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We investigate gap-closings in one- and two-dimensional tight-binding models with two bands, containing non-Hermitian hopping terms, and open boundary conditions (OBCs) imposed along one direction. We compare the bulk OBC spectra with the periodic boundary condition (PBC) spectra, pointing out that they do not coincide, which is an intrinsic characteristic of non-Hermitian systems. The non-Hermiticity, thus, results in the failure of the familiar notions of bulk-boundary correspondence found for Hermitian systems. This necessitates the search for topological invariants which can characterize gap-closings in open non-Hermitian systems correctly and unambiguously. We elucidate the behavior of two possible candidates applicable for one-dimensional slices—(1) the sum of winding numbers for the two bands defined on a generalized Brillouin zone and (2) the biorthogonal polarization (BP). While the former shows jumps/discontinuities for some of the non-Hermitian systems studied here, at points when an edge mode enters the bulk states and becomes delocalized, it does not maintain quantized values in a given topological phase. On the contrary, BP shows jumps at phase transitions, and the quantized value of one or zero, which corresponds to whether an actual edge mode exists or whether that mode is delocalized and absorbed within the bulk (not being an edge mode anymore). [ABSTRACT FROM AUTHOR]

In this paper, some new families of biorthogonal functions in one variable produced by using Fourier transforms of biorthogonal polynomials, which are suggested by Laguerre, Jacobi and Szegö–Hermite polynomials, and their biorthogonality relations obtained via Parseval identity are derived. [ABSTRACT FROM AUTHOR]

The paper proves two results involving a pair $ (A,B) $ (A , B) of P-biisometric or $ (m,P) $ (m , P) -biisometric Hilbert-space operators for arbitrary positive integer m and positive operator P. It is shown that if A and B are power bounded and the pair $ (A,B) $ (A , B) is $ (m,P) $ (m , P) -biisometric for some m, then it is a P-biisometric pair. The important case when P is invertible is treated in detail. It is also shown that if $ (A,B) $ (A , B) is P-biisometric, then there are biorthogonal sequence with respect to the inner product $ {\langle \cdot ;\cdot \rangle _P}={\langle P \cdot ;\cdot \rangle } $ 〈 ⋅ ; ⋅ 〉 P = 〈 P ⋅ ; ⋅ 〉 that have a shift-like behavior with respect to this inner product. [ABSTRACT FROM AUTHOR]

A finite family of $ R_I $ R I polynomials is introduced and studied. It consists in a set of polynomials of $ _{3}F_{2} $ 3 F 2 form whose biorthogonality to an ensemble of rational functions is spelled out. These polynomials are shown to satisfy two generalized eigenvalue problems: in addition to their recurrence relation of $ R_I $ R I type, they are also found to obey a difference equation. Underscoring this bispectrality is a triplet of operators with tridiagonal actions. The algebra associated to these operators is provided. [ABSTRACT FROM AUTHOR]

This paper presents a novel method for designing non-stationary biorthogonal wavelets using normality and biorthogonality conditions of nonstationary multiresolution analysis. Non-stationary biorthogonal wavelets are effective in reducing image size while preserving quality, and can capture localized features accurately, making them a popular choice in image compression algorithms. [ABSTRACT FROM AUTHOR]

This article introduces a unified mixed finite element framework based on a saddle-point formulation that applies to time-dependent fourth order linear and nonlinear problems with clamped, simply supported, and Cahn-Hilliard type boundary conditions. The classical mixed formulations lead to large matrix systems that demand huge storage and computational time making the schemes expensive, especially for the time-dependent problems. The proposed scheme circumvents this by employing biorthogonal basis functions that lead to sparse and positive-definite systems. The article discusses a mixed finite element method for the biharmonic problem and the time-dependent linear and nonlinear versions of the extended Fisher-Kolmogorov equations equipped with the aforementioned boundary conditions. The wellposedness of the scheme is discussed and a priori error estimates are presented for the semi-discrete and fully discrete finite element schemes. The numerical experiments validate the theoretical estimates derived in the paper. [ABSTRACT FROM AUTHOR]

We present a methodology for analyzing chemical bonds embedded in the electronic wave function of molecules, especially in terms of spin correlations or so-called "local spin." In this paper, based on biorthogonal second quantization, the spin correlation functions of molecules are naturally introduced, which enables us to extract local singlet and local triplet elements from the wave function. We also clarify the relationship between these spin correlations and traditional chemical concepts, i.e., resonance structures. Several chemical reactions, including the intramolecular radical cyclization and the formation of preoxetane, are demonstrated to verify the analysis method numerically. [ABSTRACT FROM AUTHOR]

We construct a system of solutions for the coefficients of expansion in the system of trigonometric functions for the solution of the Helmholtz equation in a cylindrical coordinate system in the form of homogeneous polynomials in two biorthogonal systems of functions. Some properties of the biorthogonal systems of functions are proved. [ABSTRACT FROM AUTHOR]

Hydrogen sulfide (H2S), one of the most important gasotransmitters, plays a critical role in endogenous signaling pathways of many diseases. However, developing H2S donors with both tunable release kinetics and high release efficiency for subcellular delivery has been challenging. Here, we describe a click and release reaction between pyrone/pyranthiones and bicyclononyne (BCN). This reaction features a release of CO2/COS with second‐order rate constants comparable to Strain‐Promoted Azide‐Alkyne Cycloaddition reactions (SPAACs). Interestingly, pyranthiones showed enhanced reaction rates compared to their pyrone counterparts. We investigated pyrone biorthogonality and demonstrated their utility in protein labeling applications. Moreover, we synthesized substituted pyranthiones with H2S release kinetics that can address the range of physiologically relevant H2S dynamics in cells and achieved quantitative H2S release efficiency in vitro. Finally, we explored the potential of pyranthiones as H2S/COS donors for mitochondrial‐targeted H2S delivery in living cells. [ABSTRACT FROM AUTHOR]

We study the phase reconstruction of signals f belonging to complex Gaussian shift-invariant spaces V ∞ (φ) from spectrogram measurements | G f (X) | where G is the Gabor transform and X ⊆ R 2 . An explicit reconstruction formula will demonstrate that such signals can be recovered from measurements located on parallel lines in the time-frequency plane by means of a Riesz basis expansion. Moreover, connectedness assumptions on |f| result in stability estimates in the situation where one aims to reconstruct f on compacts intervals. Driven by a recent observation that signals in Gaussian shift-invariant spaces are determined by lattice measurements (Grohs and Liehr in Injectivity of Gabor phase retrieval from lattice measurements. Appl. Comput. Harmon. Anal. 62, 173–193 (2023)) we prove a sampling result on the stable approximation from finitely many spectrogram samples. The resulting algorithm provides a provably stable and convergent approximation technique. In addition, it constitutes a method of approximating signals in function spaces beyond V ∞ (φ) , such as Paley–Wiener spaces. [ABSTRACT FROM AUTHOR]

We investigate anomalous localization phenomena in non-Hermitian systems by solving a class of generalized Su–Schrieffer–Heeger/Rice–Mele models and by relating their provenance to fundamental notions of topology, symmetry-breaking, and biorthogonality. We find two types of bound states in the continuum, both stable even in the absence of chiral symmetry: the first being skin bulk states, which are protected by the spectral winding number. The second type is constituted by boundary modes associated with a quantized biorthogonal polarization. Furthermore, we find an extended state stemming from the boundary state that delocalizes while remaining in the gap at bulk critical points. This state may also delocalize within a continuum of localized (skin) states. These results clarify fundamental aspects of topology and symmetry in light of different approaches to the anomalous non-Hermitian bulk-boundary correspondence and are of direct experimental relevance for mechanical, electrical, and photonic systems. [ABSTRACT FROM AUTHOR]

This article discusses the problem of Hermitian interpolation, which involves constructing a function that matches the values and derivatives of a given function at specific points. The authors focus on interpolation formulas based on window systems of functions generated by translates of the Gaussian function. They propose a method that does not require integration and reduces computational requirements. The article presents the construction of Hermitian cardinal functions and provides simple formulas for them in certain cases. The approach described is applicable to other window systems as well. [Extracted from the article]

Matrix elements between nonorthogonal Slater determinants represent an essential component of many emerging electronic structure methods. However, evaluating nonorthogonal matrix elements is conceptually and computationally harder than their orthogonal counterparts. While several different approaches have been developed, these are predominantly derived from the first-quantized generalized Slater–Condon rules and usually require biorthogonal occupied orbitals to be computed for each matrix element. For coupling terms between nonorthogonal excited configurations, a second-quantized approach such as the nonorthogonal Wick's theorem is more desirable, but this fails when the two reference determinants have a zero many-body overlap. In this contribution, we derive an entirely generalized extension to the nonorthogonal Wick's theorem that is applicable to all pairs of determinants with nonorthogonal orbitals. Our approach creates a universal methodology for evaluating any nonorthogonal matrix element and allows Wick's theorem and the generalized Slater–Condon rules to be unified for the first time. Furthermore, we present a simple well-defined protocol for deriving arbitrary coupling terms between nonorthogonal excited configurations. In the case of overlap and one-body operators, this protocol recovers efficient formulas with reduced scaling, promising significant computational acceleration for methods that rely on such terms. [ABSTRACT FROM AUTHOR]

We develop time-dependent vibrational coupled cluster with time-dependent modals (TDMVCC), where an active set of one-mode basis functions (modals) is evolved in time alongside coupled-cluster wave-function parameters. A biorthogonal second quantization formulation of many-mode dynamics is introduced, allowing separate biorthogonal bases for the bra and ket states, thus ensuring complex analyticity. We employ the time-dependent bivariational principle to derive equations of motion for both the one-mode basis functions and the parameters describing the cluster (T) and linear de-excitation (L) operators. The choice of constraint (or gauge) operators for the modal time evolution is discussed. In the case of untruncated cluster expansion, the result is independent of this choice, but restricting the excitation space removes this invariance; equations for the variational determination of the constraint operators are derived for the latter case. We show that all single-excitation parts of T and L are redundant and can be left out in the case of variationally determined constraint-operator evolution. Based on a pilot implementation, test computations on Henon–Heiles model systems, the water molecule, and a reduced-dimensionality bi-thiophene model are presented, showing highly encouraging results for TDMVCC. It is demonstrated how TDMVCC in the limit of a complete cluster expansion becomes equivalent to multiconfiguration time-dependent Hartree for the same active-space size. Similarly, it is discussed how TDMVCC generally gives better and more stable results than its time-independent-modals counterpart, while equivalent results are obtained for complete expansions and full one-mode basis sets. [ABSTRACT FROM AUTHOR]

The variational iteration method is used to deal with linear and nonlinear differential equations. The main characteristics of the method lie in its flexibility and ability to accurately and easily solve nonlinear equations. In this work, a general framework is presented for a variational iteration method for the analytical treatment of partial differential equations in fluid mechanics. The Caputo sense is used to describe fractional derivatives. The time-fractional Kaup-Kupershmidt (KK) equation is investigated, as it is the solution of the system of partial differential equations via the Boussinesq-Burger equation. By comparing the results that are obtained by the variational iteration method with those obtained by the two-dimensional Legendre multiwavelet, the optimal homotopy asymptotic method (OHAM), the q-homotopy analysis transform method, the Laplace Adomian Decomposition Method, and the homotopy perturbation method, the first method proved to be very effective and convenient. The main methodology in this work is anticipated to be applied to various fractional calculus, linear, and nonlinear problems. [ABSTRACT FROM AUTHOR]

The properties of the Pastro polynomials on the real line are studied with the help of a triplet of q-difference operators. The q-difference equation and recurrence relation these polynomials obey are shown to arise as generalized eigenvalue problems involving the triplet of operators, with the Pastro polynomials as solutions. Moreover, a discrete biorthogonality relation on the real line for the Pastro polynomials is obtained and is then understood using adjoint operators. The algebra realized by the triplet of q-difference operators is investigated. [ABSTRACT FROM AUTHOR]

The paper is concerned with the Bari basis property of a boundary value problem associated in L2([0,1];C2)$L^2([0,1]; \mathbb {C}^2)$ with the following 2 × 2 Dirac‐type equation for y=col(y1,y2)$y = \operatorname{col}(y_1, y_2)$: LU(Q)y=−iB−1y′+Q(x)y=λy,B=b100b2,b1<0

The disturbance flow field in a hypersonic boundary layer excited by particle impingement was investigated with a focus on the first stage of the laminar-to-turbulent transition process, namely the receptivity process. A previously validated direct numerical simulation approach adopting disturbance flow tracking is used to simulate the particle-induced transition process. Particle impingement generates a highly complex disturbance flow field that can be characterised by a wide range of frequencies and wavenumbers. After providing some insight about the spectral characteristics of the disturbance flow field in the frequency and wavenumber domains, biorthogonal decomposition is employed to reveal the composition of the disturbance flow field consisting of different continuous and discrete eigenmodes that are triggered through particle impingement. The disturbance flow characteristics for different frequency and wavenumber pairs are discussed where large contributions in the disturbance flow spectrum are observed in the vicinity of the impingement location. A significant amount of the disturbance energy is diverted into the free stream leading to large coefficients of projection for the slow and fast acoustic branches while contributions to the entropy and vorticity branches are negligible. In addition to the continuous acoustic spectra, the first-, second- and other higher-order Mack modes are activated and provide large contributions to the disturbance flow field inside the boundary layer. Finally, it is demonstrated that the disturbance flow field in the vicinity of the impingement location can be reconstructed with a maximum relative error of $2.3\,\%$ by employing a theoretical biorthogonal eigenfunction system expansion and by considering contributions from fast and slow acoustic waves and at most four discrete modes only. [ABSTRACT FROM AUTHOR]

A formulation is given for the spectral transformation of the generalized eigenvalue problem through the decomposition of the second‐order differential operators. This allows us to construct some Laurent biorthogonal polynomial systems with gaps in the degree of the polynomial sequence. These correspond to an exceptional‐type extension of the orthogonal polynomials, as an extension of the Laurent biorthogonal polynomials. Specifically, we construct the exceptional extension of the Hendriksen–van Rossum polynomials, which are biorthogonal analogs of the classical orthogonal polynomials. Similar to the cases of exceptional extensions of classical orthogonal polynomials, both state‐deletion and state‐addition occur. [ABSTRACT FROM AUTHOR]

In this study, we obtained a system of eigenfunctions and eigenvalues for the mixed homogeneous Sturm-Liouville problem of a second-order differential equation containing a fractional derivative operator. The fractional differentiation operator was considered according to two definitions: Gerasimov-Caputo and Riemann-Liouville-Visualizations of the system of eigenfunctions, the biorthogonal system, and the distribution of eigenvalues on the real axis were presented. The numerical behavior of eigenvalues was studied depending on the order of the fractional derivative for both definitions of the fractional derivative. [ABSTRACT FROM AUTHOR]

Let |$\mathcal {X}$| be a Banach space with a fundamental biorthogonal system, and let |$\mathcal {Y}$| be the dense subspace spanned by the vectors of the system. We prove that |$\mathcal {Y}$| admits a |$C^\infty $| -smooth norm that locally depends on finitely many coordinates (LFC, for short), as well as a polyhedral norm that locally depends on finitely many coordinates. As a consequence, we also prove that |$\mathcal {Y}$| admits locally finite, |$\sigma $| -uniformly discrete |$C^\infty $| -smooth and LFC partitions of unity and a |$C^1$| -smooth locally uniformly rotund norm. This theorem substantially generalises several results present in the literature and gives a complete picture concerning smoothness in such dense subspaces. Our result covers, for instance, every weakly Lindelöf determined Banach space (hence, all reflexive ones), |$L_1(\mu)$| for every measure |$\mu $|⁠ , |$\ell _\infty (\Gamma)$| spaces for every set |$\Gamma $|⁠ , |$C(K)$| spaces where |$K$| is a Valdivia compactum or a compact Abelian group, duals of Asplund spaces, or preduals of Von Neumann algebras. Additionally, under Martin Maximum MM , all Banach spaces of density |$\omega _1$| are covered by our result. [ABSTRACT FROM AUTHOR]

In this paper we prove that if U = {un} is a minimal sequence in a separable ℋilbert space ℋ, then multiplying each vector un by an appropriate constant cn, yields a family of functions P := {cn · un }, such that P is a Riesz-Fischer sequence in ℋ . If U is a minimal and complete sequence in ℋ, therefore having a unique biorthogonal sequence V = {vn } in ℋ, then P is a complete Riesz-Fischer sequence in ℋ and its unique biorthogonal family {vn/c̅n } is a Bessel sequence in ℋ. [ABSTRACT FROM AUTHOR]

Recently Wavelet theory is an advanced tool in science and engineering research; mainly wavelets are applicable in fast algorithms for easy implementation. In this paper, we present an innovative wavelet technique that is lifting scheme for the numerical solutions of hyperbolic differential equations come across in mathematical physics. Here we used orthogonal and biorthogonal wavelets to expose the efficiency and effectiveness of the proposed scheme. The proposed scheme speeds up the convergence in less computational time, which is illustrated through the numerical examples. [ABSTRACT FROM AUTHOR]

The main goal of the paper is to connect matrix polynomial biorthogonality on a contour in the plane with a suitable notion of scalar, multi-point Padé approximation on an arbitrary Riemann surface endowed with a rational map to the Riemann sphere. To this end we introduce an appropriate notion of (scalar) multi-point Padé approximation on a Riemann surface and corresponding notion of biorthogonality of sections of the semi-canonical bundle (half-differentials). Several examples are offered in illustration of the new notions. [ABSTRACT FROM AUTHOR]

The purpose of this research is to employ a method involving Chelyshkov wavelets to construct a numerical solution to the inverse problem of determining the right-hand side function of a non-linear fractional differential equation by utilizing over-measured data. The novelty of this research is that this type of inverse problem is studied by Chelyshkov wavelets. Firstly, the problem is reduced into a system of algebraic equations with an unknown right-hand side by means of the orthonormal base of Chelyshkov wavelets. Secondly, by choosing suitable nodes, this system is transformed into a homogenous system of algebraic equations. The solution of the homogenous system allows us to determine the coefficients of the bases vectors for the solution of the non-linear fractional differential equation. In the final step, the right-hand side is obtained by substituting the constructed solution into a non-linear fractional differential equation. The presented examples illustrate that the numerical solution, obtained by this method, is remarkably close to the exact solution. [ABSTRACT FROM AUTHOR]

In this manuscript we provide effective computational algorithms based on Legendre wavelet (LW), Bernstein wavelet (BW), and standard tau approach to approximate the solution of viscoelasticity damping motion nonlinear distributed order fractional differential equation (VDMNDOFDE) and multi-dimensional distributed order time-fractional nonlinear partial differential equation (DOT-FNPDE). To the best of our understanding, the proposed computational algorithm is new and has not been previously applied for solving VDMNDOFDE and DOT-FNPDE. The matrix representation of distributed order fractional derivatives, integer order derivatives and product term associated with the integral based on LW and BW are established to find the numerical solutions of the proposed VDMNDOFDE and DOT-FNPDE. Moreover, the association of standard tau rule and Legendre-Gauss quadrature (LGQ) techniques along with constructed matrix representation of differential and integral operators diminish VDMNDOFDE and DOT-FNPDE into system of nonlinear algebraic equations. Here, these systems of nonlinear algebraic equations are solved by the Newton's method to get the value of unknown vectors. Error bounds, convergence analysis, numerical algorithms of the VDMNDOFDE and DOT-FNPDE are rigorously investigated. For the reliability of the proposed computational algorithm, numerous test examples have been incorporated in the manuscript to ensure the robustness and theoretical results of proposed technique. Also, the proposed technique found to be more accurate with the existing scheme. [ABSTRACT FROM AUTHOR]

Wavelet expansions are a powerful tool for constructing adaptive approximations. For this reason, they find applications in a variety of fields, from signal processing to approximation theory. Wavelets are usually derived from refinable functions, which are the solution of a recursive functional equation called the refinement equation. The analytical expression of refinable functions is known in only a few cases, so if we need to evaluate refinable functions we can make use only of the refinement equation. This is also true for the evaluation of their derivatives and integrals. In this paper, we detail a procedure for computing integrals of wavelet products exactly, up to machine precision. The efficient and accurate evaluation of these integrals is particularly required for the computation of the connection coefficients in the wavelet Galerkin method. We show the effectiveness of the procedure by evaluating the integrals of pseudo-splines. [ABSTRACT FROM AUTHOR]

Neural network has good self‐learning and adaptive capabilities. In this paper, a wavelet neural network is proposed to be used to solve the value problem of fractional differential equations (FDE). We construct a wavelet neural network (WNN) with the structure 1 ×N× 1 based on the wavelet function and give the conditions for the convergence of the given algorithm. This method uses the truncated power series of the solution function to transform the original differential equation into an approximate solution, then, using WNN, update the parameters, and finally get the FDE solution. Simulation results prove the validity of WNN. [ABSTRACT FROM AUTHOR]

The fractional Fourier transform, which is a generalization of the Fourier transform, has become the focus of many research papers in recent years because of its applications in electrical engineering and optics. In this paper, we introduce the notion of fractional biorthogonal wavelets on R and obtain the necessary and sufficient conditions for the translates of a single function to form the fractional Riesz bases for their closed linear span. We also provide a complete characterization for the fractional biorthogonality of the translates of fractional scaling functions of two fractional MRA's and the associated fractional biorthogonal wavelet families. Moreover, under mild assumptions on the fractional scaling functions and the corresponding fractional wavelets, we show that the fractional wavelets can generate Reisz bases for L 2 (R). [ABSTRACT FROM AUTHOR]

In this work, an extension of Malmqvist's algorithm is efficiently implemented for the reciprocal transformation of a seniority number restricted multideterminant wave function constructed by nonorthogonal orbitals to an equivalent one constructed by biorthogonal orbitals. Test calculations show that the novel algorithm is capable of applying to transform quite large nonorthogonal orbital based wave function of distributing 22 electrons on 22 orbitals with the maximum seniority number sector and for the evaluation of a three-body operator, at an affordable memory cost of about 800 GB, which is only about one tenth of the size of the corresponding complete active space. Moreover, the algorithm displays improved computational efficiency, due to the compactness of seniority number restricted multi-determinant wave functions. [ABSTRACT FROM AUTHOR]

In this study, the Stokes flow problem in an S-shaped double lid-driven cavity filled with fluid was analyzed. Side edges of the cavity were considered as immovable walls. The flow region was divided into two sub-regions, and the streamfunction in each sub-region was considered as an extension of Papkovich–Faddle eigenfunctions. Parameters in the analytical solution were obtained using biorthogonality conditions. The Newton iteration method was used to obtain the eigenvalues of the problem, and integrals were calculated with the Gaussian quadrature method. It was ensured that solutions made separately for the two sub-regions converge on the interface, which is the intersection of these sub-regions. The two parameters controlling the flow structure were determined as the speed ratio of movable lids (S) and the aspect ratio of the cavity (A). The effects of these parameters on flow structures were shown. New eddy formation mechanisms and bifurcations were observed in the cavity by keeping the speed ratio of the lids constant and slowly changing the aspect ratio. [ABSTRACT FROM AUTHOR]

Purpose: The purpose of the present work is to develop a new wavelet method, named as Krawtchouk wavelets method, for solving both Caputo fractional and Caputo–Hadamard fractional differential equations on a semi‐infinite domain. Design/methodology/approach: We have utilized the discrete Krawtchouk orthogonal polynomial for the construction of Krawtchouk wavelets method. The supporting analysis of the method such as construction of operational matrices, procedure of implementation, and convergence analysis of the method are being provided. We have also proposed a method by combining the Krawtchouk wavelets method with the method of step for the solution of Caputo–Hadamard fractional delay differential equations. Findings: We have provided the orthogonality condition for the Krawtchouk wavelets. We have derived and constructed the Krawtchouk wavelets matrix, Krawtchouk wavelets operational matrix of Riemann–Liouville and Hadamard‐type fractional‐order integration, and Krawtchouk wavelets operational matrix of Riemann–Liouville and Hadamard‐type fractional‐order integration for boundary value problems. These matrices are successfully utilized for the solution of Caputo and Caputo–Hadamard fractional differential equations. Operational matrices contains many zero entries, which makes the present method more efficient. Furthermore, we workout on the procedure of implementation of the method for the Caputo fractional differential equations as well as for the Caputo–Hadamard fractional differential equations. We also derived the convergence analysis of the Krawtchouk wavelets method, which completes the theoretical analysis of the proposed method. We have applied the Krawtchouk wavelets method for the numerical solutions of several Caputo fractional differential equations and Caputo–Hadamard fractional differential equations and compare the obtained results with the analytical solutions. The comparison shows the effectiveness of the present numerical method. In this paper, we have considered initial value problem, boundary value problem, and delay problem. According to the numerical results, the present method is more efficient and accurate. Originality/value: Since fractional differential equation is a latest and emerging field, many engineers and scientists can utilize the present method for solving their fractional models. To the best of the author's knowledge, the present wavelets method has never been introduced and implemented for Caputo fractional and Caputo–Hadamard fractional differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we investigate Laurent biorthogonal polynomials with a weight function of three parameters, i.e. z^\alpha e^{-t_1z-\frac {t_2}{z}}, z\in (0,+\infty), (t_1>0,\ t_2>0,\ \alpha \in \mathbb {R}). First, the structure relation of the Laurent biorthogonal polynomials is found with the aid of biorthogonality. Then we derive an alternate discrete Painlevé II by considering the compatibility condition of the three-term recurrence relation and the structure relation. In addition, we make use of the relativistic Toda chains and nonlinear difference system to obtain two continuous Painlevé-type differential equations. [ABSTRACT FROM AUTHOR]

The study of greedy approximation in the context of convex optimization is becoming a promising research direction as greedy algorithms are actively being employed to construct sparse minimizers for convex functions with respect to given sets of elements. In this paper we propose a unified way of analyzing a certain kind of greedy-type algorithms for the minimization of convex functions on Banach spaces. Specifically, we define the class of Weak Biorthogonal Greedy Algorithms for convex optimization that contains a wide range of greedy algorithms. We analyze the introduced class of algorithms and establish the properties of convergence, rate of convergence, and numerical stability, which is understood in the sense that the steps of the algorithm are allowed to be performed not precisely but with controlled computational inaccuracies. We show that the following well-known algorithms for convex optimization — the Weak Chebyshev Greedy Algorithm (co) and the Weak Greedy Algorithm with Free Relaxation (co) — belong to this class, and introduce a new algorithm — the Rescaled Weak Relaxed Greedy Algorithm (co). Presented numerical experiments demonstrate the practical performance of the aforementioned greedy algorithms in the setting of convex minimization as compared to optimization with regularization, which is the conventional approach of constructing sparse minimizers. [ABSTRACT FROM AUTHOR]

The main aim of this research article is to coin a novel numerical algorithm for approximating the solutions of integro-differential equations using Haar scale 3 wavelets. Highest order derivatives involved in the integro-differential equations are approximated using Haar scale 3 functions and discretized by collocation technique. The integrals of unknown functions involved in the integro-differential equations are handled by Quadrature formula using Haar scale 3 wavelets. To explain the supermacy of the proposed algorithm. a few examples from the already published reports have been analyzed. The exact and approximate solutions are compared by calculating the errors at different collocation points. The results are presented through tables and graphs. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the unified error estimate for some weak biorthogonal greedy algorithms with respect to dictionaries in Banach spaces by using some kind of K-functional. From this estimate, we derive the sufficient conditions for the convergence and the convergence rates on sparse classes induced by the K-functional. The results on convergence and the convergence rates are sharp. [ABSTRACT FROM AUTHOR]

We consider a local projection stabilization based on biorthogonal systems for convection–diffusion–reaction differential equations with mixed boundary conditions. The approach based on biorthogonal systems is numerically more efficient than other existing approaches to obtain a uniform approximation for convection dominated problems. We prove optimal a priori error estimates for the proposed numerical technique. Numerical examples are presented to demonstrate the performance of the approach. [ABSTRACT FROM AUTHOR]

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The theory of Pólya ensembles of positive definite random matrices provides structural formulas for the corresponding biorthogonal pair, and correlation kernel, which are well suited to computing the hard edge large N asymptotics. Such an analysis is carried out for products of Laguerre ensembles, the Laguerre Muttalib–Borodin ensemble, and products of Laguerre ensembles and their inverses. The latter includes, as a special case, the Jacobi unitary ensemble. In each case, the hard edge scaled kernel permits an expansion in powers of 1/N, with the leading term given in a structured form involving the hard-edge scaling of the biorthogonal pair. The Laguerre and Jacobi ensembles have the special feature that their hard edge scaled kernel – the Bessel kernel – is symmetric and this leads to there being a choice of hard edge scaling variables for which the rate of convergence of the correlation functions is O (1 / N 2). [ABSTRACT FROM AUTHOR]

In this paper, we consider the following degenerate/singular parabolic equation [Math Processing Error] u t − (x α u x) x − μ x 2 − α u = 0 , x ∈ (0 , 1) , t ∈ (0 , T) , where [Math Processing Error] 0 ≤ α < 1 and [Math Processing Error] μ ≤ (1 − α) 2 / 4 are two real parameters. We prove the boundary null controllability by means of a [Math Processing Error] H 1 (0 , T) control acting either at [Math Processing Error] x = 1 or at the point of degeneracy and singularity [Math Processing Error] x = 0. Besides we give sharp estimates of the cost of controllability in both cases in terms of the parameters [Math Processing Error] α and [Math Processing Error] μ. The proofs are based on the classical moment method by Fattorini and Russell and on recent results on biorthogonal sequences. [ABSTRACT FROM AUTHOR]

This paper is concerned about certain generalization of Laurent biorthogonal polynomials together with the corresponding related integrable lattices. On one hand, a generalization for Laurent biorthogonal polynomials is proposed and its recurrence relation and Christoffel transformation are derived. On the other hand, it turns out the compatibility condition between the recurrence relation and the Christoffel transformation for the generalized Laurent biorthogonal polynomials yields an extension of the fully discrete relativistic Toda lattice. And also, it is shown that isospectral deformations of the generalized Laurent biorthogonal polynomials lead to two different generalizations of the continuous-time relativistic Toda lattice, one of which can reduce to the Naritaâ€"Itohâ€"Bogoyavlensky lattice. [ABSTRACT FROM AUTHOR]

This paper is devoted to the wavelet Galerkin method to solve the Fractional Riccati equation. To this end, biorthogonal Hermite cubic Spline scaling bases and their properties are introduced, and the fractional integral is represented based on these bases as an operational matrix. Firstly, we obtain the Volterra integral equation with a weakly singular kernel corresponding to the desired equation. Then, using the operational matrix of fractional integration and the Galerkin method, the corresponding integral equation is reduced to a system of algebraic equations. Solving this system via Newton's iterative method gives the unknown solution. The convergence analysis is investigated and shows that the convergence rate is O (2 − s) . To demonstrate the efficiency and accuracy of the method, some numerical simulations are provided. [ABSTRACT FROM AUTHOR]

In this paper, the Riemann–Hilbert problem, with a jump supported on an appropriate curve on the complex plane with a finite endpoint at the origin, is used for the study of the corresponding matrix biorthogonal polynomials associated with Laguerre type matrices of weights—which are constructed in terms of a given matrix Pearson equation. First and second order differential systems for the fundamental matrix, solution of the mentioned Riemann–Hilbert problem, are derived. An explicit and general example is presented to illustrate the theoretical results of the work. The non-Abelian extensions of a family of discrete Painlevé IV equations are discussed. [ABSTRACT FROM AUTHOR]