increase in cells of mesenchymal origin in the MSC-treated wounds was maintained until day 21 (The ratio of the vimentin-positive stained area to the granulation tissue per unit area was higher in the MSC-treated wounds (23.80%) than in the PBS-treated wounds . On day 14, the number of blood vessels declined and there was no significant difference in the vascularity of the MSC-treated wounds and the PBS-treated wounds  than the PBS-treated wounds (3.7 blood vessels/hpf; d wounds .7 MSC-treated wounds and the PBS-treated wounds was measured at the mRNA level by quantitative RT-PCR. On day 7, expression of bFGF was increased in the MSC-treated wounds but was decreased in the PBS-treated wounds. The expression of bFGF in the MSC-treated wounds was decreased on day 14. Meanwhile, expression of KGF-1 was higher in the PBS-treated wounds than in the MSC-treated wounds. Expression of KGF-1 also declined on day 14 postwounding in both types of wounds. Expression of MMP-2 demonstrated a slight increase over time, and the MSC-treated wounds showed increased MMP-2 expression from day 3 to 14. Expression of IL-2 and IFN-γ showed a similar change, increasing by day 7 and decreasing by day 14. The MSC-treated wounds maintained lower IL-2 and IFN-γ expression from day 3 to 14 postwounding relative to the PBS-treated wounds (The expression of wound healing-related factors in the 1 × 10d wounds .PCNA.In this study, MSC-treated wounds showed more rapid wound closure, and this was presumed to be due to MSCs promoting re-epithelialization. Faster epithelialization, following BM-derived stem cell transplantation for cutaneous wound healing, has been demonstrated in previous studies in other species.PCNA immunohistochemistry results. Other studies in other species have shown that MSCs promote fibroblast proliferation, resulting in intensified granulation tissue formation.2In the trichrome-stained sections, increased collagen deposition in MSC-treated wounds was demonstrated, which indicated the promotion of fibroblast proliferation, as shown by the vimentin and α-SMA, which is in agreement with previous studies.α-SMA is a marker of mature blood vessels.bFGF has a promoting effect on the growth and migration of endothelial cells,MMP-2, a member of the family of matrix metalloproteinases, plays a role in degradation of the extracellular matrix, consequently facilitating endothelial cell migration.bFGF in MSC-treated wounds was increased. At the same time, increased mRNA expression of MMP-2 in MSC-treated wounds was confirmed. Thus, upregulation of bFGF and MMP-2 in MSC-treated wounds can be attributed partly to increased blood vessel formation.Apart from fibroblast proliferation, the formation of new blood vessels is necessary to maintain granulation tissue.bFGF could also partly explain the promotion of fibroblast proliferation and increased production of collagen fibres. Although accumulation of larger collagen fibres increases the wound strength, excessive accumulation of collagen may induce hypertrophic scarring.MMP in MSC-treated wounds can be favourable for improving the quality of the healed wounds. Thus, MSCs seemed to influence the degree of wound maturation. Prevention of excessive wound contraction or hypertrophic scar formation is important for maintaining normal function of the skin and its cosmetic appearance. To obtain a therapeutic strategy involving less wound contraction, dermal substitutes or biomaterial have been applied.In addition, as mentioned above, fibroblasts play a major role in wound healing.IL-2 and IFN-γ, which implies that MSCs exert a suppressive effect on local inflammation. The IL-2 and IFN-γ levels do not represent the complete state of tissue inflammation. However, the results of the present study demonstrated that the MSCs exhibited a paracrine effect of decreasing postinjury inflammation. In particular, IFN-γ is known to impair re-epithelialization;IFN-γ expression may act favourably in the wound healing process.Damage of the skin inevitably induces local inflammation and recruits inflammatory cells, which act as sources of the cytokines that affect the wound healing process.in vitroin vivo.in vivo studies were inconsistent, and the persistence of the transplanted MSCs was very low in some studies.There has been no consensus explanation for the exact mechanism of action of MSCs on wound healing. One hypothesis involves transdifferentiation of MSCs into epithelial cells. The engrafted MSCs were seldom detected, and transdifferentiation of the MSCs into keratinocytes was not demonstrated in the present study, although several previous studies demonstrated transdifferentiation of MSCs both KGF, epidermal growth factor and transforming growth factor-β, in a hypoxic environment in vitro.Another hypothesis is that MSCs exert paracrine effects on wound healing, such as promoting fibroblast proliferation or exaggerating angiogenesis.6 cells/cm2 of wound was needed for a significant therapeutic effect. In contrast, the present study failed to determine the therapeutic dose of topical MSC transplantation. Confirmation of the therapeutic level of cell transplantation may be crucial in clinical cell transplantation therapy, because stem cell transplantation is associated with the underlying risks of the teratogenic and oncogenic processes; therefore, the appropriate number of cells needed for efficient wound healing must be investigated further, and the appropriate number of cells may differ according to the route of delivery.In the present study, there was no statistically significant difference in the results obtained between the different concentrations MSCs that were transplanted. This finding conflicts with a previous human clinical trial,One study reported decreased engraftment in the wound site of systemically transplanted MSCs without chemo-attractants; systemic allogenic cell transplantation is not free from the risk of a host reaction leading to transplantation failure.In conclusion, topical injection of MSCs resulted in more rapid re-epithelialization and increased collagen deposition and angiogenesis in a canine wound healing model. In addition, topically applied MSCs appeared to have a suppressive effect on local inflammation in the wounded skin, suggesting that MSCs may be applied in patients with not only large or nonhealing skin defects but also inflammatory and fibrotic skin diseases.


𝒩𝒫Σλ,δ of analytic and bi-univalent functions in the open unit disk 𝕌. For functions belonging to the class 𝒩𝒫Σλ,δ, we obtain estimates on the first two Taylor-Maclaurin coefficients |a2| and |a3|.We introduce and investigate an interesting subclass  When δ = 1, we get Sǎlǎgean's differential operator D1n = Dn, [For l-Oboudi  introducgiven by , then frz)=f(z),Dδ1f(z)=,Dδnf(z)=Dven by (1Dδnf(z)=z𝕌. In fact, the Koebe one-quarter theorem [𝕌 under every univalent function f ∈ 𝒮 contains a disk of radius 1/4. Thus every function f ∈ 𝒜 has an inverse f−1, which is defined byf−1 is given bySince univalent functions are one-to-one, they are invertible and the inverse functions need not be defined on the entire unit disk  theorem  ensures f ∈ 𝒜 is said to be bi-univalent in 𝕌 if both f and f−1 are univalent in 𝕌. Let Σ denote the class of bi-univalent functions in 𝕌 given by  if the following conditions are satisfied:β ∈ , λ ≥ 1, the function g is given byDδn is the Al-Oboudi differential operator.Let fined by , is said𝒩𝒫Σλ,δ reduces to the class denoted by 𝒩𝒫Σλ,δ which is the subclass of the functions f ∈ Σ satisfyingβ ∈ , λ ≥ 1, the function g is defined by  reduces to the class denoted by 𝒩𝒫Σλ,δ which is the subclass of the functions f ∈ Σ satisfyingβ ∈ , λ ≥ 1, the function g is defined by  reduces to the class denoted by 𝒩𝒫Σλ which is the subclass of the functions f ∈ Σ satisfyingβ ∈ , λ ≥ 1, the function g is defined by  reduces to the class denoted by 𝒩𝒫Σλ which is the subclass of the functions f ∈ Σ satisfyingβ ∈ , λ ≥ 1, and the function g is defined by  reduces to the class denoted by 𝒩𝒫Σ which is the subclass of the functions f ∈ Σ satisfyingβ ∈  and the function g is defined by  given by𝕌. Suppose also that the function φ(z) given by𝕌. If φ(z)≺h(z)  (z ∈ 𝕌), thenLet the function a2| and |a3| for functions in this new subclass 𝒩𝒫Σλ,δ of the function class Σ.The object of the present paper is to find estimates on the Taylor-Maclaurin coefficients |𝒩𝒫Σλ,δ given by In this section, we state and prove our general results involving the bi-univalent function class f(z) given by the Taylor-Maclaurin series expansion (Let the function xpansion  be in thp(z)≺h(z) and q(w)≺h(w) have the following Taylor-Maclaurin series expansions:It follows from  that(36β (z∈𝕌),eiβ((1−λ)p, q ∈ h(𝕌), according to p1, p2, q1, and q2, from the equalities (a2| as asserted in (From  and 42)42), we oso, from  and 43)(44)p1=−qrding to |pk|=|p(k)p1=−q1,2e2iβ given by the Taylor-Maclaurin series expansion (Let the function xpansion  be in thBy settingf(z) given by the Taylor-Maclaurin series expansion (Let the function xpansion  be in thBy settingf(z) given by the Taylor-Maclaurin series expansion  given by the Taylor-Maclaurin series expansion (Let the function xpansion  be in thBy settingf(z) given by the Taylor-Maclaurin series expansion  given by the Taylor-Maclaurin series expansion (Let the function xpansion  be in thBy settingf(z) given by the Taylor-Maclaurin series expansion  given by the Taylor-Maclaurin series expansion (Let the function xpansion  be in th


We characterize those metrics which are Douglasian or locally projectively flat by some equations. In particular, it shows that the known fact that β is always closed for those metrics in higher dimensions is no longer true in two-dimensional case. Further, we determine the local structures of two-dimensional -metrics which are Douglasian, and some families of examples are given for projectively flat classes with β being not closed.We study a class of two-dimensional Finsler metrics defined by a Riemannian metric  D) is an important projective invariants in projective Finsler geometry. A Finsler metric is called  Douglasian if D = 0 and  locally projectively flat if at every point, there are local coordinate systems in which geodesics are straight. It is known that a locally projectively flat Finsler metric can be characterized by D = 0 and vanishing Weyl curvature. As we know, the locally projectively flat class of Riemannian metrics is very limited, nothing but the class of constant sectional curvature (Beltrami theorem). However, the class of locally projectively flat Finsler metrics is very rich. It is known that locally projectively flat Finsler metrics must be Douglasian, but Douglas metrics are not necessarily locally projectively flat. Therefore, it is a natural problem to study and classify Finsler metrics which are Douglasian or locally projectively flat. For this problem, we can only investigate some special classes of Finsler metrics.Projective Finsler geometry studies equivalent Finsler metrics on the same manifold with the same geodesics as points . Douglasβ = bi(x)yi on a manifold M. Such metrics are called -metrics. An -metric can be expressed in the following form:ϕ(s) > 0 is a C∞ function on . It is known that F is a regular Finsler metric (defined on the whole TM − {0} and positive definite) for any  with ||β||α < bo if and only ifbo is a constant. If ϕ does not satisfy (F = αϕ(β/α) is singular.In this paper, we shall consider a special class of Finsler metrics defined by a Riemannian metric  only if        ϕα, β)-metrics. It is known that a Randers metric F = α + β is a Douglas metric if and only if β is closed , where c ≠ 0 is a constant, then we have β is not closed). Since every two-dimensional Riemann metric is locally conformally flat, we may put). Since  is equivproof in ), 59) h h109)α=α=(109)α=the formβ=beσ(x)[the formβ=beσ(x)[given by2τ~=(ξ2+η only if(ξ2+η2)-metric F on R2 byk, b are constants with k ≠ −1/b. Then F is a Douglas metric if and only if α and β can be locally defined by , s = β/α, be a two-dimensional -metric, where ϕ(0) = 1. Let ϕ(s) be given by   F = αϕ(sgiven by ). Then Ffined by  and (110 τ~=0 in . There a Next we construct some singular examples for F = αϕ(s), s = β/α, be two-dimensional -metric, where ϕ(0) = 1. Let ϕ(s) be given by , s = β/α, be two-dimensional -metric, where ϕ(0) = 1. Let ϕ(s) be given by  is a scalar function. We can obtain the local expression of u = u(x), v = v(x) are a pair of scalar functions such thatSince  β~=β by  and , , 120), agiven by  and 7),,α and β α expressed in  is defined in  is a scalar function. Then by the result in [u = u(x), v = v(x) are a pair of scalar functions such thatNow we express esult in , we haveu, v and the fact that u, v, and σ satisfy the PDEs , where B : = b2. Firstly by , , α and βgiven by .


We study the local convergence properties of inexact Newton-Gauss method for singular systems of equations. Unified estimates of radius of convergence balls for one kind of singular systems of equations with constant rank derivatives are obtained. Application to the Smale point estimate theory is provided and some important known results are extended and/or improved. In the case when m = n and f′(x) is invertible for each x ∈ D, Newton's method is a classical numerical method to find an approximation solution for such system. There are a lot of results that improve, generalize, or extend the convergence of Newton's method for solving  is not invertible, we choose its Moore-Penrose inverse f′(x)† instead of its classical inverse and call it Gauss-Newton's method given as follows:Consider the following system of nonlinear equations:f (x)=0,f (x)=0,A : ℝn → ℝm be a linear operator (or an m × n matrix). Recall that an operator (or n × m matrix) A† : ℝm → ℝn is the Moore-Penrose inverse of A, if it satisfies the following four equations:A* denotes the adjoint of A. Let kerA and im  A denote the kernel and image of A, respectively. For a subspace E of ℝn, we use ΠE to denote the projection onto E. Then, it is clear thatA is full row rank , AA† = Imℝ; when A is full column rank , A†A = Inℝ.Let x0 is an initial guess):rk satisfiesλk} is a sequence of forcing terms such that 0 ≤ λk < 1. In  → ℝ corresponding to  is defined byλ = θ = 0,  = 0, hλ,θ′(0) = −(1 + λ + θ) and hλ,θ′ is convex and strictly increasing. SetThroughout this paper, we assume that fined byhλ,θ(t)=−For the convergence analysis, we need the following useful lemma about elementary convex analysis.R > 0. If g :  → ℝ is continuously differentiable and convex, then g(t) − g(τt))/t ≤ (1 − τ)g′(t), for all t ∈  and τ ∈ ,(g(u) − g(τu))/u ≤ (g(v) − g(τv))/v, for all u, v ∈ .Let L average” was used. In the case when f′(ζ) is not surjective (see .Let γ-condition for operators in Banach spaces was introduced in −1 exists andf′(ζ) is full row rank, we have f′(ζ)f′(ζ)† = Imℝ andf′(x) is full row rank; that is, rankf′(x) = rankf′(ζ).Since xk} coincides with the sequence generated by inexact Newton-Gauss method (f′(ζ)†f(ζ)|| = ||Πkerf′(ζ)⊥|| = 1, thus, we have L-average Lipschitz condition . So, xk} converges to ζ as follows. Note that f(·) = f′(·)f′(·)†f(·); it follows that ζ is a zero of f.Let fined by1f^(x)=f′fore, by , we can ondition  on B = 0, f′(ζ) is full row rank, and f′ satisfies the L-average Lipschitz condition . Then, one hasSuppose that ondition  on B is full row rank, we have f′(ζ)f′(ζ)† = Imℝ. It follows thatInℝ − f′(ζ)†(f′(ζ) − f′(x)) is invertible for any x ∈ B. Thus, in view of the equality A†A = ΠkerA⊥, for any m × n matrix A, one has thatSince L-average condition  is defined by t0 ≤ 1/2. In fact, in view of the definition of r* given in (ϕ(t) ≤ 0. Consequently, we get thatRecall that the majorizing sequence {fined by  and the fined by . By Lemmote that  givestk} is dndition (||f′(xk)†ws from (||f′(xk)†ombining  that(85given in , for any


𝒜 be an alphabet with two elements. Considering a particular class of words (the phrases) over such an alphabet, we connect with the theory of numerical semigroups. We study the properties of the family of numerical semigroups which arise from this starting point.Let  𝒜 be a nonempty finite set called the alphabet. Elements of 𝒜 are called letters or symbols. A word is a sequence of letters, which can be finite or infinite. We denote by 𝒜*  the set of all finite  words over 𝒜. The sequence of zero letters is called the empty word and is denoted by ɛ. Any subset ℒ⊆𝒜* is called a language over 𝒜. The length of a word u is denoted by |u|. If u, v are words, we define their product or concatenation as the word uv. We say that a word u is a factor of a word v if there exist two words x, y such that v = xuy. If u is a factor of v with x = ɛ , then u is a prefix  of v.Let We have taken these definitions from . In this𝒜 = {a, ⌣}. We say that f ∈ 𝒜* is a phrase if it fulfills the following conditions: f,⌣ is not a prefix or suffix of f.⌣⌣ is not a factor of  We denote 𝒜ℱ = {f ∈ 𝒜* | f is a phrase}.Let us take If we consider that ⌣ represents a gap between two words, then we have a suitable justification for the above definition.𝒞 be a language over 𝒜 such that 𝒞⊆𝒜ℱ. We will denote by ℓ(𝒞) = {|c| | c ∈ 𝒞}. In this work we are going to deal with the structure of the set ℓ(𝒞) for particular choices of 𝒞. In fact, let {w1,…, wn}⊆𝒜* be a finite set of words such that ⌣ is not a factor of wi,  1 ≤ i ≤ n. Then 𝒞⊆𝒜ℱ is the language in which each phrase f is obtained as product of factors belonging to {w1,…, wn}∪{⌣}. Moreover, in order to achieve the results of this paper, we assume that ɛ ∈ 𝒞.Let aaaa, aaaaa}, then f1 = aaaaaaaa, f2 = aaaaaaaaa, and f3 = aaaaa⌣aaaa belong to 𝒞. However, f4 = aaaa⌣aaaaaa, f5 = aaa, and f6 = aa⌣a do not belong to 𝒞.If we take {ℕ be the set of nonnegative integers. A numerical semigroup is a subset S of  that is closed under addition, contains the zero element, and such that ℕ∖S is finite.Let ℓ) is a numerical semigroup. We will also see that there exist numerical semigroups that cannot be obtained by this procedure. This fact allows us to give the following definition.In 𝒜 be the alphabet given by the set {a, ⌣}. A numerical semigroup S is the set of lengths of a language of phrases (PL-semigroup for abbreviation) if there exists 𝒞 = 𝒞⊆𝒜ℱ such that S = ℓ(𝒞).Let S is a PL-semigroup if and only if x + y + 1 ∈ S for all x, y ∈ S∖{0}.The next aim of S be a numerical semigroup. Since ℕ∖S is a finite set, we can consider two notable invariants of S (see [Frobenius number of S is the maximum of ℕ∖S and is denoted by F(S). On the other hand, the genus of S is the cardinality of ℕ∖S and is denoted by g(S).Let f S (see ). On theFrobenius variety is a nonempty family 𝒱 of numerical semigroups that fulfills the following conditions:S, T ∈ 𝒱, then S∩T ∈ 𝒱,if S ∈ 𝒱 and S ≠ ℕ, then S ∪ {F(S)} ∈ 𝒱.if A 𝒮PL = {S | S is a PL-semigroup}. In 𝒮PL is a Frobenius variety. This fact, together with the results of [𝒮PL in a tree with root ℕ. Moreover, we will also characterize the sons of a vertex, in order to build recursively such a tree.Let us denote sults of , will almultiplicity of a numerical semigroup S, denoted by m(S), is the minimum of S∖{0}. We will study the set 𝒮PL(m) = {S ∈ 𝒮PL | m(S) = m} in S) | S ∈ 𝒮PL(m)} and {g(S) | S ∈ 𝒮PL(m)}. We will also see that the elements of 𝒮PL(m) can be ordered in a tree with root the numerical semigroup S = {0, m, →} (where the symbol → means that every integer greater than m belongs to S).The In We finish this introduction pointing out that this work admits different generalizations. Some of them are in working process and other ones have already been developed (see ).X is a nonempty subset of ℕ, we denote by 〈X〉 the submonoid of  generated by X; that is,X〉 is a numerical semigroup if and only if gcd{X} = 1 . On the other hand, every numerical semigroup S is finitely generated, and therefore there exists a finite subset X of S such that S = 〈X〉. In addition, if no proper subset of X generates S, then we say that X is a minimal system of generators of S. In [S has a unique (finite) minimal system of generators. The elements of such a system are called minimal generators of S.If 𝒜 be an alphabet and let {w1,…, wn}⊆𝒜* be a finite set of words. If 𝒞 = 𝒞⊆𝒜* is the language in which each word is obtained as product of factors belonging to {w1,…, wn}, then it is easy to see that ℓ(𝒞) is a submonoid of . In addition, if gcd{|w1|,…, |wn|} = 1, then ℓ(𝒞) is a numerical semigroup. Moreover, it is a simple exercise to show that we can get any numerical semigroup in this way.Let As we indicated in the introduction we are going to study the particular case in which we consider lengths of phrases. Consequently, we will focus our attention in a particular family of numerical semigroups.𝒜 = {a, ⌣} be an alphabet. If 𝒞 = 𝒞⊆𝒜ℱ, then ℓ(𝒞) is a numerical semigroup.Let ɛ ∈ 𝒞 and |ɛ| = 0, we have that 0 ∈ ℓ(𝒞).First of all, being that l1, l2 ∈ ℓ(𝒞), then l1 + l2 ∈ ℓ(𝒞). In effect, let f1, f2 ∈ 𝒞 such that |f1| = l1 and |f2| = l2. Then the concatenation f1f2 (of f1 and f2) is an element of 𝒞 with |f1f2| = l1 + l2.Now, let us see that, if f ∈ 𝒞 with |f| ≠ 0 . Since f⌣f ∈ 𝒞, we have that {|f|, 2|f| + 1}⊆ℓ(𝒞). By the previous step, we know that 𝒞 is closed under addition and, consequently, 〈|f|, 2|f| + 1〉⊆ℓ(𝒞). As gcd{|f|, 2|f| + 1} = 1, we have 〈|f|, 2|f| + 1〉 is a numerical semigroup. Therefore, ℕ∖ℓ(𝒞) is finite.Finally, let  We conclude that ℓ(𝒞) is a numerical semigroup.We proceed in three steps. 𝒜 = {a, ⌣}. As in the introduction, we say that a numerical semigroup S is a PL-semigroup if there exists 𝒞 = 𝒞⊆𝒜ℱ such that S = ℓ(𝒞). From S is a PL-semigroup and x ∈ S∖{0}, then 2x + 1 ∈ S. Consequently, there exist numerical semigroups which are not of this type. For example, S = 〈5,7, 9〉 is not a PL-semigroup because 2 · 5 + 1 = 11 ∉ S.From now on, unless another thing is stated, we take In the next result we give a characterization of PL-semigroups.S be a numerical semigroup. The following conditions are equivalent. S is a PL-semigroup.x, y ∈ S∖{0}, then x + y + 1 ∈ S.If Let S = ℓ) for some nonempty finite set {w1,…, wn}⊆𝒜*. If x, y ∈ S∖{0}, then there exist f, g ∈ 𝒞∖{ɛ} such that |f| = x and |g| = y. It is clear that f⌣g ∈ 𝒞 and |f⌣g| = x + y + 1. Therefore, x + y + 1 ∈ S.(1⇒2) By hypothesis, n1,…, np} be the minimal system of generators of S. Let us take the set {w1,…, wp | |wi| = ni,  1 ≤ i ≤ p}. Our aim is to show that if 𝒞 = 𝒞, then S = ℓ(𝒞).(2⇒1) Let {n1,…, np}⊆ℓ(𝒞), by applying S = 〈n1,…, np〉⊆ℓ(𝒞). Now, let l ∈ ℓ(𝒞). In order to prove that l ∈ S, we are going to use induction over l. If l = 0, then the result is trivially true. Let us assume that l > 0, and let f ∈ 𝒞 such that |f| = l. If ⌣ is not a factor of f, then the result follows immediately. In other case, there exist f1, f2 ∈ 𝒞∖{ɛ} such that f = f1⌣f2. By hypothesis of induction, |f1|, |f2| ∈ S. Thereby, l = |f| = |f1| + |f2| + 1 ∈ S.Since {numerical semigroup that admit a linear nonhomogeneous pattern. For a general study of this family of numerical semigroups see, for instance, [The previous theorem leads to the concept of nstance, , 7.S be a numerical semigroup with minimal system of generators given by {n1,…, np}. Following [s ∈ S, then we define the order of s (in S) bys).Let ollowing , if s ∈ X is the minimal system of generators of S, then every system of generators of S contains X. Consequently, the definition of ord does not depend on the considered system of generators; that is, it only depends on s and S.From [S be a numerical semigroup with minimal system of generators given by {n1,…, np} and let s ∈ S. i ∈ {1,…, p} and s − ni ∈ S, then ord(s − ni) ≤ ord(s) − 1.If s = α1n1 + ⋯+αpnp, with ord(s) = α1 + ⋯+αp and αi ≠ 0, then ord(s − ni) = ord(s) − 1.If Let s − ni = β1n1 + ⋯+βpnp, with β1 + ⋯+βp = ord(s − ni). Then s = β1n1 + ⋯+(βi + 1)ni + ⋯+βpnp, and thus ord(s − ni) + 1 = β1 + ⋯+(βi + 1)+⋯+βp ≤ ord(s).(1) Assume that s − ni = α1n1 + ⋯+(αi − 1)ni + ⋯+αpnp, we have ord(s − ni) ≥ α1 + ⋯+(αi − 1)+⋯+αp = ord(s) − 1. Thereby, ord(s) − 1 ≤ ord(s − ni). Now, by applying the previous item, we conclude that ord(s − ni) = ord(s) − 1.(2) Since In item (2) of the next proposition, it is shown a characterization of PL-semigroups in terms of minimal systems of generators. Thus, we can decide if a numerical semigroup is a PL-semigroup in an easier way.S be a numerical semigroup with minimal system of generators given by {n1,…, np}. The following conditions are equivalent. S is a PL-semigroup.i, j ∈ {1,…, p}, then ni + nj + 1 ∈ S.If s ∈ S∖{0, n1,…, np}, then s + 1 ∈ S.If s ∈ S∖{0}, then s + {0,…, ord(s) − 1}⊆S.If Let (1⇒2) It is an immediate consequence of s ∈ S∖{0, n1,…, np}, then it is clear that there exist i, j ∈ {1,…, p} and s′ ∈ S such that s = ni + nj + s′. Thus, s + 1 = (ni + nj + 1) + s′ ∈ S.(2⇒3) If s). If ord(s) = 1, then the result is trivially true. Now, let us assume that ord(s) ≥ 2 and that α1,…, αp are nonnegative integers such that s = α1n1 + ⋯+αpnp, ord(s) = α1 + ⋯+αp, and ai ≠ 0 for some i ∈ {1,…, p}. By s − ni) = ord(s) − 1. Then, by hypothesis of induction, we have that s − ni + {0,…, ord(s) − 2}⊆S. Therefore, s + {0,…, ord(s) − 2}⊆S. Moreover, (s − ni + ord(s) − 2) + ni + 1 ∈ S. Thereby, s + {0,…, ord(s) − 1}⊆S.(3⇒4) We reason by induction over ord ≥ 2. Thus, we get that x + y + 1 ∈ S. By applying S is a PL-semigroup.(4⇒1) If S = 〈4,5, 6〉; that is, let S be the numerical semigroup with minimal system of generators given by {4,5, 6}. It is obvious that 4 + 4 + 1 = 9, 4 + 5 + 1 = 10, 4 + 6 + 1 = 11, 5 + 5 + 1 = 11, 5 + 6 + 1 = 12, and 6 + 6 + 1 = 13 are elements of S. Therefore, by applying S is a PL-semigroup.Let The following result is straightforward to prove and appears in .S, T be numerical semigroups. S∩T is a numerical semigroup.S ≠ ℕ, then S ∪ {F(S)} is a numerical semigroup.If Let Having in mind the definition of Frobenius variety, which was given in the introduction, we get the next result.𝒮PL = {S | S is a PL-semigroup} is a Frobenius variety.The set ℕ ∈ 𝒮PL and, therefore, 𝒮PL is a nonempty set.First of all, let us observe that S, T ∈ 𝒮PL. In order to show that S∩T ∈ 𝒮PL, we are going to use x, y ∈ (S∩T)∖{0}, then x, y ∈ S∖{0} and x, y ∈ T∖{0}. Therefore, x + y + 1 ∈ S∩T. Consequently, S∩T ∈ 𝒮PL.Let S ∈ 𝒮PL such that S ≠ ℕ. By applying S ∪ F(S) ∈ 𝒮PL. Let x, y ∈ (S ∪ F(S))∖{0}. If x, y ∈ S, then x + y + 1 ∈ S⊆S ∪ F(S). On the other hand, if F(S)∈{x, y}, then x + y + 1 > F(S) and, thereby, x + y + 1 ∈ S⊆S ∪ F(S). We conclude that S ∪ F(S) ∈ 𝒮PL.Now, let graph  G is a pair , where V is a nonempty set and E is a subset of { ∈ V × V | v ≠ w}. The elements of V are called vertices of G and the elements of E are called edges of G. A path (of length n) connecting the vertices x and y of G is a sequence of different edges of the form , ,…,  such that v0 = x and vn = y.A G is a tree if there exist a vertex v* (known as the root of G) such that for every other vertex x of G, there exists a unique path connecting x and v*. If  is an edge of the tree, then we say that x is a son of y.We say that a graph 𝒮PL) in the following way:𝒮PL is the set of vertices of G(𝒮PL),S, S′) ∈ 𝒮PL × 𝒮PL is an edge of G(𝒮PL) if S′ = S ∪ {F(S)}.(We define the graph G(As a consequence of [𝒮PL) is a tree with root equal to ℕ. Moreover, the sons of a vertex S ∈ 𝒮PL are S∖{x1},…, S∖{xr}, where x1,…, xr are the minimal generators of S that are greater than F(S) and such that S∖{x1},…, S∖{xr} ∈ 𝒮PL.The graph G the minimal system of generators of S, then msg(S) = (S∖{0})∖((S∖{0})+(S∖{0}))  ∪ (msg(S)).Let (Necessity). If x − 1 ∉ {0} ∪ (ℕ∖S) ∪ (msg(S)), then x − 1 ∈ S∖({0} ∪ (msg(S))). Accordingly, there exist y, z ∈ S∖{0} such that x − 1 = y + z. In fact, it is clear that y, z ∈ S∖{x, 0}. Therefore, by applying S∖{x} is a PL-semigroup, we have x = y + z + 1 ∈ S∖{x}, which is a contradiction.(Sufficiency). Let y, z ∈ S∖{x, 0}. Since S is a PL-semigroup, by y + z + 1 ∈ S. As x − 1 ∈ {0} ∪ (ℕ∖S) ∪ (msg(S)), we deduce that y + z + 1 ≠ x. Thus y + z + 1 ∈ S∖{x}. By applying S∖{x} is a PL-semigroup.As a consequence of the previous proposition, we have the next result.S be a PL-semigroup such that S ≠ ℕ, and let x be a minimal generator of S greater than F(S). Then S∖{x} is a PL-semigroup if and only if x − 1 ∈ msg(S) ∪ {F(S)}.Let 𝒮PL) as is shown in the following example.By applying S = 〈4,6, 7,9〉 is a PL-semigroup with Frobenius number equal to 5. From S are S∖{6} = 〈4,7, 9,10〉 and S∖{7} = 〈4,6, 9,11〉.It is clear that 𝒮PL) such as it is shown in Let us observe that we can build recursively a tree, from the root, if we know the sons of each vertex. Therefore, we can build the tree G(𝒮PL), we are going to study the relation between the minimal generators of a numerical semigroup S and the minimal generators of S∖{x}, where x is a minimal generator of S that is greater than F(S). First of all, let us observe that if S is minimally generated by {m, m + 1,…, 2m − 1} , then S∖{m} = {0, m + 1, →} is minimally generated by {m + 1, m + 2,…, 2m + 1}. In other case we have the following result.In order to have an easier making of the tree G = {n1,…, np}. If m(S) = n1 < np and np > F(S), then S∖{np} = 〈n1,…, np−1, np + n1〉.Let i ∈ {2,…, p}. Since np > F(S) and n1 < ni, we have that np + ni − n1 ∈ S. Thus, np + ni − n1 = α1n1 + ⋯+αpnp for some α1,…, αp ∈ ℕ. Thereby, np + ni = (α1 + 1)n1 + ⋯+αpnp. By applying that {n1,…, np} is a minimal system of generators, we have that αp = 0. Therefore, np + ni ∈ 〈n1,…, np−1〉. In particular, 2np ∈ 〈n1,…, np−1〉.Let us take s ∈ S∖{np}. Then s ∈ S and, thus, there exist β1,…, βp ∈ ℕ such that s = β1n1 + ⋯+βpnp. Since 2np ∈ 〈n1,…, np−1〉, we can assume that βp ∈ {0,1}. On the one hand, if βp = 0, then s ∈ 〈n1,…, np−1〉. On the other hand, if βp = 1, then there exists i ∈ {1,…, p − 1} such that βi ≠ 0. If i = 1, then it is obvious that s ∈ 〈n1,…, np−1, np + n1〉. And if i ≠ 1, since np + ni ∈ 〈n1,…, np−1〉, we have that s ∈ 〈n1,…, np−1〉. In any case, we conclude that S∖{np} = 〈n1,…, np−1, np + n1〉.Now, let S be a numerical semigroup with msg(S) = {n1,…, np}. If m(S) = n1 < np and np > F(S), then Let S∖{np}) is {n1,…, np−1} or {n1,…, np−1, np + n1}. In addition, msg(S∖{np}) = {n1,…, np−1} if and only if np + n1 ∈ 〈n1,…, np−1〉.From np + n1 ∈ 〈n1,…, np−1〉, then there exist α1,…, αp−1 ∈ ℕ such that np + n1 = α1n1 + ⋯+αp−1np−1. Since {n1,…, np} is a minimal system of generators, we get that α1 = 0. Thus, there exists i ∈ {2,…, p − 1} such that αi ≠ 0. Consequently, np + n1 − ni ∈ S.If i ∈ {2,…, p − 1} such that np + n1 − ni ∈ S, then np + n1 − ni = β1n1 + ⋯+βpnp for some β1,…, βp ∈ ℕ. Thereby, np + n1 = β1n1 + ⋯+(βi + 1)ni + ⋯+βpnp. Since {n1,…, np} is a minimal system of generators, we have that βp = 0 and, therefore, np + n1 ∈ 〈n1,…, np−1〉.Conversely, if there exists We finish this section with an illustrative example about the above corollary.S be the numerical semigroup with msg(S) = {3,5, 7}. It is obvious that F(S) = 4. By S∖{5} = 〈3,7, 8〉. In addition, 8 − 7 = 1 ∉ S. Thereby, applying S∖{5}) = {3,7, 8}. On the other hand, applying S∖{7} = 〈3,5, 10〉. Finally, since 10 − 5 = 5 ∈ S, we conclude that msg(S∖{7}) = {3,5}.Let m be a positive integer. We will denote by Δ(m) = {0, m, →}. It is clear that Δ(m) is the greatest (with respect to set inclusion) PL-semigroup with multiplicity m. Our first aim in this section will be to show that there also exists the smallest (with respect to set inclusion) PL-semigroup with multiplicity m.Let As an immediate consequence of item (4) in S is a PL-semigroup, m ∈ S∖{0}, and k ∈ ℕ∖{0}, then km + i ∈ S for all i ∈ {0,…, k − 1}.If m ∈ ℕ∖{0}. Then the numerical semigroup generated by {(i + 1)m + i | i ∈ {0,…, m − 1}} is the smallest (with respect to set inclusion) PL-semigroup with multiplicity m.Let S = 〈m, 2m + 1,…, m2 + (m − 1)〉. From m has to contain S. In order to conclude the proof, we will show that S is a PL-semigroup. For this purpose, since {(i + 1)m + i | i ∈ {0,…, m − 1}} is a system of generators of S, it will be enough to check item (2) of i, j ∈ {0,…, m − 1}, then (i + 1)m + i + (j + 1)m + j + 1 ∈ S. We distinguish two cases. i + j + 1 ≤ m − 1, then (i + 1)m + i + (j + 1)m + j + 1 = (i + j + 2)m + (i + j + 1) ∈ S.If i + j + 1 ≥ m, then (i + 1)m + i + (j + 1)m + j + 1 = (i + j + 2)m + (i + j + 1) = (m + 1)m + (i + j − m + 2)m + (i + j − m + 1) ∈ S.If Let m) = 〈m, 2m + 1,…, m2 + (m − 1)〉 and by 𝒮PL(m) the set of all PL-semigroups with multiplicity equal to m. Let us recall that Δ(m) = max(𝒮PL(m)) and Θ(m) = min(𝒮PL(m)).We will denote by Θ is finite.The set S ∈ 𝒮PL(m), then Θ(m)⊆S⊆Δ(m). Since Δ(m) and Θ(m) are numerical semigroups, we have that Δ(m)∖Θ(m) is finite. Consequently, 𝒮PL(m) is also finite.If The previous result can be considered a particular case of [m). For that, several concepts and results are introduced.Now we are interested in computing the Frobenius number and the genus of Θ = {s ∈ S | s − m ∉ S}. It is clear  = {ω(0) = 0, ω(1),…, ω(m − 1)}, where ω(i) is, for each i ∈ {0,…, m − 1}, the least element of S that is congruent with i modulo m.If n S (see ) is Ap = max) − m,F(S) = (1/m)w) − ((m − 1)/2).g = 0, ω(1),…, ω(m − 1)}⊆ℕ such that, for each i ∈ {1,…, m − 1}, ω(i) is congruent with i modulo m. Let S be the numerical semigroup generated by X ∪ {m}. The following conditions are equivalent. S, m) = X.Ap(ω(i) + ω(j) ≥ ω((i + j)modm) for all i, j ∈ {1,…, m − 1}.Let m ∈ ℕ∖{0}, thenIf ω(i) = (i + 1)m + i is congruent with i modulo m for all i ∈ {1,…, m − 1}. Let us see now that, if i, j ∈ {1,…, m − 1}, then ω(i) + ω(j) ≥ ω((i + j)mod m). Indeed, ω(i) + ω(j) = (i + 1)m + i + (j + 1)m + j > (i + j + 1)m + (i + j)≥((i + j)mod m + 1)m + (i + j)mod m = ω((i + j)mod m). The proof follows from It is clear that m ∈ ℕ∖{0,1}, then m)) = m − 1,F(Δ(m)) = m − 1,g(Δ(m)) = m2 − 1,F(Θ(m)) = (m − 1)(m + 2)/2.g(Θ(If Items (1) and (2) are trivial. On the other hand, items (3) and (4) are immediate consequences of m) can be rewritten asm) is a numerical semigroup generated by an arithmetic sequence with first term m and common difference m + 1 .S is a numerical semigroup, then the cardinality of the minimal system of generators of S is called the embedding dimension of S and is denoted by e(S). It is well known (see [S) ≤ m(S). We say that a numerical semigroup S has maximal embedding dimension if e(S) = m(S). It is clear that {m, m + 1,…, 2m − 1} is the minimal system of generators of Δ(m). Therefore, Δ(m) is a numerical semigroup with maximal embedding dimension. Now we will show that Θ(m) has also maximal embedding dimension.If The next result is [S be a numerical semigroup with multiplicity m and assume that Ap = {ω(0) = 0, ω(1),…, ω(m − 1)}. Then S has maximal embedding dimension if and only if ω(i) + ω(j) > ω((i + j)mod m) for all i, j ∈ {1,…, m − 1}.Let ω(i) + ω(j) > ω((i + j)mod m) for all i, j ∈ {1,…, m − 1}. Therefore, by applying Let us observe that, in the proof of m ∈ ℕ∖{0}, then Θ(m) is a numerical semigroup with maximal embedding dimension.If m, 2m + 1,…, m2 + m − 1} is the minimal system of generators of Θ(m).As a consequence of this corollary, we have that {𝒮PL(m). Thus, we define the graph G(𝒮PL(m)) in the following way:𝒮PL(m) is the set of vertices of G(𝒮PL(m));S, S′) ∈ 𝒮PL(m) × 𝒮PL(m) is an edge of G(𝒮PL(m)) if S′ = S ∪ {F(S)}.) is a tree with root equal to Δ(m). Moreover, the sons of a vertex S ∈ 𝒮PL(m) are S∖{x1},…, S∖{xr}, with {x1,…, xr} = {x ∈ msg(S) | x ≠ m, x > F(S),  and S∖{x} ∈ 𝒮PL}.The graph G(𝒮PL(m)) such as is shown in the next example.By applying 𝒮PL(3)), that is, the tree of the PL-semigroups with multiplicity equal to 3.We are going to depict G is a tree, then the height of T is the maximum of the lengths of the paths that connect each vertex with the root. Let us observe that the height of G(𝒮PL(3)) is 3. In general, the height of the tree G(𝒮PL(m)) is equal toIf m.Let us study now the possible values of the Frobenius number and the genus for PL-semigroups with multiplicity m ∈ ℕ∖{0}, then S) | S ∈ 𝒮PL(m)} = {x ∈ ℕ | m − 1 ≤ x ≤ (m − 1)(m + 2)/2};{g(S) | S ∈ 𝒮PL(m)} = (Δ(m)∖Θ(m)) ∪ {m − 1}.{F(If m)∖Θ(m) = {x1 > x2 > ⋯>xρ}.Let us assume that Δ(m)∪{xi, →} ∈ 𝒮PL(m) and that g(Θ(m)∪{xi, →}) = (m − 1)(m + 2)/2 − i.(1) By m)∪{xi + 1, →} ∈ 𝒮PL(m) with Frobenius number equal to xi. Thus, (Δ(m)∖Θ(m)) ∪ {m − 1}⊆{F(S) | S ∈ 𝒮PL(m)}. For the other inclusion, let us take S ∈ 𝒮PL(m) such that S ≠ Δ(m). Then F(S) > m and, thereby, F(S) ∈ Δ(m). Since Θ(m)⊆S, we have F(S) ∉ Θ(S). Therefore, we conclude that F(S) ∈ Δ(m)∖Θ(S).(2) It is clear that Θ(S) | S ∈ 𝒮PL(3)} = {2,3, 4,5}. Since Δ(3) = 〈3,4, 5〉 and Θ(3) = 〈3,7, 11〉, we have that (Δ(3)∖Θ(3)) ∪ {2} = {2,4, 5,8}. Therefore, by applying S) | S ∈ 𝒮PL(3)} = {2,4, 5,8}.By 𝒮PL is not a numerical semigroup. For instance, ⋂n∈ℕ{0, n, →} = {0}. On the other hand, it is clear that the (finite or infinite) intersection of numerical semigroups is always a submonoid of .Let us observe that, in general, the infinite intersection of elements of M be a submonoid of . We will say that M is a 𝒮PL-monoid if it can be expressed like the intersection of elements of 𝒮PL.Let The next lemma has an immediate proof.𝒮PL-monoids is a 𝒮PL-monoid.The intersection of In view of this result, we can give the following definition.X be a subset of ℕ. The 𝒮PL-monoid generated by X (denoted by 𝒮PL(X)) is the intersection of all 𝒮PL-monoids containing X.Let M = 𝒮PL(X), then we will say that X is a 𝒮PL-system of generators of M. In addition, if no proper subset of X is a 𝒮PL-system of generators of M, then we will say that X is a minimal 𝒮PL-system of generators of M.If 𝒮PL is a Frobenius variety. Therefore, by applying [Let us recall that, by 𝒮PL-monoid has a unique minimal 𝒮PL-system of generators, which in addition is finite.Every The proof of the following lemma is also immediate.X⊆ℕ, then 𝒮PL(X) is the intersection of all PL-semigroups that contain X.If X is a nonempty subset of ℕ∖{0}, then 𝒮PL(X) is a PL-semigroup.If 𝒮PL(X) is a submonoid of . Therefore, in order to show that 𝒮PL(X) is a numerical semigroup, it will be enough to see that ℕ∖𝒮PL(X) is a finite set.We know that x ∈ X. If S is a PL-semigroup containing X, then . Since gcd{x, 2x + 1} = 1, we get that 〈x, 2x + 1〉 is a numerical semigroup and, thus, ℕ∖〈x, 2x + 1〉 is finite. Consequently, ℕ∖𝒮PL(X) is finite.Let  then by  we know 𝒮PL(X) is a PL-semigroup. Let x, y ∈ 𝒮PL(X)∖{0}. If S is a PL-semigroup containing X, from x, y ∈ S∖{0} and from x + y + 1 ∈ S. By applying again x + y + 1 ∈ 𝒮PL(X). Therefore, by applying 𝒮PL(X) is a PL-semigroup.Now, let us see that 𝒮 be the set of all numerical semigroups. It is clear that 𝒮 is a Frobenius variety. If we take X = {2}, then the intersection of all elements of 𝒮 containing {2} is exactly M = 〈2〉, which is not a numerical semigroup.Let us observe that, in general, The next result will be key for our last purpose in this section.𝒮PL = {𝒮PL(X) | X is  a  nonempty  finite  subset  of  ℕ∖{0}}. S ∈ 𝒮PL, then (by X of ℕ∖{0} such that S = 𝒮PL(X).By  then by  there ex𝒮PL is a Frobenius variety, by applying [Since M be a 𝒮PL-monoid and let x ∈ M. Then M∖{x} is a 𝒮PL-monoid if and only if x belongs to the minimal 𝒮PL-system of generators of M.Let As an immediate consequence of this proposition we have the following result.X be a nonempty subset of ℕ∖{0}. Then the minimal 𝒮PL-system of generators of 𝒮PL(X) is {x ∈ X | 𝒮PL(X)∖{x} is a PL-semigroup}.Let S = 〈3,7, 11〉 is a PL-semigroup. By applying S = 𝒮PL({3}) and {3} is the minimal 𝒮PL-system of generators of S.By 𝒮PL(X) when X is a fixed nonempty finite set of positive integers. Let us observe that by Now we want to describe n1,…, np be positive integers. We will denote by S the set {α1n1 + ⋯+αpnp + r | r, α1,…, αp ∈ ℕ, r < α1 + ⋯+αp} ∪ {0}. Our next purpose will be to show that S = 𝒮PL.Let S be a numerical semigroup, let s1,…, st ∈ S∖{0}, and let α1,…, αt ∈ ℕ. Then ord(α1s1 + ⋯+αtst) ≥ α1 + ⋯+αt.Let n1,…, np} be the minimal system of generators of S. Then, for each i ∈ {1,…, t}, there exist βi1,…, βip ∈ ℕ such that si = βi1n1 + ⋯+βipnp. Moreover, since si ≠ 0, we have that βi1 + ⋯+βip ≥ 1. Thus,Let {n1,…, np are positive integers, then S is the smallest (with respect to set inclusion) PL-semigroup containing {n1,…, np}.If x, y ∈ S∖{0}, then x + y ∈ S. In effect, we know that there exist α1,…, αp, β1,…, βp, r, r′ nonnegative integers such that x = α1n1 + ⋯+αpnp + r, y = β1n1 + ⋯+βpnp + r′, r < α1 + ⋯+αp, and r′ < β1 + …+βp. Therefore, x + y = (α1 + β1)n1 + ⋯+(αp + βn)np + r + r′ with r + r′ < (α1 + β1)+⋯+(αp + βn). Consequently, x + y ∈ S.Let us see that if ℕ∖S is finite. Since n1 = 1 · n1 + 0 · n2 + ⋯+0 · np + 0 and 2n1 + 1 = 2 · n1 + 0 · n2 + ⋯+0 · np + 1, we have that n1, 2n1 ∈ S. By applying the first step, we get that 〈n1, 2n1〉⊆S. Using the same reasoning as we did in the proof of Let us see that S is a numerical semigroup. Let us see now that S is a PL-semigroup. In order to do that, it is enough ∖{0}, then x + y + 1 ∈ S. Indeed, arguing as in the first step, we have that x + y + 1 = (α1 + β1)n1 + ⋯+(αp + βn)np + r + r′ + 1 with r + r′ + 1 < (α1 + β1)+⋯+(αp + βn). Therefore, x + y + 1 ∈ S.From the previous steps, we know that nough by  to show n1,…, np}⊆S.Following the proof of the second step, it is clear that {S is the smallest PL-semigroup that contains {n1,…, np}. In fact, we will show that if T is PL-semigroup containing {n1,…, np}, then S⊆T. Thus, let x ∈ S∖{0}. Then there exist α1,…, αp, r ∈ ℕ such that x = α1n1 + ⋯+αpnp + r with r < α1 + ⋯+αp. Since {n1,…, np}⊆T, by α1n1 + ⋯+αpnp + {0,…, ord(α1n1 + ⋯+αpnp) − 1}⊆T. By applying r < ord(α1n1 + ⋯+αpnp) and, therefore, x ∈ T.Finally, let us see that We divide the proof into five steps. In this way, we have proved the statement.The next result is an immediate consequence of the previous theorem.n1,…, np are positive integers, then 𝒮PL = S.If We finish this section with an example that illustrates its content.S = {0,4, 7,8, 9,11, →} = 〈4,7, 9〉. Therefore, 𝒮PL = 〈4,7, 9〉.It is clear that


Using the variational approach, we prove the existence and uniqueness of variational solutions to such system. Moreover, we prove that this variational solution generates a random dynamical system. The main results are applied to a general type of nonlinear SPDEs and the stochastic generalized p-Laplacian equation.This paper deals with the following type of stochastic partial differential equations (SPDEs) perturbed by an infinite dimensional fractional Brownian motion with a suitable volatility coefficient Φ:  H) of classical Brownian motion. It becomes the standard Brownian motion when H equals to 1/2. However, it was proved in  → ℋ such thatϕ = H(2H − 1) | t − s|H−22. From  → L2 such that the function Φ(·)u is contained in Lp for each u ∈ ℋ and ||Φ||Lp) < ∞. Then, we define the stochastic integral asL2. Indeed, it, from , A is (ℬ(V) ⊗ ℱ)-measurable, and Φ :  → L2 such that the function Φ(·)u is contained in Lp for each u ∈ ℋ and ||Φ||Lp) < ∞. Furthermore, we impose some conditions on A and Φ as follows.H1 hemicontinuity): for all v1, v2, v ∈ V, the map( is continuous on ℝ.H2 weak monotonicity): there exists c1 ∈ ℝ such that for all v1, v2 ∈ V(H3 coercivity): there exist c2 > 0, c3 ∈ ℝ, α > 1, and an (ℱt)-adapted process f ∈ L1 such that for all v ∈ V,  t ∈ (H4 boundedness): there exist c4 ≥ 0 and an (ℱt)-adapted process g ∈ Lα/(α−1) such that for all v ∈ V,  t ∈  = PNΦ(t) for all t ∈ .such that for the corresponding orthogonal projection Let H1)–(H4) are the standard monotonicity and coercivity conditions for SPDEs (cf.  is called a variational solution to ∩L2 andt ∈ , ℙ-a.s.A continuous ution to  if X ∈ LNow we claim and prove the main results.H1)–(H5) hold. For any given x ∈ L2, there exists a unique variational solution to  = Y(t) + Zt(ω), where Y(0) = x ∈ L2. First, we show the existence and uniqueness of solutions to (H1 to H4. Since Zt(ω) is an ℋ-valued Gaussian process, it is obvious to check the hemicontinuity and weak monotonicity for A. For the coercivity, we also have thatH3, the 2nd inequality holds using that −||v+Zt(ω)||Vα ≤ −2α1−||v||Vα + ||Zt(ω)||Vα, and the 3rd inequality holds using Young's inequality with a small value ε > 0. Next, by H4, we estimateε to be sufficiently small such that Zt(ω)||Vα and ||Zt(ω)||ℋα are all in L1, and gα/(α−1) ∈ L1 holds due to g ∈ Lα/(α−1). Thus, Let us define new PDEY(t)=Y(0)ave that2V∗〈A~ω(testimate||A(v+Zt(H4 we getg ∈ Lα/(α−1) and ||Zt(ω)||Vα ∈ L1. Thus we prove the boundedness of X(t) = Y(t) + Zt(ω), we haveIndeed, by quently,  has a unφ in such a way that φ is defined by the solution of the SPDE at time t, for a noise path ω, with initial point x.The next theorem shows that the unique solution of  generateX of  of  defines φ = x. Let us check then the cocycle property: for t, τ ∈ ℝ+ and x ∈ ℋ, we haver = s − τ leads toφ : ℝ+ × Ω × ℋ → ℋ. Note that the maps t → φ and x → φ are continuous, thus we only need to prove the measurability of ω → φ. By the proof of the existence and uniqueness of solutions to  is the weak limit of a subsequence of the Galerkin approximations φn in Lα. Let ξk ∈ C0∞(ℝ) be a Dirac sequence with supp(ξk)⊆Bk1/(0), where Bk1/(0) is an open ball of radius 1/k centered at the point 0. Then )(t) is well defined for k large enough. For each such k ∈ ℕ and h ∈ ℋ we haven → ∞. Since ω → φn ∈ Lα is measurable, so is ω → , h〉ℋ)(t). Moreover, it follows from , h〉ℋ)(t) is measurable. On the other hand, t → φ is continuous in ℋ. Therefore, , h〉ℋ)(t)→〈φ, h〉ℋ(t) and the measurability of ω → , h〉ℋ)(t) imply the measurability of ω → 〈φ, h〉ℋ(t). Since this is true for all h ∈ ℋ, we get the measurability of ω → φ. Consequently, we complete the proof that φ defines a continuous RDS.Trivially ution to , we gettions to , we know we have–(H4) hold for A(X(t)) = −X(t) | X(t)|p−2 with α : = p. Thus, by the above results, for suitable Φ(t) the SPDE (L2(Λ);, moreover, this solution defines a continuous RDS.According to [the SPDE  has a unp < ∞, Λ ⊂ ℝd, Λ is open. Denote by H0p1, the completion of C0∞(Λ) with respect to the norm defined as:d → ℝd is monotone, continuous and that for some strictly positive constants a1, a2u ∈ ℝd. Then we consider the following Gelfand tripleAgain let 2 ≤ H1)–(H4) hold for A(X(t)) = div(Ψ(∇X(t)) − X(t) | X(t)|p−2) with α : = p. Thus, by the above results, for suitable Φ(t), the SPDE (L2(Λ); moreover, this solution defines a continuous RDS.According to [the SPDE  has a unp-Laplacian equation. It is useful to note that our results for SPDEs in a Hilbert space can reduce to known results for a standard infinite dimensional Wiener process if the Hurst parameter H = 1/2 though the techniques of proof are different. Furthermore, the conditions (H1)–(H4) can be replaced by some much weaker assumptions  according to some recent results in [Using the variational approach we have studied a general type of fBm-driven nonlinear SPDEs with a suitable volatility coefficient in Hilbert space. We proved the existence and uniqueness of variational solutions to such system under some monotonicity and coercivity conditions. We further proved that this variational solution generates a random dynamical system. Finally, we applied the main results to two types of SPDEs including the stochastic generalized sults in . When th


Rm to Rn and any continuous function, which has limit at infinite place, from limitless close subset of Rm to Rn. This extends the nonlinear approximation ability of traditional BP neural network and RBF neural network.The neural network with two weights is constructed and its approximation ability to any continuous functions is proved. For this neural network, the activation function is not confined to the odd functions. We prove that it can limitlessly approach any continuous function from limited close subset of  So far, many neural networks models have been presented and applied in pattern recognition, automatic control, signal processing, and aided decision. Among these models, BP (feedforward) neural networks and RBF  neural networks are widely used because of their nonlinear ability of approximation to any continuous function. Up to now, these two classes of neural networks have successfully been applied to approach any nonlinear continuous function defined on bounded close subsets –7. Howevy is the output of neurons, ϕ is the activation function, θ is the threshold value, ωj is the direction weight value, zj is the core weight value, xj is the input, and p and s are two parameters.Wang et al.  presentezj = 0, p = 0, and s = 1, then  with one hidden layer Nn → R defined byi = 1,2,…, nSuppose In this paper, one is concerned with the neural networks with two weights and a single hidden layer:s and p, the neural network with two weights can not only approximate any continuous functions defined on a bounded close subset, namely, it has the same approximation ability as BP neural network in  to denote the space of continuous real-valued functions defined on , it is equalled by the supremum norm ||·||. Let f be a real-valued function defined on . η, the modulus of continuity of f, is defined as η = suph≤δ0<||f(·+h) − f(·)||. Then we have limδ→0η = 0, and for any real number λ ≥ 0,  η ≤ (1 + λ)η.We use the following notations: the symbols f is called Lipschitz α (0 < α ≤ 1) continuous and is written as f ∈ LipC(f)α, if there exists a constant C(f) such that η ≤ C(f)δα. C(h) denotes the positive constant dependents only on h and its value may be different at different occurrence.The modulus of continuity is usually considered as the measure of the smoothness of function and the approximation error in approximation theory. The function Our main result is as follows.ϕ is a bounded, strictly monotonously increasing, and odd function defined on R, f ∈ C, and n ∈ N. Then there exists a feedforward neural network (BP) with one hidden layer Nn → R defined byi = 1,2,…, n,Suppose a, b] into n equal intervals; each has length of (b − a)/n and let a = x0 < x1 < ⋯<xn = b. For i = 1,2,…, n, we set ci = (1/2m)(f(xi) − f(xi−1)), dn = ϕ−1((m/2n) − m), m = supx∈R|ϕ(x)|, θi = dn − ((dn − dn*)n/(b − a))[a + (i − 1)((b − a)/n)], dn* = ϕ−1(m − (m/2n)),  ωi = (dn − dn*)n/(b − a).Divide , given n, we assume that x ∈  and note thatx* ∈  and choose f, so that Nn(xi*) = f(xi*). Let Gi(xi*) = ϕ(ωix + θi); it follows thatx > xi*; (ii) x ≤ xi*.For any Case (i). When x > xi*, we have, for i = 1,2,…, n,Case (ii). When x ≤ xi*, we have, for i = 1,2,…, n,2n.From  and 13)Case (ii)n.From (|Gi(x)−GiThe activation functions are assumed to be odd functions in , while iϕ is a bounded, strictly monotonously increasing, and odd function defined on R, f ∈ C, and n ∈ N, z ∉ . Then there exists a neural network with two weights and one hidden layer Nn :  → R defined byp < s, p < 0 such thatci is defined in lis is equal to ωi in Suppose H : x → u, where u = (x−z)s/|x−z|p−s, x ∈ . Since z ∉ , thus H is a continuous transformation. Hence f = U is also a bounded close set; moreover, 0 ∉ U.We make fractional transformation H− : u → x, where x = z + us1/|u|s − p)/sp = f(x) = f(z + us1/|u|s − p)/sp is a continuous real-valued function defined on bounded close set U. For g(u), from Nn :  → R defined byThere exists an inverse transformation for fractional transformation p < s, p < 0, the network with two weights is not BP and RBF neural network; thus, n → ∞, from When ϕ is a bounded, strictly monotonously increasing, and odd function defined on R; D is an unbounded subset of R  andf ∈ C(D), z ∉ D, limx→∞f(x) = b(constant) and then there exists a neural network with two weights and one hidden layer:p < s, p < 0 such thatSuppose H : x → u, where u = (x−z)s/|x−z|p−s, x ∈ D. Since z ∉ D, thus H is a contimuous transformation on D and as x → ∞, u → 0. Let H(∞) = 0; H(x) is a continuous function on D ∪ {∞}. We prove that H(D ∪ {∞}) = H(D)∪{0} is a close set. If v is an arbitrary accumulation point of H(D)∪{0}, then v = 0 or v is an accumulation point of H(D). When v = 0, z ∈ H(D)∪{0}. When v is an accumulation point of H(D), then there exists a sequence uk ∈ H(D) and another sequence xk ∈ D, such that xk → uk,  uk → v ≠ 0  (k → ∞). From fractional transformation, there exists uk → v ≠ 0  (k → ∞) and inverse transformation H−1 : x = z + us1/|u|s − p)/sp = H(D)∪{0} is a close set. Letg(u) is a continuous function on H(D). Let g(0) = b = limx→∞f(x); it is easy to prove that g(u) is a continuous function on H(D)∪{0}. For a continuous function g(u) defined on H(D)∪{0}, by means of i=1nciϕ(lis((x−z)s/|(x−z)|s)|x−z|p − θi) such thatWe make fractional transformation p < s,  p < 0, the network with two weights is not BP and RBF neural network; thus, n → ∞, from When In this paper, we construct a new BP neural network and prove that the network has the approximation ability to any nonlinear continuous function. In our result, the threshold values and direction weight values in  are diff


We prove that the equation has infinitely many weak solutions by using the variant fountain theorem due to Zou (2001) and f, g do not need to satisfy the (P.S) or (P.S*) condition.We study the following  Moreover, limu→0g/|u|p−1 = 0 uniformly for  x ∈ Ω.There exists (F4)Assume that one of the following conditions hold:(1)u|→∞|g/|u|p−2u = 0 uniformly for  x ∈ Ω;lim(2)u|→∞|g/|u|p−2u = −∞ uniformly for  x ∈ Ω; furthermore, f/|u|p−2u and g/|u|p−2u are decreasing in  u  for  u  is large enough;lim(3)u|→∞|g/|u|p−2u = ∞ uniformly for  x ∈ Ω; g/|u|p−2u is increasing in  u  for  u  is large enough; moreover, there exists α > max{σ, δ} such thatlimwhere  G = ∫0ugdt.In this paper, we study the following  p = 2.The above conditions were given in Zou  for the r < p < s < p*.A simple example which satisfies (F1)–(F4) isSobolev spaceEquation  is posedu ∈ Wp1,(Ω), where  F = ∫0ufdt  and  ds  is the measure on the boundary. It is easy to see that Φ ∈ C1, ℝ) andu, v ∈ Wp1,(Ω). It is well-known that the weak solution of .The corresponding energy functional of  is definμ(x) ≡ 1.Under condition , it is eη = 0) and obtained the existence of infinitely many weak solutions. Moreover, the existence of three solutions for  was researched in , satisfies(A1)λ maps bounded sets to bounded sets uniformly for  λ ∈ ; furthermore, Φλ(−u) = Φλ(u) for all ∈ × E.Φ(A2)B(u) ≥ 0 for all  u ∈ E; B(u) → ∞ as ||u|| → ∞  on any finite dimensional subspace of  E.(A3)ρk > rk > 0 such thatThere exists for all  λ ∈ ,The λn → 1, u(λn) ∈ Yn, such thatThen there exist u(λn)} has a convergent subsequence for every  k, then  Φ1  has infinitely many nontrivial critical points {uk} ∈ E∖{0} satisfying Φ1(uk) → 0− as k → ∞.Particularly, if {u(λn)}  is a  (P.S*)  sequence.Obviously, the sequence  {E = Wp1,(Ω), E is a reflexive and separable Banach space; then there are  ej ∈ E  and ej* ∈ E* such thatFor our working space Xj : = span{ej}; then  Yk, Zk  can be defined as that in the beginning of λ : E → ℝ defined byWe write  B(u) ≥ 0  for all  u ∈ E;  B(u) → ∞  as  ||u|| → ∞  on any finite dimensional subspace of  E;  Φλ(−u) = Φλ(u)  for all  λ ∈ , u ∈ E.  We need the following lemmas.Then  q < p*, then one hasIf  1 ≤ First, we check the condition of Assume (F1)–(F3); then (A1)–(A3) hold.n > k > 2; we assume that σ ≤ δ  and define(A1) and (A2) are obvious. Let u ∈ Zk. Note that q < p*; there exists a constant c > 0 such that Sobolev trace imbedding inequality, we haveK  such that, for all  η < Λ*,ɛ > 0, there exists  cɛ  such thatp < q; we may choose ɛ > 0 and R > 0 sufficiently small thatu ∈ Wp1,(Ω)  with  ||u|| ≤ R. So we haveu ∈ Zk  with ||u|| ≤ R. Choosingβk(σ) → 0,  βk(δ) → 0  as  k → ∞, it follows that  ρk → 0  as  k → ∞, so there exists  k0  such that ρk ≤ R when k ≥ k0. Thus, for k ≥ k0, u ∈ Zk, and ||u|| = ρk, we have Φλ(u) ≥ ρkp/4p > 0; then ak(λ) ≥ 0 for all λ ∈ .Observe that|u|σ≤βk < 0  for all  λ ∈ .On the other hand, if u ∈ Zk with ||u|| ≤ ρk, k ≥ k0, we see thatdk(λ) → 0 as k → ∞. Thus, (A3) holds.Furthermore, if By λn → 1 and u(λn) ∈ Yn such thatThere exist u(λn)} is bounded.In order to complete our proof of u(λn)} is bounded in  Wp1,(Ω).  {λn′|Yn(u(λn)) = 0, thenSince Φη < Λ such that 1 − ηK > 0. If, up to a subsequence, ||u(λn)|| → ∞ as n → ∞, then, by (F2),n is large enough. Obviously, it is a condition if (F4)(1) holds.We can choose 0 < Λ < Λ* and if wn = u(λn)/||u(λn)||; then, up to a subsequence,w ≠ 0 in E and limu|→∞|g/|u|p−2u = −∞ in (F4)(2), then, for n is large enough, by Fatou's Lemma, we have thatu|→∞|g/|u|p−2u = ∞ in (F4)(3). Thus, w = 0.Otherwise, we set tn ∈  such thatc > 0  large enough, and n is large enough, we have thatn→∞Φλn(tnu(λn)) → ∞. Obviously,Let It follows thatp)fu − F and (1/p)gu − G are decreasing in  u  for u is large enough. Therefore,s > 0 and u ∈ ℝ; it is a contradiction.If (F4)(2) holds, we have that (1/If (F4)(3) holds, then we have thatu(λn), for n is large enough, since α > max{δ, σ}, we have thatΩ((1/p)g)u(λn) − G))dx is bounded, which contradicts } is bounded.By the above arguments, we have that {c > 0 such thats < t or t < s < 0,  ∀ x ∈ Ω, where H = (1/p)ft − F and In fact, our result still holds if we consider a weaker condition than (F4)(2); that is, there is


We study a diffusive predator-prey model with nonconstant death rate and general nonlinear functional response. Firstly, stability analysis of the equilibrium for reduced ODE system is discussed. Secondly, sufficient and necessary conditions which guarantee the predator and the prey species to be permanent are obtained. Furthermore, sufficient conditions for the global asymptotical stability of the unique positive equilibrium of the system are derived by using the method of Lyapunov function. Finally, we show that there are no nontrivial steady state solutions for certain parameter configuration. One of the dominant themes in both ecology and mathematical ecology is the dynamic relationship between predators and their prey due to its universal existence and importance in population dynamics. The investigations on predator-prey models are developed during these thirty years, and more realistic models are derived in view of laboratory experiments and observations  and P represent the population density of prey and predator at x ∈ Ω and at time t, respectively. The parameter ɛ > 0 is the specific growth rate of the prey in the absence of predation and without environment limitation; in the absence of predators, the prey population grows logistically to a carrying capacity K > 0; the functional response of the predator is a Holling type II function a(PN/(β + N)); a, β, and b are satiation coefficients or conversion rates. The specific mortality of predators in the absence of preyγ is the mortality at low density, and δ is the maximal mortality with the natural assumption γ < δ. Di > 0 are constants, i = 1,2, while Δ denotes the Laplace operator in Ω ⊂ Rn with Ω bounded and connected. The advantage of the present model over the more often used models is that here the predator mortality is neither a constant nor an unbounded function, and still it is increasing with quantity.In , Duque aφ(x) in Holling type I function and Holling type II function is monotonic in the first quadrant. It implies that, as the prey population increases, the consumption rate of prey per predator increases. But some experiments and observations indicate that a nonmonotonic response occurs at this level: when the nutrient concentration reaches a high level, an inhibitory effect on the specific growth rate may occur. To model such an inhibitory effect, Andrews , where K < l.there exists a Motivated by the above question a functional response may be monotonic or nonmonotonic; we consider the following diffusive predator-prey model with general nonlinear functional response:aϕ(N) represents Holling type II functional response or Holling type IV functional response; that is, aϕ(N) = aN/(β + N) or aϕ(N) = aN/(β + N2).Note that hypotheses (i)–(v) are satisfied if function aϕ(N) = aN/(β + N), system  and E2 =  on the boundary of Ω = { : N ≥ 0, P ≥ 0} and a set of nontrivial critical points obtained as the intersection of the curvesE3 = , if −γ + bϕ(K) > 0, whereE1 =  isJE1 has two eigenvalues λ1 = ɛ, λ2 = −γ. Hence, the equilibrium E1 =  is a saddle point and the stable and unstable manifolds lie on the P-axis and N-axis, respectively.For the reaction-diffusion predator-prey system , the reddNdt=ɛ(1−rewrite (dNdt=aϕ(N),wheref(N)=ɛ isJE2 has two eigenvalues λ1 = −ɛ, λ2 = −γ + bϕ(K). Hence the equilibrium E2 =  is a saddle point, if −γ + bϕ(K) > 0, and it is locally asymptotically stable when −γ + bϕ(K) ≤ 0.The Jacobian matrix of system  at E2 =E3 =  isJ are the roots of the polynomialThe Jacobian matrix of system  at E3 =The following result holds.N0 ∈ , then the equilibrium E3 =  is locally asymptotically stable with respect to system –(v), and (J = J11 + J22 = aϕ(N0)f′(N0) − P0M′(P0) < 0. Moreover, since J11, J22, J12 < 0 and J21 > 0, it follows that det(J) > 0, which completes the proof.Taking into account that N0 ∈ K/, K, the N, P) of ) of  lies in u is defined byt→∞u = K.Assume that t > 0. Furthermore, the nonnegative solution , P) of ) of  satisfieThe nonnegativity of the solutions of  is clearη > 0, there exists Tη(>0) such that, for any t > Tη,P. Taking into account the second equation of system (γ ≤ M(P) ≤ δ, we getT = T(M0) > 0 such thatη = 1 and taking into account thatT∗ = max{Tη, T}. Then, from (z(t) is the solution of the equationt0 ≥ T∗ such thatP → 0, t → ∞, which is a contradiction.From the first equation of system , we havef system  and the uch thatP>M0get thataϕ(N)NP≥ant that2N≤K+ies that0<N≤bϕ(N)NN,limN→0ϕ≤K+uch thatbϕ(N)=bϕ = P − M0, t > 0. We have to consider two possibilities. If there is some t0 such that Φ ≠ 0 for any t > t0, we say that the zeroes of Φ are bounded. If this is not true we say that the zeroes are unbounded; in this case, Φ = 0 for a sequence of tn tending to +∞ as n → ∞.Let us define the function Φ are bounded, then 0 < P ≤ M0 for any t ≥ t0.If the zeroes of Φ is of constant sign. Let us assume that Φ ≥ 0 for t ∈ Jn, n = 1,3, 5,…. Since P ≥ M0 on the intervals Jn−12, arguing as in the proof that  ≤ z(t) for t ∈ Jn−12, where z is the solution of the equation z′(t) = −(γ/2)z(t), with a suitable initial condition. Thus,n which is large enough. This contradiction completes the proof of our claim.If the zeroes are unbounded, the oof that  is not pη* > 0,An immediate consequence of the proof of the former result is that for a given r system . Hence, r system  is dissiNow, we will summarize some facts contained in Hale and Waltman studies, see , about tΩ is a complete metric space with Ω = Ω0 ∪ ∂Ω0 for an open set Ω0, where ∂Ω0 is the boundary of the set Ω0. We will typically choose Ω0 to be the positive cone in an ordered Banach space. A flow or semiflow on Ω under which Ω0 and ∂Ω0 are forward invariant is said to be permanent if it is dissipative and if there is a number η > 0 such that any trajectory starting in Ω0 will be at least a distance η from ∂Ω0 for all sufficiently large t. To state a theorem implying permanence, we need a few definitions. An invariant set M for the flow or semiflow is said to be isolated if it has a neighborhood U such that M is the maximal invariant subset of U. Let ω(∂Ω0)⊂∂Ω0 denote the union of the sets ω(u) over u ∈ ∂Ω0. This differs from the standard definition of the ω-limit set of a set but it is more convenient for our purposes; see , whereXΛ = Ω0 ∪ ∂Ω0. Assume that operator S(t) is compact for t > 0, which is defined by [S(t)φ](x) = U, where U is the classical solution of system (S(t)}t≥0 is pointwise dissipative.Let us set , system  can be rf system . MoreoveS(t) is positively invariant on ∂Ω0, which in turn implies that S(t) is positively invariant on Ω0.A direct application of the maximum principle shows that ω(∂Ω0) = {E1, E2}. First, we considerφ1(x) > 0 for all φ1(x) ≥ 0 for all φ1(x)≢0 and applying the strong maximum principle, we obtain that N > 0 for all t > 0 and t1 and choose φ1(x) = N.Now, let us show that u(t) and ω(t) be the solutions of the equation z′ = ɛ(1 − z/K)z such that z(t) = K attracts any positive solution, we conclude that N → K as t → ∞.Let M(P) ≥ γ, we get thatP → 0 as t → ∞. Henceforth, we conclude that ω(∂Ω0) = {E1, E2} and it is isolated and acyclic. By choosing M1 = E1 and M2 = E2, then M = M1 ∪ M2 is the covering required by Let us now considerω(∂Ω0) is positively invariant and the fact that the stable and unstable manifolds of the equilibrium E1 lie on ω(∂Ω0), we obtain that Ws(E1)∩Ω0 = ∅. In order to show that Ws(E2)∩Ω0 = ∅, let us linearize ∩Ω0 = ∅.Taking into account that inearize  about E)P.From  and by uSince all hypotheses of Permanence implies that condition  is an imq, Q, w, and W such that q ≤ N ≤ Q, w ≤ N ≤ W as time converges to infinity. If ϕ(N) = N/(β + N), system  > 0, in this case system  of the system  of the operator −Δ on Ω with no-flux boundary conditions. Let us recall that the eigenvalues {λj}j=0∞ satisfy the following inequality:Let us first briefly study the local stability of the homogeneous stationary solution U0 of  via the equation  about Ugiven by∂W∂t=DΔW+given by∂W∂t=DΔW+W = 0 is equivalent to that each matrix Bj has two eigenvalues with negative real parts for all j ≥ 0. And the eigenvalues of the matrix Bj are the roots of the polynomialHenceforth, the asymptotic stability of The following result holds.N0 ∈ , then the equilibrium U0 is locally asymptotically stable with respect to system –(v), and (Bj = A11 + A22 − λj(D1 + D2) = aϕ(N0)f′(N0) − P0M′(P0) − λj(D1 + D2) < 0. Moreover, since A11, A22, A12 < 0 and A21 > 0, it follows that detBj = (A11 − λjD1)(A22 − λjD2) − A12A21 > 0, which completes the proof.Taking into account that N0 ∈ K/, K, the U0.From Propositions U0. Biologically, our statement of the global stability of U0 means that no matter the two species diffuse, they will be spatially homogeneously distributed as time converges to infinity.Now, we will focus on the global stability of U0 is globally asymptotically stable.Assume thatE(t) ≥ 0 for all t ≥ 0. Differentiating E(t) along the solutions of the system (E(t) satisfies Lyapunov's asymptotic stability theorem, and the positive equilibrium U0 of system  + g) ≥ 0 for x ∈ Ω, ∂ω/∂υ ≤ 0 for x ∈ ∂Ω, and g) ≥ 0.If ω(x) + g) ≤ 0 for x ∈ Ω, ∂ω/∂υ ≥ 0 for x ∈ ∂Ω, and g) ≤ 0.If Suppose that The following result gives an upper estimate for classical solutions of , where cK, b such that γ < bϕ(K) < δ; then for any positive classical solution  of system  < K. This in turn implies that Let us set f system , we obtaψ2 = D2P, we get from the second equation of system  ≤ (bϕ(K) − γ)/(δ − bϕ(K)). This in turn implies thatAnalogously, by setting f system  that − γ)/(δ − bϕ(K)) and γ < bϕ(K) < δ, then  is a nonconstant positive classical solution of  such that N(x), P(x) ≤ C. Let Ω by parts, we haveAssume that (ution of . By 65)N, P) is ation of  by N-N¯  we haveD1∫Ω|∇, it fol account , we obta112, in [μ1∫Ω[D1 represents Holling type II functional response or Holling type IV functional response; that is, aϕ(N) = aN/(β + N) or aϕ(N) = aN/(β + N2). Hence, we can obtain the sufficient and necessary conditions which guaranting the predator and the prey species of the system with monotonic or nonmonotonic functional response to be permanent. If ϕ(N) = N/(β + N), system (Note that hypotheses (i)–(v) in , system  reduces , system  that was


The state transfer problem of a class of nonideal quantum systems is investigated. It is known that traditional Lyapunov methods may fail to guarantee convergence for the nonideal case. Hence, a hybrid impulsive control is proposed to accomplish a more accurate convergence. In particular, the largest invariant sets are explicitly characterized, and the convergence of quantum impulsive control systems is analyzed accordingly. Numerical simulation is also presented to demonstrate the improvement of the control performance. To avoid the confusion, we define ∠〈ψ | ψf〉 = 0° if 〈ψ | ψf〉 = 0. In addition, the impulsive control matrix Bk should be chosen to satisfy ΔV1(tk) ≤ 0; that is,ρf = |ψf〉〈ψf| is the density matrix of target state |ψf〉. Inequality (Bk which commutes with ρf. The control law satisfying (u(t) can make the system leave the initial state even if |ψ(0)〉 is an eigenstate of H0.In order that the designed control can work in the case of the initial state being orthogonal to the goal state, we rewrite  as V˙1=Klthat is,Bk∗ρfBk−ρtisfying  will be H0 with H0|ψ(0)〉 = λ0|ψ(0)〉 and 〈ψ(0) | ψf〉 = 0, then the following conclusions hold: l ∈ J such that ℑ〈ψf | Hl | ψ(0)〉≠0, then 〈ψ(t) | ψf〉 ≠ 0  (t > 0); if there exists ℑ〈ψf | Hl | ψ(0)〉 = 0, for all l ∈ J, and there exists l ∈ J such that ℜ〈ψf | Hl | ψ(0)〉≠0 and λ0 ≠ 0, then 〈ψ(t) | ψf〉≠0, t > t′ > 0, where t′ is sufficiently small; otherwise, the designed control fields cannot achieve the state steering of the closed-loop system. if For control law  and 7),,7), if tdt, as t ≠ tk, we have dt → 0, |ψ(dt)〉 = (I − iHdt) | ψ(0)〉. Since 〈ψ(0) | ψf〉 = 0, the inequality 〈ψf | ψ(dt)〉≠0 is equivalent to ∑l=1rul1〈ψf | Hl | ψ(0)〉 = ∑l=1rul1(iℑ〈ψf | Hl | ψ(0)〉+ℜ〈ψf | Hl | ψ(0)〉) ≠ 0. By (ul1ℑ〈ψf | Hl | ψ(0)〉≥0, for all l ∈ J. It follows from the assumption in case (i) that there exists l ∈ J such that ul1ℑ〈ψf | Hl | ψ(0)〉>0, and consequently, 〈ψf | ψ(dt)〉≠0. Since ψ(t) | ψf〉≠0, t ∈ . By (ψf | ψ(t1+)〉|≥|〈ψf | ψ(t1)〉| = |〈ψf | ψ(t1−)〉|>0. Hence, we obtain that 〈ψ(t) | ψf〉≠0, (t > 0).(i) For a sufficiently small  ≠ 0. By , we havet1). By  we have ul1(0) = 0, l ∈ J. For a sufficiently small t* < t1, one can obtain that |ψ(t*)〉 = eiH0t*− | ψ(0)〉 = eiλ0t*− | ψ(0)〉. Moreover, as dt → 0, we have 〈ψf | ψ(t* + dt)〉 = 〈ψf | (I − iHdt) | ψ(t*)〉 = dt[−i  cos(λ0t*) − sin(λ0t*)]∑l=1r〈ψf | Hlul1(t*) | ψ(0)〉. Noticing that there exists l ∈ J such that 〈ψf | Hl | ψ(0)〉≠0, we have ul1(t*) = Klfl[ℑ(eiλ0t*−〈ψf | Hl | ψ(0)〉)] = Klfl[−sin(λ0t*)〈ψf | Hl | ψ(0)〉] ≠ 0. Similar to the discussion in case (i), we obtain that 〈ψf | ψ(t* + dt)〉≠0, and then 〈ψ(t) | ψf〉≠0, t > t*. This completes the proof.(ii) Initially, the system evolves freely because For the characterization of invariant sets, properties of the states such that ψ0 | ψf〉 = 0 and the conditions (i) or (ii) in t ≠ tk,t ≠ tk,λl ∈ ℝ such that 〈ψf|(λlI − Hl)|ψ(t)〉 = 0, t ≠ tk, l ∈ J. there exists If 〈V1 = 0. We first present the extensive LaSalle invariance principle for impulsive systems in [stems in . 𝒟:V such that V′(x)fc(x) ≤ 0, t ≠ tk and ΔV(tk) ≤ 0, then x(t) → ℳ as t → ∞, where ℳ is the largest invariant set contained in 𝔼≜{x : V′(x)fc(x) = 0}∩{x : ΔV(x) = 0}. Consider the following differential impulsive system on an open set Bk is designed such that it commutes with H0, k ∈ ℤ+.  The following theorem presents the characterization of the invariant set for the nonideal systems under the hybrid impulsive control, by which the invariant set is smaller compared with that obtained by the conventional Lyapunov method. In the following, the unitary matrix G = 𝕊n−12⋂E1⋂E2 with E1 = {|ψ〉 : |ψ〉 ∈ Mkl, for  all  l ∈ J, k = 1,2,…}, E2 = {|ψ〉 : |ψ〉 ∈ Nk, k = 1,2,…}, andXl1, Xl2,…, Xlml constitute the basis of the set {(i)s[H0s) = 0 for the whole system trajectory will be transformed to the conditions on the initial state. By the Taylor expansion and commutativity between H0 and Bk, it yields thatt = tk + dt, the free evolution of |ψ(t)〉 is given bys = 0,1,…, ∑i=1kni. Noticing that the set {(i)s[H0s)ned thatD−V1(tk)=random, (D−V1(tk)=Mkl in , l ∈ J, ψ〉 : ΔV1(tk) = 0, k = 1,2,…}:Nk in =0 s[H0s)) = V2(t) = (1/2)〈ψ(t) − ψf | ψ(t) − ψf〉 = 1 − ℜ〈ψf | ψ(t)〉. Similar to the hybrid control design in ω is a new real scalar control field. For the convenience of the computation, the introduced ω may be used to adjust the global phase without changing the physical quantities regarding |ψ〉. While in practical implementation, it is not necessary to be implemented to the system. Similar conclusion can be drawn if there exists more than one control Hamiltonian Hl, l ≥ 2. The time derivative of V2 isu0 = λf + ω. We design the following control to ensure t ≠ tk:K1 > 0, and the function y1 = f1(·) is defined as that in  = 0, s = 1,…, m1},X1, X2,…, Xm1 are the basis of the set {I, (i)s[H0s)K = F1⋂Fω = −λf. When the system satisfies u1 = 0, the evolution of system  = 0⇔ℑ(in1〈ψf | [H0n1)〉) = 0. According to the similar method in the proof of t = tk, it yields thats = 0,…, ∑i=1kni. Denote the basis of the set {I, (i)s[H0s) = 0⇔ℑ〈ψf | Xs∏j=k−11Bj | ψ(t0)〉 = 0, s = 1,2,…, m1. This equality is denoted as Uk in (Let f system  becomes system (i|ψ˙(t)〉=)〉.From , we obta system (i|ψ˙(t)〉=〉.From (V˙2(t0)=0lds thatD−V2(tk)=s Uk in . ψ〉:ΔV2(tk) = 0, k = 1,2,…}. From the definition of V2, we have ΔV2(t1) = ℜ〈ψf | ψ(t1+)〉−ℜ〈ψf | ψ(t1)〉 = ℜ〈ψf | B1 | ψ(t1)〉−ℜ〈ψf | ψ(t1)〉. The following relations can be obtained:V2(tk) = 0⇔ℜ〈ψf | ψ(t0)〉−ℜ〈ψf | ∏j=1kBj | ψ(t0)〉 = 0, which can be denoted as Wk in  = x, i = 1,2. Choose the impulsive instant to be tk = 3k − 1, k ∈ ℤ+ and K1 = K2 = 0.2. Using the hybrid impulsive control based on the state distance, simple computation yields that the invariant set G = {|3〉} , which implies that under the hybrid impulsive control the system converges to |ψf〉. The populations of the controlled system are illustrated in Consider the five-level system with the internal Hamiltonian and impulsive control Hamiltonian given by Now we compare performance of the hybrid impulsive control with that of classical Lyapunov control. If the impulsive control is not applied to the system, then the hybrid impulsive control is reduced to the classical Lyapunov control, by which the performance of the controlled system is shown in π, 0}. The unitary operation is chosen as f1(x) = x, and K0 = 0.1, K1 = 0.2. Choose the impulsive instant as tk = 3k − 1, k ∈ ℤ+. Using the hybrid impulsive control based on the state error, simple computation yields that the invariant set K = {|5〉} , which implies that under the hybrid impulsive control the system converges to |ψf〉. Simulation results are illustrated in Consider the five-level system with the same internal Hamiltonian as the previous example. Let the target state and the initial state be When the hybrid impulsive control is reduced to the classical Lyapunov control, the trajectory of the controlled system is plotted in H0 is not strong regular. The dynamical properties of the resulted quantum impulsive control system have been discussed to facilitate the convergence analysis. Based on two kinds of Lyapunov functions, the largest invariant sets have been characterized explicitly. Consequently, more accurate convergence of the controlled system has been achieved by the extensive LaSalle invariance principle. Compared with some existing results, the improved control performance has been shown for the nonideal case. Since the practical implementation of impulsive control has been studied in known literature, we believe that it is feasible. The optimal determination of the impulsive control Hamiltonian and impulsive instants is worth to be explored in the future work. In this paper, the coherent hybrid impulsive control for closed quantum systems has been investigated for the nonideal case that


The existence of scattering operators and finite time blow-up of the solutions for the systems in higher space dimensions is also shown.The study of nonlinear Schrödinger systems with quadratic interactions has attracted much attention in the recent years. In this paper, we summarize time decay estimates of small solutions to the systems under the mass resonance condition in 2-dimensional space. We show the existence of wave operators and modified wave operators of the systems under some mass conditions in  If we let uj = eitmjc2−vj in  conservation law of solutions to  conservation law and as a result global existence in time of solutions to  is defined byH2m,s(ℝn) = Hm,s(ℝn) and Hm,0(ℝn) = Hm(ℝn) for simplicity. Lp(ℝn) denotes the usual Lebesgue space with the normp < ∞ ands = [s] + σ, 0 < σ < 1, ϕy(x) = ϕ(x + y),  1 ≤ p, q ≤ ∞, and [s] is the largest integer less than s. It is known that C be the space of continuous functions from an interval I to a Banach space E. Different positive constants might be denoted by the same letter C. The homogeneous Sobolev spaces x ∈ ℝn and δ ≠ 0 and define E = ei/2)t|ξ|2−( and M = ei/2t)|x|2−( for t ≠ 0.We now introduce some function spaces to present exact statements of our results. For any ℝn) (see ). We letUδ(t) is written asℱ, ℱ−1 are the Fourier transform and the inverse Fourier transform, respectively. We also haveJmj,k1/ = xk + i(t/mj)∂k = Umj1/(t)xkUmj1/(−t), where j ∈ {1,…, l} and k ∈ {1,…, n} is an important tool to study time decay of solutions to nonlinear Schrödinger equations satisfying the gauge invariant condition Δ, we have commutation relations such thatEvolution operator tten as1(Uδ(t)ϕ) ∈ ℝ × ℝ2, where mj is a mass of particle for j = 1,2, 3. In [The system  includes2, 3. In , the sysi∂tv1+12m2, 3. In , the stai∂tv1+12mn = 2 shown in [n = 2 based on a paper [n = 2 from a paper [This paper is organized as follows. shown in –7, 9. In a paper .  Sectio a paper . In the  a paper . In the n = 2, we start with time decay estimates of solutions to linear Schrödinger systems.To state time decay of solutions to nonlinear Schrödinger systems for j ≤ l.For , the corvj(t) of (vj(t) = Umj1/(t)ϕj. It is known that vj(t) is decomposed into a main term and a remainder one asn = 2, where Rj decays rapidly in time; indeed we have the estimatet ≠ 0. By ||Umj1/(t)vj||L2(ℝ2) = ||vj||L2(ℝ2) and L∞(ℝ2) − L1(ℝ2) time decay estimatet ≠ 0, we have the following time decay estimates through the interpolation theorem (see [As we know the solution j(t) of  is reprerem (see ).p ≤ ∞, and let p, p′ be conjugate indices, t ≠ 0. Then we have Umj1/(t) : Lp′(ℝ2) → Lp(ℝ2) which are bounded operators and satisfyj = 1,…, l.Let 2 ≤ t, x) ∈ ℝ × ℝ2, where m1 and m2 are the masses of particles and λ, μ ∈ ℂ. If we let u2 = (μ/|λμ|)v2 in the above system, then we obtain the system as belowt, x) ∈ ℝ × ℝ2, where γ = λμ/|λμ| ∈ ℂ. Therefore we survey the results on time decay of solutions to the system i∂tas belowi∂tv1+12mm1 = m2 and γ = 1. Then there exists ɛ > 0 such that  ∈ H2(ℝ2)∩H0,2(ℝ2) satisfyingt ∈ ℝ.Assume that 2uch that  with theL∞(ℝ2) time decay estimates of solutions of  conservation law such thatWhen γ = 1,  satisfieγ = 1, . Under t  time decay of small solutions for (ℱUmj1/(−t)vj||L∞(ℝ2). The similar idea has been used for construction of H2,2(ℝ2) solutions to a single nonlinear Schrödinger equation by a paper [m1 = m2 by using the factorization formulas of Schrödinger evolution group stated in ψj = Dmj1/ℱUmj1/(−t)vj for j = 1,2 and ɛ > 0. Asymptotic behavior in time of solutions of (O(tɛ−1−) which is integrable in time. This is the reason why we use the condition such that the data must be in Hβ0,(ℝ2), β > 1.In the case of the mass resonance condition 21 = m2,  satisfie1 = m2, . Global 1 = m2,  is obtaiions for  is prove a paper . Theoremtions of  is detera paper [i∂tψ1=γt−The system  is a genϕ =  ∈ H2,2(ℝ2) and Fj satisfies the conditions  ∈ H2,2(ℝ2) satisfyingt ∈ ℝ.One assumes that nditions  and 8)  ϕ =  by ϕ ∈ Hβ(ℝ2)∩Hβ0,(ℝ2) with 1 < β.t, x) ∈ ℝ × ℝ2, where m1, m2, m3 are the masses of particles and λ1, λ2, λ3, μ1, μ2, μ3 ∈ ℂ∖{0} are constants.Now we focus on the following system:Time decay problem of solutions to  is consim1 + m2 = m3 is satisfied. One also assumes that Imλj ≤ 0 for j = 1,2, 3 and κ1, κ2, κ3 > 0. Then there exists ɛ > 0 such that  ∈ Hs(ℝ2)∩Hs0,(ℝ2) satisfyings < 2. Moreover, the time decay estimatet ∈ ℝ.Assume that the mass resonance condition uch that  has a unλj | vj | vj acts as a dissipation one which requires logarithmic correction in time of solutions and the negative time is not considered. We have the following theorem.Ifv be the solution to the system  = μ2vj; then vj,μ(t) satisfies the system  = μ2ϕj(μx). We haven = 2 in the function space α, β satisfy 0 ≤ β < 1 < α ≤ 2 and can be taken to be close to 1. Therefore our function space has relation with the invariant space and the considered data are not necessarily in L2(ℝ2).Define the scaled function by e system  with the problem . Other i problem  are givee system  for n = β < 1 < α ≤ 2. Then there exists ɛ > 0 such that  satisfyingt ≠ 0.Assume that  and 8)  8) hold.ϕ=ϕ,…,ϕl∈H˙0Imλj < 0, where 0 ≤ δ < 1 < α ≤ 2. Then we have the time decay estimate such thatt ≥ 1.We note that in  we consit, x) ∈ ℝ × ℝ2, where m1, m2 are the masses of particles. It is obvious that the nonlinearities of (v1(0) = ϕ1 can be considered the Cauchy problem for the linear Schrödinger equation, we find the value of v1 explicitly by v1 = Um11/(t)ϕ1. Therefore, we haveIn the final part of this subsection, we will show that the assumption  is impori∂tv1+12mities of . Since t we havei∂tv2+12mϕ1 ∈ H0,2(ℝ2),  ϕ2 ∈ L2(ℝ2). Lett > 1.Suppose that ution of . Then thThis fact was pointed first in , 23 in tv+ = (vi+), we assume that there exists a unique solution v = (vi) of the system (Um1/(t)v+ is the solution of linear problems with the initial data v+ and ||·||X is the norm of Banach space X. Then we define the map W+ : v+ ↦ v(t) and call it the wave operator. We also call v+ the final state  since it is considered the value of Um1/(−t)v(t) at infinity if Um1/(t) is the unitary operator in X. The same problem can be considered for negative time.First, we briefly explain the definition of the wave operator (see ). For a e system  satisfyiϕ1+, ϕ2+):t, x) ∈ ℝ × ℝ2. If there exists a nontrivial solution for the above system, then we say that there exists a usual wave operator.To study existence of wave operators for , we consm1 ≠ m2 and m1 ≠ m2.We consider  under thm1 ≠ m2 and m1 ≠ m2. Then there exists ɛ > 0 such that, for anyt ≥ 1, where 1/2 < b < 1.Let 2e system  has a unm1 ≠ m2 and m1 ≠ m2 are used to obtain better time decay of solutions. Oscillating properties of nonlinear terms v1+2 are different from those of solutions to linear problem which yield an additional time decay from nonlinear terms; namely, nonlinear interactions are not critical. By combining this fact and the Strichartz type estimates, the result of The mass nonresonance conditions 2m1 = m2 which is also the mass nonresonance case and the support conditions on the data.We next consider  under thm1 = m2. Assume thatɛ > 0 such that, for any ϕ+ =  with the normt ≥ 1, where 1/2 < b < 3/4.Let e system  satisfyiFrom the result we have wave operators when the support of the Fourier transform of the Schrödinger data is restricted. Restriction on the support of the Fourier transform of the Schrödinger data was used to obtain an improved time decay estimate of the nonlinear term We turn to investigate existence of wave operators for . First, ϕ =  ∈ H2,2(ℝ2) and let v be global in time of solutions of  of the system  from a suitable function space and define a function f by v+. Then we try to find a unique solution of nonlinear problems under the asymptotic conditionX is the norm of Banach space X. Namely, the problem is solved if we can define the function f satisfying the asymptotic condition (MW+ : v+ ↦ v(t) instead of the wave operator. We call MW+ the modified wave operator since we modified the final states.In onditionlimt→∞||t, x) ∈ ℝ × ℝ2. In m1 = m2 and γ = 1. Since the nonlinearity is critical in this case, it is impossible to find a solution in the neighborhood of the free final state (Um11/(t)ϕ1+, Um21/(t)ϕ2+). Indeed we have the nonexistence of the usual scattering states.We consider  again whm1 = m2, γ = 1, and letϕ+ =  ∈ H2(ℝ2)∩H0,2(ℝ2) such that ϕ1+ ≠ 0 andt → ∞.Let 2From the result, we need to modify the final state with time dependence. We note here that the modified wave operator for nonlinear dispersive equation was first constructed in  for the m1 = m2 is determined by the solutions of the following system:In γ = −1 and the particular solutions of  > 0 and θ(ξ) is a real valued given function. We also find thatγ = ±1.By calculation we find that the particular solutions of  are(75)tions of  are(76)tions of  when γ =The following theorem shows existence of the modified wave operators of .m1 = m2 and γ = ±1. Then there exists ɛ > 0 such that, for any ω, θ ∈ H2(ℝ2) with norm ||ω||H2(ℝ2) ≤ ɛ,  ≤ ɛ,  has a unm1 = m2, we get the existence of modified wave operators by the contraction mapping principle. Since the identity Umj1/(t) = Mmj−Dt/mjℱMmj− is known for j = 1,2, we have the estimate from the above theoremt ≥ 1, where 1/2 < b < 1.Using the resonance condition, 2Existence of modified wave operator for a single nonlinear Schrödinger equation was studied in , 27. Asyn ≥ 3, the scattering theory for i∂tvW+ which maps a Banach space X into itself. Namely, for any given v+ ∈ X, we assume that there exists a unique solution v(t) ∈ C; X) of the nonlinear system such thatv(0) which are determined by the solution v(t) in the time interval t ∈ [0, ∞). If the initial value problem has a unique global solution v(t) ∈ C and we can find a unique v− ∈ X from the solution v(t) satisfyingW−−1 : v(0) ∈ X ↦ v− ∈ X. From this operator we can define S = W−−1W+ : X → X. We call the operator the scattering operator.We will explain the scattering problem (See ) brieflyn ≥ 4, existence of the scattering operator was proved in the space Hn/2)−2((ℝn) which is close to the invariant space n = 4, we have the results in the invariant space L2(ℝ4). In the case of n = 3, existence of the scattering operator was proved in the space H0,1/2(ℝ3), under the mass resonance condition 2m1 = m2, which is close to the invariant space In the case, n ≥ 4. Then there exist ε0 and C0 such that 0 < ε0 ≤ 1 ≤ C0 with the following property.Let ɛ with 0 < ɛ ≤ ε0 and any ϕ =  ∈ Bɛ,  ∈ C−2((ℝn)). Moreover, there exist unique ϕ± =  ∈ BC0ɛ such thatt → ±∞.(1) For any ) ∈ Bɛ,  has a un+ For any ɛ with 0 < ɛ ≤ ε0 and any ϕ+ =  ∈ Bɛ,  ∈ C−2((ℝn)) such that v(0) = (v1(0), v2(0)) ∈ BC0ɛ,t → +∞.(2)) ∈ Bɛ,  has a un− For any ɛ with 0 < ɛ ≤ ε0 and any ϕ− =  ∈ Bɛ,  ∈ C−2((ℝn)) such that v(0) = (v1(0), v2(0)) ∈ BC0ɛ,t → −∞.(2)) ∈ Bɛ,  has a unW± : ϕ± ↦ v(0) are defined as mappings from Bɛ to BC0ɛ  for any ɛ with 0 < ɛ ≤ ε0. The scattering operator S : ϕ+ ↦ ϕ− is defined as a mapping from BC0−1ɛ to BC0ɛ for any ɛ with 0 < ɛ ≤ ε0.To state the following theorem, we introduceThe wave operators m1 = m2. Then there exist ε0 and C0 such that 0 < ε0 ≤ 1 ≤ C0 with the following property.Let  2ɛ with 0 < ɛ ≤ ε0 and any v with Um1/(−t)v ∈ C). Moreover, there exist unique t → ±∞.(1) For any y ϕ∈B~ɛ,  has a un+ For any ɛ with 0 < ɛ ≤ ε0 and any v with Um1/(−t)v ∈ C) such that t → +∞.(2) ϕ+∈B~ɛ,  has a un− For any ɛ with 0 < ɛ ≤ ε0 and any v with Um1/(−t)v ∈ C) such that t → −∞.(2) ϕ-∈B~ɛ,  has a unW± : ϕ± ↦ v(0) are defined as mappings from ɛ with 0 < ɛ ≤ ε0. The scattering operator S : ϕ+ ↦ ϕ− is defined as a mapping from ɛ with 0 < ɛ ≤ ε0.The wave operators n ≤ 6 under the mass conditions 2m1 = m2 and γ ∈ ℝ. To state the blow-up result we need local existence in time of solutions to  > 0 such that i∂tvrsion of 100)∂t2∂t2(99)i∂ = 1 in (∂t2u1−Δu1n papers , 35 with a paper  if m2 <ator for  is an opained in  for the  = 1 in (∂t2u1−Δu1Small data blow-up for a system of nonlinear Schrodinger equations was studied in  under soThe asymptotic behavior of solutions to quadratic derivative nonlinear Schrödinger systems has been considered recently in papers , 40 unde