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In a recently published theorem on the split common fixed point problem for strict pseudocontractive and asymptotically nonexpansive mappings, Tang et al. (J. Inequal. Appl. 2015:305, 2015) studied a uniformly convex and 2-uniformly smooth real Banach space with the Opial property and best smoothness constant κ satisfying the condition $0 0 < κ < 1 2 , as a real Banach space more general than Hilbert spaces. A well-known example of a uniformly convex and 2-uniformly smooth real Banach space with the Opial property is $E=l_{p}$ E = l p , $2\leq p 2 ≤ p < ∞ . It is shown in this paper that, if κ is the best smoothness constant of E and satisfies the condition $0 0 < κ ≤ 1 2 , then E is necessarily $l_{2}$ l 2 , a real Hilbert space. Furthermore, some important remarks concerning the proof of this theorem are presented.

In a recent paper referred to in the title, the author used the concept of quasi self-adjointness to obtain conservation laws for a system of the ( 2 + 1 ) -dimensional Bogoyavlensky–Konopelchenko equation. Apart from the adjoint system of equations, all the results on the quasi self-adjointness and the subsequent conservation laws obtained are inaccurate. In this comment, we clarify these inaccuracies and also generate conservation laws for a potential form of the underlying equation through Noether theorem.

It is shown that a known asymptotic expansion of the Kummer function $U(a,b,z)$ as $a$ tends to infinity is valid for $z$ on the full Riemann surface of the logarithm. A corresponding result is also proved in a more general setting considered by Olver (1956).

We propose methods to estimate sufficient reductions in matrix-valued predictors for regression or classification. We assume that the first moment of the predictor matrix given the response can be decomposed into a row and column component via a Kronecker product structure. We obtain least squares and maximum likelihood estimates of the sufficient reductions in the matrix predictors, derive statistical properties of the resulting estimates and present fast computational algorithms with assured convergence. The performance of the proposed approaches in regression and classification is compared in simulations.We illustrate the methods on two examples, using longitudinally measured serum biomarker and neuroimaging data.

We point out errors and oversights in a paper by Giordano, Laforgia, and Pecaric [3] on inequalities involving Bessel functions.

Romberg extrapolation is a long-known method to improve the convergence rate of the trapezoidal rule on intervals. For simple regions such as the cube \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[0,1]^q$$\end{document}[0,1]q it is directly transferable to cubature in q dimensions. In this paper, we formulate Romberg extrapolation for Euler summation-based cubature on arbitrary q-dimensional regular regions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {G}\subset \mathbb {R}^q$$\end{document}G⊂Rq and derive an explicit representation for the remainder term.

We propose a complete-search algorithm for solving a class of non-convex, possibly infinite-dimensional, optimization problems to global optimality. We assume that the optimization variables are in a bounded subset of a Hilbert space, and we determine worst-case run-time bounds for the algorithm under certain regularity conditions of the cost functional and the constraint set. Because these run-time bounds are independent of the number of optimization variables and, in particular, are valid for optimization problems with infinitely many optimization variables, we prove that the algorithm converges to an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document}ε-suboptimal global solution within finite run-time for any given termination tolerance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon > 0$$\end{document}ε>0. Finally, we illustrate these results for a problem of calculus of variations.

The Domain Interface Method (DIM) is extended in this contribution for the case of mixed fields as encountered in multiphysics problems. The essence of the non-conforming domain decomposition technique consists in a discretization of a fictitious zero-thickness interface as in the original methodology and continuity of the solution fields across the domains is satisfied by incorporating the corresponding Lagrange Multipliers. The multifield DIM inherits the advantages of its irreducible version in the sense that the connections between non-matching meshes, with possible geometrically non-conforming interfaces, is accounted by the automatic Delaunay interface discretization without considering master and slave surfaces or intermediate surface projections as done in many established techniques, e.g. mortar methods. The multifield enhancement identifies the Lagrange multiplier field and incorporates its contribution in the weak variational form accounting for the corresponding consistent stabilization term based on a Nitsche method. This type of constraint enforcement circumvents the appearance of instabilities when the Ladyzhenskaya---Babuska---Brezzi (LBB) condition is not fulfilled by the chosen discretization. The domain decomposition framework is assessed in a large deformation setting for mixed displacement/pressure formulations and coupled thermomechanical problems. The continuity of the mixed field is studied in well selected benchmark problems for both mixed formulations and the objectivity of the response is compared to reference monolithic solutions. Results suggest that the presented strategy shows sufficient potential to be a valuable tool in situations where the evolving physics at particular domains require the use of different spatial discretizations or field interpolations.

In this short paper we study the d-Hermite condition about stably free modules for skew $$\textit{PBW}$$ extensions. For this purpose, we estimate the stable rank of these non-commutative rings. In addition, and closely related with these questions, we will prove Kronecker’s theorem about the radical of finitely generated ideals for some particular types of skew $$\textit{PBW}$$ extensions.

We construct a stabilized finite-element method to compute flow and finite-strain deformations in an incompressible poroelastic medium. We employ a three-field mixed formulation to calculate displacement, fluid flux and pressure directly and introduce a Lagrange multiplier to enforce flux boundary conditions. We use a low order approximation, namely, continuous piecewise-linear approximation for the displacements and fluid flux, and piecewise-constant approximation for the pressure. This results in a simple matrix structure with low bandwidth. The method is stable in both the limiting cases of small and large permeability. Moreover, the discontinuous pressure space enables efficient approximation of steep gradients such as those occurring due to rapidly changing material coefficients or boundary conditions, both of which are commonly seen in physical and biological applications.

A domain decomposition technique is proposed which is capable of properly connecting arbitrary non-conforming interfaces. The strategy essentially consists in considering a fictitious zero-width interface between the non-matching meshes which is discretized using a Delaunay triangulation. Continuity is satisfied across domains through normal and tangential stresses provided by the discretized interface and inserted in the formulation in the form of Lagrange multipliers. The final structure of the global system of equations resembles the dual assembly of substructures where the Lagrange multipliers are employed to nullify the gap between domains. A new approach to handle floating subdomains is outlined which can be implemented without significantly altering the structure of standard industrial finite element codes. The effectiveness of the developed algorithm is demonstrated through a patch test example and a number of tests that highlight the accuracy of the methodology and independence of the results with respect to the framework parameters. Considering its high degree of flexibility and non-intrusive character, the proposed domain decomposition framework is regarded as an attractive alternative to other established techniques such as the mortar approach.

We investigate the techniques and ideas used in the convergence analysis of two proximal ADMM algorithms for solving convex optimization problems involving compositions with linear operators. Besides this, we formulate a variant of the ADMM algorithm that is able to handle convex optimization problems involving an additional smooth function in its objective, and which is evaluated through its gradient. Moreover, in each iteration we allow the use of variable metrics, while the investigations are carried out in the setting of infinite dimensional Hilbert spaces. This algorithmic scheme is investigated from the point of view of its convergence properties., Comment: Updates in Section 2 concerning the derivation of the convergence rates + a unifying convergence theorem for the sequence of iterates

The origins of the alternating Schwarz method lie in the difficulty to prove the Dirichlet principle. This principle was evoked by Riemann in the proof of what is now the well known Riemann Mapping Theorem. We tell in this short paper the story of this exciting journey through the world of research mathematicians, up to the first computational Schwarz methods.

In a previous paper based on the interpolation finite difference method, a calculation system was shown for calculating 1D (one-dimensional) Laplace’s equation and Poisson’s equation using high-order difference schemes. Finite difference schemes, from the usual second-order to tenth-order differences, including odd number order differences, were systematically and instantaneously derived over equally/unequally spaced grid points based on the Lagrange interpolation function. Using the direct method with the band diagonal matrix algorithm, 1D Poisson equations were numerically calculated under double precision floating arithmetic, but it became clear that high accurate calculations could not be secured in high-order differences because “digit-loss errors” caused by the finite precision of computations occurred in the calculations when using the high-order differences. The double precision calculation corresponds to 15 (significant) digit calculation. In this paper, we systematically investigate how the calculation accuracy changes by high precision calculations (30-digit, and 45-digit calculations). Under 45-digit calculation, where the digit-loss error can be almost ignored, the high-order differences enable extremely high-accurate calculations.

In this paper, we deal with a specific type of quasilinear boundary value problem with Dirichlet boundary conditions and with p -Laplacian. We show two ways of proving the existence of nontrivial weak solutions. The first one uses the mountain pass theorem, the other one is based on our new minimax theorem. This method is novel even for p = 2 . In the paper, we also present a numerical algorithm based on the introduced approach. The suggested algorithm is illustrated on numerical examples and compared with a current approach to demonstrate its efficiency.

Let Γ be a smooth curve inside a two-dimensional rectangular region Ω. In this paper, we consider the Poisson interface problem − ∇ 2 u = f in Ω ∖ Γ with Dirichlet boundary condition such that f is smooth in Ω ∖ Γ and the jump functions [ u ] and [ ∇ u ⋅ n → ] across Γ are smooth along Γ. This Poisson interface problem includes the weak solution of − ∇ 2 u = f + g δ Γ in Ω as a special case. Because the source term f is possibly discontinuous across the interface curve Γ and contains a delta function singularity along the curve Γ, both the solution u of the Poisson interface problem and its flux ∇ u ⋅ n → are often discontinuous across the interface. To solve the Poisson interface problem with singular sources, in this paper we propose a sixth order compact finite difference scheme on uniform Cartesian grids. Our proposed compact finite difference scheme with explicitly given stencils extends the immersed interface method (IIM) to the highest possible accuracy order six for compact finite difference schemes on uniform Cartesian grids, but without the need to change coordinates into the local coordinates as in most papers on IIM in the literature. Also in contrast with most published papers on IIM, we explicitly provide the formulas for all involved stencils and therefore, our proposed scheme can be easily implemented and is of interest to practitioners dealing with Poisson interface problems. Note that the curve Γ splits Ω into two disjoint subregions Ω + and Ω − . The coefficient matrix A in the resulting linear system A x = b , following from the proposed scheme, is independent of any source term f, jump condition g δ Γ , interface curve Γ and Dirichlet boundary conditions, while only b depends on these factors and is explicitly given, according to the configuration of the nine stencil points in Ω + or Ω − . The constant coefficient matrix A facilitates the parallel implementation of the algorithm in case of a large size matrix and only requires the update of the right hand side vector b for different Poisson interface problems. Due to the flexibility and explicitness of the proposed scheme, it can be generalized to obtain the highest order compact finite difference scheme for non-uniform grids as well. We prove the order 6 convergence for the proposed scheme using the discrete maximum principle. Our numerical experiments confirm the sixth accuracy order of the proposed compact finite difference scheme on uniform meshes for the Poisson interface problems with various singular sources.

The singular value decomposition of a complex matrix is a fundamental concept in linear algebra and has proved extremely useful in many subjects. It is less clear what the situation is over a finite field. In this paper, we classify the orbits of GU m ( q ) × GU n ( q ) on M m × n ( q 2 ) (which is the analog of the singular value decomposition). The proof involves Kronecker’s theory of pencils and the Lang–Steinberg theorem for algebraic groups. Besides the motivation mentioned above, this problem came up in a recent paper of Guralnick et al. (2020) where a concept of character level for the complex irreducible characters of finite, general or special, linear and unitary groups was studied and bounds on the number of orbits was needed. A consequence of this work determines possible pairs of Jordan forms for nilpotent matrices of the form A A ∗ and A ∗ A over a finite field and A A ⊤ and A ⊤ A over arbitrary fields.

In this paper we present a pseudospectral method in the disk. Unlike the methods already known, the disk is not duplicated. Moreover, we solve the Laplace equation under nonhomogeneous Dirichlet, Neumann and Robin boundary conditions, as well as the biharmonic equation subject to nonhomogeneous Dirichlet conditions, by only using the elements of the corresponding differentiation matrices. It is worth mentioning that we do not use any quadrature, nor need to solve any decoupled system of ordinary differential equations, nor use any pole condition, nor require any lifting. We also solve several numerical examples to show the spectral convergence. The pseudospectral method developed in this paper is applied to estimate Sherwood numbers integrating the mass flux to the disk, and it can be implemented to solve Lotka–Volterra systems and nonlinear diffusion problems involving chemical reactions.

The paper concerns with three iterative regularization methods for solving a variational inclusion problem of the sum of two operators, the one is maximally monotone and the another is monotone and Lipschitz continuous, in a Hilbert space. We first describe how to incorporate regularization terms in the methods of forward-backward types, and then establish the strong convergence of the resulting methods. With several new stepsize rules considered, the methods can work with or without knowing previously the Lipschitz constant of cost operator. Unlike known hybrid methods, the strong convergence of the proposed methods comes from the regularization technique. Several applications to signal recovery problems and optimal control problems together with numerical experiments are also presented in this paper. Our numerical results have illustrated the fast convergence and computational effectiveness of the new methods over known hybrid methods.

In the third part of his famous 1926 paper ‘Quantisierung als Eigenwertproblem’, Schrödinger came across a certain parametrized family of tridiagonal matrices whose eigenvalues he conjectured. A 1991 paper wrongly suggested that his conjecture is a direct consequence of an 1854 result put forth by Sylvester. Here we recount some of the arguments that led Schrödinger to consider this particular matrix and what might have led to the wrong suggestion. We then give a self-contained elementary (though computational) proof which would have been accessible to Schrödinger. It needs only partial fraction decomposition. We conclude this paper by giving an outline of the connection established in recent decades between orthogonal polynomial systems of the Hahn class and certain tridiagonal matrices with fractional entries. It also allows to prove Schrödinger’s conjecture.

A new method is developed for solving optimal control problems whose solutions are nonsmooth. The method developed in this paper employs a modified form of the Legendre–Gauss–Radau orthogonal direct collocation method. This modified Legendre–Gauss–Radau method adds two variables and two constraints at the end of a mesh interval when compared with a previously developed standard Legendre–Gauss–Radau collocation method. The two additional variables are the time at the interface between two mesh intervals and the control at the end of each mesh interval. The two additional constraints are a collocation condition for those differential equations that depend upon the control and an inequality constraint on the control at the endpoint of each mesh interval. The additional constraints modify the search space of the nonlinear programming problem such that an accurate approximation to the location of the nonsmoothness is obtained. The transformed adjoint system of the modified Legendre–Gauss–Radau method is then developed. Using this transformed adjoint system, a method is developed to transform the Lagrange multipliers of the nonlinear programming problem to the costate of the optimal control problem. Furthermore, it is shown that the costate estimate satisfies one of the Weierstrass–Erdmann optimality conditions. Finally, the method developed in this paper is demonstrated on an example whose solution is nonsmooth.

This paper is devoted to solve weakly singular Volterra integral equations with noncompact kernels, which differ from the well-known case of Abel-type equations. We consider using the mapped Laguerre spectral method to deal with this type of equations. The construction and analysis of log orthogonal functions collocation method are presented in this paper and some numerical examples are included to show the efficiency of the proposed method.

This paper proposes a new algorithm for computing the Nash equilibrium based on an iterative approach of both the proximal and the gradient method for homogeneous, finite, ergodic and controllable Markov chains. We conceptualize the problem as a poly-linear programming problem. Then, we regularize the poly-linear functional employing a regularization approach over the Lagrange functional for ensuring the method to converge to some of the Nash equilibria of the game. This paper presents two main contributions: The first theoretical result is the proposed iterative approach, which employs both the proximal and the gradient method for computing the Nash equilibria in Markov games. The method transforms the game theory problem in a system of equations, in which each equation itself is an independent optimization problem for which the necessary condition of a minimum is computed employing a nonlinear programming solver. The iterated approach provides a quick rate of convergence to the Nash equilibrium point. The second computational contribution focuses on the analysis of the convergence of the proposed method and computes the rate of convergence of the step-size parameter. These results are interesting within the context of computational and algorithmic game theory. A numerical example illustrates the proposed approach.

Given the costs and a feasible solution for a finite-dimensional linear program (LP), inverse optimization involves finding new costs that are close to the original and that also render the given solution optimal. Ahuja and Orlin employed the absolute sum norm and the maximum absolute norm to quantify distances between cost vectors, and applied duality to establish that the inverse LP problem can be formulated as another finite-dimensional LP. This was recently extended to semi-infinite LPs, countably infinite LPs, and finite-dimensional conic optimization problems. These works provide sufficient conditions so that the inverse problem also belongs to the same class as the forward problem. This paper extends this result to conic LPs in potentially infinite-dimensional Banach spaces. Moreover, the paper presents concrete derivations for continuous conic LPs, whose special cases include continuous linear programs and continuous conic programs; normed cone programs in Banach spaces, which include second-order cone programs as a special case; and semi-definite programs in Hilbert spaces. These derivations reveal the sharper result that, in each case, the inverse problem belongs to the same specific subclass as the forward problem. Instances where existing forward algorithms can then be adapted to solve the inverse problems are identified. Results in this paper may enable the application of inverse optimization to as yet unexplored areas such as continuous-time economic planning, continuous-time job-shop scheduling, continuous-time network flow, maximum flow with time-varying edge-capacities, and wireless optimization with time-varying coverage requirements.

In this paper, we propose a new viscosity extragradient algorithm for solving variational inequality problems of pseudo-monotone and non-Lipschitz continuous operator in real Hilbert spaces. We prove a strong convergence theorem under some appropriate conditions imposed on the parameters. Finally, we give some numerical experiments to illustrate the advantages of our proposed algorithms. The main results obtained in this paper extend and improve some related works in the literature.

We develop an efficient and robust high-dimensional sparse Fourier algorithm for noisy samples. Earlier in the paper ``Multi-dimensional sublinear sparse Fourier algorithm" (2016), an efficient sparse Fourier algorithm with $\Theta(ds \log s)$ average-case runtime and $\Theta(ds)$ sampling complexity under certain assumptions was developed for signals that are $s$-sparse and bandlimited in the $d$-dimensional Fourier domain, i.e. there are at most $s$ energetic frequencies and they are in $ \left[-N/2, N/2\right)^d\cap \mathbb{Z}^d$. However, in practice the measurements of signals often contain noise, and in some cases may only be nearly sparse in the sense that they are well approximated by the best $s$ Fourier modes. In this paper, we propose a multiscale sparse Fourier algorithm for noisy samples that proves to be both robust against noise and efficient.

Purpose The purpose of this paper is to study the Hölder calmness of solutions to equilibrium problems and apply it to economics. Design/methodology/approach The authors obtain the Hölder calmness by using an effective approach. More precisely, under the key assumption of strong convexity, sufficient conditions for the Hölder continuity of solution maps to equilibrium problems are established. Findings A new result in stability analysis for equilibrium problems and applications in economics is archived. Originality/value The authors confirm that the paper has not been published previously, is not under consideration for publication elsewhere and is not being simultaneously submitted elsewhere.

At the 1900 International Congress of Mathematicians, Hilbert claimed that the Riemann zeta function is not the solution of any algebraic ordinary differential equation its region of analyticity \cite{HilbertProb}. In 2015, Van Gorder addresses the question of whether the Riemann zeta function satisfies a {\it non}-algebraic differential equation and constructs a differential equation of infinite order which zeta satisfies \cite{RHequiv}. However, as he notes in the paper, this representation is formal and Van Gorder does not attempt to claim a region or type of convergence. In this paper, we show that Van Gorder's operator applied to the zeta function does not converge pointwise at any point in the complex plane. We also investigate the accuracy of truncations of Van Gorder's operator applied to the zeta function and show that a similar operator applied to zeta and other $L$-functions does converge., Comment: This version varies from the published version in JNT in that a convergence issue has been corrected. Section 4.1 is replaced by 2 new sections to correct this. (Previous version is the published version.)

In this paper stability of numerical solution of Fredholm integral equation of the first kind is studied for radial basis kernels which possess positive Fourier transform. As a result, the equivalence relation between strong and weak forms of partial differential equations (PDEs) is proved for some special radial test functions. Also the stability of boundary elements method (BEM) is proved analytically for Laplace and Helmholtz equations by obtaining Fourier transform of singular fundamental solutions applied in BEM. Analytical result presented in this paper is an extension of stability idea of radial basis functions (RBFs) used to interpolate scattered data described by Wendland in [51] . Similar to the interpolation, it is proved here mathematically that integral operators which have radial kernels with positive Fourier transform are strictly positive definite. Thanks to the stability idea presented in [51] , a positive lower bound for eigenvalues of these integral operators is found here, explicitly.

In 1986, Proinov published an explicit lower bound for the diaphony of finite and infinite sequences of points contained in the d−dimensional unit cube [Proinov, P. D.:On irregularities of distribution, C. R. Acad. Bulgare Sci. 39 (1986), no. 9, 31–34]. However, his widely cited paper does not contain the proof of this result but simply states that this will appear elsewhere. To the best of our knowledge, this proof was so far only available in a monograph of Proinov written in Bulgarian [Proinov, P. D.: Quantitative Theory of Uniform Distribution and Integral Approximation, University of Plovdiv, Bulgaria (2000)]. The first contribution of our paper is to give a self contained version of Proinov’s proof in English. Along the way, we improve the explicit asymptotic constants implementing recent, and corrected results of [Hinrichs, A.—Markhasin, L.: On lower bounds for the ℒ2 -discrepancy, J. Complexity 27 (2011), 127–132.] and [Hinrichs, A.—Larcher, G.: An improved lower bound for the ℒ2 -discrepancy, J. Complexity 34 (2016), 68–77]. (The corrections are due to a note in [Hinrichs, A.—Larcher, G. An improved lower bound for the ℒ2 -discrepancy, J. Complexity 34 (2016), 68–77].) Finally, as a main result, we use the method of Proinov to derive an explicit lower bound for the dyadic diaphony of finite and infinite sequences in a similar fashion.

In this paper, an alternative approach is provided in the well-posedness analysis of elliptic variational–hemivariational inequalities in real Hilbert spaces. This includes the unique solvability and continuous dependence of the solution on the data. In most of the existing literature on elliptic variational–hemivariational inequalities, well-posedness results are obtained by using arguments of surjectivity for pseudomonotone multivalued operators, combined with additional compactness and pseudomonotonicity properties. In contrast, following (Han in Nonlinear Anal B Real World Appl 54:103114, 2020; Han in Numer Funct Anal Optim 42:371–395, 2021), the approach adopted in this paper is based on the fixed point structure of the problems, combined with minimization principles for elliptic variational–hemivariational inequalities. Consequently, only elementary results of functional analysis are needed in the approach, which makes the theory of elliptic variational–hemivariational inequalities more accessible to applied mathematicians and engineers. The theoretical results are illustrated on a representative example from contact mechanics.

This paper addresses the three concepts of consistency, stability and convergence in the context of compact finite volume schemes for systems of nonlinear hyperbolic conservation laws. The treatment utilizes the framework of “balance laws”. Such laws express the relevant physical conservation laws in the presence of discontinuities. Finite volume approximations employ this viewpoint, and the present paper can be regarded as being in this category. It is first shown that under very mild conditions a weak solution is indeed a solution to the balance law. The schemes considered here allow the computation of several quantities per mesh cell (e.g., slopes) and the notion of consistency must be extended to this framework. Then a suitable convergence theorem is established, generalizing the classical convergence theorem of Lax and Wendroff. Finally, the limit functions are shown to be entropy solutions by using a notion of “Godunov compatibility”, which serves as a substitute to the entropy condition.

In 2010, Andrews investigated a variety of parity questions in the classical partition identities of Euler, Rogers, Ramanujan and Gordon. In particular, he considered the Rogers-Ramanujan-Gordon partitions with some constraints on even and odd parts. At the end of this paper, he left fifteen open questions, of which the eleventh is to extend his parity consideration to overpartitions. The main purpose of this paper is to undertake that question. In 2013, Chen, Sang and Shi derived an overpartition analogue of the Rogers–Ramanujan–Gordon theorem. Motivated by their work, we define two kinds of Rogers–Ramanujan–Gordon type overpartitions with some parity constraints on even and odd parts. We then provide the generating functions for such partitions in some cases.

In this paper, we prove the weak convergence of a modified extragradient algorithm for solving a variational inequality problem involving a pseudomonotone operator in an infinite dimensional Hilbert space. Moreover, we establish further the R-linear rate of the convergence of the proposed algorithm with the assumption of error bound. Several numerical experiments are performed to illustrate the convergence behaviour of the new algorithm in comparisons with others. The results obtained in the paper have extended some recent results in the literature.

In a series of papers published since 2017 the author introduced a simple alternative of the $n$-simplex type, to enhance the accuracy of approximations of second-order boundary value problems subject to Dirichlet boundary conditions, posed on smooth curved domains. This technique is based upon trial functions consisting of piecewise polynomials defined on straight-edged triangular or tetrahedral meshes, interpolating the Dirichlet boundary conditions at points of the true boundary. In contrast, the test functions are defined by the standard degrees of freedom associated with the underlying method for polytopic domains. While the mathematical analysis of the method for Lagrange and Hermite methods for two-dimensional second- and fourth-order problems was carried out in earlier paper by the author this paper is devoted to the study of the three-dimensional case. Well-posedness, uniform stability and optimal a priori error estimates in the energy norm are proved for a tetrahedron-based Lagrange family of finite elements. Novel error estimates in the $L^2$-norm, for the class of problems considered in this work, are also proved. A series of numerical examples illustrates the potential of the new technique. In particular, its superior accuracy at equivalent cost, as compared to the isoparametric technique, is highlighted.

Ostwald ripening is a main phenomenon responsible for growth of particles manufactured using colloidal methods. This paper considers mathematical description of the process using population balance approach solved by method of moments. However, in the case of particles’ dissolution which is a part of the process a closure problem is generated. A novel closure method for the moment – transformed population balance equations based on Gram – Charlier expansions is introduced. Due to its simplicity it allows to reconstruct the full particle size distribution in a stepwise manner giving an access to the pointwise values of the density function required to close the model. The introduced numerical scheme is further used to simulate several carefully chosen test cases including dissolution of solid particles and ageing of particles’ population. The mathematical model presented in the paper is then applied to interpret experimental results for batch precipitation of silica.

In this paper, we investigate the exponential mean-square stability for both the solution of n-dimensional stochastic delay integro-differential equations (SDIDEs) with Poisson jump, as well for the split-step θ-Milstein (SSTM) scheme implemented of the proposed model. First, by virtue of Lyapunov function and continuous semi-martingale convergence theorem, we prove that the considered model has the property of exponential mean-square stability. Moreover, it is shown that the SSTM scheme can inherit the exponential mean-square stability by using the delayed difference inequality established in the paper. Eventually, three numerical examples are provided to show the effectiveness of the theoretical results.

In this paper, we use the Riemnn-Liouville fractional integrals to establish new results related to Hardy inequalities. For our results, some result of the paper [ A.Khameli et al : New Riemann-Lio...

Let f be any arithmetic function and define S f ( x ) : = ∑ n ⩽ x f ( [ x / n ] ) . When f equals the Euler totient function φ, several authors studied the upper and lower bounds of S φ ( x ) . In this paper we shall prove that S φ ( x ) has an asymptotic formula by the method of exponential sums. This result proves a conjecture proposed by Bordelles, Dai, Heyman, Pan and Shparlinski. Some other asymptotic formulas for arbitrary f are also given in this paper.

Since their introduction in 1967, Lawson methods have achieved constant interest in the time discretization of evolution equations. The methods were originally devised for the numerical solution of stiff differential equations. Meanwhile, they constitute a well-established class of exponential integrators, which has turned out to be competitive for solving space discretizations of certain types of partial differential equations. The popularity of Lawson methods is in some contrast to the fact that they may have a bad convergence behaviour, since they do not satisfy any of the stiff order conditions. The aim of this paper is to explain this discrepancy. It is shown that non-stiff order conditions together with appropriate regularity assumptions imply high-order convergence of Lawson methods. Note, however, that the term regularity here includes the behaviour of the solution at the boundary. For instance, Lawson methods will behave well in the case of periodic boundary conditions, but they will show a dramatic order reduction for, e.g., Dirichlet boundary conditions. The precise regularity assumptions required for high-order convergence are worked out in this paper and related to the corresponding assumptions for splitting schemes. In contrast to previous work, the analysis is based on expansions of the exact and the numerical solution along the flow of the homogeneous problem. Numerical examples for the Schrödinger equation are included.

In this paper we analyze a class of trace finite element methods for the discretization of vector-Laplace equations. A key issue in the finite element discretization of such problems is the treatment of the constraint that the unknown vector field must be tangential to the surface (‘tangent condition’). We study three different natural techniques for treating the tangent condition, namely a consistent penalty method, a simpler inconsistent penalty method and a Lagrange multiplier method. The main goal of the paper is to present an analysis that reveals important properties of these three different techniques for treating the tangent constraint. A detailed error analysis is presented that takes the approximation of both the geometry of the surface and the solution of the partial differential equation into account. Error bounds in the energy norm are derived that show how the discretization error depends on relevant parameters such as the degree of the polynomials used for the approximation of the solution, the degree of the polynomials used for the approximation of the level set function that characterizes the surface, the penalty parameter and the degree of the polynomials used for the approximation of the Lagrange multiplier.

In this paper, we introduce an explicit iterative algorithm for solving a (pseudo) monotone variational inequality under Lipschitz condition in a Hilbert space. The algorithm is constructed around some projections incorporated by inertial terms. It uses variable stepsizes which are generated at each iteration by some simple computations. Furthermore, it can be easily implemented without the prior knowledge of the Lipschitz constant of the operator. Theorems of weak convergence are established under mild conditions, and some numerical results are reported for the purpose of comparison with other algorithms. The obtained results in this paper extend some related works in the literature.

This paper considers some multiscale radial basis function collocation methods for solving the two-dimensional Monge–Ampère equation with Dirichlet boundary. We discuss and study the performance of the three kinds of multiscale methods. The first method is the cascadic meshfree method, which was proposed by Liu and He (2013). The second method is the stationary multilevel method, which was proposed by Floater and Iske (1996), and is used to solve the fully nonlinear partial differential equation in the paper for the first time. The third is the hierarchical radial basis function method, which is constructed by employing successive refinement scattered data sets and scaled compactly supported radial basis functions with varying support radii. Compared with the first two methods, the hierarchical radial basis function method can not only solve the present problem on a single level with higher accuracy and lower computational cost but also produce highly sparse nonlinear discrete system. These observations are obtained by taking the direct approach of numerical experimentation.

In this paper, we consider the split equality fixed point problem for quasi-pseudo-contractive mappings without a priori knowledge of operator norms in Hilbert spaces, which includes split feasibility problem, split equality problem, split fixed point problem, etc., as special cases. A unified framework for the study of this kind of problems and operators is provided. The results presented in the paper extend and improve many recent results.

In this paper, we propose a single species logistic model with feedback control and additive Allee effect in the growth of species. The basic aim of the paper is to discuss how the additive Allee effect and feedback control influence the above model’s dynamical behaviors. Firstly, the existence and stability of equilibria are discussed under three different cases, i.e., weak Allee effect, strong Allee effect, and the critical case. Secondly, we prove the occurrence of saddle-node bifurcation and transcritical bifurcation with the help of Sotomayor’s theorem. The above dynamical behaviors are richer and more complex than those in the traditional logistic model with feedback control. We find that both Allee effect and feedback control can increase the species’ extinction property. We also reveal some new bifurcation phenomena which do not exist in the single-species model with feedback control (Fan and Wang in Nonlinear Anal., Real World Appl. 11(4):2686–2697, 2010 and Lin in Adv. Differ. Equ. 2018:190, 2018).

We derive a posteriori error estimates for a semi-discrete finite element approximation of a nonlinear eddy current problem arising from applied superconductivity, known as the $p$-curl problem. In particular, we show the reliability for non-conforming N\'{e}d\'{e}lec elements based on a residual type argument and a Helmholtz-Weyl decomposition of $W^p_0(\text{curl};\Omega)$. As a consequence, we are also able to derive an a posteriori error estimate for a quantity of interest called the AC loss. The nonlinearity for this form of Maxwell's equation is an analogue of the one found in the $p$-Laplacian. It is handled without linearizing around the approximate solution. The non-conformity is dealt by adapting error decomposition techniques of Carstensen, Hu and Orlando. Geometric non-conformities also appear because the continuous problem is defined over a bounded $C^{1,1}$ domain while the discrete problem is formulated over a weaker polyhedral domain. The semi-discrete formulation studied in this paper is often encountered in commercial codes and is shown to be well-posed. The paper concludes with numerical results confirming the reliability of the a posteriori error estimate., Comment: 32 pages

In some previous works, two of the authors introduced a technique to design high-order numerical methods for one-dimensional balance laws that preserve all their stationary solutions. The basis of these methods is a well-balanced reconstruction operator. Moreover, they introduced a procedure to modify any standard reconstruction operator, like MUSCL, ENO, CWENO, etc., in order to be well-balanced. This strategy involves a non-linear problem at every cell at every time step that consists in finding the stationary solution whose average is the given cell value. In a recent paper, a fully well-balanced method is presented where the non-linear problems to be solved in the reconstruction procedure are interpreted as control problems. The goal of this paper is to introduce a new technique to solve these local non-linear problems based on the application of the collocation RK methods. Special care is put to analyze the effects of computing the averages and the source terms using quadrature formulas. A general technique which allows us to deal with resonant problems is also introduced. To check the efficiency of the methods and their well-balance property, they have been applied to a number of tests, ranging from easy academic systems of balance laws consisting of Burgers equation with some non-linear source terms to the shallow water equations—without and with Manning friction—or Euler equations of gas dynamics with gravity effects.

In this paper, we use the combined-unified hybrid censoring samples to obtain the maximum likelihood estimates of the unknown parameters, survival, and hazard functions of Pareto distribution. Next, we discuss some efficiency criteria of the maximum likelihood estimators, including, the unbiasedness, consistency, and sufficiency. Additionally, we use MCMC to obtain the Bayesian estimates of the unknown parameters. In addition, we calculate the intervals estimation of the unknown parameters. Finally, we analyze a set of real data in view of the theoretical findings of the paper.

In the paper, we introduce a new notion of reduced linear c-shift operator $L _{c}^{r}\,f$Lcrf, and with the aid of this new operator, we study the uniqueness of meromorphic functions $f(z)$f(z) and $L_{c}^{r}\,f$Lcrf sharing two or more values in the extended complex plane. The results obtained in the paper significantly improve a number of existing results. Further, using the notion of weighted sharing of sets, we deal the same problem. We exhibit a handful number of examples to justify certain statements relevant to the content of the paper. We are also able to determine the form of the function that coincides with its reduced linear c-shift operator. At the end of the paper, we pose an open question for future research.

In this paper, we consider the blow-up curve of semilinear wave equations. Merle and Zaag (Am J Math 134:581–648, 2012) considered the blow-up curve for $$\partial _t^2 u- \partial _x^2 u = |u|^{p-1}u$$ and showed that there is the case that the blow-up curve is not differentiable at some points when the initial value changes its sign. Their analysis depends on the variational structure of the problem. In this paper, we consider the blow-up curve for $$\partial _t^2 u- \partial _x^2 u = |\partial _t u|^{p-1}\partial _t u$$ which does not have the variational structure. Nevertheless, we prove that the blow-up curve is not differentiable if the initial data changes its sign and satisfies some conditions.