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In this paper, we propose a fast spectral-Galerkin method for solving PDEs involving an integral fractional Laplacian in Rd, which is built upon two essential components: (i) the Dunford-- Taylor formulation of the fractional Laplacian; and (ii) Fourier-like biorthogonal mapped Chebyshev functions (MCFs) as basis functions. As a result, the fractional Laplacian can be fully diagonalized, and the complexity of solving an elliptic fractional PDE is quasi-optimal, i.e., O((N log2 N)d) with N being the number of modes in each spatial direction. Ample numerical tests for various decaying exact solutions show that the convergence of the fast solver perfectly matches the order of theoretical error estimates. With a suitable time discretization, the fast solver can be directly applied to a large class of nonlinear fractional PDEs. As an example, we solve the fractional nonlinear Schrödinger equation by using the fourth-order time-splitting method together with the proposed MCF--spectral-Galerkin method. [ABSTRACT FROM AUTHOR]

For the derivatives of Chebyshev second-kind polynomials of a complex vafiable, a system of functions biorthogonal with them on closed curves of the complex plane is constructed. Properties of these functions and the conditions of expansion of analytic functions in series in polynomials under consideration are established. The examples of such expansions are given. In addition, we obtain some combinatorial identities of independent interest. [ABSTRACT FROM AUTHOR]

We consider the following problem (which is a generalisation of a folklore result Proposition 1 below): given a continuous linear operator T : X → Y , where Y is a Banach space with a (long) sub-symmetric basis, under which conditions can we find a continuous linear operator S : X → Y such that S (B X) contains the basis of Y. As a tool we also consider a non-separable version of Theorem 2 below: Given an infinite subset A ⊂ X ∗ , under which conditions can we find a biorthogonal system in X × A of cardinality card A ? [ABSTRACT FROM AUTHOR]

Summary: This article proposes a new numerical approach for solving fractional optimal control problems including state and control inequality constraints using new biorthogonal multiwavelets. The properties of biorthogonal multiwavelets are first given. The Riemann‐Liouville fractional integral operator for biorthogonal multiwavelets is utilized to reduce the solution of optimal control problems to a nonlinear programming one, to which existing, well‐developed algorithms may be applied. In order to save the memory requirement and computation time, a threshold procedure is applied to obtain algebraic equations. The method is computationally very attractive and gives very accurate results. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to investigate the relationship between the two order structures of constructing operators for a generalized Riesz system and (D , ℰ)-quasi bases for two fixed biorthogonal sequences {ϕ n } and {Ψ n }. In a previous paper, we have studied the order structure of the set C ϕ of all constructing operators for a generalized Riesz system {ϕ n }, and furthermore we have shown that the notion of generalized Riesz systems has a close relation with that of (D , ℰ)-quasi bases. For this reason, in this paper we define an order structure in the set M ϕ , ψ of all pairs of dense subspaces D and E in ℋ such that {ϕ n } and {Ψ n } are (D , ℰ)-quasi bases, and shall investigate the relationships between the order sets C ϕ , C Ψ and M ϕ,Ψ. These results seem to be useful to find suitable constructing operators for each physical model. [ABSTRACT FROM AUTHOR]

Tight framelet system is a recently developed tool in applied mathematics. Framelets, due to their nature, are widely used in the area of image manipulation, data compression, numerical analysis, engineering mathematical problems such as inverse problems, visco-elasticity or creep problems, and many more. In this manuscript we provide a numerical solution of important weakly singular type of Volterra - Fredholm integral equations WSVFIEs using the collocation type quasi-affine biorthogonal method. We present a new computational method based on special B -spline tight framelets and use it to introduce our numerical scheme. The method provides a robust solution for the given WSVFIE by using the resulting matrices based on these biorthogonal wavelet. We demonstrate the validity and accuracy of the proposed method by some numerical examples. [ABSTRACT FROM AUTHOR]

Content based image retrieval (CBIR) is an imperative and testing errand in numerous fields, for example, military, common, restorative and even in web applications. Here, we introduced fresh content based medical image retrieval by destroyed bi-orthogonal key wavelet filter banks. Proposed algorithm is an extension for the existing low-level feature extraction method done by using spline wavelet filters, also utilized iterative partitioning (IP) for extracting the similar patterns of query and database images. Simulation results demonstrations that the projected method is performed larger to the existing scheme in terms of accuracy and recollection. [ABSTRACT FROM AUTHOR]

Intriguing issues in one-dimensional non-reciprocal topological systems include the breakdown of usual bulk-edge correspondence and the occurrence of half-integer topological invariants. In order to understand these unusual topological properties, we investigate the topological phase diagrams and the zero-mode edge states of a generalized non-reciprocal Su-Schrieffer-Heeger model with a general form fulfilling the chiral symmetry, based on some analytical results. Meanwhile, we provide a concise geometrical interpretation of the bulk topological invariants in terms of two independent winding numbers and also give an alternative interpretation related to the linking properties of curves in three-dimensional space. For the system under the open boundary condition, we construct analytically the wavefunctions of zero-mode edge states by properly considering a hidden symmetry of the system and the normalization condition with the use of biorthogonal eigenvectors. Our analytical results directly give the phase boundary for the existence of zero-mode edge states and unveil clearly the evolution behavior of edge states. In comparison with results via exact diagonalization of finite-size systems, we find our analytical results agree with the numerical results very well. [ABSTRACT FROM AUTHOR]

Summary: Cracks with quasibrittle behavior are extremely common in engineering structures. The modeling of cohesive cracks involves strong nonlinearity in the contact, material, and complex transition between contact and cohesive forces. In this article, we propose a novel contact algorithm for cohesive cracks in the framework of the extended finite element method. A cohesive‐contact constitutive model is introduced to characterize the complex mechanical behavior of the fracture process zone. To avoid the stress oscillations and ill‐conditioned system matrix that often occur in the conventional contact approach, the proposed algorithm employs a special dual Lagrange multiplier to impose the contact constraint. This Lagrange multiplier is constructed by means of the area‐weighted average and biorthogonality conditions at the element level. The system matrix can be condensed into a positive definite matrix with an unchanged size at a very low computational cost. In addition, we illustrate solving the cohesive crack contact problem using a novel iteration strategy. Several numerical experiments are performed to illustrate the efficiency and high‐quality results of our method in contact analysis of cohesive cracks. [ABSTRACT FROM AUTHOR]

In this paper, we introduce some application related to an b-intuitionistic fuzzy normed space and we define the Cartesian product of two b-intuitionistic fuzzy normed spaces, then we prove that the Cartesian product of two b-intuitionistic fuzzy normed space is also b-intuitionistic fuzzy normed space. Finally, we prove the completeness of the Cartesian product of two complete b-intuitionistic fuzzy normed spaces. [ABSTRACT FROM AUTHOR]

This article extends a numerical method for solving the fractional delay differential equations specified in terms of Caputo derivatives. The approach is based on the second kind Chebyshev wavelet. Our investigation is concentrated on convergence analysis and we prove a theorem for the error bound of the Chebyshev wavelets of the fractional differential in Caputo sense. Also, we discuss convergence analysis of the collocation method. In the end, some examples are presented to indicate the credibility and applicability of the numerical technique. The obtained results are compared with other numerical methods which our results are much more accurate than others. [ABSTRACT FROM AUTHOR]

Abstract—A new biorthogonal system of wavelet bases has been constructed, which is oriented toward reconstructing the useful signal of a measuring system if the measurement process is represented as a convolution model. New biorthogonal wavelet bases are obtained by using a instrumental function to modify a Kravchenko orthogonal wavelet system with a finite spectrum. The properties of new biorthogonal frequency-modified wavelets are studied, and digital filters that realize fast computational algorithms are constructed. Schemes for multiresolution analysis are proposed, which, during discrete wavelet transform, immediately solve the problem of reconstructing the useful signal, as well as effective noise suppression, which can significantly speed up computations. [ABSTRACT FROM AUTHOR]

Two conformers of ClCOOH were produced upon irradiation at 355 nm of a gaseous flowing mixture of Cl2, HCOOH, and N2. A step-scan Fourier-transform infrared spectrometer coupled with a multipass absorption cell was utilized to monitor the transient spectra of ClCOOH. Absorption bands with origins at 1808.0 and 1328.5 cm-1 are attributed to the C=O stretching and COH bending modes of t-ClCOOH, respectively; those at 1883.0 and 1284.9 cm-1 are assigned as the C=O stretching and COH bending modes of c-ClCOOH, respectively. These observed vibrational wavenumbers agree with corresponding values for t-ClCOOH and c-ClCOOH predicted with B3LYP/aug-cc-pVTZ density-functional theory and the observed rotational contours agree satisfactorily with simulated bands based on predicted rotational parameters. The observed relative intensities indicate that t-ClCOOH is more stable than c-ClCOOH by ∼3 kJ mol-1. A simple kinetic model is employed to account for the production and decay of ClCOOH. [ABSTRACT FROM AUTHOR]

With low-order scaling correlated wave function theories in mind, we present second quantization formalism as well as biorthonormalization procedures for general—singular or nonsingular—bases. Of particular interest are the so-called projected atomic orbital bases, which are obtained from a set of atom-centered functions and feature a separation of occupied and virtual spaces. We demonstrate the formalism by deriving and implementing second-order Mo\ller-Plesset perturbation theory in it, and discuss the convergence and preconditioning of the iterative amplitude equations in detail. [ABSTRACT FROM AUTHOR]

We introduce the use of Fourier analysis on lattices as an integral part of a lattice-based construction. The tools we develop provide an elegant description of certain Gaussian distributions around lattice points. Our results include two cryptographic constructions that are based on the worst-case hardness of the unique shortest vector problem. The main result is a new public key cryptosystem whose security guarantee is considerably stronger than previous results (O(n1.5) instead of O(n7)). This provides the first alternative to Ajtai and Dwork's original 1996 cryptosystem. Our second result is a family of collision resistant hash functions with an improved security guarantee in terms of the unique shortest vector problem. Surprisingly, both results are derived from one theorem that presents two indistinguishable distributions on the segment [0 1). It seems that this theorem can have further applications; as an example, we use it to solve an open problem in quantum computation related to the dihedral hidden subgroup problem. [ABSTRACT FROM AUTHOR]

A biorthogonal formulation is applied to the non-Hermite transcorrelated Hamiltonian, which treats a large amount of the dynamic correlation effects implicitly. We introduce biorthogonal canonical orbitals diagonalizing the non-Hermitian Fock operator. We also formulate many-body perturbation theory for the transcorrelated Hamiltonian. The biorthogonal self-consistent field followed by the second order perturbation theory are applied to some pilot calculations including small atoms and molecules. © 2001 American Institute of Physics. [ABSTRACT FROM AUTHOR]

We consider the typical one-dimensional strongly degenerate parabolic operator Pu = ut − (xαux)x with 0 < x < ℓ and α ∈ (0, 2), controlled either by a boundary control acting at x = ℓ, or by a locally distributed control. Our main goal is to study the dependence of the so-called controllability cost needed to drive an initial condition to rest with respect to the degeneracy parameter α. We prove that the control cost blows up with an explicit exponential rate, as eC/((2−α)2T), when α → 2− and/or T → 0+. Our analysis builds on earlier results and methods (based on functional analysis and complex analysis techniques) developed by several authors such as Fattorini-Russel, Seidman, Güichal, Tenenbaum-Tucsnak and Lissy for the classical heat equation. In particular, we use the moment method and related constructions of suitable biorthogonal families, as well as new fine properties of the Bessel functions Jν of large order ν (obtained by ordinary differential equations techniques). [ABSTRACT FROM AUTHOR]

In recent years we have considered various problems related to Malmquist{Takenaka (MT) functions, which form orthogonal systems on the torus. We have introduced their discrete versions and applied them successfully for compression and representation of human ECG signals [6, 7]. Also we have shown electrostatic interpretation of the discretiza- tion points. In these investigations we have taken the MT systems on the torus. In this paper we construct discrete MT type systems by generalizing our former discretization method applied on the torus to the unit disc. [ABSTRACT FROM AUTHOR]

Ultimately aiming to estimate Dirichlet polynomials, a representation problem for special biorthogonal systems of exponentials is explored in L²(0, a). If a = +8, a method of construction of such systems through suitable Blaschke products is known, but the method ceases to operate when a is finite. It turns out that the Blaschke product cannot be even adjusted to maintain the old method for the new situation. The biorthogonal system is then represented by a single determinant of a modified Gram matrix of the original system. Bernstein- type inequalities for Dirichlet polynomials and their higher order derivatives are established. The best constants and extremal polynomials are obtained in terms of the Gram matrix. [ABSTRACT FROM AUTHOR]

With the increasingly mature medical imaging technology, super-resolution imaging and other systems have been widely applied to pathological medical image and tele-medical image transmission system has also attracted extensive heed and application. While pathological diagnosis is expedited, the transmission and storage of pathological medical image have become a problem demanding prompt solution. Due to its excellent spatial resolution and frequency resolution, wavelet analysis is particularly applicable to the analysis of non-stationary signals. As natural image has this non-stationary characteristic, it can be seen as a linear combination of signals with concentration of energy space (image edges and details) and frequency (the part of image with gradual changes). Because conventional wavelet transform requires rounding operation of wavelet coefficients, it cannot reconstruct the original lossless image. So, this paper introduces bi-orthogonal filter bank into wavelet transform, provides the conditions for accurate reconstruction of bi-orthogonal filter bank suited for wavelet transform as well as the multi-resolution analysis under bi-orthogonal condition and the construction method for bi-orthogonal wavelet, achieves the three-level orthogonal wavelet transform in the Lifting Scheme by means of symmetric and periodic extension, and concentrates the energy into a few wavelet coefficients. With the final experiment method, this paper analyzes the distribution characteristics of image wavelet coefficients, laying a foundation for selecting proper compressed encoding scheme. And the experiment result proves that the image compression conducted via bi-orthogonal wavelet transform in this paper can avoid the rounding error of computer and avails the lossless compression of medical image so as to improve image compression in both the efficiency and the accuracy. [ABSTRACT FROM AUTHOR]

Let ℍ {\mathbb{H}} be a separable Hilbert space. In this paper, we establish a generalization of Walnut's representation and Janssen's representation of the ℍ {\mathbb{H}} -valued Gabor frame operator on ℍ {\mathbb{H}} -valued weighted amalgam spaces W ℍ ⁢ (L p , L v q) {W_{\mathbb{H}}(L^{p},L^{q}_{v})} , 1 ≤ p , q ≤ ∞ {1\leq p,q\leq\infty}. Also, we show that the frame operator is invertible on W ℍ ⁢ (L p , L v q) {W_{\mathbb{H}}(L^{p},L^{q}_{v})} , 1 ≤ p , q ≤ ∞ {1\leq p,q\leq\infty} , if the window function is in the Wiener amalgam space W ℍ ⁢ (L ∞ , L w 1) {W_{\mathbb{H}}(L^{\infty},L^{1}_{w})}. Further, we obtain the Walnut representation and invertibility of the frame operator corresponding to Gabor superframes and multi-window Gabor frames on W ℍ ⁢ (L p , L v q) {W_{\mathbb{H}}(L^{p},L^{q}_{v})} , 1 ≤ p , q ≤ ∞ {1\leq p,q\leq\infty} , as a special case by choosing the appropriate Hilbert space ℍ {\mathbb{H}}. [ABSTRACT FROM AUTHOR]

We consider the properties of biorthogonal systems induced by a convolution operator with Carleman kernel for a regular triangle. This is a perturbed singular operator with fixed singularities. We describe its set of anti-invariant points. To this end, we regularize the operator using a Carleman linear convolution shift that maps each triangle side to itself and changes its orientation, with the middle points of the sides being the fixed points of the shift. We search for a solution in the form of a Cauchy-type integral with unknown density. For this, both the theory of the Carleman boundary-value problem and the method of locally conformal gluing are used in an essential manner. We also apply the theory of elliptic functions that are generated by the corresponding doubly periodic group determined by the triangle as 'half' of the fundamental set. Using the method of contracting mappings in a Banach space, we study the corresponding homogeneous Fredholm integral equation of the second kind with regard to its solvability. Its fundamental system of solutions contains a single function; the fundamental system of solutions of the conjugated equation contains only the constant function. This makes it possible to use this equation for the construction of a system of biorthogonally conjugated analytic functions. More precisely, we consider a system of successive derivatives of a certain rational function determined by the Carleman kernel for the triangle and investigate the approximating properties of this system, as well as those of the corresponding biorthogonally conjugated system. This is a system of Cauchy-type integrals over the triangle boundary with a density which is invariant under the considered Carleman shift. Nontrivial decompositions of zero are obtained using the system of successive derivatives of the given rational function. The results are applied to the representation of some classes of analytical functions by means of the corresponding biorthogonal series. [ABSTRACT FROM AUTHOR]

This paper presents biorthogonal wavelet-based multigrid (BWMG) and full approximation scheme (FAS) for the numerical solution of parabolic partial differential equations (PPDEs), which are working horse behind many commercial applications like finger print image compression. Performance of the proposed schemes is better than the existing ones in terms of super convergence with less computational time. Some of the test problems are taken to demonstrate the applicability and validity of the method. [ABSTRACT FROM AUTHOR]

In this paper, Geronimus transformations for matrix orthogonal polynomials in the real line are studied. The orthogonality is understood in a broad sense, and is given in terms of a nondegenerate continuous sesquilinear form, which in turn is determined by a quasidefinite matrix of bivariate generalized functions with a well-defined support. The discussion of the orthogonality for such a sesquilinear form includes, among others, matrix Hankel cases with linear functionals, general matrix Sobolev orthogonality and discrete orthogonal polynomials with an infinite support. The results are mainly concerned with the derivation of Christoffel-type formulas, which allow to express the perturbed matrix biorthogonal polynomials and its norms in terms of the original ones. The basic tool is the Gauss-Borel factorization of the Gram matrix, and particular attention is paid to the non-associative character, in general, of the product of semi-infinite matrices. The Geronimus transformation in which a right multiplication by the inverse of a matrix polynomial and an addition of adequate masses are performed, is considered. The resolvent matrix and connection formulas are given. Two different methods are developed. A spectral one, based on the spectral properties of the perturbing polynomial, and constructed in terms of the second kind functions. This approach requires the perturbing matrix polynomial to have a nonsingular leading term. Then, using spectral techniques and spectral jets, Christoffel-Geronimus formulas for the transformed polynomials and norms are presented. For this type of transformations, the paper also proposes an alternative method, which does not require of spectral techniques, that is valid also for singular leading coefficients. When the leading term is nonsingular, a comparison of between both methods is presented. The nonspectral method is applied to unimodular Christoffel perturbations, and a simple example for a degree one massless Geronimus perturbation is given. [ABSTRACT FROM AUTHOR]

In this work, a sequence of orthonormal rational functions leading to recurrence relations of RII-type is constructed. This sequence is proved to be biorthogonal to another sequence of rational functions as well. Two illustrations of such recurrence relations of RII-type, one through the associated linear pencil matrix leading to the -1 little Jacobi polynomials and the other through the bilinear transformation yielding the Bannai-Ito polynomials, which are orthogonal on the real line are exhibited. [ABSTRACT FROM AUTHOR]

It is well known that the system of translates in L 2 (ℝ) has been characterized as Bessel sequences, frames and Riesz bases in terms of the Fourier transform. The aim of this paper is to obtain similar types of characterization for the wavelet system emerging out of integer translations and dyadic dilations. The existence of the biorthogonal dual system and nonredundancy properties of the wavelet system are also investigated here. [ABSTRACT FROM AUTHOR]

Abstract A new construction of biorthogonal splines for isogeometric mortar methods is proposed. The biorthogonal basis has a local support and, at the same time, optimal approximation properties, which yield optimal results with mortar methods. We first present the univariate construction, which has an inherent crosspoint modification. The multivariate construction is then based on a tensor product for weighted integrals, whereby the important properties are inherited from the univariate case. Numerical results including large deformations confirm the optimality of the newly constructed biorthogonal basis. Highlights • A new set of biorthogonal splines for mortar patch coupling is proposed. • The dual basis combines local support with optimal approximation properties. • These properties ensure an efficient dual isogeometric mortar method. • The multivariate basis is deduced by a tensor product from the univariate case. • Numerical results show the optimality and advantages compared to alternative methods. [ABSTRACT FROM AUTHOR]

for a second-order ordinary differential operator with integral boundary conditions on an interval of the real line, we derive conditions for the uniform convergence of the spectral expansion of a function in a series in the system of eigenfunctions and associated functions of the operator. We obtain estimates of the rate of convergence of the series and the rate of equiconvergence of such an expansion of a function and its expansion in the trigonometric Fourier series. We also study the uniform convergence of the expansion of a function in the biorthogonal system. [ABSTRACT FROM AUTHOR]

Abstract This study deals with cross-products of Bessel functions of the first and second kinds χ ν (p , w) = Y ν (w) J ν (p) − Y ν (p) J ν (w) , ν , p , w ∈ C. Combinations χ ν (p , w) are shown to be generating functions for associated Legendre functions. This fact permits to establish new integral and series representations of χ ν (p , w). Under special choices of parameters, the representations become integral transforms or coefficients of orthogonal series whose inverting leads to new addition formulas for Bessel functions and their continuous analogues. On other hand, integral representations in some cases are integral operators with known inversion formulas and can be treated as operator relations between two systems of functions. Hence theorems on expanding functions into χ ν (p , w) series are proved and biorthogonal systems of functions are found in special cases. As examples, it is applied to evaluate new integrals and series. [ABSTRACT FROM AUTHOR]

We consider a mixed finite element method for an obstacle problem with the p-Laplace differential operator for p ∈ (1 , ∞) {p\in(1,\infty)} , where the obstacle condition is imposed by using a Lagrange multiplier. In the discrete setting the Lagrange multiplier basis forms a biorthogonal system with the standard finite element basis so that the variational inequality can be realized in the point-wise form. We provide a general a posteriori error estimate for adaptivity and prove an a priori error estimate. We present numerical results for the adaptive scheme (mesh-size adaptivity with and without polynomial degree adaptation) for the singular case p = 1.5 {p=1.5} and the degenerated case p = 3 {p=3}. We also present numerical results on the mesh independency and on the polynomial degree scaling of the discrete inf-sup constant when using biorthogonal basis functions for the dual variable defined on the same mesh with the same polynomial degree distribution. [ABSTRACT FROM AUTHOR]

Examines a second order Moller-Plesset perturbation theory free of the basis set superposition errors. Use of biorthogonal perturbation theory to obtain the first order wave function; Calculation of matrix elements in generalized Hylleraas functionals; Indication of differences between the Chemical Hamiltonian Approach and the counterpoise schemes.

In this paper, we report the use of extended coupled cluster functional of Arponen, Bishop, and co-workers to implement a stationary biorthogonal response approach. The objective of this is to calculate nonlinear molecular properties like hyperpolarizability, etc. in a more convenient way. [ABSTRACT FROM AUTHOR]

Formulas for the renormalized full two-particle one hole-Tamm-Dancoff approximation (2ph-TDA) decoupling of the dilated biorthogonal electron propagator based on complex scaled bivariational self-consistent-field (SCF) procedure are derived and the diagonal 2ph-TDA approximation is implemented for the first time. The 2P shape resonances in e-Be, e-Mg, and e-Ca scattering are characterized using the diagonal 2ph-TDA, the second-order and the zeroth-order (bivariational SCF) approximations to the dilated electron propagator. A comparative investigation of these different decouplings reveals that although the resonance energies and widths depend on the level of correlation employed, greater correlation need not lead to sharper resonances. [ABSTRACT FROM AUTHOR]

PHOTO (COLOR) The papers in this special issue all have orthogonal polynomials, special functions and their applications as a common theme. The special issue OPSFA15 contains the proceedings of the 15th International Symposium on I Orthogonal Polynomials, Special Functions and Applications i , which was held in Hagenberg, Austria, from 22 July to 26 July 2019. [Extracted from the article]

For a mixed-type equation, we examine the second boundary-value problem and by using the spectral method prove the uniqueness and existence of solutions. The uniqueness criterion is proved based on the completeness property of the biorthogonal system of functions corresponding to the onedimensional spectral problem. A solution of the problem is constructed as the sum of a biorthogonal series. [ABSTRACT FROM AUTHOR]

Abstract Let Sym denote the algebra of symmetric functions and P μ (⋅ ; q , t) and Q μ (⋅ ; q , t) be the Macdonald symmetric functions (recall that they differ by scalar factors only). The (q , t) -Cauchy identity ∑ μ P μ (x 1 , x 2 , ... ; q , t) Q μ (y 1 , y 2 , ... ; q , t) = ∏ i , j = 1 ∞ (x i y j t ; q) ∞ (x i y j ; q) ∞ expresses the fact that the P μ (⋅ ; q , t) 's form an orthogonal basis in Sym with respect to a special scalar product 〈 ⋅ , ⋅ 〉 q , t. The present paper deals with the inhomogeneous interpolation Macdonald symmetric functions I μ (x 1 , x 2 , ... ; q , t) = P μ (x 1 , x 2 , ... ; q , t) + lower degree terms. These functions come from the N -variate interpolation Macdonald polynomials, extensively studied in the 90s by Knop, Okounkov, and Sahi. The goal of the paper is to construct symmetric functions H μ (⋅ ; q , t) with the biorthogonality property 〈 I μ (⋅ ; q , t) , H ν (⋅ ; q , t) 〉 q , t = δ μ ν. These new functions live in a natural completion Sym ˆ ⊃ Sym. As a corollary one obtains a new Cauchy-type identity in which the interpolation Macdonald polynomials are paired with certain multivariate rational symmetric functions. The degeneration of this identity in the Jack limit (q , t) = (q , q k) → (1 , 1) is also described. [ABSTRACT FROM AUTHOR]

Abstract A discrete-time integrable system, called the DSF chain, is derived from the compatibility condition of spectral equations for biorthogonal polynomials associated with the discrete Schur flow. The DSF algorithm for computing eigenvalues of a type of matrix eigenvalue problem is designed. A discrete analogue of the Miura transformation between the discrete Schur flow and the qd-algorithm is constructed. Two special solutions to the DSF chain related to q -special functions are presented. [ABSTRACT FROM AUTHOR]

This paper mainly talks about the Cauchy two-matrix model and its corresponding integrable hierarchy with the help of orthogonal polynomial theory and Toda-type equations. Starting from the symmetric reduction in Cauchy biorthogonal polynomials, we derive the Toda equation of CKP type (or the C-Toda lattice) as well as its Lax pair by introducing time flows. Then, matrix integral solutions to the C-Toda lattice are extended to give solutions to the CKP hierarchy which reveals the time-dependent partition function of the Cauchy two-matrix model is nothing but the τ-function of the CKP hierarchy. At last, the connection between the Cauchy two-matrix model and Bures ensemble is established from the point of view of integrable systems. [ABSTRACT FROM AUTHOR]

Currently, novel approaches of deep learning are being investigated with respect to their potential for self-improvement in clinical applications. This article presents a deep learning algorithm to handle big data acquired from an electrocardiogram system. Based on this measurement modality, electrical activity of the heart can be monitored over a period of time in various sitting scenarios, such as in the hospital, at home, or during travel. The automatic classification of arrhythmia is carried out based on a biorthogonal wavelet filter bank and a convolutional neural network in order to categorize the ECG signals into normal, left and right bundle branch block, and premature ventricular contraction on a beatby- beat basis, based on supervised machine learning using the standard MIT-BIH database. An averaged sensitivity of 99.2% was achieved. With the proposed system architecture of deep structured learning, the potential of learning mechanisms can be implemented in batch processing. The benefits of the proposed technique are to save the diagnostic time, to reduce the overall cost in healthcare, and to promote individual cardiac monitoring. [ABSTRACT FROM AUTHOR]

Rosengren and Schlosser introduced notions of RN-theta functions for the seven types of irreducible reduced affine root systems, RN = AN−1, BN, B N ∨ , CN, C N ∨ , BCN, DN, N ∈ N , and gave the Macdonald denominator formulas. We prove that if the variables of the RN-theta functions are properly scaled with N, they construct seven sets of biorthogonal functions, each of which has a continuous parameter t ∈ (0, t*) with given 0 < t* < ∞. Following the standard method in random matrix theory, we introduce seven types of one-parameter (t ∈ (0, t*)) families of determinantal point processes in one dimension, in which the correlation kernels are expressed by the biorthogonal theta functions. We demonstrate that they are elliptic extensions of the classical determinantal point processes whose correlation kernels are expressed by trigonometric and rational functions. In the scaling limits associated with N → ∞, we obtain four types of elliptic determinantal point processes with an infinite number of points and parameter t ∈ (0, t*). We give new expressions for the Macdonald denominators using the Karlin–McGregor–Lindström–Gessel–Viennot determinants for noncolliding Brownian paths and show the realization of the associated elliptic determinantal point processes as noncolliding Brownian brides with a time duration t*, which are specified by the pinned configurations at time t = 0 and t = t*. [ABSTRACT FROM AUTHOR]

The spectral problem -u"(x) + αn"(-x) = λu(x), -1 < x < 1, with nonlocal boundary conditions u(-1) = βu(1), u'(-1) = u'(1), is studied in the spaces Lp(-1,1) for any α ≠ (-1,1) and β = ±1. It is proved that if r = √(1-α)/(1+α) is irrational then the system of its eigenfunctions is complete and minimal in Lp(-1,1) for any p > 1, but does not form a basis. In the case of a rational value of r, the way of supplying this system with associated functions is specified to make all the root functions a basis in Lp(-1,1). [ABSTRACT FROM AUTHOR]

Highlights • IDP is used to solve the optimization of ASP flooding based on modeling. • A biorthogonal spatial-temporal Hammerstein modeling method is developed. • The necessary condition for the existence of basis function is inferred. • The proposed method can get the optimal injection strategy well. Abstract In this paper, an iterative dynamic programming (IDP) based on biorthogonal spatial-temporal Hammerstein modeling is developed to solve the enhanced oil recovery for alkali-surfactant-polymer (ASP) flooding. At first, the biorthogonal spatial-temporal Hammerstein model is presented to build the relation between the inputs and states, in which the Hammerstein model is expanded on a set of spatial basis functions and temporal basis functions. After inferring the necessary condition of solutions, these basis functions are determined by the snapshots method. Then, the least square estimation and singular value decomposition is used to identify the parameters in Hammerstein model. In addition, Auto-Regressive and Moving Average (ARMA) model is applied to build the model between states and outputs, whose parameters are identified by recursive least squares. At last, IDP algorithm is applied to solve the enhanced oil recovery problem for ASP flooding based on the identification model. Simulation verifies the accuracy and effectiveness of proposed method. [ABSTRACT FROM AUTHOR]

Highlights • 14 more data records were added. • 800 more experiments were conducted for revision. • 2 more CS based approaches were tested. • A new subsection "Sparsity of ECG in different domains" under results and discussion section. • Detailed abstract and conclusion sections. Abstract This paper elaborates the design details of a new set of bi orthogonal wavelet filters derived from double sided exponential splines. The designed wavelets are applied in compressed sensing (CS) scenario and results were quite promising. CS is a signal acquisition paradigm, which surpasses the traditional limit of Nyquist sampling. Increasing the reconstruction quality with minimum number of samples in CS is always challenging. We have addressed this challenging task of increasing the reconstruction quality within a minimum number of measurements in CS by developing this new set of biorthogonal wavelet filters. Biorthogonal wavelets have several advantages such as linear phase as compared to orthogonal wavelets. This wavelet which we prefer to call as dew1 (double exponential wavelet 1) is applied in CS based ECG reconstruction scenarios and experimented over 21 data records from MIT arrhythmia data base. A total of 950 experiments were conducted in three CS based methodologies for ECG reconstruction and the results were noted. Over all we were able to get nearly 30% improvement in the reconstruction quality. This paper elaborates the design of these bi orthogonal filters and its application in CS based ECG reconstruction scenario. Other than endorsing the results, we also aim to familiarize this newly designed wavelet so it can be further experimented in different domains. [ABSTRACT FROM AUTHOR]

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SN‐38, the active metabolite of irinotecan, is released upon liver hydrolysis to mediate potent antitumor activity. Systemic exposure to SN‐38, however, also leads to serious side effects. To reduce systemic toxicity by controlling where and when SN‐38 is generated, a new prodrug was specifically designed to be metabolically stable and undergo rapid palladium‐mediated activation. Blocking the phenolic OH of SN‐38 with a 2,6‐bis(propargyloxy)benzyl group led to significant reduction of cytotoxic activity (up to 44‐fold). Anticancer properties were swiftly restored in the presence of heterogeneous palladium (Pd) catalysts to kill colorectal cancer and glioma cells, proving the efficacy of this novel masking strategy for aromatic hydroxyls. Combination with a Pd‐activated 5FU prodrug augmented the antiproliferative potency of the treatment, while displaying no activity in the absence of the Pd source, which illustrates the benefit of achieving controlled release of multiple approved therapeutics—sequentially or simultaneously—by the same bioorthogonal catalyst to increase anticancer activity. Palladium‐functionalized microdevices were used to uncage newly developed prodrug that releases SN‐38 (irinotecan′s active metabolite) in combination with previously known anticancer prodrug of 5FU. [ABSTRACT FROM AUTHOR]

We study spectral properties of an essentially nonself-adjoint problem generated by nonlocal multipoint conditions for the operator of differentiation of order 2n and analyze the cases of regular and irregular Birkhoff two-point boundary conditions. A system of root functions of the problem and elements of biorthogonal systems are constructed. We also establish sufficient conditions under which these systems are complete and conditions under which they form a Riesz basis (under certain additional assumptions). [ABSTRACT FROM AUTHOR]