To prove the existence of traveling wave solutions, an invariant cone is constructed by upper and lower solutions and Schauder's fixed point theorem is applied. The nonexistence of traveling wave solutions is proved by two-sided Laplace transform. However, to apply two-sided Laplace transform, the prior estimate of exponential decrease of traveling wave solutions is needed. For this aim, a new method is proposed, which can be applied to reaction-diffusion systems consisting of more than three equations.Based on Codeço's cholera model (2001), an epidemic cholera model that incorporates the pathogen diffusion and disease-related death is proposed. The formula for minimal wave speed  Vibrio cholera. An estimated 3–5 million cases and over 100,000 deaths occur each year around the world /c, a1 = −(1 + κ) < 0, and a2 = (κ − c2)/c. To investigate distribution of roots of =λ3+ang lemma .0 > 0, ((a) If Δ0 > 0,  has one 0 = 0, ((b) If Δ0 = 0,  has a mu0 < 0, ((c) If Δ0 < 0,  has threThen, we get the following lemma about the distribution of eigenvalues.U0 > 1/g′(0). Then, there exists a constant c* > 0 which is the only positive root of  < 0 if 0 < ε < λ2 − λ1.one can assume roots of ; then H = 0 has only one positive root c* such that Δ < 0 for c > c* and Δ > 0 for 0 < c < c*. Direct calculations show that Δ0 = Δ/(108c4). Since a0 > 0 and a1 < 0, Descartes' rule of signs shows that (H(λ) is a cubic polynomial.Obviously, ows that  has onlyows that  has rootU0 > 1/g′(0) and c > c* unless other conditions are specified. Denote λ1 < λ2 to be the two positive roots of .In this section, we always suppose roots of  and defiξ ≠ lnV0/λ1.The function ξ < lnV0/λ1 and, therefore, g′′(V) ≤ 0 for any V ≥ 0, we haveθ < 1.Firstly, assume atisfies  and g′′.For σ > 0 be sufficiently large to ensure 1/αln(U0/σ) < lnV0/λ1. When ξ > 1/αln(U0/σ), then ξ < 1/αln(U0/σ). Then, θ < 1. Let σ = 1/α. Sinceα > 0 sufficiently small and σ > 0 sufficiently large such thatLet ε = min{α, λ1, λ2 − λ1}/2. Then, for M > 0 large enough, the function ξ ≠ 1/εln(1/M).Let ξ = 1/αln(U0/σ), that ξ = 1/εln(1/M), and that 1/εln(1/M) < 1/αln(U0/σ) if and only if M > (σ/U0)ε/α)(. Let M > (σ/U0)ε/α)(. When ξ > 1/εln(1/M), then eλ1ξ(1 − Meεξ) < 0, It is clear that ξ < 1/εln(1/M). Then, ξ < 1/αln(U0/σ), θ < 1. Sinceξ < 1/αln(U0/σ), thenH(λ1 + ε) < 0, inequality . Let μ be a positive constant which will be specified in the following. For Φ(ξ) = (ϕ1(ξ), ϕ2(ξ)), defineC. Firstly, we change system (β1 > limV→+∞g(V),1 < 0 < Λ2 are the two roots of equation Λ2 − cΛ − 1 = 0. Furthermore, define F =  : Γ → C byμ < min{β1/c, κ/c, −Λ1, Λ2}.To apply Schauder's fixed point theorem, we will introduce a topology in e system  into theF(Γ) ⊂ Γ.Consider U(·), V(·)) ∈ Γ; that is Let (β1 > limV→+∞g(V). Then,First of all, we haveF2(U(·), V(·))(ξ). If ξ ≥ ξ0≜1/εln(1/M), then ξ < ξ0. From ξ ∈ R.Now, we study Similarly, we can show F =  : Γ → C is continuous with respect to the norm |·|μ in Bμ.Map 1(·) = (U1(·), V1(·)), Φ2(·) = (U2(·), V2(·)) ∈ Γ, we haveM1 = β1 + J + U0g′(0), and V* is between V1(ξ) and V2(ξ). Therefore, we haveξ ≤ 0, we getFor Φξ > 0, it follows thatξ ∈ R, whereF1 : Γ → C is continuous with respect to the norm |·|μ in Bμ.When F2(U(ξ), V(ξ)). Firstly, we haveM3 = (J + U0g′(0))κ/(κ − cμ). Therefore,ξ < 0, it holds thatξ ≥ 0, we haveF2 : Γ → C is continuous with respect to the norm |·|μ in Bμ. The proof is completed.In addition, consider F =  : Γ → Γ is compact with respect to the norm |·|μ in Bμ.Map ξ) = (U(ξ), V(ξ)) ∈ Γ, it is clear thatH1(ξ)| = |[β1 − g(V(ξ))]U(ξ)| ≤ β1U0, we haved/dξ)F1(Φ(·))(·)|μ < 2β1U0/c. Since Φ(·) = (U(·), V(·)) ∈ Γ, we getd/dξ)F2(Φ(·))(·)|μ < 2(β1U0 + V0)/(Λ2 − Λ1). Consequently, |(d/dξ)F1(Φ(·))(·)|μ and |(d/dξ)F2(Φ(·))(·)|μ are bounded, which shows that F(Γ) is uniformly bounded and equicontinuous with respect to the norm |·|μ in Bμ.For any Φ is uniformly bounded and equicontinuous with respect to the norm |·|μ in Bμ, which implies that Fn : Γ → Γ is a compact operator. SinceF2n(U(·),V(·))(·)−F2(U(·),V(·))(·)|μ → 0 when n → +∞. Thus, |Fn(U(·),V(·))(·)−F(U(·),V(·))(·)|μ → 0 when n → +∞. By Proposition 2.1 in Zeilder (λ) is increasing in (λ) = +∞ or λ* = +∞.Assume  has a trondition . From thondition , it follG(V): = g′(0)V − g(V), from which it follows that G′(0) = 0. Since g(V) satisfies hypothesis (A1), we have G(V) ≥ 0 for any V ≥ 0.Furthermore, we haveality of  can be rλ* < +∞, it is clear that L[V(·)](λ*−) = +∞. SinceG′(0) = 0 imply L[Q(·)](λ*) < +∞. However, λ*)<+∞, which contradicts  = +∞, it is impossible that the above equality holds, again a contradiction.Taking two-sided Laplace transform of , we haveΘ(λ)L[V(·Θ(λ)L[V(·U(y + cτ), W(y + cτ), V(y + cτ)) is a traveling wave solution of system (∞−∞+V(ξ)dξ = ∫∞−∞+H2(ξ)dξ. Furthermore,Suppose (f system  satisfyif system . Then, b


Scientific Reports6: Article number: 28657; 10.1038/srep28657 published online: 06282016; updated: 08032016.st–3rd Qu’ and ‘Range’ “−3.07–1.80” and “−5.06–0.42” should read: “of “−3.07– −1.80” and “−5.06– −0.42” respectively. In addition, the Girls-pre values for ‘1st–3rd Qu’ and ‘Range’ “−2.54–0.85” and “−6.01–0.42” should read: “−2.54– −0.85” and “−6.01– −0.42” respectively.This Article contains typographical errors in Table 1. In the column ‘Pubertal age’ the values for Boys-pre ‘1


This paper mainly studies the Laplacian-energy-like invariants of the modified hexagonal lattice, modified Union Jack lattice, and honeycomb lattice. By utilizing the tensor product of matrices and the diagonalization of block circulant matrices, we derive closed-form formulas expressing the Laplacian-energy-like invariants of these lattices. In addition, we obtain explicit asymptotic values of these invariants with software-aided computations of some integrals. G, which is studied in chemistry and used to approximate the total electron energy of a molecule [Eπ is closely related to the energy releasing from the formation progress of the conjugated carbon oxides and could be approximately calculated by Hückel molecular orbital theory. And in the method of HMO, the calculation of Eπ can be attributed to the sum of the absolute values of all the eigenvalues of its molecular graph [Molecular structure descriptors or topological indices are used for modelling information of molecules, including toxicologic, chemical, and other properties of chemical compounds in theoretical chemistry. Topological indices play a very important role in mathematical chemistry, especially in the quantitative structure-property relationship (QSPR) and quantitative structure activity relationship (QSAR). Many topological indices have been introduced and investigated by mathematicians, chemists, and biologists, which contain energy , the Lapmolecule . During ar graph –20.Compared with adjacency matrix, the definition of Laplacian matrix added to all vertices degrees. As Mohar said, the Laplacian eigenvalues can reflect more the combination properties of graphs. Cvetković and Simić –23 pointG, the vertex set and edge set of G will be denoted by V(G) = {v1, v2,…, vn} and E(G) = {e1, e2,…, em}, respectively [G are, respectively, A(G) and D(G); then the matrix L(G) = D(G) − A(G) is called the Laplacian matrix of the graph G [G are χG(λ) = det(λI − A(G)) and μG(λ) = det(λI − L(G)) [A(G) and L(G) are symmetric matrices; their eigenvalues are real numbers [G as λ1 ≥ λ2 ≥ ⋯≥λn, and the Laplacian eigenvalues are μ1 ≥ μ2 ≥ ⋯≥μn [G is a connected graph, then μi > 0, i = 1,2,…, n − 1, μn = 0  [All the graphs discussed in this paper are simple, finite, and undirected. For a graph ectively . The adj graph G , 28. The − L(G)) . Both A, is defined asLet A = m×n, B = s×t, the tensor product of A and B, denoted by A ⊗ B, is defined asFor two matrices Gn} be a sequence of finite simple graphs with bounded average degree such that Let {Hn} be a sequence of spanning subgraphs of {Gn} such that Let {Gn and Hn have the same asymptotic Laplacian-energy-like invariant.That is, In what follows, we will explore the Laplacian-energy-like invariants formulas of the modified hexagonal lattice, modified Union Jack lattice, and honeycomb lattice.t.The modified hexagon lattice with toroidal boundary condition is denoted by MHαi = 2πi/n1, βj = 2πj/n2. ThenLet In1, In2 are the unit matrices and M ⊗ N is tensor product of matrices M and N. ConsiderWith the proper labelling of the vertices of the modified hexagonal lattice, its Laplacian matrix isL) can be defined as follows: The matrix g0, g1,…, gn−1} be a cyclic group of order n. Obviously, ρ : gi → Bni can express the group. The cyclic group of order n has linear values of n  χi  ,  χi(g) = ωni, where ωn are said n-times unit roots.Let {1 = Therefore, there is a reversible matrix In fact, So In2 ⊗ (6In1 − Dn1 − Dn1−1) − Dn2 ⊗ (In1 + Dn1−1) − Dn2−1 ⊗ (In1 + Dn1) is a diagonal matrix whose diagonal elements are i ≤ n1 − 1 and  0 ≤ j ≤ n2 − 1.It is not difficult to find that L are μ = 6 − 2cosαi − 2cosβj − 2cos(αi − βj), 0 ≤ i ≤ n1 − 1, and  0 ≤ j ≤ n2 − 1, where αi = 2πi/n1 and βj = 2πj/n2.This means that the eigenvalues of the matrix By formula , the LapSo The numerical integration value in last line is calculated with the software MATLAB . As suchBy Theorems MHt, MHc, and MHf with toroidal, cylindrical, and free boundary conditions, then,For the modified hexagonal lattices St.The modified Union Jack lattice with toroidal boundary condition is denoted by αi = 2πi/n1;  βj = 2πj/n2. Then Let With a proper labelling of the vertices of the modified Union Jack lattice, its Laplacian matrix can be represented as Based on LetActually, So i ≤ n1 − 1 and  0 ≤ j ≤ n2 − 1.It is not difficult to find that L are μ = 8 − 2cosαi − 2cosβj − 4cosαiβj, 0 ≤ i ≤ n1 − 1, and  0 ≤ j ≤ n2 − 1, where αi = 2πi/n1 and βj = 2πj/n2.This means that the eigenvalues of the matrix By formula , the LapSo By Theorems St, Sc, and Sf with toroidal, cylindrical, and free boundary conditions, then, For the modified Union Jack lattices t, can be constructed by starting with an m × n square lattice and adding two diagonal edges to each square.The honeycomb lattice with toroidal boundary condition, denoted by HCαi = 2πi/n1 and βj = 2πj/n2. Then Let M = n1n2 and F is an M × M matrix. The matrix F can be written in the following form: I represents the unit matrix of n1 × n1 and IM represents the unit matrix of M × M, respectively.Similarly, the Laplacian matrix of the honeycomb lattice is F can be written as Based on Let Similarly, hence, So In2 ⊗ (In1 + Dn1−1) + Dn2−1 ⊗ In1 is a diagonal matrix whose diagonal elements are 1 + ωn1i− + ωn2j−, so matrix L) can be reduced to the following form: It is not difficult to find that μI − L)) = 0, we can get By det) characteristic eigenvalues are i ≤ n1 − 1 and 0 ≤ j ≤ n2 − 1.Therefore, the αi = 2πi/n1 and βj = 2πj/n2. By formula , HCc, and HCf with toroidal, cylindrical, and free boundary conditions, then, For the honeycomb lattices HCIn this paper, we mainly studied the Laplacian-energy-like invariants of the modified hexagonal lattice, modified Jack lattice, and honeycomb lattice. The Laplacian-energy-like invariants formulas of these lattices are obtained. The proposed results imply that the asymptotic Laplacian-energy-like invariants of those lattices are independent of the three boundary conditions.The problems on the various topological indices of lattices have much important significance in the mathematical theory, chemical energy, statistical physics, and networks science. This paper investigated the Laplacian-energy-like invariants of some lattices. However, the other topological indices of the general lattices remain to be studied.


The “Hydraulically weighted diameter (mm)—average” and “Hydraulically weighted diameter (mm) –standard deviation” columns in S2 Table(XLSX)Click here for additional data file.


GL(3) and discuss the analytic behavior of the test functions on both sides. Applications to Weyl’s law, exceptional eigenvalues, a large sieve and L-functions are given.We develop a fairly explicit Kuznetsov formula on  GL(2) and the cornerstone for the investigation of moments of families of L-functions, including striking applications to subconvexity and non-vanishing. It can be viewed as a relative trace formula for the group G=GL(2) and the abelian subgroup U2×U2⊆G×G where U2 is the group of unipotent upper triangular matrices. The Kuznetsov formula in the simplest case is an equality of the shape n,m∈ℤ∖{0},δn,m is the Kronecker symbol,uj for the group SL2(ℤ) having spectral parameter tj and Hecke eigenvalues λj(n) for n∈ℕ (and λj(−n):=±λj(n) depending on whether uj is even or odd),the sum on the left-hand side runs over an orthogonal basis of Hecke–Maaß cusp forms σt(n) is the Fourier coefficient of an Eisenstein series defined by dspect=π−2ttanh(πt) dt is the spectral measure,h is some sufficiently nice, even test function, andh± is a certain integral transform of h, the sign being sgn(nm), described in , SO and SU; see also  with compact support. Letε>0 and letIfX1=X2=1, thenMoreover, there is a constantcdepending onf1,f2such that the following holds: ifthenand ifν1,ν2are given by  an anLetA,given by  and in at1,t2,X1,X2 as above, I as a function of ν1,ν2∈{z∈ℂ:|ℑz|≤1/2} is under some technical assumptions a function with a bump at ℑν1=τ1 and ℑν2=τ2 of size X1=X2=1. Only if we need a test function that blows up at exceptional eigenvalues we will choose X2 to be large. The asymmetry in X1,X2 in >1/50. It is now easy to see that the remaining u1-integral for A→∞ vanishes, and we are left with the sum over the residues. Next we shift in the same way the u2-integral to −∞, and express  whenever j=3, r∈{1,2}  αj,αr≤3, τ1,τ2≥0,ℑν2≥ℑν1≥0 and show that they all satisfy the bound  or the term j=3, r=2 (if ρ<0) satisfies . By Stilarge by  +O(1) and can therefore be bounded by  and can therefore be bounded by X1=X2=1. It follows from  \documnated by . Similarnated by  the 4 tenated by . We procnated by  for X1=ows from  and  into three pairs ν1|<ε the second line can be bounded by the mean value theorem. Then we use the functional equation sΓ(s)=Γ(s+1) of the Gamma-function in connection with Stirling’s formula as before and bound the preceding display by ν1, this can again be estimated by the mean value theorem giving the crude bound α1, α2, α3 are pairwise distinct, that is ν1=0 follows by continuity. □Finally we need to treat the case ondition  is emptyondition  is also ondition  for X1=eturn to  and partible for . This coτ1, τ2 sufficiently large one has ν1,ν2∈iℝ in a neighborhood of iτ1, iτ2, respectively, and it is very small outside this region.An inspection of the proof, in particular –3.13), , 3.13), Γ=SL3(ℤ). We denote by Let ν1,ν2  in the notation of 3 is contained in a fundamental domain for ν1,ν2 is sufficiently large. □We conclude from Lemma 1 that as in  the uppesee e.g. ), we obtSL2(ℤ) and spectral parameter ν∈iℝ ). A cusp form u has a Fourier expansion Wν was defined in . Again we see that an arithmetically normalized cusp form u is essentially L2-normalized, and Wν has roughly norm one with respect to the inner product We briefly discuss cusp forms fined in . Similarhence by   the evows from , 6.576.4There are three types of Eisenstein series on the space minimal parabolic Eisenstein series: for ν1,ℜν2 sufficiently large we define the minimal parabolic Eisenstein series ν1 and ν2, and its non-zero Fourier coefficients are given by I-function.The first term gives rise to fined in . It has 5.1 . Then we define s, and as the minimal parabolic Eisenstein series it is an eigenform of all Hecke operators; in particular for s=1/2+μ it is an eigenform of T with eigenvalue c. This can be seen by setting m1=m2=1 and comparing with  denotes the least common multiple. In particular, ifAll implied constants depend only onε.It remains to show  which isD1∣D2, we put m1,n1 (mod D1) and n2 (mod D2/D1), and for p1q1≡1 (mod D1), p2q2≡1 (mod D2) we have . Analytic properties of We choose now With the notation developed so far, we haveLetC1,C2≥0, ε>0. ThenandIn the special case whenA1,A2≤1, X1=1, X2=X≥1, R1+R2≍τ1+τ2this can be improved toLetgbe a fixed smooth function with compact support in . Then forR1,R2≫1 sufficiently large we haveOn the left hand side we have suppressed the dependence ofonτ1, τ2.A1,A2 from below, and therefore D1,D2 in (X1=X2=1. It is instructive to compare this with the GL(2) situation: one can construct a sufficiently nice test function h on the spectral side with essential support on  such that the integral transforms h± in , , 8.8),  are not 1,D2 in  from abos h± in  are neglτ1,τ2 can be performed at almost no cost, in other words, we save a factor (R1R2)ε1− compared to trivial integration.The bound  shows thy-integrals in 2. Hence the second part of  defined by  instead of ε. To this end, we combine as before −1/2≍(R1+R2)−1/2. Our present assumption X1=1, X2=X≥1, A1,A2≤1 together with the size constraints ((y)≪j1 uniformly in all other variables. We can assume that ϵ1=sgn(x1) and |x1|≍τ1+τ2, otherwise we can save as many powers of τ1+τ2 as we wish by repeated partial integration. In that case we make another change of variables and re-write (τ1+τ2)−1/2: we cut out smoothly the region y1=1+O((τ1+τ2)−1/2) which we estimate trivially. For the rest we apply integration by parts. The treatment of the y2 integral is very similar. Here our assumptions imply that the integral is of the form τ1+τ2)−1/2.The proof of  is a smae C1 in  by C1+ηhe bound  follows s before  and the egral in  are bothstraints –8.16) i iR1+R2τ1,τ2. In the situation (ε+η(C1+C2) instead of ε. In order to show , this condition is empty unless A≥B≥0. Since we are assuming that Ξ1,Ξ2 are sufficiently large, we can deduce from (x3-region is x2) and hence the total contribution under the assumption −100) by the first part of (T1T2(T1+T2). On the spectral side, we drop the Eisenstein spectrum and large parts of the cuspidal spectrum to conclude by  balls. To prove the lower bound, we choose  K so large that O((T1+T2)ε) by known bounds for the zeta function on the line ℜs=1, the third line contributes similarly O((T1+T2)ε1+) by Weyl’s law for SL2(ℤ) and lower bounds for the L-functions in the denominator [T1,T2 sufficiently large. □For the  combine , 8.2), , proof oX2=1 in , fix a f part of  and  is O(T2). In order to prove Theorem 2, it is therefore enough to consider those Maaß forms with |ρj|≥ε. Moreover, by symmetry it is enough to bound only Maaß forms satisfying  The long Weyl element contributes at most X=T2 completes the proof of Theorem 2. □The tisfying . In . The left hand side of  and using  by an approximate functional equation. As we are summing over the archimedean parameters of the L-functions, we need an approximate functional equation whose weight function is essentially independent of the underlying family. This has been obtained in [ϕj as in Theorem 4 put n|=n(1)+⋯+n(6) and Finally we prove LetG0:→ℝ be a smooth function with functional equationG0(x)+G0(1/x)=1 and derivatives decaying faster than any negative power ofxasx→∞. LetM∈ℕ and fix a Maaß formϕas above. There are explicitly computable rational constantscn,ℓ∈ℚ depending only onn, ℓ, Msuch that the following holds forFor anyε>0 one haswhere |κj|=1 and the implied constant depends at most onε, M, and the functionG0.T is sufficiently large. Let s=σ>0. By (Cj≪ηjT2≪T3. This is at most O(Tε5+). By Theorem 1 or (O(Tε5+). This completes the proof of Theorem 4. □It is now a simple matter to prove Theorem 4. We can assume that =σ>0. By  and Mellem 3 and  the firsrem 1 or  it is ea


Then we study some properties about approximation of lattice ordered effect algebras.Many authors have studied roughness on various algebraic systems. In this paper, we consider a lattice ordered effect algebra and discuss its roughness in this context. Moreover, we introduce the notions of the interior and the closure of a subset and give some of their properties in effect algebras. Finally, we use a Riesz ideal induced congruence and define a function  Quantum effects play a basic role in the foundations of quantum mechanics. In 1994, Foulis and Bennett introduced effect algebras for modeling unsharp measurement in a quantum mechanical system . It is aThe theory of rough sets was first introduced by Pawlak  as a tooE can be regarded as (possibly) unsharp experimentally testable propositions about a physical system. A subset X of E can be regarded as (possibly) some unsharp experimentally testable propositions about a physical system. We think about a rough approximation of X to be the set of propositions which is indiscernible with some propositions in X. It means that two propositions p and q are observable simultaneously in terms of quantum measurement theory that p, q ∈ E are indiscernible.Events of quantum logics do not describe “unsharp measurements” since unsharp measurements do not have a “yes-no” character. To include such events another algebraic structure was introduced by Foulis and Bennett (1994), called an effect algebra. Hence elements of an effect algebra In this paper, we discuss the roughness of lattice ordered effect algebra and introduce the notions of the interior and the closure of a subset. We give some properties of the interior and the closure of a subset in lattice ordered effect algebras.In this section, we recall some definitions and results which will be used in the sequel.E with partial binary operation ⊕ and two nullary operations 0 and 1 satisfying the following axioms.(E1)a ⊕ b = b ⊕ a if a ⊕ b is defined.(E2)a ⊕ b) ⊕ c = a ⊕ (b ⊕ c) if one side is defined.((E3)a ∈ E there exists a unique b ∈ E such that a ⊕ b = 1.For every (E4)a is defined then a = 0.If 1 ⊕ An effect algebra is a partial algebra R, where a⊥b is defined by a + b ≤ 1, in which case a ⊕ b = a + b.(1) A simple example of an effect algebra is ⊆n-chain, Cn = {0, a, 2a,…, na = 1}, where ia⊥ja if and only if i + j ≤ n for i, j = 0,1, 2,…, n, in which case ia ⊕ ja = (i + j)a.(2) Another example is an H be a complex Hilbert space and let Φ(H) be the set of all bounded self-adjoint operators on H. The positive cone Φ(H)+ in Φ(H) is the set of all A ∈ Φ(H) that satisfy 〈Ax, x〉 ≥ 0 for all x ∈ H. We then write A ≤ B if B − A ∈ Φ(H)+. Letting 0 and 1 be the zero and identity operators, respectively, we have that 1 ∈ Φ(H)+ and (Φ(H), +, 0, ≤) is a partially ordered Abelian group. It can be checked that ℘(H) = Φ(H)+ is an effect algebra but not a Boolean effect algebra.(3) Let E, we can introduce a partial order ≤ on E: a ≤ b iff there exists c : a ⊕ c = b. We denote b ⊖ a = c iff a ⊕ c = b. It is easy to check that ⊖ is a well defined partial operation. In θ⊆X}. Here X0 is called the interior (or the lower rough approximation) of X, X, and X. Apr(X) is called the rough approximation of X (or the rough set) in  (see ,∀θ is a congruence and [0]θ = [p]θ = [q]θ = I and [p′]θ = [q′]θ = [1]θ = {p′, q′, 1}. Let X = {0,1}; then X0 = ∅ and Clearly U, θ) be an approximation space and let X be a subset of U.X is called a crisp set.If X0 = ∅, then X is called having an empty interior with respect to θ.If X. If X is called having an empty exterior with respect to θ.We define Let  be an approximation space. Clearly, ∅ and U are definable with respect to θ. The family of all crisp sets is denoted by Cri(Apr).Let  instead of the approximation space . Also, in this case we use the symbols XI0 instead of X0.Let E be an effect algebra, let I be a Riesz ideal of E, and let X be a subset of E. Then the following statements are equivalent:X is a crisp set;XI0 = X iff X is a union of some equivalence classes.Let The proof is trivial.The following properties of approximation spaces are well known and obvious. They are similar to Proposition  3.1 in  and PropE be an effect algebra and let I be a Riesz ideal of E. For the approximation space , for arbitrary subsets X, Y⊆E, and for each x ∈ X, one hasE and ∅ are crisp sets,XI0, X]I are crisp sets with respect to I,X⊆Y, then XI0⊆YI0 and if X∩Y)I0 = XI0∩YI0,(X ∪ Y)I0⊇XI0 ∪ YI0, denoted by Aprc(X) is defined by The rough complement of Apr(The proof is similar to the I = [a]I = {0, a}, [b]I = [1]I = {b, 1}. Therefore, we have XI0 = ∅, YI0 = {b, 1}, X ∪ Y)I0 = E, X ∪ Y)I0 ⊈ XI0 ∪ YI0, Then . We also can check that I = {0, a} is a Riesz ideal. Let X = {a, 1} and Y = {0, b, c} be subsets of E. Then [0]I = [a]I = {0, a}, [b]I = [c]I = {b, c}, [d]I = [1]I = {d, 1}, [e]I = {e}. Therefore, we have YI0 = {b, c}, X ∪ Y)I0 = {0, a, b, c}, X ∪ Y)I0 ⊈ XI0 ∪ YI0, Let X be a subset of an effect algebra E and let X∧ be the annihilator of X in E defined by X∧ = {a ∈ E : a∧x = 0, ∀x ∈ X}. Denote X⊥ = (X′)∧, where X′ = {x′ ∈ E : x ∈ X}.Let I be a Riesz ideal of effect algebra E and let X be a subset of E. ThenX⊥)I0 ⊆ (XI0)⊥,⊥ and again by X⊥)I0⊆(XI0)⊥. Similarly, we can prove (2), (3), and (4).(1) By The following example shows that inclusion symbols ⊆ in E; ⊕, 0,1) be a lattice ordered effect algebra in X = {a}, Y = {0, a} be subsets of E and let I = {0, a} be a Riesz ideal of E. It is easy to check that X⊥ = {0, a}, Y⊥ = {0}, [0]I = [a]I = I, [1]I = [b]I = {b, 1}. So we have (Y⊥)I0 = ∅, (YI0)⊥ = {0},  YI0)⊥ ⊈ (Y⊥)I0, in a lattice ordered effect algebra E and build a relationship between it and congruence classes. Then we study some properties about approximation in lattice ordered effect algebras.In this section, we define a function We give some examples of lattice ordered effect algebras.R is also a lattice ordered effect algebra.(1) In G be a partially ordered abelian group with an operator “+”. Let u ∈ G with u > 0, and let L : = G+ = {g ∈ G∣0 ≤ g ≤ u}. Then L can be organized into a lattice ordered effect algebra  by defining p ⊕ q iff p + q ≤ u, in which case p ⊕ q = p + q. In the effect algebra L we have a′ = u − a and the effect algebra partial order on L coincides with the restriction to L of the partial order on G. An effect algebra of the form G+  is called an interval effect algebra or, for short, an interval algebra.(2) Let E in (3) Effect algebra E, we define e = (a ⊖ (a∧b))∨(b ⊖ (a∧b)).In a lattice ordered effect algebra E be a lattice ordered effect algebra. Then the following properties hold for every a, b, c ∈ E:e = e,e = 0 if and only if a = b,e = e,e ≤ e∨e, if c ≤ a∧b.Let e = (a ⊖ (a∧b))∨(b ⊖ (a ⊖ b)) = (a∨b)⊖(a∧b), e = (a′∨b′) ⊖ (a∨b)′ = (a∧b)′ ⊖ (a∨b)′ = (1 ⊖ (a∧b))⊖(1 ⊖ (a∨b)), by e = (a∨b)⊖(a∧b). (iv): e = (a ⊖ (a∧b))∨(b ⊖ (a∧b)) = (a∨b)⊖(a∧b), e = (a ⊖ (a∧c))∨(c ⊖ (a∧c)) = (a∨c)⊖(a∧c), e = (b ⊖ (b∧c))∨(c ⊖ (b∧c)) = (b∨c)⊖(b∧c); if c ≤ a∧b, we have e∨e ≥ e.(i) and (ii) are obvious. (iii): I be a Riesz ideal in a lattice ordered effect algebra E. Then I is a lattice ideal and ~I is a lattice congruence. Moreover for all a, b ∈ E, a ~ Ib if and only if (a ⊖ (a∧b))∨(b ⊖ (a∧b)) ∈ I.Let I is a Riesz ideal in a lattice ordered effect algebra E, then for all a, b ∈ E, a ~ Ib if and only if e ∈ I.From the lemma, we have that if E be an effect algebra and let X be a subset of E. Then X is called convex if for every x, y ∈ X and z ∈ E, x ≤ z ≤ y implies z ∈ X.Let I be a Riesz ideal of effect algebra E. Then [a]I is convex for each a ∈ E.Let a ∈ E, take t, s ∈ [a]I with t ≤ s and let t ≤ z ≤ s. Since z ⊖ t ≤ s ⊖ t ~ I0, we have z ⊖ t ~ I0, too. Hence, z ~ It. That is z ∈ [a]I. So [a]I is convex for each a ∈ E.Given I be a Riesz ideal of linear ordered effect algebra E. If a ≤ b and [a]I ≠ [b]I, then for each t ∈ [a]I and s ∈ [b]I, t ≤ s.Let t ∈ [a]I and s ∈ [b]I such that s < t. We show that it is a contradiction. First, let t ≤ b. So we obtain s ≤ t ≤ b and by t ∈ [b]I which is a contradiction. Now let b ≤ t. Thus a ≤ b ≤ t and by b ∈ [a]I which is a contradiction again. Therefore for each t ∈ [a]I and s ∈ [b]I, t ≤ s.Suppose that there exist X is a subset of lattice ordered effect algebra E, it is easy to check that for every X, Y⊆E if we have X⊆Y, then X′⊆Y′.If E be a lattice ordered effect algebra and let I be a Riesz ideal of E. Let X be a subset of E. ThenXI0)′ = (X′)I0. ∈ I and y ∈ X. By e ∈ I and y′ ∈ X′, so y′ ∈ [x]I∩X′ and this implies that x]I∩X′ ≠ ∅. Suppose that y ∈ [x]I∩X′; then y′ ∈ [x′]I∩X. Similarly, (i) Let x ∈ (XI0)′, so x′ ∈ XI0 by x′]I⊆X. Hence for each y ∈ [x′]I, we have e ∈ I and hence e ∈ I by y′ ∈ [x]I⊆X′. So x ∈ (X′)I0. Conversely, let x ∈ (X′)I0, so [x]I⊆X′. Suppose that y ∈ [x]I⊆X′; then y′ ∈ [x′]I⊆X by x′ ∈ XI0. By X′ = {x′ : x ∈ X} again, we have x ∈ (XI0)′ and this proves part (ii).(ii) Let E be a lattice ordered effect algebra and let X be a subeffect algebra of E. ThenE,X is an ideal and I⊆X, then XI0 is a subeffect algebra of E.if Let X, then x ∈ X such that x ∈ [a]I; then x′ ~ Ia′, so [a′]I∩X ≠ ∅, therefore a⊥b, then there are x ∈ [a]I, y ∈ [b]I, such that x, y ∈ X and x⊥y. Since X is subeffect algebra of E, we have x ⊕ y ∈ X. Note that x ~ Ia and y ~ Ib; we have x ⊕ y ~ Ia ⊕ b and hence x ⊕ y ∈ [a ⊕ b]I∩X. That is, (i) Since 0,1 ∈ I⊆X. First we prove 0 ∈ XI0 and 1 ∈ XI0. For all x ∈ [0]I, by x ⊖ i = 0 ⊖ 0, i ≤ x, i ∈ I, thus x = i ∈ I⊆X, so [0]I⊆X, therefore 0 ∈ XI0. Similarly, 1 ∈ XI0 can be proved. Then we prove that x ∈ XI0 implies x′ ∈ XI0. For all y ∈ [x′]I, we have y′ ~ Ix. It follows that y′ ∈ X, then y ∈ X. Therefore x′ ∈ XI0. Let x, y ∈ XI0 and x⊥y. For all z ∈ [x ⊕ y]I, by z ⊖ i = (x ⊕ y) ⊖ j ≤ x ⊕ y ∈ X, i ≤ z, j ≤ x ⊕ y, i ∈ I. This yields that z ⊖ i ∈ X, thus z ∈ X. Therefore x ⊕ y ∈ XI0.(ii) Let II0 are subeffect algebras of E.By the above theorem, we note that E be a lattice ordered effect algebra. ThenE,E.Let E such that E = C(E) and an orthomodular effect algebra to be a lattice ordered effect algebra in which every element is principal.We define a Boolean effect algebra to be an effect algebra I be a Riesz ideal of a linear ordered effect algebra E and let X be a convex subset of E. Then XI0 and Let x, y ∈ XI0 and x ≤ z ≤ y. We divide the proof into three cases. First let x~Iy, we have x, y ∈ [a]I for some a ∈ E. By a]I is convex, so that z ∈ [a]I. It follows that [z]I = [a]I = [x]I⊆X; therefore z ∈ XI0. Next let [x]I ≠ [y]I and z ∈ [x]I. Then [z]I = [x]I. Since x ∈ XI0, [z]I = [x]I⊆X. So z ∈ XI0. Similarly for [x]I ≠ [y]I and z ∈ [y]I, we can prove that z ∈ XI0 and we omit it. Finally let [x]I ≠ [y]I, z]I⊆X. So let t ∈ [z]I. By u ∈ [x]I and v ∈ [y]I we have u ≤ t ≤ v. Since [x]I, [y]I⊆X, we have u, v ∈ X. Note that X is convex; we obtain t ∈ X and hence [z]I⊆X. It follows that z ∈ XI0. This shows that XI0 is convex.Let x ≤ z ≤ y. We divide the proof into three cases. First let x~Iy; we have x, y ∈ [a]I for some a ∈ E. By a]I is convex, so that z ∈ [a]I. It follows that [z]I = [a]I = [x]I and [x]I∩X ≠ ∅; therefore [z]I∩X ≠ ∅. Hence x]I ≠ [y]I and z ∈ [x]I. Then [z]I = [x]I. Since x]I∩X ≠ ∅ and hence [z]I∩X ≠ ∅. Therefore x]I ≠ [y]I and z ∈ [y]I, we can prove that x]I ≠ [y]I, z]I∩X ≠ ∅. Since x]I∩X ≠ ∅ and [y]I∩X ≠ ∅. So there exist u ∈ [x]I∩X and v ∈ [y]I∩X. By u ≤ z ≤ v. So z ∈ X since X is convex. This implies that [z]I∩X ≠ ∅ and hence Now, let E be a lattice ordered effect algebra, I and J two Riesz ideals of E, and X a subset of E.X is an ideal of E and I, J⊆X, then XI0∩XJ0 = XI∩J0,If X is crisp with respect to I or J, then If Let XI0∩XJ0⊆XI∩J0. Conversely, assume x ∈ XI∩J0. Then [x]I∩J⊆X and hence x ∈ X. Now, let y ∈ [x]I, so e ∈ I⊆X. Because X is an ideal and x ∈ X, so x∧y ∈ X. Hence we have e⊕(x∧y) ∈ X. On the other hand, by e = (x∨y)⊖(x∧y) and so y ≤ x∨y = e⊕(x∧y). This means that y ∈ X, which implies x ∈ XI0. Similarly, we can obtain x ∈ XJ0 and so x ∈ XI0∩XJ0. Therefore XI0∩XJ0 = XI∩J0.(i) By X is crisp with respect to I, so (ii) First, assume that E such that E = C(E) and an orthomodular effect algebra to be a lattice ordered effect algebra in which every element is principal.We define a Boolean effect algebra to be an effect algebra X and Y be subsets of E. We define X∨Y = {a∨b : a ∈ X, b ∈ Y} and define the downset X↓ = {c ∈ E : ∃a ∈ X, c ≤ a}.Let E be an orthomodular effect algebra and let I be a Riesz ideal of E, and let X, Y be subsets of E. Then Let a]I∩(X∨Y)↓ ≠ ∅. Thus there exists b ∈ [a]I∩(X∨Y)↓ and consequently there exist x ∈ X and y ∈ Y such that b ≤ x∨y. On the other hand, since [a]I = [b]I, by i, j ∈ I, i ≤ a, j ≤ b such that a = (b ⊖ j) ⊕ i ≤ b∨i ≤ x∨y∨i = (x∨i)∨y. By x∨i]I = [x]I, Let E be an orthomodular effect algebra and let I be a Riesz ideal of E, and let X, Y be subsets of E. Then (XI0∨YI0)↓⊆((X∨Y)↓)I0.Let a ∈ (XI0∨YI0)↓, so there exist x ∈ XI0 and y ∈ YI0 such that a ≤ x∨y. Now, let b ∈ [a]I; thus by by i, j ∈ I, i ≤ a, j ≤ b such that b = (a ⊖ i) ⊕ j ≤ a∨j ≤ x∨y∨j = (x∨j)∨y. By x∨j ∈ [x∨j]I = [x]I⊆X and y ∈ Y. Hence b ∈ X∨Y, this implies that a ∈ ((X∨Y)↓)I0.Suppose that The following example shows that we can not replace the inclusion symbol ⊆ by equal sign in E = {0, a, b, c, d, e, 1}, as the lattice-ordered effect algebra in X = {0}, Y = {0, b, c, 1} be the subsets of E and let I = {0, a} be the Riesz ideal of E. We have (X∨Y)↓ = E, XI0 = ∅ and YI0 = {b, c}. Therefore ((X∨Y)↓)I0 = E and (XI0∨YI0)↓ = ∅. This shows that (XI0∨YI0)↓ ≠ ((X∨Y)↓)I0.Consider E is a Boolean effect algebra.In what follows, that is, Propositions E be a Boolean effect algebra and let I, J be two Riesz ideals of E. Then (I∨J)↓ is also a Riesz ideal of E.Let a, b ∈ (I∨J)↓ and a⊥b; then a ≤ x1∨y1, b ≤ x2∨y2, where xi ∈ I, yi ∈ J, i = 1,2. Since a ⊕ b = a∨b, then a ⊕ b ≤ (x1∨y1)∨(x2∨y2) = (x1∨x2)∨(y1∨y2)∈(I∨J)↓. So (I∨J)↓ is an ideal. Let x ∈ (I∨J)↓; then x ≤ i∨j, where i ∈ I, j ∈ J. For a, b ∈ E, a⊥b and x ≤ a ⊕ b, we have x ≤ (a∨b)∧(i∨j). Taking a1 = a∧(i∨j) = (a∧i)∨(a∧j), b1 = (b∧i)∨(b∧j), we have a1, b1 ∈ (I∨J)↓ and a1 ≤ a, b1 ≤ b. Since E is a distributive lattice ordered effect algebra and  is an orthomodular lattice, it follows that a1 ⊕ b1 = (a∧(i∨j))⊕(b∧(i∨j)) = (a∧(i∨j))∨(b∧(i∨j)) = (a∨b)∧(i∨j). Therefore, x ≤ a1 ⊕ b1. Hence, (I∨J)↓ is a Riesz ideal.Let E be a Boolean effect algebra, and let I, J be two Riesz ideals of E and let X be a subset of E. Then Let X is a subset of E. Let x]I∨J)↓(∩X ≠ ∅. Thus there exists s ∈ [x]I∨J)↓∈(I∨J)↓ and s ∈ X; hence by h, k ∈ (I∨J)↓, h ≤ s, k ≤ x, such that x = (s ⊖ h) ⊕ k and also there exist i ∈ I, j ∈ J such that k ≤ i∨j. Then x = (s ⊖ h) ⊕ k ≤ s∨k ≤ s∨(i∨j) = (s∨i)∨(s∨j). On the other hand by s∨i]I = [s]I and [s∨j]J = [s]J. Therefore, Assume that The following example shows that the symbol inclusion in E = {0, a, b, 1} as the Boolean effect algebra in I = {0, a}, J = {0} are two ideals of E. And obviously, (I∨J)↓ = {0, a} is an ideal of E too. We have [0]I = I, [a]I = I, [b]I = {1, b}, [1]I = {b, 1}, [0]J = {0}, [a]J = {a}, [b]J = {b}, [1]J = {1}. Assume X = {b}; then Consider E be a Boolean effect algebra, I, J two Riesz ideals of E, and X a subset of E. Then XI∨J)↓(0⊆(XI0∨XJ0)↓. Furthermore, if a ∈ (XI0∨XJ0)↓, then one obtains [a]I∨J)↓(⊆(XI0∨XJ0)↓.Let a ∈ XI∨J)↓↓↓↓.Let a ∈ (XI0∨XJ0)↓. There exist x1 ∈ XI0 and x2 ∈ XJ0 such that a ≤ x1∨x2. Suppose that b ∈ [a]I∨J)↓↓, i ≤ a, j ≤ b such that b = (a ⊖ i) ⊕ j. Also, there are h ∈ I and k ∈ J such that j ≤ h∨k. Then b ≤ a∨j ≤ x1∨x2∨h∨k ≤ (x1∨h)∨(x2∨k). By x1∨h]I = [x1]I and [x2∨k]I = [x2]I. This implies that [a]I∨J)↓(⊆(XI0∨XJ0)↓.Now, suppose that From the above proposition, we have the following corollary.E be a Boolean effect algebra, and let I, J be two Riesz ideals of E and let X be a lattice ideal of E. Then one has XI∨J)↓(0 = (XI0∨XJ0)↓.Let XI∨J)↓(0⊆(XI0∨XJ0)↓. Since X is an ideal, we can show easily, (XI0∨XJ0)↓⊆X. Assume a ∈ (XI0∨XJ0)↓, so [a]I∨J)↓(⊆(XI0∨XJ0)↓⊆X from a ∈ XI∨J)↓(0. It yields that (XI0∨XJ0)↓⊆XI∨J)↓↓(0 = (XI0∨XJ0)↓.By E be a lattice ordered effect algebra and F a Boolean effect algebra. If I is a Riesz ideal of E and h : E → F is a monomorphism, then h(I) is a Riesz ideal of F.Let h(I) is an ideal. Let h(x), h(y) ∈ h(I) and h(x)⊥h(y), where x, y ∈ I. Since h is a monomorphism, then h(x) ⊕ h(y) = h(x ⊕ y) ∈ h(I). So h(I) is an ideal of F. Next, we prove that h(I) is a Riesz ideal. For any h(x) ∈ h(I), let h(x) ≤ a ⊕ b, a, b ∈ F, a⊥b. Take a1 = h(x)∧a, b1 = h(x)∧b; then a1, b1 ∈ h(I) and a1 ≤ a, b1 ≤ b. Since F is a distributive lattice ordered effect algebra, and  is an orthomodular lattice, it follows that a1 ⊕ b1 = (h(x)∧a)⊕(h(x)∧b) = h(x)∧(a∨b). Therefore, h(x) ≤ a1 ⊕ b1. Hence, h(I) is a Riesz ideal of F.First, we prove that E, F be lattice ordered effect algebras, let h : E → F be a morphism, and let Ih : = {a ∈ E : h(a) = 0} be the kernel of h.X is a subset of E, then If h is a monomorphism and  is a Boolean effect algebra. If X is a subset of E and I is a Riesz ideal of E containing Ih, then Assume that Let h(x) = y. So we have [x]Ih∩X ≠ ∅. Let a ∈ [x]Ih∩X. We have a ∈ [x]Ih and a ∈ X, so there exist i, j ∈ Ih, i ≤ a, j ≤ x such that a = (x ⊖ j) ⊕ i. Since h is a morphism, we get that h(a) = h(x ⊖ j) ⊕ h(i) = h(x) ⊖ h(j) ⊕ h(i) = h(x). It follows that h(a) = y ∈ h(X), which proves the theorem.(1) Since y = h(x). Since [x]I∩X ≠ ∅, suppose z ∈ [x]I∩X, so we have [z]I = [x]I and z ∈ X. Hence there exist i, j ∈ I, i ≤ z, j ≤ x such that z = (x ⊖ j) ⊕ i. Since h a monomorphism, then [h(z)]h(I) = [h(x)]h(I) and h(z) ∈ h(X). Therefore, (2) Let y]h(I)∩h(X) ≠ ∅. Suppose z ∈ [y]h(I)∩h(X); it implies that [y]h(I) = [z]h(I) and z ∈ h(X). Since h is a monomorphism, there exists x ∈ E such that y = h(x) and there exist t ∈ X and s ∈ I such that z = h(t) and e(h(t), h(x)) = h(s), so h) = h(s) and it implies that h ⊖ s) = 0. Hence e ⊖ s ∈ I. So e ∈ I and t ∈ [x]I by x]I∩X ≠ ∅, so Conversely, let In order to well illustrate the above theorem, we give the following example.E = {0, a, b, 1}, as the lattice ordered effect algebra in F = {0, y1, y2, 1}. For all y ∈ F, define y1 ⊕ y2 = y2 ⊕ y1 = 1, y ⊕ 0 = 0 ⊕ y = y; otherwise ⊕ is undefined. Then  is a Boolean effect algebra.Consider ϕ : E → F is given by ϕ(1) = 1, ϕ(a) = y1, ϕ(b) = y2, ϕ(0) = 0. We can easily check that ϕ is a monomorphism. Since kerϕ = {0}, so [0]kerϕ = {1}, [a]kerϕ = {a}, [b]kerϕ = {b}, [1]kerϕ = {1}. Suppose that X = {a, b}; then ϕ(X) = {y1, y2} and The map The earliest research on rough set theory of algebraic structures are on semigroups, groups, and modules. We set up the rough structure on partial algebraic system. The core foundation of rough set theory and application is a pair of approximation operators induced from the approximation space, namely, the closure operator and interior operator . We try to define rough approximation operators by a congruence relation on effect algebras and thus induce rough structure of effect algebras. Because effect algebras are incomplete algebras, not any two elements can compute, so we give a full operation on an effect algebra to construct a distance function. Moreover by use of Riesz ideals and distance functions we induced a congruence relation and then we obtain rough approximation operators and rough effect algebra system.


S = −ΔV/ΔT, the determination of the temperature plays an important role. We present a method for the determination of the temperature difference using a combination of a finite element simulation, which reveals the temperature distribution of the sample, and the measurement of the resistance change due to laser heating of sensor leads on both sides next to the junction. Our results for the measured thermopower are in agreement with recent reports in the literature.We report the development of a novel method to determine the thermopower of atomic-sized gold contacts at low temperature. For these measurements a mechanically controllable break junction (MCBJ) system is used and a laser source generates a temperature difference of a few kelvins across the junction to create a thermo-voltage. Since the temperature difference enters directly into the Seebeck coefficient  S = −ΔV/ΔT, where ΔV is the thermo-voltage and ΔT is the temperature difference. In general S is a function of energy and temperature:The energy and heat management in electronic devices has become a challenge in recent years due to the down-scaling of electronic components to the nanoscale, where the transport is governed by quantum-mechanical properties, which are partially not explored thoroughly yet. This includes solid-state semiconducting devices  and orgaEF is the Fermi energy, τ(E) is the transmission function, e is the electron charge, kB is the Boltzmann constant and T the temperature of the system  + c, where b = 36 µm and s = 7.6 µm. The formula to calculate the spot size leads to  All other parameters, A, b0, and c are not relevant for the determination.To estimate the diameter of the laser spot we used the knife edge technique. Thereby the laser spot is moved across a 36 µm wide  stripe of gold while the reflected intensity is detected with a photodiode. Due to the different reflectivity of gold and Kapton Cirlex there is an intensity profile corresponding to the convolution of a box function for the stripe and a 2D Gauss profile for the laser. It can be easily shown that the direction parallel to the stripe does not play any role, so the convolution reduces to a 1D integral, which leads to a difference of two error functions, which is used to fit the dataset. −1 K−1, specific heat cp = 128 J kg−1 K−1, mass density ρm = 19300 kg m−3, electrical resistivity ρ(T) = 8.73575 × 10−9 Ωm + 0.08325 × 10−9 Ωm K−1·T; polyimide: κ = 0.15 W m−1 K−1, cp = 1100 J kg−1 K−1, ρm = 1300 kg m−3; Kapton: κ = (−1.372384 × 10−3 + 5.601653 × 10−3·T + 2.082966 × 10−6·T2 − 5.05445 × 10−9·T3) W m−1 K−1 (for 5 K ≤ T < 140 K), κ = (−7.707532 × 10−3 + 5.769136 × 10−3·T + 5.622796 × 10−7·T2 − 4.329984 × 10−10·T3) W m−1 K−1 (for 140 K ≤ T ≤ 300 K) [cp = (2.809666 − 1.394349·T−1 + 0.2106639·T−2 + 4.752016 × 10−3·T−3 − 3.279998 × 10−4·T−4 + 4.282249 × 10−6·T−5) J kg−1 K−1 (for 4 K ≤ T < 30 K), cp = −86.86946 + 7.816917·T − 0.03664788 × 10−2·T−2 + 9.9128 × 10−5·T−3 − 1.08441 × 10−7·T−4 J kg−1 K−1 (for 30 K ≤ T ≤ 300 K) [m = 1420 kg m−3.For the simulations we used the following material parameters: Gold: Thermal conductivity κ = 320/1.85 W m


Let {Xn;n ≥ 1} be a sequence of independent copies of a real-valued random variable X and set Sn = X1+⋯+Xn, n ≥ 1. We say X satisfies the -Chover-type law of the iterated logarithm ) if X ∈CTLIL. We obtain sets of necessary and sufficient conditions for X∈CTLIL for the five cases: α = 2 and 0 < β <∞, α = 2 and β = 0, 1<α<2 and −∞<β<∞, α = 1 and −∞ <β <∞, and 0 < α <1 and −∞ <β <∞. As for the case where α = 2 and −∞ <β <0, it is shown that X∉CTLIL for any real-valued random variable X. As a special case of our results, a simple and precise characterization of the classical Chover law of the iterated logarithm ) is given; that is, X∈CTLIL if and only if α ≤ 2.Let 0 < Primary: 60F15; Secondary: 60G50 Xn; n ≥ 1} is a sequence of independent copies of a real-valued random variable X. As usual, the partial sums of independent identically distributed (i.i.d.) random variables Xn, n ≥ 1 will be denoted by Lx= log(e∨x), x ≥ 0.Throughout, {X has a symmetric stable distribution with exponent α∈, i.e., t∈, Chover When X is in the domain of normal attraction of a nonnormal stable law, Pakshirajan and Vasudeva X is in the domain of attraction of a nonnormal stable law, Scheffler X is in the generalized domain of operator semistable attraction of some nonnormal law, Chen and Hu This is what we call the classical Chover law of iterated logarithm (LIL). Since then, several papers have been devoted to develop the classical Chover LIL. See, for example, Hedye Motivated by the previous study of the classical Chover LIL, we introduce a general Chover-type LIL as follows.α ≤ 2 and −∞ <β <∞. Let {X,Xn; n ≥ 1} be a sequence of real-valued i.i.d. random variables. We say X satisfies the -Chover-type law of the iterated logarithm ) ifLet 0 < X∈CTLIL  holds with β=1/α) when X has a symmetric stable distribution with exponent α∈.From the classical Chover LIL and Definition 1.1, we see that X∈CTLIL. The main results are stated in Section 2. We obtain sets of necessary and sufficient conditions for X∈CTLIL for the five cases: α = 2 and 0 < β <∞ (see Theorem 2.1), α=2 and β = 0 (see Theorem 2.2), 1 < α < 2 and −∞<β <∞ (see Theorem 2.3), α = 1 and −∞ < β < ∞ (see Theorem 2.4), and 0 < α < 1 and −∞ < β < ∞ (see Theorem 2.5). The proofs of Theorems 2.1-2.5 are given in Section 4. For proving Theorems 2.1-2.5, three preliminary lemmas are stated in Section 3. Some llustrative examples are provided in Section 5This paper is devoted to a characterization of α = 2 and 0 < β <∞.The main results of this paper are the following five theorems. We begin with the case where β <∞. Let {X,Xn;n ≥ 1} be a sequence of i.i.d. real-valued random variables. ThenLet 0 < if and only ifα = 2 and β = 0, we have the following result.For the case where X,Xn;n ≥ 1} be a sequence of i.i.d. non-degenerate real-valued random variables. ThenLet {if and only ifIn either case, we havec be a constant. Note thatLet ∞ <β <0, X ∉CTLIL for any real-valued random variable X.Thus, from Theorem 2.2, we conclude that, for any −X ∈CTLIL for the three cases where 1 < α < 2 and −∞ < β < ∞, α = 1 and −∞ <β <∞, and 0 < α <1 and −∞ <β <∞ respectively.In the next three theorems, we provide necessary and sufficient conditions for α<2 and −∞ <β <∞. Let {X,Xn;n ≥ 1} be a sequence of i.i.d. real-valued random variables. ThenLet 1<if and only if∞ <β <∞. Let {X,Xn;n ≥ 1} be a sequence of i.i.d. real-valued random variables. ThenLet −if and only ifIn particular, α <1 and −∞ <β <∞. Let {X,Xn;n ≥ 1} be a sequence of i.i.d. real-valued random variables. ThenLet 0 < if and only ifX∈CTLIL) is obtained as follows. For 0 < α≤ 2, we haveFrom our Theorems 2.1, 2.3, 2.4, and 2.5, a simple and precise characterization of the classical Chover LIL ∞<β<∞ such that There exists a constant −Then we haveif and only if∞<β<∞ such that(ii) There exists a constant −if and only if∞<β<∞ such that(iii) There exists a constant −if and only ifProof We prove the sufficiency part of Part (i) first. It follows from (3.3) that the setNote thatWe thus conclude thatwhich ensures (3.2).b≠β, let h=(b+β)/2. ThenWe now prove the necessity part of Part (i). For all It follows from (3.2) that the setNote thatWe thus have thatNote that (3.1) implies thatThus it follows from (3.4) thati.e., (3.3) holds.We leave the proofs of Parts (ii) and (iii) to the reader since they are similar to the proof of Part (i). This completes the proof of Lemma 3.1. □The following result is a special case of Corollary 2 of Einmahl and Li b>0. Let {X,Xn;n ≥ 1} be a sequence of i.i.d. random variables. ThenLet if and only ifx ≥ 0.where The following result is a generalization of Kolmogorov-Marcinkiewicz-Zygmund strong law of large numbers and follows easily from Theorems 1 and 2 of Feller α<2 and −∞<b<∞. Let {X,Xn;n ≥ 1} be a sequence of i.i.d. random variables. ThenLet 0 < if and only ifα<2 or α=1 and −∞<b≤ 0.where In this section, we only give the proofs of Theorems 2.1-2.2. By applying Lemmas 3.1 and 3.3, the proofs of Theorems 2.3-2.5 involves only minor modifications of the proof of Theorem 2.1 and will be omitted.Proof of Theorem 2.1 We prove the sufficiency part first. Note that the second part of (2.2) implies thatb>β,We thus see that for all h=(b+β)/2. Since β<h<b, it follows from (4.1) thatwhere We thus conclude from (2.2) thatwhich, by applying Lemma 3.2, ensures thatβ>0, the second part of (2.2) implies thatSince which ensures thatcn=LLn, n ≥ 1. It then follows from (4.2) and (4.3) thatLet By Lemma 3.1, we see that (4.4) is equivalent toi.e., (2.1) holds.We now prove the necessity part. By Lemma 3.1, (2.1) is equivalent to (4.4) which ensures that (4.2) holds. By Lemma 3.2, we conclude from (4.2) thatβ<∞, it follows from (4.5) thatSince 0 < β1<β then, using the argument in the proof of the sufficiency part, we have thatIf Hence, by Lemma 3.1, (4.6) implies thatwhich is in contradiction to (2.1). Thus (2.2) holds. The proof of Theorem 2.1 is complete. □Proof of Theorem 2.2 Using the same argument used in the proof of the sufficiency part of Theorem 2.1, we have from (2.4) thatX is a non-degenerate random variable, by the classical Hartman-Wintner-Strassen LIL, we have thatSince which implies thatcn=LLn, n ≥ 1. It then follows from (4.7) and (4.8) thatLet By Lemma 3.1, we see that (4.9) is equivalent toi.e., (2.5) holds, so does (2.3).Using the same argument used in the proof of the necessity part of Theorem 2.1, we conclude from (2.3) thatClearlyThus (2.4) holds. The proof of Theorem 2.2 is therefore complete. □In this section, we provide the following examples to illustrate our main results. By applying Theorems 2.3-2.5, we rededuce the classical Chover LIL in the first example.α≤ 2. Let X be a symmetric real-valued stable random variable with exponent α. Clearly, α≤ 2.Let 0 < α<2, we haveFor 0 < it follows thatand hence thatX∈CTLIL .Thus, by Theorems 2.3-2.5, α=2, we have X∉CTLIL but X∈CTLIL.However, for X∈CTLIL for some certain α and β even if the distribution of X is not in the domain of attraction of the stable distribution with exponent α.From our second example, we will see that α≤ 2. Let dn= exp(2n), n ≥ 1. Given −∞<λ<∞. Let X be a symmetric i.i.d. real-valued random variable such thatLet 0 < whereThenX is not in the domain of attraction of the stable distribution with exponent α. Also α≤ 2 or α=1 and λ<0. It is easy to see thatThus the distribution of and hence thatα = 2 and 0 < λ<∞, then X∈CTLIL.If α=2 and −∞<λ≤ 0, then X∈CTLIL.If α<2, then X∈CTLIL.If 0 < Thus, by Theorems 2.1-2.5, we haveX may satisfy the other Chover-type LIL studied by Chen and Hu X∉CTLIL.Our third example shows that f(x) of X byDefine the density function p≠0, 0 < γ<1, c=c is a positive constant such that ∞<b<∞where α = 2 and p < 0, then X∈CTLIL.If α = 2 and p > 0, then X∉CTLIL for any 0≤ β <∞.If α < 2, then X ∉CTLIL for any −∞ <β <∞. However, for 0 < α < 2, by Theorem 3.1 in Chen and Hu If 0 < From Theorem 2.1-2.5, we haveB(x) is the inverse function of xα/ exp(p(logx)γ), x ≥e.where


We propose a quite concrete approach based on the notion of Köthe-Toeplitz duals with respect to the non-Newtonian calculus. Finally, we derive some inclusion relationships between Köthe space and solidness.The important point to note is that the non-Newtonian calculus is a self-contained system independent of any other system of calculus. Therefore the reader may be surprised to learn that there is a uniform relationship between the corresponding operators of this calculus and the classical calculus. Several basic concepts based on non-Newtonian calculus are presented by Grossman (1983), Grossman and Katz (1978), and Grossman (1979). Following Grossman and Katz, in the present paper, we introduce the sets of bounded, convergent, null series and  It is certainly not unusual to measure deviations by ratios rather than differences. For instance, during the Renaissance, many scholars, including Galileo, discussed the following problem. Two estimates, 10 and 1000, are proposed as the value of a horse, which estimates, if any, deviates more from the true value of 100? The scholars who maintained that deviations should be measured by differences concluded that the estimate of 10 was closer to the true value. However, Galileo eventually maintained that the deviations should be measured by ratios, and he concluded that two estimates deviated equally from the true value. From the story, the question comes out this way, what if we measure by ratios? The answer is the main idea of non-Newtonian calculus which consists of many calculuses such as the classical, geometric, anageometric, and bigeometric calculus.Bashirov et al. , 2 have bs*, cs*, and cs0* consisting of the sets of all bounded, convergent, null series based on the non-Newtonian calculus, as follows:ω* = {x = (xk) : xk ∈ C*for all  k ∈ N}. One can conclude that the sets bs*, cs*, and cs0* are complete non-Newtonian metric spaces with the metric d∞N defined byFollowing Tekin and Başar  we can cbv*, bvp*, and bv∞* of bounded variation sequences in the sense of non-Newtonian calculus, as follows:bv*, bvp*, and bv∞* are complete with corresponding metrics on the right-hand side with (Δx)k = xk ⊖ xk−1, x)k′ = xk ⊖ xk+1 for all k ∈ N.Secondly, we introduce several sets  generator is a one-to-one function whose domain is R and whose range is a subset of B⊆R, the set of real numbers. Each generator generates exactly one arithmetic, and conversely each arithmetic is generated by exactly one generator. As a generator, we choose the function β such that its basic algebraic operations are defined as follows:x, y ∈ R(N), where the non-Newtonian real field R(N)≔{β{x} : x ∈ R} as in [AR} as in .β-positive real numbers, denoted by R+(N), are the numbers x in R such that β-negative real numbers, denoted by R−(N), are those for which beta-zero, beta-one, β(0) and β(1). Further, β-integers turns out to be the following:The X be a nonempty set and let d* : X × X → R(N) be a function such that, for all x, y, z ∈ X, the following axioms hold:  x = y,(NM1)   d* = d*,(NM2)   (NM3)   Then, the pair  and d* are called a non-Newtonian metric space and a non-Newtonian metric on X, respectively.Let X =  be a non-Newtonian metric space. Then the basic notions can be defined as follows. x = (xk) is a function from the set N into the set R(N). The β-real number xk denotes the value of the function at k ∈ N and is called the kth term of the sequence.A sequence xn) in a metric space X =  is said to be ∗-convergent if for every given n0 = n0(ε) ∈ N and x ∈ X such that n > n0 and is denoted by *limn→∞xn = x or n → ∞.A sequence (xn) in X =  is said to be non-Newtonian Cauchy (∗-Cauchy) if for every n0 = n0(ε) ∈ N such that m, n > n0.A sequence  is defined as β(|β−1(x)|) and is denoted by |x|β. For each number x in B ⊂ R(N), x1 and x2 is defined by alpha-generator in  also are the ordered pair of arithmetics (β-arithmetic and α-arithmetic). The sets beta-generator generates beta-arithmetics, respectively. Definitions given for β-arithmetic are also valid for α-arithmetic.Suppose that α-arithmetic is used for arguments and β-arithmetic is used for values; in particular, changes in arguments and values are measured by α-differences and β-differences, respectively. The operators of this calculus type are applied only to functions with arguments in A and values in B. The ∗-limit of a function with two generators α and β is defined byf is ∗-continuous at a point a in A if and only if a is an argument of f and *limx→af(x) = f(a). When α and β are the identity function I, the concepts of ∗-limit and ∗-continuity are identical with those of classical limit and classical continuity.The important point to note here is that α-arithmetic to β-arithmetic is the unique function ι (iota) that possesses the following three properties:(i)ι is one to one.(ii)ι is from A onto B.(iii)u and v in A,For any numbers,  It turns out that ι(x) = β{α−1(x)} for every x in A and that n. Since, for example, α-arithmetic can readily be transformed into a statement in β-arithmetic.The isomorphism from C*; that is,Let C*, ⊕, ⊙) is a field. of ∗-complex numbers, the formal notation,N-series. Also, for integers, n ∈ N, the finite ∗-sums sn* = ∗∑∑k=0nzk* are called the partial sums of complex N-series. If the sequence ∗-converges to a complex number s*, then we say that the series ∗-converges and write s* = ∗∑∑k=0nzn*. The number s* is then called the ∗-sum of this series. If (sn)  ∗-diverges, we say that the series ∗-diverges or it is ∗-divergent.Given a sequence -triangle inequality).(ii)(iii)Let  and zk*, tk* ∈ C* for k ∈ {0,1, 2,3,…, n}. Then, For any Following Tekin and Başar , we can z* ∈ C* be an arbitrary element. The distance function d* is called ∗-norm of z* and is denoted by z1*, z2* ∈ C* we have d* is the induced metric from Let z* by z* but leaves the real part the same. ThusLet (i) Let α and β be the same generator functions and z* ∈ C*. Then the following condition holds(ii) Let C*, d*) is a complete metric space, where d* is defined by l∞*, c*, c0*, and lp*, p ≥ 1, are sequence spaces.The sets (b)λ* denote any of the spaces l∞*, c*, and c0* and z = (zk*), t = (tk*) ∈ λ*. Define d∞* on the space λ* by λ*, d∞*) is a complete metric space.Let (c)l∞*, c*, and c0* are Banach spaces with the norm ||z||∞* defined byThe spaces (d)lp* is Banach spaces with the norm ||z||p* defined byThe space The following statements hold.bs*, cs*, cs0* and bv*, bvp*, bv∞* consisting of all bounded, convergent, null series and the sets of bounded variation sequences in the sense of non-Newtonian calculus which correspond to the sets bs, cs, cs0 and bv, bvp, bv∞ over the complex field C, respectively.In the present section, we introduce the sets μ* denote any of the spaces bs*, cs*, and cs0* and z = (zk*), t = (tk*) ∈ μ*. Define d∞N on the space μ* byα, β operators and corresponding function ι = βα−1. Then,  is a complete metric space.Let cs* and cs0*, we prove the theorem only for the space bs*. Let the ∗-sums ∗∑∑k=0nzk*, ∗∑∑k=0ntk* ∈ C*, where z = (zk*), t = (tk*) ∈ C*. Then the following metric axioms in  (NM1) From  it can b d∞N = d∞N holds.(NM2) It is trivial that the condition  z = (zk*), t = (tk*), w = (wk*) ∈ C*. In fact by taking into account (NM3) We show that ∗triangle inequality in  Since the axioms (NM1)–(NM3) are satisfied,  is a non-Newtonian metric space. It remains to prove the completeness of the space bs*.Since the proof is similar to the spaces xm) be a ∗-Cauchy sequence in bs*, where xm = {x1m) = {xk(1), xk(2),…, xkm) and show that x ∈ bs*. From (r → ∞ and m > n0 we havexm) ∈ bs*, there exists k ∈ N. Thus, (m > n0k ∈ N whose right-hand side does not involve k. Hence (xk) is a bounded sequence of geometric complex numbers; that is, x = (xk) ∈ bs*. Also from (m > n0m → ∞ for an arbitrary ∗-Cauchy sequence (xm). Hence bs* is complete.Let , y = (yk) ∈ bv*, and (Δx)k′ = xk ⊖ xk+1. Then,  is a complete metric space. Similarly, one can conclude that the other bounded variation sets  and  are also complete.Let bv* is a Banach space with the norm ||x||bv* defined byThe space The idea of dual sequence space which plays an important role in the representation of linear functionals and the characterization of matrix transformations between sequence spaces was introduced by Köthe and Toeplitz , whose mλ*, μ* ∈ C*, the set S, defined byλ* and μ* for all k ∈ N. One can easily observe for a sequence space ν* of ∗-complex numbers that the inclusions S ⊂ S if ν* ⊂ λ* and S ⊂ S if μ* ⊂ ν* hold.In this section, we focus on the alpha-, beta-, and gamma-duals of the classical sequence spaces over non-Newtonian complex field. For λ* ⊂ ω* which are, respectively, denoted by {λ*}α, {λ*}β, and {λ*}γ, as follows:wk⊙zk) is the coordinatewise product of ∗-complex numbers w and z for all k ∈ N. Then {λ*}β is called beta-dual of λ* or the set of all convergence factor sequences of λ* in cs*. Firstly, we give a remark concerning the ∗-convergence factor sequences.Firstly, we define the alpha-, beta-, and gamma-duals of a set Throughout the text, we also use the notation “<” for a non-Newtonian linear subspace which was created in .∅ ≠ λ* ⊂ ω*. Then the following statements are valid. λ*}β is a sequence space if φ* < {λ*}β < ω*, where {λ* ⊂ μ* ⊂ ω*, then {μ*}β < {λ*}β.If λ* ⊂ {λ*}ββ : = ({λ*}β)β.φ*}β = ω* and {ω*}β = φ*.{Let w = (wk), m = (mk), and n = (nk)∈{λ*}β. (a)λ*}β < ω* holds from the hypothesis. We show that m ⊕ n ∈ {λ*}β for m, n ∈ {λ*}β. Suppose that l ∈ λ*. Then (mk⊙lk) ∈ cs* and (nk⊙lk) ∈ cs* for all l ∈ λ*. We can deduce thatIt is trivial that { m ⊕ n ∈ {λ*}β. Now, we show that t⊙w ∈ {λ*}β for any t ∈ C* and w = (wk)∈{λ*}β, since (wk⊙lk) ∈ cs* for all l ∈ λ*. Combining this with ((tk⊙wk)⊙lk) = tk⊙(wk⊙lk) ∈ cs* for all l ∈ λ* we get t⊙w ∈ {λ*}β. Therefore, we have proved that {λ*}β is a subspace of the space ω*.Hence, (d)ω*}β ⊂ φ*. Suppose that w = (wn)∈{ω*}β and z = (zn) are given with ∗-division by zn⊙wn = 1* if wn ≠ 0* and zn : = 1* otherwise. By taking into account the set φ* from inclusion (a), then there exists an integer N ∈ N for all n ≥ N such that Using (a) we need only to show { w⊙z ∈ cs* implies that w ∈ φ*. The rest is an immediate consequence of this part and we omitted the details.Further, Since the proof is trivial for the conditions (b) and (c), we prove only (a) and (d). Let c0*}β = {c*}β = {l∞*}β = l1*.{l1*}β = l∞*.{The following statements hold: l∞*}β ⊂ {c*}β ⊂ {c0*}β by l1* ⊂ {l∞*}β and {c0*}β ⊂ l1*. Now, consider that w = (wk) ∈ l1* and z = (zk) ∈ l∞* are given. By taking into account the cases (c)-(d) of w⊙z ∈ cs*. So the condition l1* ⊂ {l∞*}β holds.(a) Obviously {y = (yk) ∈ ω*∖l1*, we prove the existence of an x ∈ c0* with y⊙x ∉ cs*. According to y ∉ l1* we may confirm an index sequence (np) which is strictly increasing with x = (xk) ∈ c0* by xk : = (sgn*yk⊘p), the non-Newtonian complex signum function is defined byy = (yk) ∈ C*. Finally, by using α = β, we getnp−1 ≤ k < np. Therefore y⊙x ∉ cs* and thus y ∉ {c0*}β. Hence {c0*}β ⊂ l1*.Conversely, for a given l∞* ⊂ ({l∞*}β)β = {l1*}β since {l∞*}β = l1*. Now we assume the existence of a w = (wn)∈{l1*}β∖l∞*. Since w is unbounded there exists a subsequence (wnk) of (wn) and we can find a real number (k + 1)2 such that k ∈ N1. The sequence (xn) is defined by xn : = (sgn*(wnk)⊘(k + 1)2) if n = nk and x ∈ l1*. However,w ∉ {l1*}β, which contradicts our assumption and {l1*}β ⊂ l∞*. This step completes the proof.(b) From the condition (c) of ζ-duals .In addition to the statements in ∅ ≠ λ* ⊂ ω*. Then the following statements are valid:φ* < {λ*}α < {λ*}β < {λ*}γ < ω*; in particular, {λ*}ζ is a sequence space over C*;λ* < μ* < ω*, then {μ*}ζ < {λ*}ζ;if I is an index set, if λi* are sequence spaces, and if λ* : = ⋃i∈Iλi*, then 〈λ*〉ζ = ⋂i∈I{λi*}ζ, where the notation "〈〉" stands for the span of linear subspace over C*;λ* ⊂ {λ*}ζζ : = ({λ*}ζ)ζ.Let l∞* < cs* < bs*. We only show the cases (c) and (d) taking ζ = alpha. The rest of the parts can be obtained in a similar way.(c)λi* ⊂ 〈λ*〉 the following 〈λ*〉α ⊂ {λi*}α and 〈λ*〉α ⊂ ⋂i∈I{λi*}α hold by (b). On the other hand, if y ∈ ⋂i∈I{λi*}α, that is, y ∈ {λi*}α, then x⊙y ∈ l1* for all x ∈ λ* and therefore y ∈ {λ*}α ⊂ 〈λ*〉α.Now, as an immediate consequence (d)λ* ⊂ {λ*}ζζ. Let w ∈ λ*; then w⊙z ∈ l1* for all z ∈ {λ*}α; thus w ∈ {λ*}ζζ and λ* ⊂ {λ*}ζζ by (a).We can deduce The case (b) obviously is true, and (a) follows from λ* ≠ {λ*}ζζ as we get from ζ = β and λ* : = c0*. We have {c0*}ββ = l∞* ≠ c0*. This remark gives rise to the following definition.Here ζ ∈ {α, β, γ} and let λ* be a sequence space over the field C*. λ* is called ζ-space if λ* = {λ*}ζζ. Further, an α-space is also called a Köthe space or perfect sequence space.Let From λ* is a sequence space over the field C* and , then {λ*}ζ is a ζ-space; that is, {λ*}ζ = {λ*}ζζζ.If λ*}α = {λ*}β = {λ*}γ. This gives rise to the notion of solidity.Now we look for sufficient conditions for {X be a sequence space. Then X is called solid ifLet λ* be a sequence space over the field C*. Then λ* is solid if and only ifLet u = (uk) ∈ ω* and x = (xk) ∈ λ* with ξk, ηk ∈ R. Obviously the condition ψk, δk ∈ R. We obtain ψk2 + δk2 ≤ ξk2 + ηk2 since k ∈ N. We may choose a real number γk ∈ R with ψk2 + γk2 = ξk2 + ηk2 and consider w = (wk) defined by w ∈ λ* and ζk ∈ R such that ζk2 + δk2 = ξk2 + ηk2 it follows similarly that Let λ* < ω* is any sequence space over the field C*; then the following statements hold. λ* is a Köthe space, then λ* is solid.If λ* is solid, then {λ*}α = {λ*}β = {λ*}γ.If λ* is a Köthe space, then λ* is a ζ-space.If Consider that λ* < ω* be a sequence space over the field C*.(a)λ* is a Köthe space and u ∈ ω*, then u ∈ {λ*}αα if and only if the condition u⊙z ∈ l1* holds for all z ∈ {λ*}α. Besides this we obtain u = (uk) ∈ λ* and v = (vk) ∈ ω* and the statementIf  z ∈ {λ*}α. Therefore v⊙z ∈ l1*. Hence w ∈ λ* and λ* is solid over the field C*.holds for each (b)λ* is a solid sequence space over C*. To show {λ*}α = {λ*}β = {λ*}γ, it is sufficient condition to verify {λ*}γ < {λ*}α in v = (vk) ∈ {λ*}γ; that is,Consider that  λ*, for z = (zk) ∈ λ*, the condition u = (uk) ∈ λ* for all k ∈ N. Therefore by combining this with the inclusion (By taking into account solidness of nclusion  we deduc u⊙v ∈ l1* < cs*. Hence v ∈ {λ*}α and {λ*}γ < {λ*}α.holds and (c)This is an obvious consequence of Let φ*, ω*, lp*, c0*, and l∞* of sequences are solid.The sets c* and bv* of sequences are not solid; therefore none of them is a Köthe space.The sets ζ ∈ {α, β, γ}, thenFor each (i)l1*}ζ = l∞* and {l∞*}ζ = l1*;{(ii)ω*}ζ = φ* and {φ*}ζ = ω*.{ζ ∈ {α, β, γ} and c0* < μ* < l∞*, then {μ*}ζ = l1* and μ* ⊂ {μ*}ζζ = l∞*. In particular, {c0*}ζ = {c*}ζ = l1*, and each of c*, c0* is not a ζ-space.If The following statements hold. c0* and l∞* are solid, we know that {c0*}ζ = {l∞*}ζ = l1*. So the statements in (d) are obtained from That the specified spaces in cases (a-b) are solid is an immediate consequence of their definition. Additionally, the cases (i-ii) of (c) can be obtained ζ-duals of the spaces cs*, bs*, bv*, and bv0*. We will find that none of these sequence spaces is solid; in particular, none of them is a Köthe space.Next, we determine the cs*}α = {bv*}α = {bv0*}α = l1*; {bv0*} = bv*∩c0*,{cs*}β = bv*, {bv*}β = cs*, {bv0*}β = bs*, {bs*}β = bv0*;{cs*}γ = bv*, {bv*}γ = bs*, {bv0*}γ = bs*, {bs*}γ = bv*.{The following statements hold: cs*, bs*, bv*, and bv0* of sequences are β-spaces (ζ = beta), but they are not Köthe spaces. Moreover, the sets bs* and bv* of sequences are γ-spaces (ζ = gamma), whereas both cs* and bv0* are not γ-spaces. None of the spaces cs*, bs*, bv*, and bv0* is solid.In particular, the sets cs*}ζ, ζ ∈ {α, β, γ}, and the proofs of all other cases are quite similar.We prove the cases for the spaces {x = (xk) ∈ cs* and y = (yk) ∈ l1*. Then,x||cs* is defined by (y ∈ {cs*}α which gives the fact that l1* ⊂ {cs*}α.(a) Let fined by . Therefoy = (yk) ∈ {cs*}α∖l1*. Then to every natural number p we can construct an index sequence (np) with np < np+1 and p ∈ N. Define x = (xk) byβ-division β{(−1)k2p−} for np < k ≤ np+1. Then x = (xk) ∈ cs*. According to the choice of np, the inequalitiesp2p diverges. Thus, x⊙y ∉ l1*, which implies y ∉ {cs*}α. This contradicts the fact that y ∈ {cs*}α. Therefore {cs*}α ⊂ l1*.Conversely, suppose that bv0*}α ⊂ l1* holds as well if we take the sequence (xk) byThe condition {u = (uk) ∈ {cs*}β and w = (wk) ∈ c0*. Define the sequence v = (vk) ∈ cs* by vk = (wk ⊖ wk+1) for all k ∈ N. Therefore, ∗∑∑k⊙  vk  ∗-converges, butl1* ⊂ cs* yields (uk) ∈ {cs*}β ⊂ {l1*}β = l∞*. Then we derive by passing to the ∗-limit in  ∈ {c0*}β = {c0*}α = l1*; that is, u ∈ bv*. Therefore, {cs*}β⊆bv*.(b) Let ges, but∑∗ k=0 n u = (uk) ∈ bv*. Then, (uk ⊖ uk−1) ∈ l1*. Further, if v = (vk) ∈ cs*, the sequence (wn) defined by wn = ∗∑∑k=0nvk for all k ∈ N is an element of the space c*. Since {c*}α = l1*, the N-series ∗∑∑kwk⊙(uk ⊖ uk−1) is ∗-convergent. Also, we havewn) ∈ c* and (uk) ∈ bv* ⊂ c*, the right-hand side of inequality ⊙uk or ∗∑∑k=0∞uk⊙vk  ∗-converges. Hence bv*⊆{cs*}β.Conversely, suppose that  we have∑∗ k=m n ∈ {cs*}γ and v = (vn) ∈ c0*. Then, for the sequence (wn) ∈ cs* defined by wn = (vn ⊖ vn+1) for all n ∈ N, we can find a non-Newtonian real number n ∈ N. Since (vn) ∈ c0* and (un)∈{cs*}γ ⊂ l∞*, there exists a non-Newtonian real number n ∈ N. Therefore,uk ⊖ uk−1) ∈ {c0*}γ = {c0*}α = l1*; that is, (un) ∈ bv*. Therefore, since the inclusion {cs*}γ ⊂ bv* holds, we conclude that {cs*}γ = bv*, as desired.(c) By using (a), it is known that Although all arithmetics are isomorphic, only by distinguishing among them, we do obtain suitable tools for constructing all the non-Newtonian calculi. But the usefulness of arithmetics is not limited to the construction of calculi; we believe there is a more fundamental reason for considering alternative arithmetics; they may also be helpful in developing and understanding new systems of measurement that could yield simpler physical laws. average speed. The definition “distance traveled per unit time” is incomplete because it fails to provide a method of determining the average speed of an accelerated particle. The definition “distance divided by time,” though not incorrect, is a gross oversimplification that fails to reveal the underlying issues. Fortunately there is a completely satisfactory definition, which undoubtedly was known to Galileo. Then we isolate a constant in each given uniform motion by defining speed to be the distance traveled in any unit time-interval. Finally, for a particle that moves nonuniformly in a distance d in time t, we define the average speed to be the speed that a particle in uniform motion must have in order to travel a distance d in time t. In our opinion, neither the simplicity nor the obviousness of the answer, d/t, justifies its use as the definition of average speed .Of course, we can only speculate as to future applications of the non-Newtonian calculi. Perhaps they can be used to define new scientific concepts, to yield new or simpler scientific laws, to solve heretofore unsolved problems, or to formulate and solve new problems. For example, one constructs non-Newtonian calculi of functions of two or more real variables by choosing an arithmetic for each axis. It might even be profitable to seek deeper connections among the corresponding operators of the calculi. Like as, some of them are postponed to our future works.


We present a local convergence analysis of inexact Newton method for solving singular systems of equations. Under the hypothesis that the derivative of the function associated with the singular systems satisfies a majorant condition, we obtain that the method is well defined and converges. Our analysis provides a clear relationship between the majorant function and the function associated with the singular systems. It also allows us to obtain an estimate of convergence ball for inexact Newton method and some important special cases. F : Ω ⊂ Rn → Rm is a nonlinear operator with its Fréchet derivative denoted by DF; Ω is open and convex. In the case m = n, the inexact Newton method was introduced in  → R is continuously differentiable and convex, theng(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 DF(ζ) is not surjective (see .Let DF(x) is of full rank for each x ∈ Ω. Note that, for surjective-underdetermined systems, the fixed points of the Newton operator NF(x)∶ = x − DF(x)†F(x) are the zeros of F, while for injective-overdetermined systems, the fixed points of NF are the least square solutions of  = 0, DF(ζ) ≠ 0 and that DF satisfies the modified majorant condition , where r is given in (DF(x) ≤rank DF(ζ) for any x ∈ B and thatθ satisfies 0 ≤ θ < 1. Let {xk} be sequence generated by inexact Newton's method with any initial point x0 ∈ B∖{ζ} and the conditions for the residual rk and the forcing term λk:κ(A): = ||A†||||A|| denotes the condition number of A ∈ Rm×n. Then, {xk} converges to a zero ζ of DF(·)†F(·) in tk} is defined by  is given byλ = 0 (in this case λk = 0 and rk = 0) in If DF(x) is full column rank for every x ∈ B, then we have DF(x)†DF(x) = IRn. Thus,θ = 0. We immediately have the following corollary.If DF(x) ≤rank DF(ζ) and thatx ∈ B. Suppose that F(ζ) = 0, DF(ζ) ≠ 0 and that DF satisfies the modifed majorant condition ∖{ζ} and the condition †F(·) in tk} is defined by (θ = 0.Suppose thatrank ondition . Let {xondition  for the fined by  for θ = DF(ζ) is full row rank, the modified majorant condition (In the case when ondition  can be rondition .F(ζ) = 0 and DF(ζ) is full row rank and that DF satisfies the majorant condition , where r is given in (DF(x) ≤rank DF(ζ) for any x ∈ B and that condition ∖{ζ} and the conditions for the residual rk and the forcing term λk:xk} converges to a zero ζ of F(·) in tk} is defined by  is given by  = 0, DF(ζ) is full row rank and that DF satisfies the majorant condition , where r is given in (DF(x) ≤rank DF(ζ) for any x ∈ B and that condition ∖{ζ} and the conditions for the control residual rk and the forcing term λk:xk} converges to a zero ζ of f(·) in tk} is defined by  is invertible in hλ,θ is given by u R2 be endowed with the l1-norm. Consider the operator F : R2 → R2 defined byF is analytic on R2 and its derivative is given byDF(x)) = 1 and the Moore-Penrose inverse isk = 1,2,…, we can obtain thatx1, x2)T ∈ Ω, we haveζ = T. Then, ζ ∈ Ω satisfies F(ζ) = 0 andλk} such that 0 ≤ λk ≤ λmax = 1/2; see −1 exists andDF(ζ) is full row rank, we have DF(ζ)DF(ζ)† = IRm andDF(x) is full row rank; that is, rank DF(x) = rank DF(ζ). The proof is complete.Since xk} coincides with the sequence generated by inexact Newton's method (DF(ζ)†F(ζ)|| = ||ΠkerDF(ζ)⊥|| = 1, we haveB. So, xk} converges to ζ follows. Note that F(·) = DF(·)DF(·)†F(·); it follows that ζ is a zero of F.Let s method  for F^. fore, by , we can ondition  on B = 0, DF(ζ) is full row rank, and DF satisfies the majorant condition . Then we haveSuppose that ondition  on B is full row rank, we have DF(ζ)DF(ζ)† = IRm. It follows thatIRn − DF(ζ)†(DF(ζ) − DF(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 Using


C1-almost periodic solutions for this class of networks with time-varying delays are established. Two examples are given to show the effectiveness of the proposed method and results.On a new type of almost periodic time scales, a class of BAM neural networks is considered. By employing a fixed point theorem and differential inequality techniques,some sufficient conditions ensuring the existence and global exponential stability of  C1-almost periodic function is an important subclass of almost periodic functions. However, to the best of our knowledge, few authors have studied problems of C1-almost periodic solutions of BAM neural networks.It is well known that bidirectional associative memory (BAM) neural networks have been extensively applied within various engineering and scientific fields such as pattern recognition, signal and image processing, artificial intelligence, and combinatorial optimization –3. SinceC1-almost periodic solutions for continuous and discrete systems, respectively. Motivated by the above, our purpose of this paper is to study the existence and stability of C1-almost periodic solutions for the following BAM neural networks on time scales:T is an almost periodic time scale which will be defined in the next section; xi(t) and yj(t) correspond to the activation of the ith neurons and the jth neurons at the time t, respectively; ai(t), bj(t) are positive functions and they denote the rates with which the cells i and j reset their potential to the resting state when isolated from the other cells and inputs at time t; pji(t) and qij(t) are the connection weights at time t; γji(t), ρij(t) are nonnegative, which correspond to the finite speed of the axonal signal transmission at time t; Ii(t), Jj(t) denote the external inputs at time t; and fj and gi are the activation functions of signal transmission. For each interval J of R, we denote JT = J∩T.On the other hand, the theory of calculus on time scales  and there exist positive constants αj, βi such that u | , |v | ∈ R, i = 1,2,…, n, j = 1,2,…, m;where |H2)(ai(t) > 0, bj(t) > 0, γji(t) ≥ 0, ρij(t) ≥ 0, pji(t), qij(t), Ii(t), Jj(t) are bounded almost periodic functions on T, i = 1,2,…, n, j = 1,2,…, m.Throughout this paper, we assume the following:φk(·) denotes a real-valued bounded rd-continuous function defined on T, andSystem  is supplIn this section, we will first recall some basic definitions and lemmas which are used in what follows.T be a nonempty closed subset  of R. The forward and backward jump operators σ, ρ : T → T and the graininess μ : T → R+ are defined, respectively, byLet t ∈ T is called left-dense if t > infT and ρ(t) = t, left-scattered if ρ(t) < t, right-dense if t < supT and σ(t) = t, and right-scattered if σ(t) > t. If T has a left-scattered maximum m, then Tk = T∖{m}; otherwise, Tk = T. If T has a right-scattered minimum m, then Tk = T∖{m}; otherwise, Tk = T.A point f : T → R is right-dense continuous provided it is continuous at right-dense point in T and its left-side limits exist at left-dense points in T. If f is continuous at each right-dense point and each left-dense point, then f is said to be continuous function on T.A function y : T → R and t ∈ Tk, we define the delta derivative of y(t), yΔ(t), to be the number (if it exists) with the property that for a given ε > 0 there exists a neighborhood U of t such thats ∈ U.For y is continuous, then y is right-dense continuous, and if y is delta differentiable at t, then y is continuous at t.If y be right-dense continuous. If YΔ(t) = y(t), then we define the delta integral byLet r : T → R is called regressive ift ∈ Tk. The set of all regressive and rd-continuous functions r : T → R will be denoted by R = R(T) = R. We define the set R+ = R+ = {r ∈ R : 1 + μ(t)r(t) > 0, ∀t ∈ T}.A function r is a regressive function, then the generalized exponential function er is defined byp, q : T → R be two regressive functions; we defineIf p, q : T → R are two regressive functions; then,e0 ≡ 1 and ep ≡ 1;ep(σ(t), s) = (1 + μ(t)p(t))ep;ep) = ep/(1 + μ(s)p(s));ep = ep⊖;1/ep⊖)Δ = (⊖p)(t)ep⊖. is called an almost periodic function in t ∈ T uniformly for x ∈ D if the ε-translation set of fT for all ε > 0 and for each compact subset S of D; that is, for any given ε > 0 and each compact subset S of D, there exists a constant l > 0 such that each interval of length l contains a τ ∈ E{ε, f, S} such thatτ is called the ε-translation number of f and and l is called the inclusion length of E{ε, f, S}.Let α = {αn} and β = {βn} be two sequences. Then β ⊂ α means that β is a subsequence of α. We introduce the translation operator T, and Tαf = g means that For convenience, we introduce some notations. Let f ∈ C, and if for any given sequence α′ ⊂ Π and each compact subset S of D there exists a subsequence α ⊂ α′ such that Tαf exists uniformly on T × S, then f is called an almost periodic function in t uniformly for x ∈ D.Let f ∈ C1 is said to be a C1-almost periodic function, if f, fΔ are two almost periodic functions on T.A function x ∈ Rn, and let A(t) be an n × n rd-continuous matrix on T; the linear systemT if there exist positive constants k and α, projection P, and the fundamental solution matrix X(t) of (0 is a matrix norm on T.Let  X(t) of , satisfyA(t) is an almost periodic matrix function and f(t) is an almost periodic vector function.Consider the following linear almost periodic system:x(t):X(t) is the fundamental solution matrix of (If the linear system  admits eatrix of .ci(t) be an almost periodic function on T, where ci(t) > 0, −ci(t) ∈ R+, ∀t ∈ T, andT.Let t0 ∈ T, then F defined byf.Every rd-continuous function has an antiderivative. In particular, if p ∈ R and a, b, c ∈ T, thenIf By Lemmas f(t) is an rd-continuous function and c(t) is a positive rd-continuous function which satisfies that −c(t) ∈ R+. Lett0 ∈ T; then,Suppose that g is a real-valued almost periodic function on T and f : R → R is a Lipschitz function, then t → f(g(t)) is an almost periodic function on T.If f, g : T → R are almost periodic functions, then the following hold:f + g is almost periodic function;fg is almost periodic function.If x = T ∈ Rn+m to denote a column vector, in which the symbol T denotes the transpose of vector. We let |x| denote the absolute-value vector given by |x | = T and define ‖x‖ = maxi≤n+m1≤|xi|.First, for convenience, we introduce some notations. We will use AP(T) = {c(t) : c(t) be a bounded real-valued, almost periodic function on T}, AP1(T) = {c(t) : c(t), cΔ(t) ∈ AP(T)}, andφ ∈ B, if we define induced modulusB is a Banach space.Let H1), (H2), and the following hold:H3), t − ρij(t) ∈ T, ∀t ∈ T, i = 1,2,…, n, j = 1,2,…, m;−H4), φ ∈ B, we consider the following almost periodic differential equation:i≤n,1≤j≤m1≤{inft∈Tai(t), inft∈Tbj(t)} > 0, it follows from T. Thus, by T; that is,  ⊂ E. For any given φ ∈ E, it suffices to prove that ‖Φ(φ)‖B ≤ r0. By conditions (H1)–(H4), we haveE) ⊂ E.For any given quation:xiΔt−aitxThus, by xφitW=∫−∞quation:xiΔt−aitxquation:xiΔt−aitx we havesupt∈Txφφ, ψ ∈ E and combining conditions (H1) and (H4), we obtain thatTaking ain thatsupt∈TxφE to E. Since E is a closed subset of B, Φ has a fixed point in E, which means that  = (x1*(t), x2*(t),…, xn*(t), y1*(t),…, ym*(t))T of system (φ*(t) = (φ1*(t), φ2*(t),…, φn*(t), φn+1*(t),…, φn+m*(t))T is said to be globally exponentially stable. There exist a positive constant λ with ⊖λ ∈ R+ and M > 1 such that every solutionThe H1)–(H4) hold and supt∈Tμ(t) < +∞; then, system  hoC1-almost periodic solutionφ*(t) = (φ1*(t), φ2*(t),…, φn*(t), φn+1*(t),…, φn+m*(t))T. Suppose thatui(s) = xi(s) − xi*(s), vj(s) = yj(s) − yj*(s) and i = 1,2,…, n, j = 1,2,…, m, and the initial conditions of , we get∞) and Hi(ϵ) → −∞, ϵ → +∞, there exist Hi(ϵi) = 0, Hi(ϵ) > 0 for ϵ ∈  and eai−) and integrating on T, by ebj−) and integrating on T, we haveH4) we have M > 1. Thus, there exists λ ≤ λ0,λ ∈ R+ andt1 ∈ T, p > 1 and some k such thatH2)–(H4), we obtainC1-almost periodic solution of system  : R → R be arbitrary almost periodic functions; then, (H2)-(H3) hold. Let α1 = α2 = β1 = β2 = 1/4; then, (H1) holds. Next, let us check (H4); if we take r0 = 1, thenH4) holds for r0 = 1. By Theorems C1-almost periodic solution in the regionIn , take T T = Z:γji, ρij   : Z → Z be arbitrary almost periodic functions; then, (H2)-(H3) hold. Let α1 = α2 = β1 = β2 = 1/4; then, (H1) holds. Next, let us check (H4); if we take r0 = 1, thenH4) holds for r0 = 1. By Theorems C1-almost periodic solution in the regionIn , take T C1-almost periodic solutions for a class of neural networks with time-varying delays on a new type of almost periodic time scales are established. To the best of our knowledge, this is the first time to study the existence of C1-almost periodic solutions of BAM neural networks on time scales. Our methods that are used in this paper can be used to study other types of neural networks, such as Cohen-Grossberg neural networks and fuzzy cellular neural networks.In this paper, by using calculus theory on time scales, a fixed point theorem, and differential inequality techniques, some sufficient conditions ensuring the existence and global exponential stability of


We consider a new iterative method due to Kadioglu and Yildirim (2014) for further investigation. We study convergence analysis of this iterative method when applied to class of contraction mappings. Furthermore, we give a data dependence result for fixed point of contraction mappings with the help of the new iteration method. Recent progress in nonlinear science reveals that iterative methods are most powerful tools which are used to approximate solutions of nonlinear problems whose solutions are inaccessible analytically. Therefore, in recent years, an intensive interest has been devoted to developing faster and more effective iterative methods for solving nonlinear problems arising from diverse branches in science and engineering.D is a nonempty convex subset of a Banach space B, T is a self map of D, and {αn}n=0∞, {βn}n=0∞ are real sequences in .Very recently the following iterative methods are introduced in  and 2],,2], respWhile the iterative method  fails toS  for all n ∈ Nk=0∞αkβk = ∞ leads ton→∞||xn − un|| = 0. SinceDefineUsing the same argument as above one can easily show the implication (ii)⇒(i); thus it is omitted here.D, B, and T with fixed point x∗ be as in αn}n=0∞, {βn}n=0∞ be real sequences in  satisfyingn→∞αn = limn→∞βn = 0. limLet x0 = u0 ∈ D, consider iterative sequences {xn}n=0∞ and {un}n=0∞ defined by (un}n=0∞ converges to x∗ faster than {xn}n=0∞ does.For given fined by  and 2),,x0 = uThe following inequality comes from inequality  of TheorThe following inequality is due to . Let T : D → D be a mapping and for all T is a contraction with contractivity factor x∗ = 3; see B = R andes S [S* , Noor [1es S [S* , and NorS [S* [S  iteratioS [S* [S .We are now able to establish the following data dependence result.T satisfying condition  1/2 ≤ αn, (ii) βn ≤ αn for all n ∈ N, and (iii) ∑n=0∞αn = ∞. If Tx∗ = x∗ and n → ∞, then we haveε > 0 is a fixed number and δ ∈ .Let ondition . Let {xrated by  for T anIt follows from , 3), , , and assFrom assumption (i) we haveequality  becomesn→∞xn = x∗. Thus, using this fact together with the assumption Denote that


We define a new class of multivalent meromorphic functions using the generalised hypergeometric function. We derived this class related to conic domain. It is also shown that this new class of functions, under certain conditions, becomes a class of starlike functions. Some results on inclusion and closure properties are also derived. Mp denote the class of functions of the form p-valent in the punctured unit disc centred at origin E∗ = {z : 0 < |z| < 1} = E∖{0}. Also by f(z)≺g(z) we mean f(z) is subordinate to g(z) which implies the existence of an analytic function, called Schwartz function w(z) with |w(z)| < 1, for z ∈ E∗ such that f(z) = g(w(z)), where f(z) and g(z) are multivalent meromorphic functions. Note that if g is univalent in E then the above subordination is equivalent to f(0) = g(0) and f(E) ⊂ g(E).Let γ < 1 and k ∈ [0, ∞, qk,γ(z) for conic regions are convex univalent and given by R(t) is Legendre's complete elliptic integral of the first kind with t ∈ , and z ∈ E is chosen in such a way that k = cosh(πR′(t)/R(t)).The set of points, for 0 < qFs for complex parameters α1,…, αq and β1,…, βs with βj ≠ 0, −1, −2, −3,…, for j = 1,2, 3,…, s, is defined as q ≤ s + 1, q, s ∈ ℕ0 = ℕ ∪ {0}, and (α)n is the well-known Pochhammer symbol related to the factorial and the Gamma function by the relation The generalised hypergeometric function fined as Fsqα1,…,αHp : Mp → Mp as follows: Hp,q,s(α1) = Hp then the following identity holds for this operator:  Liu and Srivastava  defined MQp of meromorphic function associated with conic domain, for k ≥ 0, 0 ≤ λ < 1, and p ≥ 1, as follows: Shareef  defined MQp of meromorphic function associated with conic domain, for k ≥ 0, 0 ≤ λ < 1, p ≥ 1, and b ≥ 1, as follows: qk,γ is a convex and univalent function, for h(z)≺qk,γ(z) it means h(E∗) is contained in qk,γ(E∗), where Hp,q,s(α1).We now define a new subclass h2(z) be convex in E and ℜ(λh2(z) + μ) > 0, where μ ∈ ℂ, λ ∈ ℂ∖{0}, and z ∈ E. If h1(z) is analytic in E, with h1(0) = h2(0), then Let h(z) = 1 + ∑n=1∞cnzn and H(z) = 1 + ∑n=1∞dnzn and h≺H. If H(z) is univalent and convex in E, then n ≥ 1.Let qk,γ(z) = 1 + q1z + q2z2 + ⋯ then If MQp.In this section we explore some of the geometric properties exhibited by the class MQp.We begin by discussing an inclusion property for the class ℜ(α1) > pℜ − 1) then If f ∈ MQp and set z we get f ∈ MQq, therefore ℜ + α1 + p) > 0 or equivalently ℜ(α1) > pℜ − 1). Hence, f ∈ MQp.Let ntiating  with resz we get zHα1fz′=α and set −1p1+1bzHPutting  is closed under a certain integral.We now show that the class f(z) ∈ MQp, then the integral f(z) into MQp.If g(z)≺qk,γ(z) which implies From  we have zηBηHαfz=tiating .Now we get coefficient estimates of the class f(z) ∈ MQp and f(z) is given by (k ≤ n.If f(z) ∈ MQp; then, by definition, we have Let H(α1)f(z) = f(z); then h(z) = 1 + ∑n=1∞cnzn, then  is univalent and qk,γ(E) is convex, applying Rogosinski's theorem we have q1 is given in  giv giv(44)A


The eigenvalue complementarity problem (EiCP) is a kind of very useful model, which is widely used in the study of many problems in mechanics, engineering, and economics. The EiCP was shown to be equivalent to a special nonlinear complementarity problem or a mathematical programming problem with complementarity constraints. The existing methods for solving the EiCP are all nonsmooth methods, including nonsmooth or semismooth Newton type methods. In this paper, we reformulate the EiCP as a system of continuously differentiable equations and give the Levenberg-Marquardt method to solve them. Under mild assumptions, the method is proved globally convergent. Finally, some numerical results and the extensions of the method are also given. The numerical experiments highlight the efficiency of the method. A ∈ Rn×n and the matrix B ∈ Rn×n, which are positive definite matrix, then we consider to find a scalar λ ∈ R and a vector x ∈ Rn∖{0}, such thatEigenvalue complementarity problem (EiCP) is proposed in the study of the problems in mechanics, engineering, and economics. The EiCP is also called cone-constrained eigenvalue problem in –4. The EThis paper is organized as follows. In Mn(R) denotes a real matrix of order n and Mn,m(R) denotes a real matrix of order n × m. 0n =  and en = .Throughout the paper, In this section, firstly, we reformulate  as a sysf : Rn+1 → Rn is a continuously differentiable function and the matrix A ∈ Rn×n. Using the nonlinear complementarity problem (NCP) function ϕ : R2 → R, which satisfied the following basic propertyω solves the EiCP operty, . Thus, xample, [F(ω)=Ψ(ω)Now, we give the Levenberg-Marquardt method for solving . The gloα < 1, 0 < β < 1, p > 2, ρ > 0, 0 < δ ≤ 2. P > 0, ϵ > 0, ω0 ∈ Rn+1, μ0 = ||F(ω0)||δ, and k : = 0.Given 0 < ω)||≤ϵ, then stop. Otherwise, compute dk byIf ||∇Φ(ωk)Tdk + ρ||dk||P > 0, let dk = −∇Φ(ωk); otherwise, dk is computed by ||δ and k : = k + 1, and go to Let Now, we give the global convergence of ωk}, k = 1,2,… generated by Suppose that {ωk}K → ω⋆, {ωk}K is a subsequence of {ωk}, and k = 1,2,…. When there are infinitely many k ∈ K such that dk = −∇Φ(ωk), by Proposition  1.16 in [ωk}K is a convergent subsequence of {ωk}, then dk is always computed by (ωk}K for whichωk → ω⋆. Suppose that ω⋆ is not a stationary point of Φ. By (ωk)|| = 0. Then, the algorithm has stopped. On the other hand, we know that there exists a constant ζ > 0 such thatωk)Tdk ≤ −ρ||dk||P and the fact that the gradient ∇Φ(ωk) is bounded on the convergent sequence {ωk}, we get or whichlimk∈Kk→k→∞dk<∞,lim supuch thatF'ωkTF'ωkm such thatm(0) = 0, m(k) = min{M0, m(k − 1) + 1}, and M0 > 0 is a integer.In dk in dk ∈ Rn of the equationrk is the residuals and satisfiesαk ≤ a < 1 for every k.In ρ = 10, P = 3, α = 0.1, and ϵ = 10−4.We give some numerical experiments for the method. And we compare We considerBy , we knowWe considerFrom , we alsoWe considerFrom , we alsoDiscussion. In this section, we study the numerical behaviors of λ, μ) ∈ R × R and  ∈ Rn∖{0} × Rm∖{0} such thatA ∈ Mn(R), B ∈ Mn,m(R), B ∈ Mm,n(R), and D ∈ Mm(R).We also can use f(ω) = x, g(ω) = y, P(ω) = λx − Ax − By, and Q(ω) = μy − Cx − Dy. We can write the above bieigenvalue complementarity problems as f(ω) ≥ 0, g(ω) ≥ 0, P(ω) ≥ 0, Q(ω) ≥ 0, fT(ω)P(ω) = 0, gT(ω)Q(ω) = 0, ω = 1, and ω = 1. By using F-B function, similar to rewriting the EiCP, we can rewrite the above bieigenvalue complementarity problems as the following equations:ρ = 10, P = 3, α = 0.1, and ϵ = 10−4.Let A = 0, D = 0,We consider the BECP, where We use M standing for the pencil associated to a finite collection {A0, A1,…, Ar} of real matrices of order n andWe consider using We consider the quadratic pencil model, whereIn this paper, we reformulate the EiCP as a system of continuously differentiable equations and use the Levenberg-Marquardt method to solve them. The numerical experiments show that our method is a promising method for solving the EiCP. The numerical experiments of the extensions confirm the efficiency of our method.