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The determination of a Galois group is an important question in computational algebraic number theory. One approach is based on the inspection of resolvents. This article reports on this method and on the performance of the current magma [W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (1997) 235-265]. implementation. [ABSTRACT FROM AUTHOR]

We study in this paper the monotonicity properties of the numerical solutions to Volterra integral equations with nonincreasing completely positive kernels on nonuniform meshes. There is a duality between the complete positivity and the properties of the complementary kernel being nonnegative and nonincreasing. Based on this, we propose the “complementary monotonicity” to describe the nonincreasing completely positive kernels, and the “right complementary monotone” (R-CMM) kernels as the analogue for nonuniform meshes. We then establish the monotonicity properties of the numerical solutions inherited from the continuous equation if the discretization has the R-CMM property. Such a property seems weaker than log-convexity and there is no restriction on the step size ratio of the discretization for the R-CMM property to hold. [ABSTRACT FROM AUTHOR]

We propose the novel spectral properties of the Neumann Laplacian in a two-dimensional bounded domain perturbed by an infinite number of compact sets with zero Lebesgue measure, so-called curved holes. These holes consist of segments or parts of curves enclosed in small spheres such that the diameters of holes tend to zero as the number of holes approaches infinity. Specifically, we rigorously demonstrate that the spectrum of the Neumann Laplacian on the perturbed domain converges to that of the original operator on the domain without holes under specific geometric assumptions and an appropriate selection of hole sizes. Furthermore, we derive sophisticated estimates on the convergence rate in terms of operator norms and estimate the Hausdorff distance between the spectra of the Laplacians. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to explore the split variational inclusion problem in Hilbert spaces. A splitting algorithm is constructed for solving the split variational inclusion with the help of self-adaptive techniques. Convergence analysis of the proposed algorithm is provided under additional conditions. [ABSTRACT FROM AUTHOR]

Green function of a 2n-th order differential equation with normal coefficients on the half-axis is studied. We first consider the Green function of our equation with “frozen” coefficients. Using Levi’s method, we obtain a Fredholm-type integral equation for the Green function of our problem, whose kernel is a Green function of a problem with constant coefficients. We prove an existence and uniqueness theorem for this integral equation in some Banach spaces of operator-valued functions. The main result of this paper is a theorem stating that the solution of the obtained integral equation is a Green function of our problem. [ABSTRACT FROM AUTHOR]

Keywords: Navier-Stokes equations; resolvent; free surface; solution formula; Lp estimates.GraphBy Yongting Huang and Tao LuoReported by Author; Author [Extracted from the article]

Sufficient conditions are given for the unique solvability of nonlocal problems for abstract singular equations that are formulated in terms of the zeros of the modified Bessel function and the resolvent of the operator coefficient of the equations under consideration. Examples are presented. [ABSTRACT FROM AUTHOR]

We generalize the theory of homotopic deviation of square complex matrices to regular matrix pencils. To this end, we study the existence and the analyticity of the resolvent of the matrix pencils whose matrices are under homotopic deviation with the deviation parameter $ t \in \mathbb {C} $ t ∈ C . Moreover, we investigate and identify the limits of both the resolvent and the spectrum of the deviated matrix pencils, as $ | t | \to \infty $ | t | → ∞. We also study the special cases where t tends to the eigenvalues of the related matrix pairs. We use the notions and the results of the generalized homotopic deviation theory to analyze the Weierstrass structure of the deviated matrix pencils under two different assumptions, in particular, either the eigenvalues of the deviated matrix pencils are independent parameters, or the deviation parameter t is an independent parameter. Numerical examples illustrate and support the theoretical results. [ABSTRACT FROM AUTHOR]

We consider a Sturm-Liouville-type differential operator defined on a separable Hilbert space with bounded operator coefficient and periodic boundary conditions given on a finite interval. We obtain an explicit formula which we call the second regularized trace formula of the mentioned operator. [ABSTRACT FROM AUTHOR]

We obtain a regularized trace formula for -order differential operator perturbed by a quasi-differential perturbation and with periodic boundary conditions. [ABSTRACT FROM AUTHOR]

In this work, we study the solvability and maximal regularity questions for the singular third‐order differential equation with unbounded coefficients and some applications. Unlike previously studied cases, the leading and intermediate coefficients of this equation can grow independently. We obtain sufficient coefficient conditions for correctness of this equation and compactness for inverse of the corresponding differential operator. We also prove the maximal regularity estimate for a generalized solution. Using these results, we obtain upper and lower estimates for the number of Kolmogorov k‐diameters of the set associated with the linear Korteweg–de Vries equation's solutions. We give an example and show that from the above inequalities follow two‐sided estimates of the Kolmogorov k‐diameters. [ABSTRACT FROM AUTHOR]

In this paper, we present a technique for constructing generalized solutions of the Cauchy problem for abstract integro-differential equations with degeneration in Banach spaces. A generalized solution is constructed as the convolution of the fundamental operator function (fundamental solution, influence function) of the integro-differential operator of the equation with a generalized function of a special form, which involves all input data of the original problem. Based on the analysis of the representation for the generalized solution, we obtain sufficient solvability conditions for the original Cauchy problem in the class of functions of finite smoothness. Under these sufficient conditions, the generalized solution constructed turns out to be a classical solution with the required smoothness. The abstract results obtained in the paper are applied to the study of applied initial-boundary-value problems from the theory of oscillations in viscoelastic media. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to revisit the proof of the Gearhardt–Prüss–Huang–Greiner theorem for a semigroup S(t), following the general idea of the proofs that we have seen in the literature and to get an explicit estimate on the operator norm of S(t) in terms of bounds on the resolvent of the generator. In Helffer and Sjöstrand (From resolvent bounds to semigroup bounds. , math. FA, 2010) by the first two authors, this was done and some applications in semiclassical analysis were given. Some of these results have been subsequently published in three books written by the two first authors Helffer (Spectral theory and its applications. Cambridge University Press, Cambridge, 2013) and Sjöstrand (Lecture notes : Spectral properties of non-self-adjoint operators. Journées équations aux dérivées partielles (2009), article no. 1), (Non self-adjoint differential operators, spectral asymptotics and random perturbations. Pseudo-differential Operators and Applications, Birkhäuser (2018)). A second work Helffer and Sjöstrand (Integral Equ Oper Theory 93(3), 2021) presents new improvements partially motivated by a paper of Wei (Sci China Math 64:507–518, 2021). In this third paper, we continue the discussion on whether the aforementioned results are optimal, and whether one can improve these results through iteration. Numerical computations will illustrate some of the abstract results. [ABSTRACT FROM AUTHOR]

Based on the resolvent decomposition theorems presented very recently by Chen (J Theor Probab 33:2089–2118, 2020), in this paper we focus on investigating the fundamental problems of existence and uniqueness criteria for Denumerable Markov Processes with finitely many instantaneous states. Some elegant sufficient and necessary conditions are obtained for this less-investigated topic. A few important examples including the generalized Kolmogorov models are presented to illustrate our general results. [ABSTRACT FROM AUTHOR]

More than 40 years ago, Lions and Mercier introduced in a seminal paper the Douglas–Rachford algorithm. Today, this method is well-recognized as a classic and highly successful splitting method to find minimizers of the sum of two (not necessarily smooth) convex functions. Whereas the underlying theory has matured, one case remains a mystery: the behavior of the shadow sequence when the given functions have disjoint domains. Building on previous work, we establish for the first time weak and value convergence of the shadow sequence generated by the Douglas–Rachford algorithm in a setting of unprecedented generality. The weak limit point is shown to solve the associated normal problem, which is a minimal perturbation of the original optimization problem. We also present new results on the geometry of the minimal displacement vector. Funding: The research of H. H. Bauschke and W. M. Moursi was partially supported by Discovery Grants of the Natural Sciences and Engineering Research Council of Canada [Grants RGPIN-2018-03703 and RGPIN-2019-04803], respectively. [ABSTRACT FROM AUTHOR]

In this study, impulsive singular dynamic Dirac equations are considered. Firstly, a spectral function for these equations is constructed. Later, an integral representation of the resolvent operator is obtained. [ABSTRACT FROM AUTHOR]

In the paper we find sufficient conditions on the coefficients of a class of third order quasi-elliptic operator pencils providing the completeness of the part of eigen and associated vectors in a separable Hilbert space. The completeness of the system of decreasing elementary solutions of the corresponding homogeneous equation in the space of solutions of this equation is proved. All these conditions are expressed by the coefficients of the operator pencil. [ABSTRACT FROM AUTHOR]

We revisit the operator splitting schemes proposed in a recent work of [Some extensions of the operator splitting schemes based on Lagrangian and primal-dual: A unified proximal point analysis, Feng Xue, Optimization, 2022, doi: 10.1080/02331934.2022.2057309], and further analyze the convergence of the generalized Bregman distance and the primal-dual gap of these algorithms within a unified proximal point framework. The possibility of reduction to a simple resolvent is also discussed by exploiting the structure and possible degeneracy of the underlying metric. [ABSTRACT FROM AUTHOR]

As you know, the third order partial differential equation is one of the basic equations of wave theory. For example, in particular, a linearized Korteweg-de Vries type equation with variable coefficients models ion-acoustic waves into plasma and acoustic waves on a crystal lattice. In this paper, the properties of solutions of а class of the third order degenerate partial differential equations with variable coefficients given in a rectangle were studied. Sufficient conditions for the existence and uniqueness of a strong solution have been established. Note that the solution of the degenerate equation does not retain its smoothness, therefore, these difficulties in turn affect the coercive estimates.

We consider a metric graph consisting of two edges, one of which has length ε which we send to zero. On this graph we study the resolvent and spectrum of the Laplacian subject to a general vertex condition at the connecting vertex. Despite the singular nature of the perturbation (by a short edge), we find that the resolvent depends analytically on the parameter ε . In contrast, the negative eigenvalues escape to minus infinity at rates that could be fractional, namely, ε 0 , ε - 2 / 3 or ε - 1 . These rates take place when the corresponding eigenfunction localizes, respectively, only on the long edge, on both edges, or only on the short edge. [ABSTRACT FROM AUTHOR]

In a finite dimensional vector space, the inverse of an invertible operator is a polynomial in the operator. We study the connection between an invertible operator and its inverse in infinite dimensional Banach space. [ABSTRACT FROM AUTHOR]

We construct a Ces\`{a}ro-type operator acting on Dirichlet space of the upper half plane using the approach of strongly continuous semigroups of composition operators on Banach spaces. We then determine the spectral and norm properties of the obtained Ces\`{a}ro-type operator.

The resolvent approach and the using of the idea of A. N. Krylov on the acceleration of convergence of Fourier series, the properties of a formal solution of a mixed problem for a homogeneous wave equation with a summable potential and a zero initial function are studied. This method makes it possible to obtain deep results on the convergence of a formal series with arbitrary boundary conditions and without overestimating the requirements for the smoothness of the initial data. The different-order boundary conditions considered in the article are such that the operator corresponding to the spectral problem may have an infinite set of multiple eigenvalues and their associated functions. A classical solution is obtained without overstating the requirements for the initial velocity $u'_t(x,0) = \psi(x)$. It is shown that for $\psi(x) \in L[0,1]$ the formal solution, being the uniform limit of the classical ones, is a generalized solution, and when $\psi(x) \in L_p[0,1], ~ 1

For any pseudo-Riemannian hyperbolic space X over R , C , H or O , we show that the resolvent R (z) = (□ − z Id) − 1 of the Laplace–Beltrami operator −□ on X can be extended meromorphically across the spectrum of □ as a family of operators C c ∞ (X) → D ′ (X). Its poles are called resonances and we determine them explicitly in all cases. For each resonance, the image of the corresponding residue operator in D ′ (X) forms a representation of the isometry group of X , which we identify with a subrepresentation of a degenerate principal series. Our study includes in particular the case of even functions on de Sitter and Anti-de Sitter spaces. For Riemannian symmetric spaces analogous results were obtained by Miatello–Will and Hilgert–Pasquale. The main qualitative differences between the Riemannian and the non-Riemannian setting are that for non-Riemannian spaces the resolvent can have poles of order two, it can have a pole at the branching point of the covering to which R (z) extends, and the residue representations can be infinite-dimensional. [ABSTRACT FROM AUTHOR]

In this paper, we investigate and analyse the strong convergence of the sequence generated by an inexact proximal point method with possible unbounded errors for finding zeros of monotone operators in Hadamard spaces. We show that the boundedness of the generated sequence is equivalent to the zero set of the operator to be nonempty. In this case, we prove the strong convergence of the generated sequence to a zero of the operator. We also provide some applications of our main results and give a numerical example to show the performance of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Let Ω a bounded domain in Rn with smooth boundary ∂Ω. In this article, in view of our earlier paper [12], let the conditions made on this paper be sufficiently more general than [12]. We investigate the spectral properties of a non-selfadjoint elliptic differential operator (Pu)(x) = -Σni,j=1 (ρ2α(x)aij(x)Q(x)u'xi(x))'xj, acting on Hilbert space H∂ = L²(Ω)∂. Here c|s|² ≤ Σni,j=1 aij(x)sisj (s = (s1,. . ., sn) ∊ Cn, x ∊ Ω), ρ(x) = dist{x, ∂Ω}, aij(x) = aji(x) ∊ C²(Ω), 0 ≤ α < Ω. Fur-thermore, suppose that Q(x) ∊ C²(Ω, End C∂) such that for each x ∊ Ω the matrix function Q(x) has non-zero simple eigenvalues μj (x) ∊ C²(Ω) (1 ∊ j ℹ ∂) lie in the C\ɸ, where ɸ = {z ⅂ C: |arg z| ∊ φ}, φ 2 (0, π). [ABSTRACT FROM AUTHOR]

We construct a Cesaro-type operator acting on Dirichlet space of the upper half plane using the approach of strongly continuous semigroups of composition operators on Banach spaces. We then determine the spectral and norm properties of the obtained Cesaro-type operator. [ABSTRACT FROM AUTHOR]

We investigate the spectral properties of two different boundary value problems on a compact star graph in which the vertex conditions are dependent on the spectral parameter. We treat these boundary value problems as eigenvalue problems in some extended Hilbert spaces by associating them with vector-valued operators. We prove that the corresponding operators are self-adjoint. We construct the characteristic functions of these eigenvalue problems and prove that the corresponding operators have discrete spectrum. Moreover, we present some examples where we construct fundamental solutions and derive the resolvent operators. [ABSTRACT FROM AUTHOR]

As one of the important factors affecting the stability of slide valves, the analysis and research of flow force are of great significance. In recent years, more and more experts and scholars have conducted research in this field, attempting to find methods to reduce or utilize the flow force of hydraulic spool valves. Flow force includes steady-state flow force and transient flow force, with steady-state flow force having the most significant impact on spool valves. The influencing factors of flow force are complex and diverse, including the cavitation phenomenon, shape of the throttling groove, and jet angle. At present, the main ways to reduce flow force are to design the structure of the spool valve, the structure of the valve sleeve, and the flow channel of the valve body. This article mainly reviews the definition, calculation methods, influencing factors, and methods for reducing the flow force of slide valves. This provides a new approach to reducing the flow force in hydraulic spool valves. [ABSTRACT FROM AUTHOR]

The integro-differential equation of heat conduction with the time-convolution integral on the right side is considered. The direct problem is the initial-boundary problem for this integro-differential equation. Two inverse problems are studied for this direct problem consisting in determining a kernel of the integral member on two given additional conditions with respect to the solution of the direct problems, respectively. The problems are replaced with the equivalent system of the integral equations with respect to unknown functions and on the basis of contractive mapping the unique solvability inverse problem.

In this article, we discuss equilibrium problems and their resolvents on complete geodesic spaces. In particular, we consider asymptotic behavior and continuity of resolvents with positive parameter in a complete geodesic space whose curvature is bounded above. Furthermore, we apply these results to resolvents of convex functions.

For the Frobenius matrix accompanying an algebraic (differential) equation in a complex Banach algebra, the Cayley–Hamilton theorem is proved, which is used to obtain a representation of the resolvent. [ABSTRACT FROM AUTHOR]

We consider regularized traces for perturbations by Hilbert–Schmidt operators of self-adjoint operators with compact resolvent. Formulas for regularized traces of arbitrary order are obtained under the condition that the spectrum of an unperturbed self-adjoint operator is sparse. Various generalizations and examples are given. [ABSTRACT FROM AUTHOR]

Finding a zero of a sum of maximally monotone operators is a fundamental problem in modern optimization and nonsmooth analysis. Assuming that the resolvents of the operators are available, this problem can be tackled with the Douglas-Rachford algorithm. However, when dealing with three or more operators, one must work in a product space with as many factors as there are operators. In groundbreaking recent work by Ryu and by Malitsky and Tam, it was shown that the number of factors can be reduced by one. A similar reduction was achieved recently by Campoy through a clever reformulation originally proposed by Kruger. All three splitting methods guarantee weak convergence to some solution of the underlying sum problem; strong convergence holds in the presence of uniform monotonicity. In this paper, we provide a case study when the operators involved are normal cone operators of subspaces and the solution set is thus the intersection of the subspaces. Even though these operators lack strict convexity, we show that striking conclusions are available in this case: strong (instead of weak) convergence and the solution obtained is (not arbitrary but) the projection onto the intersection. To illustrate our results, we also perform numerical experiments. [ABSTRACT FROM AUTHOR]

This paper is concerned with a mixed type differential operator Lu=kyuxx−uyy+byux+qyu,$$ Lu=k(y){u}_{xx}-{u}_{yy}+b(y){u}_x+q(y)u, $$which is initially defined with C0,π∞Ω‾$$ {C}_{0,\pi}^{\infty}\left(\overline{\Omega}\right) $$, where Ω‾={x,y:−π≤x≤π,−∞

We present conditions for a family (A(x))x∫ℝd of self-adjoint operators in Hr = ...r ⊕ H for a separable complex Hilbert space H, such that the Callias operator D = ic∇ + A(X) satisfies that (D*D + 1)-N -(DD* + 1)-N is trace class in L²(ℝd, Hr). Here, c∇ is the Dirac operator associated to a Clifford multiplication c of rank r on ℝd, and A(X) is fibre-wise multiplication with A(x) in L² (ℝd, Hr). [ABSTRACT FROM AUTHOR]

In this work, we propose and study a framework of generalized proximal point algorithms associated with a maximally monotone operator. We indicate sufficient conditions on the regularization and relaxation parameters of generalized proximal point algorithms for the equivalence of the boundedness of the sequence of iterations generated by this algorithm and the non-emptiness of the zero set of the maximally monotone operator, and for the weak and strong convergence of the algorithm. Our results cover or improve many results on generalized proximal point algorithms in our references. Improvements of our results are illustrated by comparing our results with related known ones. [ABSTRACT FROM AUTHOR]

For a bounded linear operator on a Banach space, we study approximation of the spectrum and pseudospectra in the Hausdorff distance. We give sufficient and necessary conditions in terms of pointwise convergence of appropriate spectral quantities.

In this study, we consider the Oseen structure of the linearization of a compressible fluid–structure interaction (FSI) system for which the interaction interface is under the effect of material derivative term. The flow linearization is taken with respect to an arbitrary, variable ambient vector field. This process produces extra "convective derivative" and "material derivative" terms, which render the coupled system highly nondissipative. We show first a new well‐posedness result for the full incorporation of both Oseen terms, which provides a uniformly bounded semigroup via dissipativity and perturbation arguments. In addition, we analyze the long time dynamics in the sense of asymptotic (strong) stability in an invariant subspace (one‐dimensional less) of the entire state space, where the continuous semigroup is uniformly bounded. For this, we appeal to the pointwise resolvent condition introduced in Chill and Tomilov [Stability of operator semigroups: ideas and results, perspectives in operator theory Banach center publications, 75 (2007), Institute of Mathematics Polish Academy of Sciences, Warszawa, 71–109], which avoids an immensely technical and challenging spectral analysis and provides a short and relatively easy‐to‐follow proof. [ABSTRACT FROM AUTHOR]

We show that strongly bounded operators in locally convex spaces and topological vector spaces under some additional conditions have the spectral properties similar to those of bounded operators in normed spaces. Such operator enjoys the Berling–Gelfand formula in a sequentially complete locally convex space. [ABSTRACT FROM AUTHOR]

Topological insulators (TIs) are renowned for their remarkable electronic properties: quantized bulk Hall and edge conductivities, and robust edge wave-packet propagation, even in the presence of material defects and disorder. Computations of these physical properties generally rely on artificial periodicity (the supercell approximation, which struggles in the presence of edges), or unphysical boundary conditions (artificial truncation). In this work, we build on recently developed methods for computing spectral properties of infinite-dimensional operators. We apply these techniques to develop efficient and accurate computational tools for computing the physical properties of TIs. These tools completely avoid such artificial restrictions and allow one to probe the spectral properties of the infinite-dimensional operator directly, even in the presence of material defects, edges and disorder. Our methods permit computation of spectra, approximate eigenstates, spectral measures, spectral projections, transport properties and conductances. Numerical examples are given for the Haldane model, and the techniques can be extended similarly to other TIs in two and three dimensions. [ABSTRACT FROM AUTHOR]

In this paper, we study a Yosida variational inclusion problem with its corresponding Yosida resolvent equation problem. We mention some schemes to solve both the problems, but we focus our study on discussing convergence criteria for the Yosida variational inclusion problem in real Banach space and for the Yosida resolvent equation problem in q-uniformly smooth Banach space. For faster convergence, we apply an inertial extrapolation scheme for both the problems. An example is provided. [ABSTRACT FROM AUTHOR]

For a bounded linear operator on a Banach space, we study approximation of the spectrum and pseudospectra in the Hausdorff distance. We give sufficient and necessary conditions in terms of pointwise convergence of appropriate spectral quantities. [ABSTRACT FROM AUTHOR]

Until now, the correspondence between the Alexander–Kolmogorov complex, and the de Rham one, by means of a small scale parameter, has not gone that far as passing to the limit of the resolvent of the associated Laplacian, when the small parameter tends towards zero. Along these lines, a result proving a complete Hodge decomposition was missing. We bridge this gap by means of our own rescaled h-cohomology, h being a very small parameter. Passing to the limit of the resolvent enables us to consider the extension to singular spaces, in particular, our h-differential operators also enable us to also make the connection with those of analysis on fractals, as introduced by Jun Kigami, and taken up by Robert S. Strichartz. [ABSTRACT FROM AUTHOR]

In this paper, we present a method for finding the zero of the sum of finite family of maximal monotone operators on real Hilbert spaces. In the case where the number of maximal monotone operators is three, we define a function such that its fixed points are solutions of our problem. Some illustrative examples are given at the end of this paper. [ABSTRACT FROM AUTHOR]

In this article, we study an extended split equality variational inclusion and fixed point problems which is an extension of the split equality variational inclusion problems and fixed point problems. We propose a simultaneous inertial type iterative algorithm with a self adaptive stepsizes such that there is no need for a prior information about the operator norm. We further stated and prove that the proposed algorithm weakly converges to a solution of the extended split equality variational inclusion and fixed point problems. Finally, we give some numerical examples to demonstrate the performance and the applicability of the proposed algorithm. The results of this paper complements and extends results on split equality variational inclusion and fixed point problems. [ABSTRACT FROM AUTHOR]