Covariance estimation is essential yet underdeveloped for analysing multivariate functional data. We propose a fast covariance estimation method for multivariate sparse functional data using bivariate penalized splines. The tensor‐product B‐spline formulation of the proposed method enables a simple spectral decomposition of the associated covariance operator and explicit expressions of the resulting eigenfunctions as linear combinations of B‐spline bases, thereby dramatically facilitating subsequent principal component analysis. We derive a fast algorithm for selecting the smoothing parameters in covariance smoothing using leave‐one‐subject‐out cross‐validation. The method is evaluated with extensive numerical studies and applied to an Alzheimer's disease study with multiple longitudinal outcomes. Let x(k)k=1,…,p be a set of p random functions with each function in L2(T). Assume that the p‐dimensional vector x(t)=x(1),…,x(p)⊤∈Rp has a p‐dimensional smooth mean function,μ(t)=E{x(t)}=Ex(1)(t),…,Ex(p)(t)⊤=μ(1)(t),…,μ(p)(t)⊤. Define the covariance function asC=E(x(s)−μ(s))(x(t)−μ(t))⊤=Ckk′1≤k,k′≤p and Ckk'=Covx(k)(s),x(k')(t). Then, the covariance operator Γ:H→H associated with the kernel C can be defined such that for any f∈H, the kth element of Γf is given by Ck=,…,Ckp)⊤. Note that Γ is a linear, self‐adjoint, compact, and non‐negative integral operator. By the Hilbert–Schmidt theorem, there exists a set of orthonormal bases {Ψℓ}ℓ≥1∈H, Ψℓ=Ψℓ(1),…,Ψℓ(p)⊤, and <Ψℓ,Ψℓ′>H=∑k=1p∫Ψℓ(k)(t)Ψℓ′(k)(t)dt=1ℓ=ℓ′, such that dℓ is the ℓth largest eigenvalue corresponding to Ψℓ. Then, the multivariate Mercer's theorem gives Ckk′=∑ℓ=1∞dℓΨℓ(k)(s)Ψℓ(k′)(t). As shown in Saporta (x(t) has the multivariate Karhunen–Loève representation,x(t)=μ(t)+∑ℓ=1∞ξℓΨℓ(t), where ξℓ=<x−μ,Ψℓ>H are the scores with E(ξℓ)=0 and E(ξℓξℓ′)=dℓ1ℓ=ℓ′. The covariance operator Γ has the positive semidefiniteness property; that is, for any a=⊤∈Rp, the covariance function of a⊤x, denoted by Ca, satisfies that for any sets of time points ⊂T with an arbitrary positive integer q, the square matrix [Ca]{1≤i,j≤q}∈Rq×q is positive semidefinite.Let  Saporta , x(t) ha2.2yij(k),tij(k):i=1,…,n;k=1,…,p;j=1,…,mik, where tij(k)∈T is the observed time point, yij(k) is the observed kth response, n is the number of subjects, and mik is the number of observations for subject i's kth response. The model is xi(t)=xi(1)(t),…,xi(p)(t)⊤∈H, ϵij(k) are random noises with zero means and variances σk2 and are independent across i,j, and k.Suppose that the observed data take the form Ckk′. We adopt a three‐step procedure. In the first step, empirical estimates of the covariance functions are constructed. Let rij(k)=yij(k)−μ(k)tij(k) be the residuals and Cij1j2(kk′)=rij1(k)rij2(k′) be the auxiliary variables. Note thatECij1j2(kk′)=Ckk′tij1(k),tij2(k′)+σk21{k=k′,j1=j2} for 1 ≤ j1 ≤ mik,1 ≤ j2 ≤ mik′. Thus, Cij1j2(kk′) is an unbiased estimate of Ckk′tij1(k),tij2(k′) whenever k≠k′ or j1≠j2. In the second step, the noisy auxiliary variables are smoothed to obtain smooth estimates of the covariance functions. For smoothing, we use bivariate P‐splines . Let r^ij(k)=yij(k)−μ^(k)tij(k) and Ĉij1j2(kk′)=r^ij1(k)r^ij2(k′), the actual auxiliary variables.The goal is to estimate the covariance functions P‐splines model Ckk′ uses tensor‐product splines Gkk′ for 1 ≤ k,k′ ≤ p. Specifically, Gkk′=∑1≤γ1,γ2≤cθγ1γ2(kk′)Bγ1(s)Bγ2(t), where Θkk′=θγ1γ2(kk′)1≤γ1,γ2≤c∈Rc×c is a coefficient matrix, {B1(·),…,Bc(·)} is the collection of B‐spline basis functions in T, and c is the number of equally spaced interior knots plus the order (degree plus 1) of the B‐splines. Because Ckk′=Ck′k=Covx(k)(s),x(k′)(t), it is reasonable to impose the assumption that Gkk′=Gk′k. Therefore, in the rest of the section, we consider only k ≤ k′.The bivariate D∈R(c−2)×c denote a second‐order differencing matrix such that for a vector a=⊤∈Rc,Da=⊤∈Rc−2. Also let ‖·‖F be the Frobenius norm. For the cross‐covariance function Ckk′ with k<k′, the bivariate P‐splines estimate the coefficient matrix Θkk′ by Θ^kk′, which minimizes the penalized least squares λkk′1 and λkk′2 are two non‐negative smoothing parameters that balance the model fit and smoothness of the estimate and will be determined later. Indeed, the column penalty ‖DΘkk′‖F2 penalizes the second‐order consecutive differences of the columns of Θkk′ and similarly, the row penalty ‖DΘkk′⊤‖F2 penalizes the second‐order consecutive differences of the rows of Θkk′. The two penalty terms are essentially penalizing the second‐order partial derivatives of Gkk′ along the s and t directions. The two smoothing parameters are allowed to differ to accommodate different levels of smoothing along the two directions.Let Ckk with k=1,…,p, we conduct bivariate covariance smoothing by enforcing the following constraint on the coefficient matrix Θkk  is a symmetric function. Then, the coefficient matrix Θkk and the error variance σk2 are jointly estimated by Θ^kk and σ^k2, which minimize the penalized least squares Θkk and λk is a smoothing parameter. Note that the two penalty terms in Equation Θkk is symmetric, and thus, only one smoothing parameter is needed for auto‐covariance estimation.It follows that 2.2.1k and k′ with k ≤ k′. Let θkk′=vec(Θkk′)∈Rc2 be a vector of the coefficients and b(t)={B1(t),…,Bc(t)}⊤∈Rc denotes the B‐spline base. Then, Ĉij1j2(kk′) for each pair of k and k′. Let ri(k)=ri1(k),…,rimik(k)⊤∈Rmik, C^i(kk′)=ri(k)⊗ri(k′)∈Rmikmik′ and C^(kk′)=C^1(kk′),⊤,…,C^n(kk′),⊤⊤∈RNkk′, where Nkk′=∑i=1nmikmik′ is the total number of auxiliary responses for the pair of k and k′. As for the B‐splines, let bi(k)=bti1(k),…,btimik(k)∈Rc×mik, Bi(kk′)=bi(k′)⊗bi(k)⊤∈R(mikmik′)×c2, and B(kk′)=B1(kk′),⊤,…,Bn(kk′),⊤⊤∈RNkk′×c2.We first introduce the notation. Let vec(·) be an operator that stacks the columns of a matrix into a column vector and denote by ⊗ the Kronecker product. Fix Ckk′ with k<k′, the penalized least squares in Equation P1=Ic⊗D⊤D and P2=D⊤D⊗Ic. The expression in Equation θkk′. Therefore, we derive that Ckk′ is Ĉkk′={b(t)⊗b(s)}⊤θ^kk′.For estimation of the cross‐covariance functions χk∈Rc(c+1)/2 be a vector obtained by stacking the columns of the lower triangle of Θkk, and let Gc∈Rc2×c(c+1)/2 be a duplication matrix such that θkk=Gcηk =vec(Imik)∈Rmik2 and Z(k)=Z1(k),⊤,…,Zn(k),⊤⊤∈RNkk. Finally, let βk=χk⊤,σk2⊤∈Rc˜ with c˜=c(c+1)/2+1. It follows that the penalized least squares in Equation X(k)=B(kk),Z(k)∈RNkk×c˜ and Q=blockdiagGc⊤(Ic⊗DD⊤)Gc⊤,0∈Rc˜×c˜. Therefore, we obtain θ^kk=Gcχ^k, and the estimate of the auto‐covariance function Ckk is Ĉkk={b(t)⊗b(s)}⊤θ^kk.Let The above estimates of covariance functions may not lead to a positive semidefinite covariance operator and thus have to be refined. We pool all estimates together, and we shall use the following proposition.Proposition 1Ckk′=b(s)⊤Θkk′b(t). Let G=∫b(t)b(t)⊤dt∈Rc×c and assume that G is positive definite ,⊤,…,uℓ(p),⊤⊤∈Rpc is the associated eigenvector with uℓ(k)∈Rc and such that Ψℓ(k)(t)=b(t)⊤G−12uℓ(k).Assume that Ψℓ(k)(t) are linear combinations of the B‐spline basis functions, which means that they can be straightforwardly evaluated, an advantage of spline‐based methods compared with other smoothing methods for which eigenfunctions are approximated by spectral decompositions of the covariance functions evaluated at a grid of time points.The proof is provided in Appendix A. Proposition Θ^kk′, the estimate of the coefficient matrix Θkk′, the spectral decomposition of [G12Θ^kk′G12]kk′ gives us estimates d^ℓ and u^ℓ=u^ℓ(1),⊤,…,u^ℓ(p),⊤⊤. We discard negative d^ℓ to ensure that the multivariate covariance operator is positive semidefinite, and this leads to a refined estimate of the coefficient matrix Θkk′, Θ˜kk′=G−12∑ℓ:d^ℓ>0d^ℓu^ℓ(k)u^ℓ(k′),⊤G−12. Then, the refined estimate of the covariance functions is C˜kk′=b(s)⊤Θ˜kk′b(t). Proposition Ψ˜ℓ(k)(t)=b(t)⊤G−12u^ℓ(k).Once we have ion of G2Θ^kk′G12For principal component analysis or curve prediction in practice, one may select further the number of principal components by either the proportion of variance explained (PVE) −1B⊤∈RN×N, Si=Bi(B⊤B+λ1P1+λ2P2)−1B⊤∈Rmi2×N, and Sii=Bi(B⊤B+λ1P1+λ2P2)−1Bi⊤∈Rmi2×mi2.We shall also now suppress the subscript (Imi2−Sii)−2=Imi2+2Sii, which results in the generalized cross‐validation, denoted by iGCV, Then, a shortcut formula for Equation nd Huang  and Xiaond Huang , the iCVS is of dimension 2,500×2,500 if n=100 and mi=m=5 for all i. Thus, we need to further simplify the formula.Although iGCV is much easier to compute than iCV, the formula in Equation Gn=B⊤B, B˜=BGn−1/2∈RN×c2, B˜i=BiGn−1/2∈Rmi2×c2, f=B˜⊤C^∈Rc2, fi=B˜i⊤C^i∈Rc2, and Li=B˜i⊤B˜i∈Rc2×c2. Also let P˜1=Gn−1/2P1Gn−1/2∈Rc2×c2, P˜2=Gn−1/2P2Gn−1/2∈Rc2×c2, and ∑=Ic2+λ1P˜1+λ2P˜2. Then, Equation Let Σ has two smoothing parameters. Following Wood (∑=I+ρ{wP˜1+(1−w)P˜2}, where ρ=λ1+λ2 represents the overall smoothing level and w=λ1ρ−1∈ is the relative weight of λ1. We conduct a two‐dimensional grid search of , as follows. For a given w, let Udiag(s)U⊤ be the eigendecompsition of wP˜1+(1−w)P˜2, where U∈Rc2×c2 is an orthonormal matrix and s=∈Rc2 is the vector of eigenvalues. Then, ∑−1=Udiag(d˜)U⊤ with d˜=1/(1+ρs)∈Rc2.Note that ing Wood , we use Proposition 2f˜i=U⊤fi∈Rc2, f˜=U⊤f∈Rc2, g=f˜⊙f˜−∑i=1nf˜i⊙f˜i∈Rc2, L˜i=U⊤LiU∈Rc2×c2, and F=∑i=1n(f˜if˜⊤)⊙L˜i∈Rc2×c2.Let ⊙ stand for the point‐wise multiplication. Then, w, note that only d˜ depends on ρ and needs to be calculated repeatedly, and all other terms need to be calculated only once. The entire algorithm is presented in Algorithm 1. We give an evaluation of the complexity of the proposed algorithm. Assume that mi=m for all i. The first initialization (Step 1) requires O(nm2c2+nc4+c6) computations. For each w, the second initialization (Step 2) also requires O{nc4min+c6} computations. For each ρ, Steps 3–8 require O(nc4) computations. Therefore, the formula in Proposition mis are small.The proof is provided in Appendix A. For each 2.3xi(t) is generated from a multivariate Gaussian process. Suppose that we want to predict the ith multivariate response xi(t) at {si1,…,sim} for m ≥ 1. Let yi(k)=yi1(k),…,yimik(k)⊤ be the vector of observations at ti1(k),…,timik(k) for the kth response. Let μi(k),o=μ(k)ti1(k),…,μ(k)timik(k)⊤ be the vector of the kth mean function at the observed time points. Let yi=yi(1),⊤,…,yi(p),⊤⊤ and μio=μi(1),o,⊤,…,μi(p),o,⊤⊤. Let μin=μ(1)(si1),…,μ(1)(sim),…,μ(p)(si1),…,μ(p)(sim)⊤ be the vector of mean functions at the time points for prediction.For prediction, assume that the smooth curve bi(k),o=bti1(k),…,btimik(k)⊤ and bin=bsi1,…,bsim⊤. Next, let Bio=blockdiagbi(1),o,…,bi(p),o and Bin=Ip⊗bin. Then, Cov(yi) given by BioΘBio,⊤+blockdiag, and Cov(xi) and Cov are given by BinΘBin,⊤ and BioΘBin,⊤, respectively. Let Θ˜=Θ˜kk′1≤k,k′≤p∈Rpc×pc. Plugging in the estimates, we predict xi by μ^io=μ^(1)ti1(1),…,μ^(1)timi1(1),…,μ^(p)ti1(p),…,μ^(p)timip(p)⊤ is the estimate of μio, μ^in=μ^(1)(si1),…,μ^(1)(sim),…,μ^(p)(si1),…,μ^(p)(sim)⊤ is the estimate of μin, V^i=BioΘ˜Bio,⊤+blockdiagσ^12Imi1,…,σ^p2Imip. An approximate covariance matrix for x^i is % point‐wise confidence interval for the kth response is given by Var^xi(k)(sij)|yi can be extracted from the diagonal of Cov^(xi|yi).It follows that L ≥ 1 scores ξi=⊤ for the ith subject. Note that ξiℓ=∫Ψℓ(t)⊤{xi(t)−μ(t)}dt. With a similar derivation as above, xi(t)−μ(t) can be predicted by Ip⊗b(t)⊤Θ˜Bio,⊤V^i−1(yi−μ^io). By Proposition Ψℓ(k)(t) are estimated by b(t)⊤G−12u^ℓ(k), and thus, Ψℓ(t)⊤=u^ℓ⊤{Ip⊗G−12b(t)}. It follows that Finally, we predict the first 3We evaluate the finite sample performance of the proposed method (denoted by mFACEs) against mFPCA via a synthetic simulation study and a simulation study mimicking the Alzheimer's Disease Neuroimaging Initiative (ADNI) data in the real data example. Here, we report the details and results of the former as the conclusions remain the same for the latter, and details are provided in the 3.1p=3 responses. The mean functions are μ(t)=⊤. We first specify the auto‐covariance functions. Let Φ1(t)=2sin(2πt),2cos(4πt),2sin(4πt)⊤, Φ2(t)=2cos(πt),2cos(2πt),2cos(3πt)⊤, andΦ3(t)=2sin(πt),2sin(2πt),2sin(3πt)⊤. Also let Ckk=Φk(s)⊤ΛkkΦk(t),k=1,2,3. For the cross‐covariance functions, letCkk′=ρΦk(s)⊤Λkk12Λk′k′12Φk′(t) for k≠k′, where ρ∈ is a parameter to be specified. The induced covariance operator from the above specifications is proper; see Lemma ρkk′=Ckk′/CkkCk′k′ is bounded by ρ. Hence, ρ controls the overall level of correlation between responses: if ρ=0, then the responses are uncorrelated from each other.We generate data by model  (2πt),5t−2]⊤. We fℓ=1,…,9, we simulate the scores ξiℓ from N, where dℓ are the induced eigenvalues. Next, we simulate the white noises ϵij(k) fromN, where σϵ2 is determined according to the signal‐to‐noise ratio SNR=∑ℓdℓ/(pσϵ2). Here, we let SNR=2. For each response, the sampling time points are drawn from a uniform distribution in the unit interval, and the number of observations for each subject, mik, is generated from a uniform discrete distribution on {3,4,5,6,7}. Thus, the sampling points vary not only from subject to subject but also across responses within each subject.The eigendecomposition of the multivariate covariance function gives nine nonzero eigenvalues with associated multivariate eigenfunctions; hence, for n and the correlation parameter ρ. We let n=100,200, or 400. We let ρ=0.5, which corresponds to a weak correlation between responses as the average absolute correlation between responses is only 0.36. Another value of ρ is 0.9, which corresponds to a moderate correlation between responses as the average absolute correlation between responses is about 0.50.We use a factorial design with two factors: the number of subjects In total, we have six model conditions, and for each model condition, we generate 200 datasets. To evaluate the prediction accuracy of the various methods, we draw 200 additional subjects as testing data. The true correlation functions and a sample of the simulated data are shown in the Ĉkk′ be an estimate of Ckk′, and then RISE are given by ℓth eigenfunction, we use the integrated square errors (ISE), which are defined as d^ℓ/dℓ.We compare mFACEs and mFPCA in terms of estimation accuracy of the covariance functions, the eigenfunctions and eigenvalues, and prediction of new subjects. For covariance function estimation, we use the relative integrated square errors (RISE). Let For predicting new curves, we use the mean integrated square errors (MISE), which are given by 3.2ρ=0.9), the advantage of mFACEs is substantial even for the small sample size n=100.Figure % of the total variation in the functional data for ρ=0.5, and it is 80% for ρ=0.9. Figure Figures R package face . We define the relative efficiencies of different methods as the ratios of MISEs with respect to those of univariate FPCA; see Figure In summary, mFACEs shows competing performance against alternative methods.4http://ida.loni.ucla.edu/.The ADNI is a two‐stage longitudinal observational study launched in year 2003 with the primary goal of investigating whether serial neuroimages, biological markers, clinical and neuropsychological assessments can be combined to measure the progression of AD =Ckk′/CkkCk′k′. The plot indicates two groups of biomarkers, ADAS‐Cog 13 and FAQ, in one group and RAVLT.imme, RAVLT.learn, and MMSE in another group. The biomarkers within the groups are positively correlated and negatively correlated between groups, which makes sense as high values of ADAS‐Cog 13 and FAQ and low values for the other biomarkers suggest of AD. Next, we display in Figure % and 11% of the total variance in the functional part of the data. The eigenfunctions reveal how the five biomarkers covary and how a subject's trajectories of biomarkers deviate from the population mean. Indeed, we see from Figure % point‐wise confidence bands for three subjects. We focus on predicting the trajectories over the first 4 years as there are more observations. We can see that the confidence bands are getting wider at the later time points because of fewer observations.We analyse the five longitudinal biomarkers using mFACEs. For better visualization, we plot in Figure 4.2ŷij(k) is the predicted value of the kth biomarker for the ith subject at time tij(k). We conduct two types of validation: an internal validation and an external validation. For the internal validation, we perform a 10‐fold cross‐validation to the combined data of ADNI‐1 and ADNI‐2. For the external validation, we fit the model using only the ADNI‐1 data and then predict ADNI‐2 data. Figure We compare the proposed mFACEs with mFPCA and mFPCA(CE) for predicting the five longitudinal biomarkers. The prediction performance is evaluated by the average squared prediction errors (APE), 5The prevalence of multivariate functional data has sparked much research interests in recent years. However, covariance estimation for multivariate sparse functional data remains underdeveloped. We proposed a new method, mFACEs, and its features include the following: (a) A covariance smoothing framework is proposed to tackle multivariate sparse functional data; (b) an automatic and fast fitting algorithm is adopted to ensure the scalability of the method; (c) eigenfunctions and eigenvalues can be obtained through a one‐time spectral decomposition, and eigenfunctions can be easily evaluated at any sampling points; and (d) a multivariate extension of the conditional expectation approach −1/2 as in Chiou et al. dt−1/2 as in Happ and Greven . As such, the investigators within the ADNI contributed to the design and implementation of ADNI and/or provided data but did not participate in analysis or writing of this paper. A complete listing of ADNI investigators can be found at: http://adni.loni.usc.edu/wp‐content/uploads/how_to_apply/ADNI_Acknowledgement_List.pdf.Data used in preparation of this article were obtained from the Alzheimer's Disease Neuroimaging Initiative (ADNI) database (mfaces (https://github.com/cli9/mfaces) contains an open‐source implementation of the proposed method described in the article. Demo zip file contains codes to demonstrate the proposed method with a simulated dataset in Section 3.The Supporting info itemClick here for additional data file.


Scientific Reports 10.1038/s41598-018-24153-0, published online 12 April 2018Correction to: The original version of this Article contained errors in Equation 7, which was incorrectly given as:The correct Equation 7 appears below.As a result of the changes to Equation 7, the Abstract,−5 RI unit (RIU) obtained in Zone II for a 1-cm sensor length.”“The sensors can be employed over a very wide dynamic RI range from 1.316 to over 1.608 at a wavelength of 1550 nm, with the best resolution of 2.2447 × 10now reads:−5 RI unit (RIU) obtained in Zone II for a 1-cm sensor length.”“The sensors can be employed over a very wide dynamic RI range from 1.316 to over 1.608 at a wavelength of 1550 nm, with the best resolution of 2.2406 × 10Additionally, in Figure 5a, the unit of fiber diameter “μm” was incorrectly given as “mm”.The original Figure In the final paragraph of the Discussion,−5 RIU and 3.2634 × 10−5 RIU, respectively, compared to the other two Zones. For Zone I, the best resolution is achieved by the 4-cm long sensor with a minimum detection level of 1.5438 × 10−3 RIU while the 1-cm and 2.5-cm sensors are capable of resolutions of 5.1952 × 10−3 RIU and 1.7462 × 10−3 RIU, respectively.”“It is also in this Zone that the 2.5-cm and 4-cm sensors have the best relative resolutions of 2.9919 × 10now reads:−5 RIU and 3.2517 × 10−5 RIU, respectively, compared to the other two Zones. For Zone I, the best resolution is achieved by the 4-cm long sensor with a minimum detection level of 1.6116 × 10−3 RIU while the 1-cm and 2.5-cm sensors are capable of resolutions of 5.5905 × 10−3 RIU and 1.7528 × 10−3 RIU, respectively.“It is also in this Zone that the 2.5-cm and 4-cm sensors have the best relative resolutions of 2.9847 × 10In the second paragraph of the Conclusions,−5 RIU is achieved for the 1-cm sensor.”“For Zone II, the best sensor resolution of 2.2447 × 10now reads:−5 RIU is achieved for the 1-cm sensor.”“For Zone II, the best sensor resolution of 2.2406 × 10Lastly, as a result of the changes to Equation 7, the data in Table S1, S2, S3, S5, S6 and S7 in the Supplementary Information was incorrect.The original The original Article and accompanying Supplementary Information file has now been corrected.Supplementary Information.


Scientific Reportshttps://doi.org/10.1038/s41598-019-54197-9, published online 28 November 2019Correction to: σ is conductivity, R is electrical resistance measured, A is cross-sectional area of the cylinder, and l is length of the cylinder. As a result, the electric conductivity values in the original version of the Article were about 100 times smaller than the correct values.The original version of this Article contained a calculation error in electric conductivity values. As illustrated in the Methods section, the inner diameter of the glass tube for the measurement was 1.0 mm. However, the authors incorrectly used 10.0 mm for the calculations. In addition, an incorrect equation Consequently, in the Abstract,2 carbon components, and exhibits a conductivity above 10−2 S cm−1 and a specific surface area of 650 m2 g−1.”“The BDND comprises BDD and spnow reads:2 carbon components, and exhibits a conductivity above 1 S cm−1 and a specific surface area of 650 m2 g−1.”“The BDND comprises BDD and spIn addition, in the Results and Discussion section, under the subheading ‘Preparation of the boron-doped nanodiamond’,−9 S cm−1 to 3.5 × 10−4 S cm−1 with a deposition time of 1 h. The BDND prepared with 8 h deposition time exhibited a conductivity sufficient for an electrochemical electrode material (2.0 × 10−2 S cm−1).”“The electrical conductivity of the as-deposited BDND was increased rapidly from 1.8 × 10now reads:−7 S cm−1 to 1.4 × 10−2 S cm−1 with a deposition time of 1 h. The BDND prepared with 8 h deposition time exhibited a conductivity sufficient for an electrochemical electrode material (4.0 S cm−1).”“The electrical conductivity of the as-deposited BDND was increased rapidly from 7.2 × 10Finally, the y-axis scale and conductivity value in Figure 1 was incorrect. The original Figure The original Article has been corrected.


Nature Communications 10.1038/s41467-021-22353-3, published online 23 April 2021.Correction to: The original version of this Article contained errors in inline Equations in the last paragraph of the “Steering of GHZ states” subsection, which incorrectly read:ρ = p|GHZϕN+1〉〈GHZϕN+1| + (1−p)N+1, using the same measurements we obtain FQB|A  ≥ p2N2/[p +  2(1−p)/2N], 4Var QB|A  ≤ (1−p)N/2N + p(1−p)N2. Whenever p ≫ 2N−, the criterion witnesses steering.For a mixture The correct form should read:ρ = p|GHZϕN+1〉〈GHZϕN+1| + (1−p)N+1, using the same measurements we obtain FQB|A  ≥ p2N2/[p +  2(1−p)/2N], 4Var QB|A  ≤ (1−p)N + p(1−p)N2. For large N, whenever p ⪆ 1/√N, the criterion witnesses steering.For a mixture This has been corrected in the PDF and HTML versions of the Article.The original version of the σz by Alice, Bob’s conditional states are easily found to gived = 2N . With a σx measurement, (3) results inFor a measurement of p ≫ 1/d = 2N−, we can neglect the terms involving d. Then FQB|A  ⪆ pN2, VarQB|A  ⪅ p(1−p)N2< pN2 for p < 1.When The correct version reads insteadσz by Alice, Bob’s conditional states are easily found to giveFor a measurement of σx measurement, (3) results ind = 2N . When p ≫ 1/d = 2N−, we can neglect the term involving d. Then FQB|A  ⪆ pN2, and the differenceN > (1−p)/p2. For large N, this condition approximates to p ⪆ 1/√N.With a The HTML has been updated to include a corrected version of the Updated Supplementary Information


Similarity measures (SM) and correlation coefficients (CC) are used to solve many problems. These problems include vague and imprecise information, excluding the inability to deal with general vagueness and numerous information problems. The main purpose of this research is to propose an m-polar interval-valued neutrosophic soft set (mPIVNSS) by merging the m-polar fuzzy set and interval-valued neutrosophic soft set and then study various operations based on the proposed notion, such as AND operator, OR operator, truth-favorite, and false-favorite operators with their properties. This research also puts forward the concept of the necessity and possibility operations of mPIVNSS and also the m-polar interval-valued neutrosophic soft weighted average operator (mPIVNSWA) with its desirable properties. Cosine and set-theoretic similarity measures have been proposed for mPIVNSS using Bhattacharya distance and discussed their fundamental properties. Furthermore, we extend the concept of CC and weighted correlation coefficient (WCC) for mPIVNSS and presented their necessary characteristics. Moreover, utilizing the mPIVNSWA operator, CC, and SM developed three novel algorithms for mPIVNSS to solve the multicriteria decision-making problem. Finally, the advantages, effectiveness, flexibility, and comparative analysis of the developed algorithms are given with the prevailing techniques. Multicriteria decision-making (MCDM) is an essential condition for decision scientific discipline. The decision-maker should judge the choices stated by the diverse forms of distinguishing perspectives. Though, in quite a lot of situations, it is tough for someone to undertake it because of numerous uncertainties in the data. One is due to lack of expertise or ravishment of policies. Thus, to measure the given disadvantages and thinking tools, a succession of philosophies had been projected. Zadeh introduced the notion of the fuzzy set (FS)  to resolMaji et al.  expandedCorrelation plays a significant part in statistics as well as engineering science. The joint association of two variable quantities can be utilized to appraise the interdependency of the correlation qualitative analysis. In addition to using probabilistic strategies for noticeably pragmatic engineering science complications, you also can locate various boundaries to probabilistic methods. However, the bodily structure has numerous exceptions, the improvement is challenging, and it is difficult to obtain exact consequences. Thus, due to the wide variety of incomprehensible info, the consequences of probability theory are not able to provide professionals with suitable information. In addition, in natural world concerns, there is not any priggish reason out to deal along with distinguished statistical information. Due to the preliminary limitations, the outcomes of probability theory are not conducive to specialists. So, probability theory is not very adequate to resolve the insecurity explicit in the data. Several assessors around the world have prearranged and suggested different strategies to solve anxiety-related complications. Wang et al.  developeGerstenkorn and Mańko  proposedIn the past few years, quite a lot of mathematicians have progressed numerous methodologies such as similarity measures, CC, and aggregation operators (AOs) along with their applications in DM. Harish  offered In this epoch, specialists believe that real life is touching the track of multipolarization. Therefore, there is no distrust that the multipolarization of information has played a significant part in the prosperity of many arenas of science and technology. In neurobiology, multipolar neurons in the brain accumulate a lot of info from other neurons. In the whole manuscript, the motivation for expanding and mixing this research work is gradually given. We prove that under any appropriate circumstances, different hybrid structures comprehending FS will be transformed into distinct privileges of mPIVNSS. This study will be the utmost versatile form that can be used to merge data in daily life complications. The organization of the current research is such as follows: some basic concepts are presented in In the following section, we recalled some fundamental concepts which help us to construct the structure of the following study.(See ).𝒰 be a universe, and 𝒜 be an NS on 𝒰 defined as 𝒜={<u, u𝒜(u), v𝒜(u), w𝒜(u) > :u ∈ 𝒰}, where u, v, w: 𝒰⟶]0−, 1+ and 0 ≤ uα(u)+vα(u)+wα(u) ≤ 3, for all α=1,2,…, m; and u ∈ 𝒰.Let (See ).𝒰 be the universal set and ℰ be the set of attributes concerning 𝒰. Let 𝒫(𝒰) be the power set of 𝒰 and 𝒜⊆ℰ. A pair  is called an SS over 𝒰 and its mapping is given asLet It is also defined as(See ).𝒰 be the universal set and ℰ be the set of attributes concerning 𝒰. Let 𝒫(𝒰) be the set of neutrosophic sets over 𝒰 and  𝒜⊆ℰ. A pair  is called an NSS over 𝒰, and its mapping is given asLet (See ).𝒰 be a universal set; then, the interval-valued neutrosophic set can be expressed by the set 𝒜={u, (u𝒜(u), v𝒜(u), w𝒜(u)) : u ∈  𝒰}, where u𝒜, v𝒜, and w𝒜 are truth, indeterminacy, and falsity membership functions for 𝒜, respectively, u𝒜, v𝒜, and w𝒜⊆ for each u ∈ 𝒰, whereLet u ∈ 𝒰, 0 ≤ u𝒜(u)+v𝒜(u)+w𝒜(u) ≤ 3, and IVN (𝒰) represent the family of all interval-valued neutrosophic sets on 𝒰.For each point, (See ).𝒰 be a universe of discourse and ℰ be a set of attributes, and an m-polar neutrosophic soft set (mPNSS) ℘ℜ over 𝒰 defined asuα(u), vα(u), and wα(u) represent the truthiness, indeterminacy, and falsity, respectively, uα(u), vα(u), wα(u)⊆ and 0 0 ≤ uα(u)+vα(u)+wα(u) ≤ 3, for all α=1,2,3,…, m; e ∈ ℰ and u ∈ 𝒰. Simply an m-polar neutrosophic number (mPNSN) can be expressed as ℘={〈uα, vα,  wα〉}, where 0 ≤ uα+vα+wα ≤ 3 and α=1,2,3,…, m.Let (See ).𝒰 be a universe of discourse and ℰ be a set of attributes, and IVNSS ℘ℜ over 𝒰 defined asuℜ(u)=, vℜ(u)=, and wℜ(u)= represents the interval truthiness, indeterminacy, and falsity, respectively, uℜ(u), vℜ(u), wℜ(u)⊆ and 0 ≤ uℜ𝔲(u)+vℜ𝔲(u)+wℜ𝔲(u) ≤ 3, for each e ∈ ℰ and u ∈ 𝒰.Let m, representing m criteria of the object, but mPFS cannot handle indeterminacy and falsity objects. NS bargains with truth, falsity, one any choice specifications containing indeterminacy, but are not able to deal with multicriteria, multiple sources, and multiple polarities information fusion of possible choices. To conquer this question, Deli et al. , vα(u)=, and wα(u)= represent the interval truthiness, indeterminacy, and falsity, respectively; uα(u), vα(u), wα(u)⊆ and 0 ≤ uα𝔲(u)+vα𝔲(u)+wα𝔲(u) ≤ 3 for all α=1,2,3,…, m; e ∈ ℰ and u ∈ 𝒰. Simply, an m-polar interval-valued neutrosophic soft number (mPIVNSN) can be expressed as uα𝔲(u)+vα𝔲(u)+wα𝔲(u) ≤ 3 and α=1,2,3,…, m.Let ℘ℜ and ℘ℒ be two mPIVNSSs over 𝒰. Then, ℘ℜ is called an m-polar interval-valued neutrosophic soft subset of ℘ℒ, ifα=1,2,3,…, m; e ∈ ℰ and u ∈ 𝒰.Let ℘ℜ and ℘ℒ be two mPIVNSSs over 𝒰. Then, ℘ℜ=℘ℒ, ifα=1,2,3,…, m; e ∈ ℰ and u ∈ 𝒰.Let ℘ℜ be an mPIVNSS over 𝒰. Then, empty mPIVNSS can be represented as Let ℘ℜ be an mPIVNSS over 𝒰. Then, universal mPIVNSS can be represented as Let ℘ℜ be an mPIVNSS over 𝒰. Then, the complement of mPIVNSS is defined as follows:Let ℘ℜ be an mPIVNSS over 𝒰, thenIf LetThen, by using Similarly, we can prove 2 and 3.℘ℜ and ℘ℒ be two mPIVNSSs over 𝒰. Then,Let ℘ℜ and ℘ℒ be two mPIVNSSs over 𝒰. Then,Let ℘ℜ and ℘ℒ be two mPIVNSSs over 𝒰. Then,Let As we know,By using Now, by using Now,By using Hence,Similar to assertion 1.℘ℜ, ℘ℒ, and ℘ℋ be three mPIVNSSs over 𝒰. Then,Let As we know,Hence,Similarly, we can prove other results.℘R and ℘ℒ be two mPIVNSSs over 𝒰. Then, their extended union is defined asLet 𝒰={u1, u2} be a universe of discourse and E={e1, e2, e3, e4} be a set of attributes, and ℜ={e1, e2} and ℒ=Unsupported{e3, e4}⊆E. Consider 3-PIVNSSs ℘ℜ and ℘ℒ over 𝒰 can be represented as follows:Assume Then,℘R and ℘ℒ be two mPIVNSSs over 𝒰. Then, their extended intersection is defined asLet Generally, if ℘ℜ and ℘ℒ be two mPIVNSSs over 𝒰. Then, their difference is defined as follows:Let ℘ℜ and ℘ℒ be two mPIVNSSs over 𝒰. Then, their addition is defined as follows:Let ℘ℜ be an mPIVNSS over 𝒰. Then, its scalar multiplication is represented as ℘ℜ. Let ℘ℜ be the mPIVNSS over 𝒰. Then, its scalar division is represented as Let ℘ℜ be an mPIVNSS over 𝒰. Then, the truth-favorite operator on ℘ℜ is denoted by Let ℘ℜ be an mPIVNSS over 𝒰. Then, the false-favorite operator on ℘ℜ is denoted by Let ℘ℜ and ℘ℒ be two mPIVNSSs over 𝒰. Then, their AND operator is represented by ℘ℜ∧℘ℒ and defined as follows:Let ℘ℜ and ℘ℒ be two mPIVNSSs over 𝒰. Then, their OR operator is represented by ℘ℜ∨℘ℒ and defined as follows:Let Reconsider ℘ℜ be an mPIVNSS. Then, necessity operation on mIVPNSS is represented by ⊕℘ℜ and defined as follows:Let ℘ℜ be the mPIVNSS over 𝒰. Then, possibility operation on mIVPNSS is represented by ⊗℘ℜ and defined as follows:Let ℘ℜ and ℘ℒ be two mPIVNSSs over 𝒰. Then,Let 𝒰.As we know,℘ℜ∪ϵ℘ℒ=℘ℋ:Let By using ℘ℒ∪ϵ ⊕ ℘ℜ=ℵ, where ⊕℘ℜ and ⊕℘ℒ are given as follows by using the definition of necessity operation.Assume ⊕By using ℘ℜ ∪ϵ ℘ℒ)=⊕℘ℒ∪ϵ ⊕ ℘ℜ.Therefore,  represents the correlation among them. Then, the subsequent estates hold.𝒞mPIVNSS=ςmPIVNSS(℘ℜ)𝒞mPIVNSS=ςmPIVNSS(℘ℒ)Let The proof is trivial.℘ℜ and ℘ℒ be two mPIVNSSs over 𝒰. Then, the CC between them is given as δmPIVNSS and can be stated as follows:Let ℘ℜ and ℘ℒ be two mPIVNSSs over 𝒰. Then, CC between them satisfies the following properties.δmPIVNSS ≤ 10 ≤ δmPIVNSS=δmPIVNSS℘ℜ=℘ℒ, that is, ∀j, α, uαℓℜ(uj)=uαℓℒ(uj), uα𝔲ℜ(uj)=uα𝔲ℒ(uj), vαℓℜ(uj)=vαℓℒ(uj), vα𝔲ℜ(uj)=vα𝔲ℒ(uj), wαℓℜ(uj)=wαℓℒ(uj), wα𝔲ℜ(uj)=wα𝔲ℒ(uj), then δmPIVNSS=1.If Let The proof is obvious.℘ℜ and ℘ℒ be two mPIVNSSs over 𝒰. Then, their CC has also been given as δmPIVNSS1 and is expressed as follows:Let ℘ℜ and ℘ℒ be two mPIVNSSs over 𝒰. Then, CC between them satisfies the following properties.δIVIFSS1 ≤ 10 0 ≤ δIVIFSS1=δIVIFSS1℘ℜ=℘ℒ, that is, ∀j, α, uαℓℜ(uj)=uαℓℒ(uj), uα𝔲ℜ(uj)=uα𝔲ℒ(uj), vαℓℜ(uj)=vαℓℒ(uj), vα𝔲ℜ(uj)=vα𝔲ℒ(uj), wαℓℜ(uj)=wαℓℒ(uj), wα𝔲ℜ(uj)=wα𝔲ℒ(uj), then δIVIFSS1=1.If Let The proof is obvious.γ={γ1, γ2, γ3,…,γn}T, where γi > 0, ∑i=1nγi=1.These days, it is important to discuss the weight of mPNSS for practical life. When professionals set different weights for each alternative, the decision may be different. Therefore, it is perfectly correct for experts to weigh the recent decision. Suppose the weight of professionals can be stated as ℘ℜ and ℘ℒ are two mPIVNSS over 𝒰. Then, their WCC is given as δWmPIVNSS and stated as follows:Let ℘ℜ and ℘ℒ be two mPIVNSSs over 𝒰. Then, their WCC also given as δWmPIVNSS1 is defined as follows:Let γ={(1/n), (1/n),…, (1/n)}, then δWmPIVNSS and δWmPIVNSS1 are reduced to δmPIVNSS and δmPIVNSS1, respectively, defined in Definitions If we consider ℘ℜ and ℘ℒ be two mPIVNSSs over 𝒰. Then, WCC between them satisfies the following properties.δWmPIVNSS ≤ 10 ≤ δWmPIVNSS=δWmPIVNSS℘ℜ=℘ℒ, that is, ∀j, α, uαℓℜ(uj)=uαℓℒ(uj), uα𝔲ℜ(uj)=uα𝔲ℒ(uj), vαℓℜ(uj)=vαℓℒ(uj), vα𝔲ℜ(uj)=vα𝔲ℒ(uj), wαℓℜ(uj)=wαℓℒ(uj), wα𝔲ℜ(uj)=wα𝔲ℒ(uj), then δWmPIVNSS=1.If Let The proof is obvious.β={β1, β2, β3,…, βs} for assessment under the team of experts such as 𝒰={u1, u2, u3,…, un} with weights Ω=T, such that Ωi > 0, ∑i=1nΩi=1. Let ℰ={e1, e2,…, em} be a set of attributes with weights, and γ=T be a weight vector for parameters such as γi > 0, ∑j=1mγj=1. The team of experts {ui: i=1,2,…, n} evaluate the alternatives {βz)(=(uαijz), where uαij(z)=, vαijz)(=, and wαijz)(=, where 0 ≤ uαℓ(u), uα𝔲(u), vαℓ(u), vα𝔲(u), wαℓ(u), wα𝔲(u) ≤ 1 and 0 ≤ uαij𝔲(u)+vαij𝔲(u)+wαij𝔲(u) ≤ 3. So, ℒij(z)=(, ) for all i, j.Assume a set of “s” alternatives such as The flowchart of the offered algorithm is shown in In the past few years, many mathematicians developed various methodologies to solve MCDM problems, such as aggregation operators for different hybrid structures, CC, similarity measures, and decision-making applications. Some operational laws and mPIVNSWA with its decision-making approach have been established for mPIVNSS which is an extension of the interval-valued neutrosophic weighted aggregation operator . The ide℘ℜ=, ,  , ℘ℜ1=, ,  , and ℘ℜ2=, ,   be three mPIVNSNs, and the basic operators for mPIVNSNs are defined as when δ > 0.℘ℜ1 ⊕ ℘ℜ2=〈, ,  〉℘ℜ1 ⊗ ℘ℜ2=〈, , 〉δ℘ℜ=〈, ,  〉℘ℜ)δ=〈, ,  〉=δ℘ℜ2 ⊕ δ℘ℜ1℘ℜ1 ⊗ ℘ℜ2)δ=(℘ℜ1)δ ⊗ (℘ℜ2)δ(δ1℘ℜ1 ⊕ δ2℘ℜ1=(δ1 ⊕ δ2)℘ℜ1℘ℜ1)δ1 ⊗ (℘ℜ1)δ2=(℘ℜ1)δ1+δ2(℘ℜ ⊕ ℘ℜ1) ⊕ ℘ℜ2=℘ℜ ⊕ (℘ℜ1 ⊕ ℘ℜ2)(℘ℜ ⊗ ℘ℜ1) ⊗ ℘ℜ2=℘ℜ ⊗ (℘ℜ1 ⊗ ℘ℜ2)(Let The proof of the above laws is straightforward by using ℘ℜeij=, ,  be a collection of mPIVNSNs. Ωi and γj are the weight vectors for expert's and parameters, respectively, with given conditions Ωi > 0, ∑i=1nΩi=1, γj > 0, ∑j=1mγj  = 1, where . Then, the mPIVNSWA operator defined as mPIVNSWA: Δn⟶Δ is defined as follows:Let ℘ℜeij=, ,  be a collection of mPIVNSNs, where , and the aggregated value is also an interval-valued neutrosophic soft number, such asLet We can prove this easily by using IFSWA .℘ℜ=, ,   be an mPIVNSN; then the score, accuracy, and certainty functions for an mPIVNSN, respectively, defined as follows:𝕊(℘ℜ)=(1/6m)(uαℓ(u)+uα𝔲(u)+1 − vαℓ(u)+1 − vα𝔲(u)+1 − wαℓ(u)+1 − wα𝔲(u))(℘ℜ)𝔸(℘ℜ)=(1/4m)(4+uαℓ(u)+uα𝔲(u) − wαℓ(u) − wα𝔲(u))(℘ℜ)(2)ℂ(℘ℜ)=(1/2m)(2+uαℓ(u)+uα𝔲(u))(℘ℜ), where α=1,2,…, m(3)Let ℘ℜ and ℘ℜ1 be two mPIVNSSs. Then, the comparison approach is presented as follows:℘ℜ) > (℘ℜ1), then ℘ℜ is superior to ℘ℜ1(1)If (℘ℜ)=(℘ℜ1) and (℘ℜ) > 𝔸(℘ℜ1), then ℘ℜ is superior to ℘ℜ1(2)If (𝕊(℘ℜ)=𝕊(℘ℜ1), 𝔸(℘ℜ)=𝔸(℘ℜ1), and ℂ(℘ℜ) > ℂ(℘ℜ1), then ℘ℜ is superior to ℘ℜ1(3)If 𝕊(℘ℜ)=𝕊(℘ℜ1), 𝔸(℘ℜ) > 𝔸(℘ℜ1), and ℂ(℘ℜ)=ℂ(℘ℜ1), then ℘ℜ is indifferent to ℘ℜ1 and can be denoted as ℘ℜ ~ ℘ℜ1If Let β={β1, β2, β3,…, βs} for assessment under the team of experts such as 𝒰={u1, u2, u3,…, un} with weights Ω=T, such that Ωi > 0, ∑i=1nΩi=1. Let ℰ={e1, e2,…, em} be a set of attributes with weights γ=T be a weight vector for parameters such as γi > 0, ∑j=1mγj=1. The team of experts {ui: i=1,2,…, n} evaluate the alternatives {βz)(=(uαijz), where uαij(z)=, vαijz)(=, and wαijz)(=, where 0 ≤ uαℓ(u), uα𝔲(u), vαℓ(u), vα𝔲(u), wαℓ(u), wα𝔲(u) ≤ 1 and 0 ≤ uαij𝔲(u)+vαij𝔲(u)+wαij𝔲(u) ≤ 3. So, Δk=(, , ) for all i, j. Experts give their preferences for each alternative in term of mPIVNSNs by using the mPIVNSWA operator in the form ofΔk=(, , ). Compute the score values for each alternative and analyze the ranking of the alternatives.Assume a set of “s” alternatives such as The flowchart of the offered algorithm is shown in ℘R1 and ℘R2 be two mPIVNSSs over the universe of discourse 𝒰={u1,  u2,…,  uj}. Then, a cosine similarity measure between ℘R1 and ℘R2 is defined asLet ℘ℜ, ℘ℒ, and ℘Q be three mPIVNSSs. Then, the following properties hold.𝒮mPIVNSS1 ≤ 10 ≤ 𝒮mPIVNSS1=𝒮mPIVNSS1℘ℜ⊆℘ℒ⊆℘Q, then 𝒮mPIVNSS1 ≤ 𝒮mPIVNSS1 and 𝒮mPIVNSS1 ≤ 𝒮mPIVNSS1If Let By using the above definition, the proof of these properties can be done easily.℘R1 and ℘R2 be two mPIVNSSs over the universe of discourse 𝒰={u1,  u2,…,  uj}. Then, the set-theoretic similarity measure between ℘R1 and ℘R2 is defined asLet ℘ℜ and ℘ℒ be two mPIVNSSs over 𝒰. Then, the following properties hold.𝒮mPIVNSS2 ≤ 10 0 ≤ 𝒮mPIVNSS2=𝒮mPIVNSS2℘ℜ⊆℘ℒ⊆℘Q, then 𝒮mPIVNSS2 ≤ 𝒮mPIVNSS2 and 𝒮mPIVNSS2 ≤ 𝒮mPIVNSS2If Let By using the above definition, the proof of these properties can be done easily.In this section, we utilized the developed approaches based on correlation coefficient, mPIVNSWA operator, and similarity measures for decision making.β(1), β(2), β(3), β(4)}. A team of three experts has been hired by the president of the institution {u1, u2, u3} with weights T for final scrutiny and provide the selection criteria, as given in e1 = experience, e2 = publications, and e3 = research quality with weights T. Each expert gives his preferences for each alternative in the form of mPIVNSNs under the considered parameters given in Tables A university calls for the appointment of a vacant position of associate professor. For further assessment, four candidates  choose after preliminary review such as {β(1), β, β(3), ββ(1), β(2), β(3), β(4)} be a set of alternatives who are shortlisted for interview and ℰ={e1=experience,e2=publications,e3=research quality} be a set of parameters for the selection of associate professor. Let ℜ and ℒ⊆ℰ; then, we construct the 3-PIVNSS ℘ℜ(e) according to the requirement of university management such as follows:Assume {℘ℒt)((e) for each alternative according to experts, where t = 1, 2, 3, 4.Construct 3-PIVNSS 6.2.1.δmPIVNSS(℘ℜ(e), ℘ℒ(1)(e)), δmPIVNSS(℘ℜ(e), ℘ℒ(2)(e)), δmPIVNSS(℘ℜ(e), ℘ℒ(3)(e)), and δmPIVNSS(℘ℜ(e), ℘ℒ(4)(e)) by using equation , ℘ℒ(2)(e)), δmPIVNSS(℘ℜ(e), ℘ℒ(3)(e)), and δmPIVNSS(℘ℜ(e), ℘ℒ(4)(e)) given as. δmPIVNSS(℘ℜ(e), ℘ℒ(2)(e))=(25.04/28.6727)=0.87330, δmPIVNSS(℘ℜ(e), ℘ℒ(3)(e))=(23.73/29.4968)=0.80449, and δmPIVNSS(℘ℜ(e), ℘ℒ(4)(e))=(24.58/28.7433)=0.85516. This shows that δmPIVNSS(℘ℜ(e), ℘ℒ(1)(e)) >  δmPIVNSS(℘ℜ(e), ℘ℒ(2)(e)) > δmPIVNSS(℘ℜ(e), ℘ℒ(4)(e))  > δmPIVNSS(℘ℜ(e), ℘ℒ(3)(e)). The above-obtained ranking shows that β(1) is the best alternative. So, the ranking of other alternatives is given as β(1) > β(2) > β(4) > β(3). Graphical results are shown in Similarly, we can find the CC between 6.2.2.  Step 1: experts evaluate the scores for each alternative in the form of mPIVNSNs given in Tables 21=, , , Δ2=, , , Δ3=, , , and Δ4=, , .   Step 2: utilizing equation , the opi𝕊(Δ1)=0.2045, 𝕊(Δ2)=0.2004, 𝕊(Δ3)=0.1709, and 𝕊(Δ4)=0.1828.   Step 3: utilizing equation , compute𝕊(Δ1) > 𝕊(Δ2) > 𝕊(Δ4) > 𝕊(Δ3). So, β(1) > β(2)  > β(4) > β(3); hence, the alternative β(1) is the most suitable alternative for the position of associate professor. Graphical representation of the obtained results is shown in    Step 4: so, alternatives' ranking is as follows: 6.2.3.δmPIVNSS1(℘ℜ(e), ℘ℒ(1)(e)), δmPIVNSS1(℘ℜ(e), ℘ℒ(2)(e)), δmPIVNSS1(℘ℜ(e), ℘ℒ(3)(e)), and δmPIVNSS1(℘ℜ(e), ℘ℒ(4)(e)) by using equation , ℘ℒ(2)(e)), δmPIVNSS1(℘ℜ(e), ℘ℒ(3)(e)), and δmPIVNSS1(℘ℜ(e), ℘ℒ(4)(e)) given as δmPIVNSS1(℘ℜ(e), ℘ℒ(2)(e))=(1/9)(46.77/28.6727)=0.18124, δmPIVNSS1(℘ℜ(e), ℘ℒ(3)(e))=(1/9)(45.11/29.4968)= 0.16992, and δmPIVNSS1(℘ℜ(e), ℘ℒ(4)(e))=(1/9)(46.45/28.7433)=0.17956. This shows that δmPIVNSS1(℘ℜ(e), ℘ℒ(1)(e)) >  δmPIVNSS1(℘ℜ(e), ℘ℒ(2)(e)) > δmPIVNSS1(℘ℜ(e), ℘ℒ(4)(e)) > δmPIVNSS1(℘ℜ(e), ℘ℒ(3)(e)), which shows that alternative β(1) is the most appropriate and similar to ℘ℜ(e). So, alternatives ranking is given as β(1) > β(2)  > β(4) > β(3).Similarly, we can find the cosine similarity measure between δmPIVNSS2(℘ℜ(e), ℘ℒ(1)(e)), δmPIVNSS2(℘ℜ(e), ℘ℒ(2)(e)), δmPIVNSS2(℘ℜ(e), ℘ℒ(3)(e)), and δmPIVNSS2(℘ℜ(e), ℘ℒ(4)(e)). From Tables δmPIVNSS2(℘ℜ(e), ℘ℒ(1)(e))=0.17889, δmPIVNSS2(℘ℜ(e), ℘ℒ(2)(e))=0.17548, δmPIVNSS2(℘ℜ(e), ℘ℒ(3)(e))=0.16735, and δmPIVNSS2(℘ℜ(e), ℘ℒ(4)(e))=0.17766. This shows that δmPIVNSS2(℘ℜ(e), ℘ℒ(1)(e))  > δmPIVNSS2(℘ℜ(e), ℘ℒ(4)(e)) > δmPIVNSS2(℘ℜ(e), ℘ℒ(2)(e)) > δmPIVNSS2(℘ℜ(e), ℘ℒ(3)(e)). So, β(1) is the best alternative using the set-theoretic similarity measure, and the ranking of other alternatives is given as β(1) > β(4)  > β(2) > β(3). Graphical representation of results is shown in Now, we compute the set-theoretic similarity measure by using ℘ℜ(e), ℘ℒ(e)), δmPIn the next section, we are going to talk about utility, ease, and management with the help of a planned method. We also made a tentative assessment of the following with planned techniques and some existing methods.Through this study and comparison, it can be determined that the results obtained from the proposed approach are either more general than the methods available. Although, on the whole, the DM method associated with the usual DM methods adjusts the additional information to overcome the hesitation. Also, the various hybrid structures of FS are becoming a special feature of mPIVNS, with some suitable conditions being added. General information related to the object can be described accurately and analytically, as given in It turns out to be a contemporary problem. Why do we have to particularize novel algorithms according to the present novel structure? There are several indications that the recommended methodology can be exceptional compared to other existing methods. We remember the fact that IFS, picture fuzzy set, FS, hesitant fuzzy set, NS, and other fuzzy sets have been restricted by the mixed structure and cannot provide complete information regarding the situation. But, the proposed model in this study be the utmost appropriate for MCDM because it can handle three types of information such as truth, falsity, and indeterminacy. Comparative analysis with some common methods is given in Chen et al.'  multipolβ(1) is the finest alternative for the position of associate professor. The proposed approach can be compared to other available methods and observed that our proposed methodologies deliver the most reliable results comparative to available techniques. We observe one most interesting fact in our obtained results that our proposed methodologies deliver the same optimal and worst choices. The comparison of our proposed methodologies with some existing approaches is given in We recommend some novel algorithms under mPIVNSS by utilizing the developed mPIVNSS such as the mPIVNSWA operator, correlation coefficient, and similarity measures in the following section. Subsequently, we utilize the suggested algorithms to a realistic problem, namely, for the selection of an appropriate associate professor. It can be observed that The research concludes that the results obtained from the planned point of view exceed the results of the prevailing theories. Therefore, compared to existing techniques, established AOs, similarity measures, and CC handled uncertain and confusing information efficiently. However, under the current DM strategy, the main advantage of the planned method is that it can accommodate additional information in the data compared to existing techniques. This is a useful tool for resolving misinformation and vagueness in the DM method. The advantage of a planned approach with measures related to the current approach is avoiding the consequences based on negative reasons.In this study, a novel hybrid structure has been established by merging two independent structures m-polar fuzzy set and interval-valued neutrosophic soft set which is known as mPIVNSS. Some fundamental operations with their properties have been introduced for mPIVNSS. We have developed the CC and WCC with their properties in the content of mPIVNSS and also defined some operational laws for mPIVNSS and established a novel operator such as m-polar interval-valued neutrosophic weighted aggregation operator based on developed operational laws. To compute the similarity measure between two mPIVNSS, the idea of cosine and set-theoretic similarity measures have been established. Three novel algorithms based on mPIVNSS have been constructed to solve MCDM problems, correlation coefficient, mPIVNSWA operator, and similarity measures. A comparative analysis was also performed to demonstrate the proposed method. Finally, the projected ideas presented high constancy and functionality for decision-makers in the decision-making process. Based on the results acquired, it has been terminated and the above approach is extremely appropriate for finding the problem of MCDM in today's life. In the future, anyone can be introduced to the multipolar interval-valued neutrosophic weighted geometric operator with its decision-making approach. Furthermore, the concept of mPIVNSS will be extended to a multipolar interval-valued neutrosophic hypersoft set with their basic operators. The proposed impression can be functional to moderately a lot of problems in real life, including the therapeutic career, computing, artificial intelligence, pattern recognition, and finances.


Pythagorean fuzzy soft set (PFSS) is the most powerful and effective extension of Pythagorean fuzzy sets (PFS) which deals with the parametrized values of the alternatives. It is also a generalization of intuitionistic fuzzy soft set (IFSS) which provides us better and precise information in the decision-making process comparative to IFSS. The core objective of this work is to construct some algebraic operations for PFSS such as OR-operation, AND-operation, and necessity and possibility operations. Furthermore, some fundamental properties have been established for PFSS utilizing the developed operations. Moreover, a decision-making technique has been offered for PFSS based on a score matrix. To demonstrate the validity of the proposed approach, a numerical example has been presented. Finally, to ensure the practicality of the established approach, a comprehensive comparative analysis has been presented. The obtained results show that our developed approach is most effective and delivers better information comparative to prevailing techniques. Zadeh  introduc2 + NMG2 ≤ 1. As a generalized set of IFS, PFS has a close relationship with IFS. The PFS accurately access the uncertain facts than IFS. Muhammad Zulqarnain et al.  are of membership grade and nonmembership functions respectively with 0 ≤ ′ϒA(P)+ϑA(P) ≤ 1 and A ⊂ ℕ.Let X be a collection of objects, then a PFS A over X is defined asA(P), ϑA(P) : X⟶ are membership and nonmembership grade functions, respectively. Furthermore, 0 ≤ ′ϒA(P)2+ϑA(P)2 ≤ 1 and I=1 − ′ϒ(P)2 − ϑA(P)2 is called degree of indeterminacy.Let A(P)+ϑA(P) ≤ 1 and I=1 − ′ϒA(P) − ϑA(P), whereas in PFS, we have condition 0 ≤ ′ϒA(P)2+ϑA(P)2 ≤ 1 and I=1 − ′ϒA(P)2 − ϑA(P)2. We can say that a PFS is the general case of IFS.We can see from the above definitions that the only difference is in the conditions, i.e., in IFS, we deal with the condition 0 ≤ ϒX be a universal set and ℕ be set of attributes, then a pair  is called a PFSS over X where Ω : ℕ⟶℘KX is a mapping and ℘KX is known as the collection of all PFS subsets of universal set X.A(P), ϑA(P) : A⟶ are of membership grade and nonmembership functions respectively with 0 ≤ ′ϒA(P)2+ϑA(P)2 ≤ 1, degree of indeterminacy A ⊂ ℕ.Let ij=〈′ϒij, ϑij〉. For calculating the ranking of alternatives, Zulqarnain et al. . It is notified that the score function is unable to differentiate the PFSNs in some cases. For example, let ℋ11 = 0.3162,  0.4472 and ℋ12 = 0.5477,  0.6324, then according to the definition of score function, we have S(ℋ11) = −0.1 and S(ℋ12) = −0.1. So, in this case, it is impossible to find the finest alternative utilizing the score function. To handle this drawback, an accuracy function has been developed in which the sum of squares of membership and nonmembership such asA(ℋij) ∈ .For the sake of readers' convenience, we express the PFSN as ℋn et al.  introducij and ℛij, the following comparison laws are defined:(1)S(ℋij) > S(ℛij), then ℋij > ℛijIf (2)S(ℋij)=S(ℛij), thenA(ℋij) > A(ℛij), then ℋij > ℛijIf A(ℋij)=A(ℛij), then ℋij=ℛijIf If Thus, to compare two PFSNs ℋMatrices play a vital role in numerous areas of life such as calculation strategies, managing the magnitude of several engineering complications, medical science, and social science.FA, E) becomes a soft Pythagorean set softer than X, then subset X ∈ E is defined differently by RA={, e ∈ a, P ∈ FA}. Let RA be identified by its MG and NMG functions such as ′ϒRA : X × E⟶ and ϑRA : X × E⟶.α × β order.If  = {P, (′ϒA(P), ϑA(P))|P ∈ A} and  = {P, (′ϒB(P), ϑB(P))|P ∈ B} be two PFSS, where ′ϒA(P), ′ϒB(P), ϑA(P), ϑB(P) ∈ . Then, OR-operation between them is written as follows:Let  = {P, (′ϒA(P), ϑA(P))|P ∈ A} and  = {P, (′ϒB(P), ϑB(P))|P ∈ B} be two PFSS, where ′ϒA(P), ′ϒB(P), ϑA(P), ϑB(P) ∈ . Then, AND-operation between them is written as follows:Let  = {P, (′ϒA(P), ϑA(P))|P ∈ A},  = {P, (′ϒB(P), ϑB(P))|P ∈ B}, and  = {P, (′ϒC(P), ϑC(P))|P ∈ C} be three PFSS ′ϒA(P), ′ϒB(P), ′ϒC(P), ϑA(P), ϑB(P), ϑC(P) ∈ .K, A)∨ = ∨∨∨) = ∨)∨ = {P, (′ϒA(P), ϑA(P))|P ∈ A} and  = {P, (′ϒB(P), ϑB(P))|P ∈ B} be two PFSS, where ′ϒA(P), ′ϒB(P), ϑA(P), ϑB(P) ∈ . Then, utilizing (1) Let  = {P, (′ϒA(P), ϑA(P))|P ∈ A},  =  {P, (′ϒB(P), ϑB(P))|P ∈ B}, and  =  {P, (′ϒC(P), ϑC(P))|P ∈ C} be three PFSS ′ϒA(P), ′ϒB(P), ϑA(P), ϑB(P) ∈ .(2) Let  = {P, (′ϒA(P), ϑA(P))|P ∈ A},  = {P, (′ϒB(P), ϑB(P))|P ∈ B}, and  = {P, (′ϒC(P), ϑC(P))|P ∈ A} be three PFSS μA(P), μB(P), ϑA(P), ϑB(P) ∈ .K, A)∧ = ∧∧∧) = ∧)∧ = {P, (′ϒA(P), ϑA(P))|P ∈ A} and  = {P, (′ϒB(P), ϑB(P))|P ∈ B} be two PFSS, where ′ϒA(P), ′ϒB(P), ϑA(P), ϑB(P) ∈ . Then, using (1) Let  = {P, (′ϒA(P), ϑA(P))|P ∈ A},  = {P, (′ϒB(P), ϑB(P))|P ∈ B}, and  = {P, (′ϒC(P), ϑC(P))|P ∈ C} be three PFSS ′ϒA(P), ′ϒB(P), ′ϒC(P), ϑA(P), ϑB(P), ϑc(P) ∈ . Then, same as the above utilizing (2) Let  = {P, (′ϒA(P), ϑA(P))|P ∈ A} and  = {P, (′ϒB(P), ϑB(P))|P ∈ B} be two PFSS. Then, De Morgan Laws are given as follows:K, A)∨)0=O∧O∧)0=O∨O = {P, (′ϒA(P), ϑA(P))|P ∈ A} and  = {P, (′ϒB(P), ϑB(P))|P ∈ B} be two PFSS. Then,(1) Let  = {P, (′ϒA(P), ϑA(P))|P ∈ A} and  = {P, (′ϒB(P), ϑB(P))|P ∈ B} be two PFSS. Then,(2) Let  = {P, (′ϒA(P), ϑA(P))|P ∈ A} be a PFSS. Then, the necessity operation on PFSS is denoted by   ⊕   and defined as follows:Let  = {P, (′ϒA(P), ϑA(P))|P ∈ A} be a PFSS, then the possibility operation on PFSS is denoted by ⊗ and written as follows:Let  = {P, (′ϒA(P), ϑA(P))|P ∈ A} be a PFSS. Then, the following properties are satisfied.K, A)O]O = ⊗[  ⊕  O]O = ⊕  [⊗] = ⊕    ⊕  [  ⊕  ] = ⊗  ⊕  [⊗] = ⊕  ⊗[  ⊕  ] = ⊗⊗ = ⊕  ∧  ⊕    ⊕  [∨]=  ⊕  ∨  ⊕    ⊕  =⊗∧⊗⊗[∨]=⊗∨⊗⊗ to choose the best bike available in the market. Then, PFSM can be written by using parameters as follows:Let These four PFSS are represented by the following PFSM, respectively:By using score matrix definition,By using the definition of utility matrix,Now,x1, the 1st brand having maximum value, in this fashion, we conclude from the judgment of four experts that Honda is the best brand for business.From the above results g1 > g2 > g5 > g6 > g3 > g4.Ranking of alternative For comparative analysis, we made the comparison of our proposed method with different methods by using their algorithm. Through this process, we can easily show the credibility of our proposed method.Here, we use the method of Rathika and Subramanian , in whic  Step 1: input IFSSs and obtain intuitionistic fuzzy soft matrices (IFSM)  Step 2: take complement of IFSS and obtain complement IFSM from these setsA − B − C − D), (Ac − Bc − Cc − Dc), and value matrix of these IFSMs  Step 3: find , (Ac+Bc+Cc+Dc), and value matrix of these IFSMs  Step 3: find  from these sets  Step 3: evaluate the cross-product of FSM  Step 4: enumerate the optimum subscript matrix  Step 5: compute the best alternative that is having max valueWe haveg1 > g3 > g2 > g5 > g4 > g6.Ranking of alternative Furthermore, we use the algorithm of Rathika et al. , in whic  Step 1: input IFSSs and obtain IFSMs  Step 2: take complement of IFSS and obtain complement IFSM from these setsA − B − C − D), (Ac − Bc − Cc − Dc), and value matrix of these IFSMs  Step 3: find (  Step 4: compute the score of IFSMs and complement IFSMsSi for xi in-universe  Step 5: calculate the total score xi of the total score matrix Si has maximum value which is the best alternative  Step 6: we conclude that element Si has more values besides one value, then rerun the development by repeating the parameter.If max g1 > g3 > g4 > g2 > g5 > g6.Ranking of alternative Also, we use the technique of Chetia and Das , in whicStep 1: select the FSS set of parametersStep 2: obtain FSMs from these setsStep 3: find the union of these FSMsStep 4: enumerate the weight along with the individual item by taking rowwise sum membership valuesStep 5: compute the best alternative that is having max valueNow, weights matrix of the bikes of different brands,g1 > g4 > g3=g5 > g6 > g2.Ranking of alternative The results obtained through existing methodologies with the proposed technique with their score values are given in With the ongoing research and comparison, the results obtained by our proposed method and existing procedures are in In this paper, we developed some logical operators such as OR-operation and AND-operation for PFSS with their fundamental characteristics. Also, some novel operations such as necessity and possibility operations have been presented with their fundamental properties. A decision-making approach has been constructed for PFSM based on a score matrix and utility matrix. To confirm the validity of our established approach, a comprehensive numerical example has been developed. To express the legitimacy, effectiveness, and efficiency of our proposed method, a logical comparison between the current work and the proposed approach is also provided. The obtained consequences show that the developed technique is more reliable compared to existing techniques. Future research will focus on offering several other operators under the PFSS environment to address decision-making issues. Many other structures such as topological, algebra, and orderly structures can be developed and discussed in the environment under consideration. The proposed idea can be applied to many issues in real life, including the medical profession, pattern recognition, and economics.


The purpose of this paper is to define the concept of -fuzzy sets and discuss their relationship with other kinds of fuzzy sets. We describe some of the basic set operations on -fuzzy sets. -Fuzzy sets can deal with more uncertain situations than Pythagorean and intuitionistic fuzzy sets because of their larger range of describing the membership grades. Furthermore, we familiarize the notion of -fuzzy topological space and discuss the master properties of -fuzzy continuous maps. Then, we introduce the concept of -fuzzy points and study some types of separation axioms in -fuzzy topological space. Moreover, we establish the idea of relation in -fuzzy set and present some properties. Ultimately, on the basis of academic performance, the decision-making approach of student placement is presented via the proposed -fuzzy relation to ascertain the suitability of colleges to applicants. The concept of fuzzy sets was proposed by Zadeh . The theThe idea of intuitionistic fuzzy sets suggested by Atanassov  is one oThe concept of fuzzy topological spaces was introduced by Chang . He studThe main purpose of this paper is to introduce the concept of -fuzzy sets and compare them with the other types of fuzzy sets. We introduce the set of operations for the -fuzzy sets and explore their main features. Following the idea of Chang, we define a topological structure via -fuzzy sets as an extension of fuzzy topological space, intuitionistic fuzzy topological space, and Pythagorean fuzzy topological space. We discuss the main topological concepts in -fuzzy topological spaces such as continuity and compactness. In addition, the concept of relation to -fuzzy sets is investigated. Finally, an improved version of max-min-max composite relation for -fuzzy sets is proposed.In this section, we initiate the notion of -fuzzy sets and study their relationship with other kinds of fuzzy sets. Then, we furnish some operations to -fuzzy sets.X be a universal set. Then, the -fuzzy set -FS) D is defined by the following:αD(r) : X⟶ is the degree of membership and βD(r) : X⟶ is the degree of non-membership of r ∈ X to D, with the conditionLet r ∈ X to D is defined byThe degree of indeterminacy of αD(r))3+(βD(r))2+(πD(r))5=1, and πD(r)=0 whenever (αD(r))3+(βD(r))2=1. In the interest of simplicity, we shall mention the symbol D= for the -FS D={〈r, αD(r), βD(r)〉 :  r ∈ X}.It is clear that   is the degree of membership and βK(r) : X⟶ is the degree of non-membership of every r ∈ X to K.Let et (IFS)  (resp. Pet (IFS)  and Fermet (IFS) ) is defiαD(r)=0.9 and βD(r)=0.5 for X={r}. We obtain 0.9+0.5=1.40 > 1 and (0.9)2+(0.5)2=1.06 > 1 which means that D= neither follows the condition of IFS nor follows the condition of PFS. On the other hand, (0.9)3+(0.5)2=0.979 < 1 which means we can apply the -FS to control it. That is, D= is a -FS.To illustrate the importance of -FS to extend the grades of membership and non-membership degrees, assume that The set of -fuzzy membership grades is larger than the set of intuitionistic membership grades and Pythagorean membership grades.r1, r2 ∈ , we haveIt is well known that for any two numbers Then, we getHence, the space of -fuzzy membership grades is larger than the space of intuitionistic membership grades and Pythagorean membership grades. This development can be evidently recognized in X={rj :  j=1,…, k} be a universal set and D be -FS. If πD(rj)=0, then Let D is -FS and πD(rj)=0 for rj ∈ X; then,Presume that D be -FS and r ∈ X such that βD(r)=0.82 and πD(r)=0. Then, Let δ be a positive real number (δ > 0). If D1= and D2= are two -FSs, then their operations are defined as follows:D1∩D2=.D1 ∪ D2=.D1c=.Let We will use supremum “sup” instead of maximum “max” and infimum “inf” instead of minimum “min” if the union and the intersection are infinite.D1= and D2= are both -FSs. Then,D1∩D2===.D1 ∪ D2===.D1c=.δ=4.δ=4.Assume that L1= and L2= be two -FSs; then, the following properties hold:L1∩L2=L2∩L1.L1 ∪ L2=L2 ∪ L1.L1∩L2) ∪ L2=L2.(L1 ∪ L2)∩L2=L2.==L2∩L1.The proof is similar to (1).L1∩L2) ∪ L2= ∪ ===L2.(The proof is similar to (3).From L1=, L2= and L3= be three -FSs and δ > 0; then,L1∩(L2∩L3)=(L1∩L2)∩L3.L1 ∪ (L2 ∪ L3)=(L1 ∪ L2) ∪ L3.δ(L1 ∪ L2)=δL1 ∪ δL2.L1 ∪ L2)δ=L1δ ∪ L2δ.(2)The proof is similar to (1).(3)(4)The proof is similar to (3).For the three -FSs Lc is -FS for any -FS L.In the following result, we claim that L1= and L2= be two -FSs such that L1c and L2c are -FSs. Then,L1∩L2)c=L1c ∪ L2c.(L1 ∪ L2)c=L1c∩L2c.(2)The proof is similar to (1).For the two -FSs D1= and D2= be two -FSs; then,D1=D2 if and only if αD1=αD2 and βD1=βD2.D1 ≥ D2 if and only if αD1 ≥ αD2 and βD1 ≤ βD2.D2 ⊂ D1 or D1⊃D2 if D1 ≥ D2.Let D1= and D2= for X={x}, then D1=D2.If D1= and D2= for X={x}, then D2 ≤ D1 and D2 ⊂ D1.If In this section, we formulate the concept of -fuzzy topology on the family of -fuzzy sets whose complements are -fuzzy sets and scrutinize main properties. Then, we define -fuzzy continuous maps and give some characterizations. Finally, we establish two types of -fuzzy separation axioms and reveal the relationships between them.τ be a family of -fuzzy subsets of a non-empty set X. IfX, 0X ∈ τ where 1X= and 0X=,1D1∩D2 ∈ τ, for any D1, D2 ∈ τ,i∈IDi ∈ τ, for any {Di}i∈I ⊂ τ,∪then τ is called a -fuzzy topology on X and  is a -fuzzy topological space. We call D an open -FS if it is a member of τ and call its complement a closed -FS.Let τ={1X, 0X} the indiscreet -fuzzy topology on X. If τ contains all -fuzzy subsets, then we call τ the discrete -fuzzy topology on X.We call τ={1X, 0X, D1, D2, D3, D4, D5} be the family of -fuzzy subsets of X={x1, x2}, whereLet τ is -fuzzy topology on X.Hence, D on a set X is a -fuzzy set having the form D={〈r, αD(r), 1 − αD(r)〉 :  r ∈ X}. Then, every fuzzy topological space  in the sense of Chang is obviously a -fuzzy topological space in the form τ={D :  αD ∈ τ1} whenever we identify a fuzzy set in X whose membership function is αD with its counterpart D={〈r, αD(r), 1 − αD(r)〉 :  r ∈ X}. Similarly, one can note that every intuitionistic fuzzy topology (Pythagorean fuzzy topology) is -fuzzy topology. The following examples explain this note.We showed that every fuzzy set τ={1X, 0X, D1, D2} as family of fuzzy subsets of X={x}, whereConsider τ is fuzzy topology on X, and hence it is -fuzzy topology.Then, τ={1X, 0X, D1, D2} be the family of -fuzzy subsets on X={x1, x2} whereLet τ is -fuzzy topology. On the other hand, τ is neither intuitionistic fuzzy topology nor Pythagorean fuzzy topology.Hence, X, τ) be a -fuzzy topological space and D={〈x, αD(x), βD(x)〉 :  x ∈ X} be a -FS in X. Then, the -fuzzy interior and -fuzzy closure of D are, respectively, defined byD)=∩{H : H is a closed -FS in X and D ⊂ H}.cl(D)=∪{G : G is an open -FS in X and G ⊂ D}.int be a -fuzzy topological space and D be any -FS in X. Then,D) is an open -FS.int(D) is a closed -FS.cl(X)=cl(1X)=1X and int(0X)=cl(0X)=0X.int in D={〈c1, 0.67, 0.81〉, 〈c2, 0.75, 0.74〉}, then int(D)=0X and cl(D)=1X.Consider the -fuzzy topological space  be a -fuzzy topological space and D1, D2 be -FSs in X. Then, the following properties hold:D1) ⊂ D1 and D1 ⊂ cl(D1).int ⊂ int(D2) and cl(D1) ⊂ cl(D2).If D1 is an open -FS if and only if D1=int(D1).D1 is a closed -FS if and only if D1=cl(D1).Let ((1) and (2) are obvious.(3) and (4) follow from X, τ) be a -fuzzy topological space and D1, D2 be -FSs in X. Then, the following properties hold:D1) ∪ int(D2) ⊂ int(D1 ∪ D2).int(D1∩D2) ⊂ cl(D1)∩cl(D2).cl(D1∩D2)=int(D1)∩int(D2).int(D1) ∪ cl(D2)=cl(D1 ∪ D2).cl(Let ((1) and (2) follows from (1) of the above theorem.D1∩D2) ⊂ int(D1) and int(D1∩D2) ⊂ int(D2), we obtain int(D1∩D2) ⊂ int(D1)∩int(D2). On the other hand, from the facts int(D1) ⊂ D1 and int(D2) ⊂ D2, we have int(D1)∩int(D2) ⊂ D1∩D2 and int(D1)∩int(D2) ∈ τ; we see that int(D1)∩int(D2) ⊂ int(D1∩D2), and hence int(D1∩D2)=int(D1)∩int(D2).(3): since int((4) can be proved similar to (3).X, τ) be a -fuzzy topological space and D be -FS in X. Then, the following properties hold:Dc)=int(D)c.cl(Dc)=cl(D)c.int(Dc)c=int(D).cl(Dc)c=cl(D).int(Let (We only prove (1); the other parts can be proved similarly.D={〈x, αD(x), βD(x)〉 :  x ∈ X} and suppose that the family of open -fuzzy sets contained in D is indexed by the family {〈x, αUi(x), βUi(x)〉 :  i ∈ J}. Then, int(D)={〈x, ∨αUi(x), ∧βUi(x)〉}. Therefore, int(D)c={〈x, ∧βUi(x), ∨αUi(x)〉}. Now, Dc={〈x, βD(x), αD(x)〉} such that αUi ≤ αD, βUi ≥ βD for each i ∈ J. This implies that {〈x, βUi(x), αUi(x)〉 :  i ∈ J} is the family of all closed -fuzzy sets containing Dc. That is, cl(Dc)={〈x, ∧βUi(x), ∨αUi(x)〉}. Hence, cl(Dc)=int(D)c.Let f : X⟶Y be a map and A and B be -fuzzy subsets of X and Y, respectively. The functions of membership and non-membership of the image of A, denoted by f[A], are, respectively, calculated byLet B, denoted by f−1[B], are, respectively, calculated byThe functions of membership and non-membership of preimage of f[A] and f−1[B] are -fuzzy subsets, consider γA(z))5=αA(z))3+(βA(z))2. If f−1(y) is non-empty, then we obtainTo show that f−1(y)=ϕ leads to the fact that (αf[A](y))3+(βf[A](y))2=1.In contrast, f−1[B].It is easy to prove the case of f : X⟶Y be a map s.t. A and B are -fuzzy subsets of X and Y, respectively. Then, we havef−1[Bc]=f−1[B]c.f[A]c⊆f[Ac].B1⊆B2, then f−1[B1]⊆f−1[B2] where B1 and B2 are -fuzzy subsets of Y.If A1⊆A2, then f[A1]⊆f[A2] where A1 and A2 are -fuzzy subsets of X.If f[f−1[B]]⊆B.A⊆f−1[f[A]].Let (1)v ∈ X and let B be a -fuzzy subset of Y. Then,Consider βf−1[Bc](v)=βf−1[B]c(v). Therefore, f−1[Bc]=f−1[B]c, as required.Similarly, one can have (2)w ∈ Y such that f−1(w) ≠ ϕ and for any -fuzzy subset A of X, we can writeFor any Now from , we havef−1(w)=ϕ. Following a similar technique, we obtain βf[Ac](w) ≤ βf[A]c(w), which means that f[A]c⊆f[Ac].The proof is easy when (3)B1⊆B2. Then, for each v ∈ X, αf−1[B1](v)=αB1(f(v)) ≤ αB2(f(v))=αf−1[B2](v). Also, βf−1[B1](v) ≥ βf−1[B2](v). Hence, we obtain the desired result.Assume that (4)A1⊆A2 and w ∈ Y. The proof is easy when f(w)=ϕ. So, presume that f(w) ≠ ϕ. Then,Assume that αf[A1] ≤ αf[A2] follows. Similarly, we have βf[A1] ≥ βf[A2].Thus, (5)w ∈ Y s.t. f(w) ≠ ϕ, we find thatFor any αf[f−1[B]](w)=0 ≤ αB(w) when f(w)=ϕ. Similarly, we have βf[f−1[B]](w)=0 ≥ βB(w).On the other hand, we have (6)v ∈ X, we haveFor any βf−1[f[A]] ≤ βA.Similarly, we have The proof of the following result is easy, and hence it is omitted.X and Y be two non-empty sets and f : X⟶Y be a map. Then, the following statements are true:f[∪i∈IAi]=∪i∈If[Ai] for any -fuzzy subset Ai of X.f−1[∪i∈IBi]=∪i∈If−1[Bi] for any -fuzzy subset Bi of Y.f[A1∩A2] ⊂ f[A1]∩f[A2] for any two -fuzzy subsets A1 and A2 of X.f−1[∩i∈IBi]=∩i∈If−1(Bi) for any -fuzzy subset Bi of Y.Let A and U are two -fuzzy subsets. We call U a neighborhood of A, briefly nbd, if there exists an open -fuzzy subset E such that A⊆E⊆U.In a -fuzzy topological space, consider that A is open iff it contains a nbd of its each subset.A -fuzzy subset The proof is easy.f : ⟶ is said to be -fuzzy continuous if for any -fuzzy subset A of X and for any nbd V of f[A] there is a nbd U of A s.t. f[U]⊆V.A map f : ⟶:f is -fuzzy continuous.A of X and each nbd V of f[A], there is a nbd U of A s.t. for each B⊆U, we obtain f[B]⊆V.For each -FS A of X and for each nbd V of f[A], there is a nbd U of A s.t. U⊆f−1[V].For each -FS A of X and for each nbd V of f[A], f−1[V] is a nbd of A.For each -FS The following statements are equivalent for a map f be a -fuzzy continuous map. Consider A as a -FS of X and V as a nbd of f[A]. Then, there is a nbd U of A s.t. f[U]⊆V. If B⊆U, we obtain f[B]⊆f[U]⊆V. (1)⇒(2): let A as a -FS of X and V as a nbd of f[A]. According to (2), there is a nbd U of A s.t. for each B⊆U, we find f[B]⊆V. Therefore, B⊆f−1[f[B]]⊆f−1[V]. Since B is chosen arbitrarily, U⊆f−1[V]. (2)⇒(3): assume A is a -FS of X and V is a nbd of f[A]. According to (3), there is a nbd U of A s.t. U⊆f−1[V]. Since U is a nbd of A, there is an open -FS K of X s.t. A⊆K⊆U. On the other hand, we obtain A⊆K⊆f−1[V] because U⊆f−1[V]. This means that f−1[V] is a nbd of A. (3)⇒(4): presume A is a -FS of X and V is a nbd of f[A]. By hypothesis, f−1[V] is a nbd of A. So, there is an open -FS K of X s.t. A⊆K⊆f−1[V] which means f[K]⊆f[f−1[V]]⊆V. Moreover, K is an open -FS, so it is a nbd of A. Hence, we obtain the proof that f is -fuzzy continuous. (4)⇒(1): suppose that f : ⟶ is -fuzzy continuous iff f−1[B] is an open -FS of X for each open -FS B of Y.A map f as a -fuzzy continuous map. Consider an open -FS B of Y s.t. A⊆f−1[B]. This directly gives that f[A]⊆B. It follows from V of f[A] satisfying V⊆B. Now, f is -fuzzy continuous, so by (4) of f−1[V] is a nbd of A. Also, it follows from (3) of f−1[V]⊆f−1[B]. So, f−1[B] is a nbd of A. Since A is an arbitrary subset of f−1[B], then by f−1[B] is open.Necessity: presume A is a -FS of X and V is a nbd of f[A]. Then, τ2 contains a -FS L of s.t. f[A]⊆L⊆V. By hypothesis, f−1[L] is an open -FS. Also, we have A⊆f−1[f[A]]⊆f−1[L]⊆f−1[V]. Thus, f−1[V] is a nbd of A which demonstrates that f is -fuzzy continuous.Presume We build the following two examples such that the first one provides a -fuzzy continuous map, whereas the second one presents a fuzzy map that is not -fuzzy continuous.X={a1, a2} with the -fuzzy topology τ1={1X, 0X, A1} and Y={b1, b2} with the -fuzzy topology τ2={1Y, 0Y, B1}, whereConsider f : X⟶Y be defined as follows:Let Y, 0Y, and B1 are open -fuzzy subsets of Y, thenX. Thus, f is -fuzzy continuous.Since 1X={a1, a2} with the -fuzzy topology τ1={1X, 0X} and Y={b1, b2} with the -fuzzy topology τ2={1Y, 0Y, B1}, where B1={〈b1, 0.82, 0.62〉, 〈b2, 0.52, 0.90〉}.Consider f : X⟶Y be defined as follows:Let B1 is an open -fuzzy subset of Y, but f−1[B1]={〈a1, 0.82, 0.62〉, 〈a2, 0.52, 0.90〉} is not an open -fuzzy subset of X, f is not -fuzzy continuous.Since f : ⟶ is -fuzzy continuous.B of Y we have that f−1[B] is a closed -fuzzy subset of X.For each closed -fuzzy subset f−1[B])⊆f−1[cl(B)] for each -fuzzy set in Y.cl(f−1[int(B)]⊆int(f−1[B]) for each -fuzzy set in Y.The following are equivalent to each other:They can be easily proved using Theorems Y, τ) be a -fuzzy topological space and f : X⟶Y be a map. Then, there is a coarsest -fuzzy topology τ1 over X such that f is -fuzzy continuous.Let -fuzzy subsets τ1 is the coarsest -fuzzy topology over X such that f is -fuzzy continuous.(1)x ∈ X thatWe can write for any βf−1[0Y](x)=βX0(x) for any x ∈ X which implies f−1[0Y]=0X. Now, as 0Y ∈ τ, we have 0X=f−1[0Y] ∈ τ1. In a similar manner, it is easy to see that 1X=f−1[1Y] ∈ τ1.Similarly, we immediately have (2)D1, D2 ∈ τ1. Then, for i=1,2, there exists Bi ∈ τ such that f−1[Bi]=Di which implies αf−1[Bi]=αDi and βf−1[Bi]=βDi. Thus, we obtain for any x ∈ X thatAssume that βD1∩D2=βf−1[B1∩B2]. Hence, we get D1∩D2 ∈ τ1.Similarly, it is not difficult to see that (3)Di}i∈I is an arbitrary subfamily of τ1. Then, for any i ∈ I, there exists Bi ∈ τ1 such that f−1[Bi]=Di which implies αf−1[Bi]=αDi and βf−1[Bi]=βDi. Therefore, one can get for any x ∈ X thatAssume that {We prove that βi∈IDi∪=βf−1[∪i∈IBi]. Thus, we have ∪i∈IDi ∈ τ1.On the other hand, it is easy to see that f is trivial. Now, we prove that τ1 is the coarsest -fuzzy topology over X such that f is -fuzzy continuous. Let τ2⊆τ1 be a -fuzzy topology over X such that f is -fuzzy continuous. If B ∈ τ1, then there is V ∈ τ such that f−1[V]=B. Since f is -fuzzy continuous with respect to τ2, we have B=f−1[V] ∈ τ2. Hence, τ2=τ1, as required.From Separation axioms are one of the most important and popular notions in topological studies. They have been studied and applied to model some real-life issues in soft setting as explained in , 17.X ≠ ∅ and x ∈ X be a fixed element in X. Suppose that r1 ∈  are two fixed real numbers such that r13+r22 ≤ 1. Then, a -fuzzy point pr1, r2), βp(x)〉} is defined to be a -fuzzy set of X as follows.y ∈ X. In this case, x is called the support of pr1, r2)-fuzzy point pr1, r2)-fuzzy set D={〈x, αD(x), βD(x)〉} of X denoted by pr1, r2)(x ∈ D if r1 ≤ αD(x) and r2 ≥ βD(x). Two -fuzzy points are said to be distinct if their supports are distinct.Let D1={〈x, αD1(x), βD1(x)〉} and D2={〈x, αD2(x), βD2(x)〉} be two -fuzzy sets of X. Then, D1⊆D2 if and only if pr1, r2)-fuzzy point pr1, r2), and x, y ∈ X. A -fuzzy topological space  is said to be(1)T0 if for each pair of distinct -fuzzy points pr1, r2)-fuzzy sets L and K such that(2)T1 if for each pair of distinct -fuzzy points pr1, r2)-fuzzy sets L and K such thatLet X, τ) be a -fuzzy topological space. If  is T1, then  is T0.Let -fuzzy topology τ={1X, 0X, D}, where D={〈c1, 1,0〉, 〈c2, 0,1〉}. Then,  is T0 but not T1 because there does not exist an open -fuzzy set K such that K={〈x, 0,1〉, 〈y, 1,0〉}.Consider X, τ) be a -fuzzy topological space, r1, r3 ∈ . If  is T0, then for each pair of distinct -fuzzy points pr1, r2)(x) ≠ cl(y).Let  be T0 and pr1, r2)-fuzzy points of X. Then, there exist two open -fuzzy sets L and K such thatLet -fuzzy set which does not contain pr1, r2)(y) is the smallest closed -fuzzy set containing pr3, r4)(y)⊆Lc, and therefore pr1, r2)(y). Consequently, cl(x) ≠ cl(y).Let X, τ) be a -fuzzy topological space. If px is closed -fuzzy set for every x ∈ X, then,  is T1.Let x is a closed -fuzzy set for every x ∈ X. Let pr1, r2)-fuzzy points of X; then, x ≠ y implies that pxc and pyc are two open -fuzzy sets such thatSuppose X, τ) is T1.Thus, -FSs. Moreover, we provide a numerical example to elaborate on how we can apply the composite relations to obtain the optimal choices.X and Y be two (crisp) sets. The -fuzzy relation R -FR) from X to Y is a -FS of X × Y characterized by the degree of membership function αR and degree of non-membership function βR. The -FR R from X to Y will be denoted by R(X⟶Y). If D is a -FS of X, then(1)R(X⟶Y) with D is a -FS C of Y denoted by C = R o D and is defined byThe max-min-max composition of the -FR (2)R(X⟶Y) with D is a -FS C of Y denoted by C = R o D, such thatThe improved composite relation of Let Q(X⟶Y) and R(Y⟶Z) be two -FRs. Then, for all  ∈ X × Z and n ∈ Y,(1)R o Q is the -fuzzy relation from X to Z defined byThe max-min-max composition (2)R o Q is the -fuzzy relation from X to Z such thatThe improved composite relation Let The improved composite and max-min-max composite relations for -fuzzy sets are calculated by the following:D1 and D2 be two -fuzzy sets for X={x1, x2, x3, x4}. Assume thatLet D1 and D2 as follows:By using Definitions x1, x2, x3, x4) in D1 and D2, respectively, is 0.7, 0.5, 0.6, and 0.8. Also, the maximum value of the non-membership values of the elements  in D1 and D2, respectively, is 0.79, 0.87, 0.85, and 0.69. From It is obvious that the minimum value of the membership values of the elements 45), we oWe localize the idea of -FR as follows.S={r1,…, rl} be a finite set of subjects related to the colleges, C={b1,…, bm} be a finite set of colleges, and A={t1,…, tn} be a finite set of students. Suppose that we have two -FRs, U(A⟶S) and R(S⟶C), such thatαU denotes the degree to which the student (t) passes the related subject requirement (r). βU denotes the degree to which the student (t) does not pass the related subject requirement (r). αR denotes the degree to which the related subject requirement (r) determines the college (b). βR denotes the degree to which the related subject requirement (r) does not determine the college (b). Let T=RoU is the composition of R and U. This describes the state in which the students, ti, with respect to the related subject requirement, rj, fit the colleges, bk. Thus,ti ∈ A and bk ∈ C, where i, j, and k take values from 1,…, n.αRoU and βRoU of the composition T = R o U are as follows , College of Medicine (M), College of Agricultural Engineering Sciences (AE), College of Sport Sciences (Sp), College of Science (S)} be the set of colleges the students are vying for -FS. We illustrated that this type produces membership grades larger than intuitionistic and Pythagorean fuzzy sets which are already defined in the literature. However, Fermatean fuzzy sets give a larger space of membership grades than -FS. We summarize the relationships in terms of the space of membership and non-membership grades in the following figure.Regarding topological structure, we illustrated that every fuzzy topology in the sense of Chang (intuitionistic fuzzy topology and Pythagorean fuzzy topology) is a -fuzzy topology. In contrast, every -fuzzy topological space is a Fermatean fuzzy topological space because every -fuzzy subset of a set can be considered as a Fermatean fuzzy subset. The next example elaborates that Fermatean fuzzy topological space need not be a -fuzzy topological space.X={x1, x2}. Consider the following family of Fermatean fuzzy subsets τ={1X, 0X, D1, D2}, whereLet X, τ) is a Fermatean fuzzy topological space, but  is not a -fuzzy topological space.Observe that -fuzzy sets and studied their relationship with intuitionistic fuzzy, Pythagorean fuzzy, and Fermatean fuzzy sets. In addition, some operators on -fuzzy sets are defined and their relationships have been proved. The notions of -fuzzy topology, -fuzzy neighborhood, and -fuzzy continuous mapping were studied. Furthermore, we introduced the concept of -fuzzy points and studied separation axioms in -fuzzy topological space. We also introduced the concept of relation to -fuzzy sets, called -FR. Moreover, based on academic performance, the application of -FSs was explored on student placement using the proposed composition relation.In future work, more applications of -fuzzy sets may be studied; also, -fuzzy soft sets may be studied. In addition, we will try to introduce the compactness and connectedness in -fuzzy topological spaces. The motivation and objectives of this extended work are given step by step in this paper.


Wymioty są często spotykanym objawem w populacji pediatrycznej. Ich etiologia jest zróżnicowana, począwszy od łagodnych zaburzeń czynnościowych, aż do ciężkich zagrażających życiu chorób układowych. Najczęściej występują w przebiegu chorób przewodu pokarmowego, ale mogą dotyczyć także wielu innych schorzeń zlokalizowanych poza nim. Z uwagi na zróżnicowaną etiologię obejmującą różne układy i narządy niekiedy mogą sprawiać trudności diagnostyczne. W niniejszym artykule przedstawiono przypadek zespołu Panayiotopou/osa – dziecięcej padaczki potylicznej o wczesnym początku . Charakterystyczną cechą tego zespołu są napady z objawami ze strony układu autonomicznego lub występowanie autonomicznych stanów padaczkowych. Dominującymi objawami są wymioty i nudności, które w pierwszej kolejności najczęściej sugerują zapalenie żołądka czy jelit, bądź migrenę lub proces rozrostowy w ośrodkowym układzie nerwowym. Rzadko w diagnostyce różnicowej bierze się pod uwagę możliwość występowania wymiotów jako elementu napadu padaczkowego. Celem naszej pracy jest zwrócenie uwagi na trudności w zdefiniowaniu właściwej przyczyny nawracających wymiotów. Niejednokrotnie, pomimo szczegółowo zebranego wywiadu oraz szerokiej diagnostyki, dopiero bezpośrednie stwierdzenie charakterystycznych objawów dla danego zespołu pozwala na ustalenie ostatecznego rozpoznania. Helicobacter pylori, zapalenie wątroby, zapalenie trzustki, ostre schorzenia chirurgiczne  czy zespół cyklicznych wymiotów. W ustaleniu ich etiologii podstawowe znaczenie ma dokładnie zebrany wywiad zawierający pytania o początek, częstotliwość, czas trwania wymiotów oraz objawy towarzyszące [Wymioty są nieswoistym, często występującym u dzieci objawem, o zróżnicowanej etiologii, począwszy od łagodnych zaburzeń czynnościowych, aż do ciężkich zagrażających życiu chorób układowych. W większości przypadków problem ustępuje szybko i samoistnie. Wymioty najczęściej występują w przebiegu chorób przewodu pokarmowego, ale mogą dotyczyć także wielu innych schorzeń zlokalizowanych poza nim . Etiologrzyszące ,2.Celem pracy jest przedstawienie trudności diagnostycznych u pacjentki, u której przyczyną uporczywych wymiotów ostatecznie okazała się być padaczka. Po prawie 2 latach występowania nawracających wymiotów, kilkukrotnych hospitalizacjach, przeprowadzeniu wielokierunkowej diagnostyki, dopiero po zaobserwowaniu charakterystycznych objawów w trakcie pobytu w Klinice rozpoznano zespół Panayiotopoulosa, należący do padaczek o początku w dzieciństwie .Dziewczynka 11-letnia została przyjęta do Kliniki z powodu nawracających wymiotów i bólów brzucha celem diagnostyki i leczenia. Z wywiadu wiadomo, że sporadyczne wymioty pojawiły się u dziewczynki w 9 rż. W pierwszym roku ich występowania nie obserwowano cykliczności ani towarzyszących dolegliwości, natomiast w 10 rż. ich częstotliwość uległa nasileniu, a rodzice zaobserwowali, że wymioty poprzedzały: złe samopoczucie, lęk , zawroty głowy, senność oraz silny ból głowy. Najczęściej epizody trwały 1 dzień, ustępowały samoistnie. W ciągu miesiąca poprzedzającego hospitalizację w Klinice dolegliwości występowały średnio co tydzień. Wymiociny nigdy nie zawierały niepokojących domieszek, nie było konieczności wyrównywania zaburzeń jonowych oraz podaży płynów infuzyjnych.Dziewczynka z tego powodu była wielokrotnie hospitalizowana w różnych ośrodkach w Polsce. W wyniku przeprowadzonej diagnostyki wykluczono: proces rozrostowy OUN w oddziale neurologii , zaburzenia gospodarki węglowodanowej w oddziale endokrynologii, boreliozę oraz padaczkę w kolejnym oddziale neurologii (na podstawie prawidłowego 24-godzinnego zapisu EEG nie potwierdzono padaczki). Dziewczynkę dwukrotnie konsultował psycholog, nie stwierdzając istotnych nieprawidłowości. Z wywiadu wynika również, że rodzicom kilkukrotnie sugerowano konieczność konsultacji psychiatrycznej. Poza tym rodzice zgłosili pojawianie się grudkowych, swędzących zmian skórnych w zgięciach łokciowych i na twarzy.Przy przyjęciu do Kliniki stan dziecka był dobry. Z odchyleń od normy w badaniu przedmiotowym stwierdzono: pojedyncze zmiany grudkowe w zgięciach łokciowych. W badaniach laboratoryjnych bez nieprawidłowości. Ze względu na zgłaszane w trakcie epizodów zawroty głowy dziewczynka była konsultowana przez laryngologa, który nie potwierdził cech ostrego uszkodzenia części obwodowej zmysłu równowagi. Konsultujący psychiatra stwierdził możliwość występowania zachowań hipochondrycznych.W kolejnym dniu hospitalizacji dziewczynka od rana zgłaszała pogorszenie samopoczucia i zawroty głowy. W godzinach popołudniowych, w pozycji leżącej, wystąpiły jakościowe zaburzenia świadomości pod postacią pogorszenia kontaktu logicznego, z towarzyszącymi automatyzmami w formie głośnego przełykania śliny utrzymujące się około 10 minut oraz wymioty. Następnie doszło do utraty przytomności i wystąpił uogólniony napad padaczkowy toniczno-kloniczny trwający około 2 minut. Po odzyskaniu przytomności obserwowano stopniowe ustępowanie ilościowych zaburzeń świadomości (senność ponapadową).Na podstawie przebiegu klinicznego i otrzymanych wyników badań wysunięto podejrzenie zespołu Panayiotopoulosa. Dziewczynkę przekazano do Oddziału Neurologii Dziecięcej celem dalszej diagnostyki i leczenia. Ponownie wykonano EEG, w którym w stanie czuwania w odprowadzeniach potylicznych stwierdzono czynność napadową o charakterze iglic, w pozostałych odprowadzeniach, bez nieprawidłowości. Do leczenia włączono na stałe preparat kwasu walproinowego. Od tej pory, tj. przez kolejne 8 miesięcy, do dziś nie wystąpił żaden epizod drgawkowy, nie obserwowano zaburzeń świadomości ani wymiotów.Zespół Panayiotopoulosa to dziecięca padaczka potyliczna o wczesnym początku . Po raz Zespół Panayiotopoulosa został stosunkowo niedawno wyodrębniony, dlatego początkowo częstość występowania nie była jednoznacznie ustalona , 10. ObePoczątek objawów występuje między 1-14 rokiem życia, ze szczytem miedzy 3-6 rż. (76% przypadków) , 14, 15.Do napadów autonomicznych koniecznych do rozpoznania zespołu Panayiotopoulosa należą: nudności,wymioty, hipersaliwacja, sinica, bladość powłok skórnych, zaburzenia termoregulacji, bóle głowy, zaburzenia rytmu serca i oddechu ,16. WymiNudności i wymioty występujące u pacjentów z zespołem Panayiotopoulosa często początkowo nie są traktowane, jako element napadu padaczkowego, ale sugerują podejrzenie zapalenia żołądka i jelit, choroby lokomocyjnej bądź migreny, alergii pokarmowej, natomiast dołączenie się napadów połowiczych i uogólnionych nasuwa podejrzenie zapalenia mózgu lub procesu rozrostowego w ośrodkowym układzie nerwowym , 12.W diagnostyce różnicowej u przedstawionej pacjentki wzięto pod uwagę wszystkie powyższe przyczyny, jak również uwzględniono dziecięce zespoły epizodyczne, takie jak: zespół cyklicznych wymiotów, migrenę brzuszną i przedsionkową, łagodne napadowe zawroty głowy, łagodny napadowy kręcz szyi . PacjentWynik badania neurologicznego pacjentów z zespołem Panayiotopoulosa jest prawidłowy. Choroba najczęściej nie powoduje zaburzeń rozwoju psychoruchowego oraz intelektualnego dziecka. Czasami w okresie nasilenia napadów u części dzieci mogą wystąpić niewielkie zaburzenia funkcji wykonawczych i językowych .W zapisach EEG międzynapadowych występują zmiany napadowe wieloogniskowe pod postacią iglic i zespołów iglicy z falą wolną o zmiennej lokalizacji w kolejnych badaniach. U 75% pacjentów z zespołem Panayiotopoulosa w badaniu rejestrowano zmiany w okolicy potylicznej, u 25% zmiany w okolicy innej niż potyliczna, tylko u 2% występują wyładowania uogólnione, natomiast u 9% nie stwierdzono zmian napadowych , 16, 19.EEG stwierdzono uogólnione wyładowania zespołów iglicy z falą wolną, zarówno w zapisach wykonanych w stanie czuwania, jak i we śnie, które były na początku jedyną manifestacją zespołu . W zdecyZespół Panayiotopoulosa uważa się za padaczkę zwykle niewymagającą leczenia p/padaczkowego, napady mają tendencję do ustępowania , ale opiZa rozpoznaniem zespołu Panayiotopoulosa u omawianej pacjentki przemawia: typowy wiek występowania objawów, typowe objawy autonomiczne pod postacią: wymiotów, bólu głowy, zwrotu gałek ocznych, zaburzeń świadomości (w naszym przypadku dziewczynka była „jakby nieobecna”), lęku oraz głośnego przełykania śliny, zmiany w EEG (w odprowadzeniach potylicznych czynność napadowa o charakterze iglic), prawidłowy rozwój dziecka oraz efekt kliniczny zastosowanego leczenia p/padaczkowego. Zespół Panayiotopoulosa jest chorobą o łagodnym przebiegu, zwykle niezaburzającą prawidłowego rozwoju psychoruchowego, dobrze znaną neurologom, ale jeszcze słabo pediatrom.Nawracające wymioty u dzieci mogą być objawem różnych schorzeń, w tym padaczki.Wystąpienie zaburzeń orofaryngealnych i uogólnionych drgawek w trakcie hospitalizacji w Klinice pozwoliło na ustalenie właściwego rozpoznania.


Deficyt aktywności transaldolazy należy do wrodzonych błędów metabolizmu na szlaku przemiany pentoz, który do tej pory rozpoznano i opisano u 33 pacjentów, w tym 4 z Polski.W artykule przedstawiono obraz kliniczny, patogenezę i diagnostykę choroby. Autorzy przedstawili ponadto własną propozycję algorytmu diagnostyki deficytu transaldolazy. Szlak pentozofosforanowy obejmuje ciąg reakcji biochemicznych, zachodzących w cytozolu, których produktami końcowymi są rybozo-5-fosforan i NADPH. Sumaryczne równanie reakcji cyklu pentozofosforanowego jest następujące:Deficyt aktywności transaldolazy (ang. + + H2O → rybozo-5-fosforan + 2NADPH + 2H+ + CO2.glukozo-6-fosforan + 2NADPTa droga przemian biochemicznych spełnia dwie role:wytwarzanie NADPH (zredukowany dinukleotyd nikotynoamidoadeninowy), który spełnia rolę donora protonów i elektronów w redukcyjnych procesach biosyntezy.dostarczanie reszt rybozy do biosystezy nukleotydowi kwasów nukleinowych.Szczególną intensywność wykazuje on w okresie wzrastania płodu oraz w pierwszych latach życia , 2, 3.Przemiany na szlaku pentozofosforanowym dzielimy na dwie fazy: nieodwracalną fazę oksydacyjną i odwracalną fazę nieoksydacyjną. Transaldolaza (EC 2.2.1.2), obok transketolazy, jest enzymem odwracalnej fazy cyklu pentozofosforanowego i katalizuje następującą reakcję:sedoheptulozo-7-fosforan + gliceraldehydo-3-fosforan → erytrozo-4-fosforan + fruktozo-6-fosforanW wyniku deficytu transaldolazy dochodzi do nagromadzenia polioli – erytritolu, arabitolu, rybitolu, sedoheptitolu, perseitolu oraz siedmiowęglowych cukrów − sedoheptulozy, mannoheptulozy i fosfosedoheptulozy.Celem pracy jest analiza obrazu klinicznego, patogenezy i diagnostyki deficytu aktywności transaldolazy na podstawie danych z piśmiennictwa oraz doświadczeń własnych. Jak do tej pory, w dostępnych publikacjach nie został przedstawiony algorytm diagnostyczny, stąd autorzy przedstawili propozycję własną algorytmu diagnostyki TALDO.2Deficyt aktywności transaldolazy został po raz pierwszy opisany przez Verhoeven i wsp., w 2001 r., u kilkumiesięcznego niemowlęcia z wewnątrzmacicznym opóźnieniem wzrastania, hepatosplenomegalią i zaburzeniami krzepnięcia . Biopsja), w tym 4 z Polski [4-18]. Większość (79%) z nich pochodziła od spokrewnionych rodziców, w tym 12 z jednego arabskiego rodu [Dotychczas, TALDO opisano u 33 pacjentów , w tym 4ego rodu .early-onset TALDO) – wystąpienie objawów w okresie prenatalnym lub w ciągu pierwszych 3 miesięcy życia, stanowiącą chorobę o ciężkim przebiegu i niepomyślnym rokowaniu – oraz postać objawiającą się później (ang. late-onset TALDO) – powyżej 3. miesiąca życia, stanowiącą powoli postępującą chorobę z możliwym długoletnim przeżyciem [Obraz kliniczny TALDO jest zróżnicowany, jednak zawsze obejmuje postępujące uszkodzenie wątroby oraz nerek (dysfunkcja cewek nerkowych) . Wyróżnizeżyciem .Większość przypadków opisanych w literaturze stanowiło TALDO o wczesnym początku. U noworodków i niemowląt stwierdzano małą masę urodzeniową – odnotowywano już wewnątrzmaciczne opóźnienie wzrastania, powiększenie wątroby i śledziony, niedokrwistość, małopłytkowość, zaburzenia krzepnięcia, a w kilku przypadkach obrzęk płodowy. W niektórych przypadkach przebieg ciąży charakteryzował się masywnym przyrostem masy ciała u matek oraz dużym i nieprawidłowym łożyskiem. Blisko połowa tych pacjentów zmarła w okresie niemowlęcym, najczęściej w wyniku krwawienia, jako efektu koagulopatii w przebiegu uszkodzenia wątroby , 17, 18.Nieco odmiennie przedstawia się przebieg deficytu transaldolazy o wczesnym początku u polskich pacjentów, którzy mimo początkowo złego rokowania , wkroczyli w drugą dekadę życia (3 z 4 pacjentów) z objawami skompensowanej marskości wątroby i kontrolowanej tubulopatii (poprzez leczenie substytucyjne) .TALDO o późnym początku wystąpienia objawów, mimo potencjalnie lepszego rokowania, prowadzi powoli do postępującego uszkodzenia wątroby i nerek. LeDuc i wsp. opisali 8-letniego chłopca, u którego TALDO zostało wykryte przypadkowo (skrining rodzinny) . Al-ShamW przebiegu TALDO w wątrobie stwierdza się charakterystyczne drobnoguzkowe włóknienie, a ostatecznie marskość wątroby. Z czasem u chorych z dłuższym okresem przeżycia pojawiają się objawy tubulopatii – hiperkalciuria, białkomocz cewkowy (kłębuszkowo-cewkowy) – zwykle jako pierwsze objawy, aminoaciduria, glukozuria, kwasica kanalikowa – oraz stopniowo rozwija się przewlekła choroba nerek . W obrazhaemangioma cavernosum). W literaturze znajdują się ponadto opisy pacjentów z cutis laxa oraz hipertrychozą [Charakterystyczne dla TALDO są zmiany skórne – poszerzona siatka naczyń i teleangiektazje na skórze tułowia (zwłaszcza pleców) oraz tendencja do tworzenia naczyniaków jamistych , a badania hormonalne wykazały u części z nich hipogonadyzm hipergonadotropowy.W niektórych przypadkach opisywane są wrodzone wady serca, do najczęstszych należą drożny przewód tętniczy oraz ubytek przegrody międzyprzedsionkowej , 17, 18.Rozwój psychoruchowy i umysłowy pacjentów są prawidłowe , 17, 18.3Deficyt aktywności transaldolazy prowadzi do kumulacji w płynach ustrojowych polioli − erytritolu, arabitolu, rybitolu, sedoheptitolu, perseitolu i siedmiowęglowych cukrów − sedoheptulozy, mannoheptulozy i fosfosedoheptulozy , 21, 22.Diagnostyka opiera się na stwierdzeniu wydalania z moczem, jak i kumulacji w surowicy (metodą spektrometrii mas) rybitolu, arabitolu, erytritolu, oraz o najwyższej czułości sedoheptulozy i sedoheptulozo-7-fosforanu , 21, 22.TALDO1)  .W patogenezie schorzenia podkreśla się toksyczny efekt działania gromadzonych metabolitów (ufosforylowanych cukrów) na hepatocyty i kanaliki nerkowe. Inna hipoteza podkreśla znaczenie stresu oksydacyjnego, który jest efektem braku NADPH koniecznego dla odtwarzania zredukowanej formy glutationu – kofaktora peroksydazy glutationowej przeciwdziałającej stresowi oksydacyjnemu , 4. Redu4Leczenie pacjentów z TALDO jest wyłącznie objawowe. Upośledzenie czynności syntetycznej wątroby może wiązać się z koniecznością okresowej podaży albumin czy osoczowych czynników krzepnięcia. Podobnie, upośledzenie czynności cewek nerkowych może wymagać suplementacji nieorganicznych fosforanów, podaży alfa-kalcydiolu czy wyrównywania zaburzeń elektrolitowych.Dyskusyjne są wskazania do ewentualnego przeszczepienia wątroby. Niektórzy autorzy sugerują możliwość uszkodzenia przeszczepionej wątroby , 15, 16.Przeszczepienie wątroby u polskich pacjentów wobec względnie stabilnego ich stanu i równoczesnego uszkodzenia nerek nie jest obecnie brane pod uwagę . Ze wzgl5U pacjentów z niewyjaśnionym postępującym uszkodzeniem wątroby i nerek wskazane jest wykonanie badania kumulacji polioli. Potwierdzenia/wykluczenia choroby wymaga również każdy przypadek noworodka/niemowlęcia z hepatosplenomegalią, niedokrwistością, małopłytkowością i zaburzeniami krzepnięcia, zwłaszcza urodzonego z małą urodzeniową masą ciała (oraz dużym przyrostem masy ciała matki w trakcie trwania ciąży).Włóknienie, marskość wątroby o nieustalonej etiologii zawsze wymaga badania stężenia polioli.