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In this note we offer some criticism on the paper “On an identity for zeros of Bessel functions” by Baricz et al. [3] . The paper gives identities of type Stieltjes–Calogero for the sums of reciprocals of differences of fourth powers of zeros of Bessel functions. Although interesting in principle, by containing one too many sums of similar complexity the identities fail to convey the true spirit of the work of Stieltjes and Calogero. We rectify this by providing what we think is the correct type of identity for the above-said sums, in the general setup of entire functions of order

In Fritz John's famous paper (1979), he discovered that for the wave equation $\Box u=|u|^p$, where $1, Comment: 11 pages, 3 figures

Congruences mod p for a prime p and partial products of the numbers 1,…, p − 1 are obtained. Bibliography: 2 titles.

The unified transform method (UTM) provides a novel approach to the analysis of initial-boundary value problems for linear as well as for a particular class of nonlinear partial differential equations called integrable. If the latter equations are formulated in two dimensions (either one space and one time, or two space dimensions), the UTM expresses the solution in terms of a matrix Riemann-Hilbert (RH) problem with explicit dependence on the independent variables. For nonlinear integrable evolution equations, such as the celebrated nonlinear Schr\"odinger (NLS) equation, the associated jump matrices are computed in terms of the initial conditions and all boundary values. The unknown boundary values are characterized in terms of the initial datum and the given boundary conditions via the analysis of the so-called global relation. In general, this analysis involves the solution of certain nonlinear equations. In certain cases, called linearizable, it is possible to bypass this nonlinear step. In these cases, the UTM solves the given initial-boundary value problem with the same level of efficiency as the well-known inverse scattering transform solves the initial value problem on the infinite line. We show here that the initial-boundary value problem on a finite interval with $x$-periodic boundary conditions (which can alternatively be viewed as the initial value problem on a circle), belongs to the linearizable class. Indeed, by employing certain transformations of the associated RH problem and by using the global relation, the relevant jump matrices can be expressed explicitly in terms of the so-called scattering data, which are computed in terms of the initial datum. Details are given for NLS, but similar considerations are valid for other well-known integrable evolution equations, including the Korteweg-de Vries (KdV) and modified KdV equations., Comment: 14 pages, includes announcements of results in arXiv:2103.04728

We have obtained graph-theoretically based topological indices for the characterization of certain graph theoretical networks of biochemical interest. We have derived certain distance, degree and eccentricity based topological indices for various k-level hypertrees and corona product of hypertrees. We have also pointed out errors in a previous study. The validity of our results is supported by computer codes for the respective indices. Several biochemical applications are pointed out.

In an attempt to understand the soliton resolution conjecture, we consider the Sine-Gordon equation on a spherically symmetric wormhole spacetime. We show that within each topological sector (indexed by a positive integer degree $n$) there exists a unique linearly stable soliton, which we call the $n$-kink. We give numerical evidence that the $n$-kink is a global attractor in the evolution of any smooth, finite energy solutions of degree $n$. When the radius of the wormhole throat $a$ is large enough, the convergence to the $n$-kink is shown to be governed by internal modes that slowly decay due to the resonant transfer of energy to radiation. We compute the exact asymptotics of this relaxation process for the $1$-kink using the Soffer-Weinstein weakly nonlinear perturbation theory., Comment: 19 pages, 10 figures, final version

We propose a model for the transmission of perturbations across the amino acids of a protein represented as an interaction network. The dynamics consists of a Susceptible-Infected (SI) model based on the Caputo fractional-order derivative. We find an upper bound to the analytical solution of this model which represents the worse-case scenario on the propagation of perturbations across a protein residue network. This upper bound is expressed in terms of Mittag-Leffler functions of the adjacency matrix of the network of inter-amino acids interactions. We then apply this model to the analysis of the propagation of perturbations produced by inhibitors of the main protease of SARS CoV-2. We find that the perturbations produced by strong inhibitors of the protease are propagated far away from the binding site, confirming the long-range nature of intra-protein communication. On the contrary, the weakest inhibitors only transmit their perturbations across a close environment around the binding site. These findings may help to the design of drug candidates against this new coronavirus., 21 pages, 2 figures

There are claims in the literature on Clifford algebras that every Lie group can be represented as a spin group. In a letter, Pertti Lounesto emphasized, by explicit counter-examples, that this statement is false. This self-contained short paper intends to present a survey of publications of well-known scientists where explicitly developed counter-examples prove the importance of Lounesto’s letter.

In 1985, Odoni showed that in characteristic 0 0 the Galois group of the n n -th iterate of the generic polynomial with degree d d is as large as possible. That is, he showed that this Galois group is the n n -th wreath power of the symmetric group S d S_d . We generalize this result to positive characteristic, as well as to the generic rational function. These results can be applied to prove certain density results in number theory, two of which are presented here. This work was partially completed by the late R.W.K. Odoni in an unpublished paper.

International audience; In this short paper we prove that the almost Schur theorem, introduced by De Lellis and Topping, is true on 4-dimensional Riemannian manifolds of nonnegative scalar curvature and discuss some related problems on other dimensional manifolds.

In this short note we show that non-negative Ricci curvature is not preserved under Ricci flow for closed manifolds of dimensions four and above, strengthening a previous result of Knopf in \cite{K} for complete non-compact manifolds of bounded curvature. This brings down to four dimensions a similar result B\"ohm and Wilking have for dimensions twelve and above, \cite{BW}. Moreover, the manifolds constructed here are \Kahler manifolds and relate to a question raised by Xiuxiong Chen in \cite{XC}, \cite{XCL}., Comment: New version with added references and corrected typos

In this paper we consider a spatial discretization scheme with an adaptive grid for the Nagumo PDE. In particular, we consider a commonly used time dependent moving mesh method that aims to equidistribute the arclength of the solution under consideration. We assume that the discrete analogue of this equidistribution is strictly enforced, which allows us to reduce the effective dynamics to a scalar non-local problem with infinite range interactions. We show that this reduced problem is well-posed and obtain useful estimates on the resulting nonlinearities. In the sequel papers (Hupkes and Van Vleck in Travelling waves for adaptive grid discretizations of reaction diffusion systems II: linear theory; Travelling waves for adaptive grid discretizations of reaction diffusion systems III: nonlinear theory) we use these estimates to show that travelling waves persist under these adaptive spatial discretizations.

In this paper, we introduce the two-sided fractional quaternion Fourier transform (FrQFT) and give some properties of it. The main results of this paper are divided into three parts. Firstly we give a definition of the FrQFT. Secondly based on properties of the two-sided QFT, we study the relationship between the two-sided QFT and the two-sided FrQFT, and give some differential properties of the two-sided FrQFT and the Parseval identity. Finally, we give an example to illustrate the application of the two-sided FrQFT and its inverse transform in solving partial differential equations.

This paper is a survey of results on countable families with natural degree spectra. These results were obtained by a modification of the methodology proposed by Wechner, who first found a family of sets with the spectrum consisting precisely of nonzero Turing degrees. Based on this method, many researchers obtained examples of families with other natural spectra. In addition, in this paper we extend these results and present new examples of natural spectra. In particular, we construct a family of finite sets with the spectrum consisting of exactly non-K-trivial degrees and also we find new sufficient conditions on $$ {\Delta}_2^0 $$ -degree a, which guarantees that the class {x : x ≰ a} is the degree spectrum of some family. Finally, we give a survey of our recent results on the degree spectra of α-families, where α is an arbitrary computable ordinal.

In this paper, we consider boundary-value problems for a new class of hybrid systems of differential equations whose coefficients contain the Dirac delta-function. Hybrid systems are systems that contain both ordinary and partial differential equations; such systems appear, for example, when equations of motion of mechanical systems of rigid bodies attached to a beam by elastic bonds are derived from the Hamilton–Ostrogradsky variational principle. We present examples that lead to such systems and introduce the notions of generalized solutions and eigenvalues of a boundary-value problem. We also compare results of numerical simulations based on methods proposed in this paper with results obtained by previously known methods and show that our approach is reliable and universal.

Perfect matchings in chemical graphs correspond to Kekule structures in underlying chemical structure, which play an important role in the analysis of resonance energy and stability of certain chemical compounds. In 2007, Vukicevic and Trinajstic introduced the concept of the anti-Kekule number to be the smallest number of edges that have to be removed from a graph such that it remains connected but without any perfect matching. Thereafter, this parameter attracted much more attention of researchers both in mathematics and theoretical chemistry. In the present paper, we report on the results related to the anti-Kekule of graphs. Some new results on the anti-Kekule number of nanotubes, polyomino system and cactus chains are also proven. We conclude the paper with some open problems.

In this paper we compute the rates of convergence of power series statistical convergence of sequences of positive linear operators. We also investigate some Korovkin type approximation properties of the $$q$$ -Meyer–Konig and Zeller operators and Durrmeyer variant of the $$q$$ -Meyer–Konig and Zeller operators via power series statistical convergence. We show that the approximation results obtained in this paper expand some previous approximation results of the corresponding operators.

A dynamical system $$ {\displaystyle \begin{array}{ll}{u}_{tt}-\Delta u-\nabla 1\mathrm{n}\;\rho \cdot \nabla u=0& in\kern0.6em {\mathrm{\mathbb{R}}}_{+}^3\times \left(0,T\right),\\ {}{\left.u\right|}_{t=0}={\left.{u}_t\right|}_{t=0}=0& in\kern0.6em \overline{{\mathrm{\mathbb{R}}}_{+}^3},\\ {}{\left.{u}_z\right|}_{z=0}=f& for\kern0.36em 0\le t\le T,\end{array}} $$ is under consideration, where ρ = ρ(x, y, z) is a smooth positive function; f = f(x, y, t) is a boundary control; u = uf (x, y, z, t) is a solution. With the system one associates a response operator R : f ↦ uf|z = 0. The inverse problem is to recover the function ρ via the response operator. A short representation of the local version of the BC-method, which recovers ρ via the data given on a part of the boundary, is provided. If ρ is constant, the forward problem is solved in explicit form. In the paper, the corresponding representations for the solutions and response operator are derived. A way to use them for testing the BC-algorithm, which solves the inverse problem, is outlined. The goal of the paper is to extend the circle of the BC-method users, who are interested in numerical realization of methods for solving inverse problems.

This paper proposes a new stochastic transport model formalized in differential equations with random parameters. Explicit formulas for the mathematical expectation and the second-moment function for solving the corresponding equations are given. The estimation of the influence of random factors on the system, in the case of replacing the random coefficient of the equation by its mathematical expectation, is determined. An example with the Gaussian distribution of the random coefficients is also presented and discusser. This paper shows that the proposed model is applicable to the description of heat and moisture transfer processes in the surface layer of the atmosphere.

The Monte Carlo method to solve the Cauchy problem for large systems of linear differential equations is proposed in this paper. Firstly, a quick overview of previously obtained results from applying the approach towards the Fredholm-type integral equations is made. In the main part of the paper, the method is applied towards a linear ODE system that is transformed into an equivalent system of the Volterra-type integral equations, which makes it possible to remove the limitations due to the conditions of convergence of the majorant series. The following key theorems are stated. Theorem 1 provides the necessary compliance conditions that should be imposed upon the transition propability and initial distribution densities that initiate the corresponding Markov chain, for which equality between the mathematical expectation of the estimate and the functional of interest would hold. Theorem 2 formulates the equation that governs the estimate’s variance. Theorem 3 states the Markov chain parameters that minimize the variance of the estimate of the functional. Proofs are given for all three theorems. In the practical part of this paper, the proposed method is used to solve a linear ODE system that describes a closed queueing system of ten conventional machines and seven conventional service persons. The solutions are obtained for systems with both constant and time-dependent matrices of coefficients, where the machine breakdown intensity is time dependent. In addition, the solutions obtained by the Monte Carlo and Runge–Kutta methods are compared. The results are presented in the corresponding tables.

In this paper, we obtain sufficient conditions for the existence and uniqueness results of the pantograph fractional differential equations (FDEs) with nonlocal conditions involving Atangana–Baleanu–Caputo (ABC) derivative operator with fractional orders. Our approach is based on the reduction of FDEs to fractional integral equations and on some fixed point theorems such as Banach’s contraction principle and the fixed point theorem of Krasnoselskii. Further, Gronwall’s inequality in the frame of the Atangana–Baleanu fractional integral operator is applied to develop adequate results for different kinds of Ulam–Hyers stabilities. Lastly, the paper includes an example to substantiate the validity of the results.

In this paper, we consider two classes of boundary value problems for nonlinear implicit differential equations with nonlinear integral conditions involving Atangana–Baleanu–Caputo fractional derivatives of orders$00<ϑ≤1and$11<ϑ≤2. We structure the equivalent fractional integral equations of the proposed problems. Further, the existence and uniqueness theorems are proved with the aid of fixed point theorems of Krasnoselskii and Banach. Lastly, the paper includes pertinent examples to justify the validity of the results.

In this paper, we consider the operator sheaf $$ -\Delta +V+\varepsilon {\mathrm{\mathcal{L}}}_{\varepsilon}\left(\lambda \right)+{\lambda}^2 $$ in the space L2(ℝ2), where the real-valued potential V depends only on the first variable x1, e is a small positive parameter, λ is the spectral parameter, $$ {\mathrm{\mathcal{L}}}_{\varepsilon}\left(\lambda \right) $$ is a localized operator bounded with respect to the Laplacian −Δ, and the essential spectrum of this operator is independent of e and contains certain critical points defined as isolated eigenvalues of the operator $$ -\frac{d^2}{dx_1^2}+V\left({x}_1\right) $$ in L2(ℝ). The basic result obtained in this paper states that for small values of e, in neighborhoods of critical points mentioned, isolated eigenvalues of the sheaf considered arise. Sufficient conditions for the existence or absence of such eigenvalues are obtained. The number of arising eigenvalues is determined, and in the case where they exist, the first terms of their asymptotic expansions are found.

Variants of the Newton method are very popular for solving unconstrained optimization problems. The study on global convergence of the BFGS method has also made good progress. The q-gradient reduces to its classical version when q approaches 1. In this paper, we propose a quantum-Broyden–Fletcher–Goldfarb–Shanno algorithm where the Hessian is constructed using the q-gradient and descent direction is found at each iteration. The algorithm presented in this paper is implemented by applying the independent parameter q in the Armijo–Wolfe conditions to compute the step length which guarantees that the objective function value decreases. The global convergence is established without the convexity assumption on the objective function. Further, the proposed method is verified by the numerical test problems and the results are depicted through the performance profiles.

In the spread of infectious diseases, media reports have played a positive role. However, in the process of game communication, there are not only positive but also negative media reports. Therefore, in this paper, we establish a model with positive and negative media reports to analyze the role of media reports in the process of game communication. First, we study its positivity and boundedness, and calculate the basic reproduction number $$R_{0}$$ under three different incidence rates. The existence and stability of the equilibria are proved. Secondly, the optimal control problem is studied by adding two dynamic variables of media reports. Finally, in the simulation we simulate the stability of the equilibria, so as to verify the correctness of the theory. Then the influence of media parameters on $$R_{0}$$ is analyzed. The numerical results of optimal control are simulated by forward and backward sweep method. By comparing the results of the optimal control and without control, the media coverage should be controlled according to the optimal control measures shown in this paper. The serious situation of game addiction will be greatly alleviated, which can reduce the infection rate by at least 90%.

In this paper, a stochastic susceptible-infective-recovered (SIRS) epidemic model with vaccination, nonlinear incidence and white noises under regime switching and Lévy jumps is investigated. A new threshold value is determined. Some basic assumptions with regard to nonlinear incidence, white noises, Markov switching and Lévy jumps are introduced. The threshold conditions to guarantee the extinction and permanence in the mean of the disease with probability one and the existence of unique ergodic stationary distribution for the model are established. Some new techniques to deal with the Markov switching, Lévy jumps, nonlinear incidence and vaccination for the stochastic epidemic models are proposed. Lastly, the numerical simulations not only illustrate the main results given in this paper, but also suggest some interesting open problems.

In this paper, we investigate a mixed fractional integral boundary value problem with p(t)-Laplacian operator. Firstly, we derive the Green function through the direct computation and obtain the properties of Green function. For $$p(t)\ne $$ constant, under the appropriate conditions of the nonlinear term, we establish the existence result of at least one positive solution of the above problem by means of the Leray–Schauder fixed point theorem. Meanwhile, we also obtain the positive extremal solutions and iterative schemes in view of applying a monotone iterative method. For $$p(t)=$$ constant, by using Guo–Krasnoselskii fixed point theorem, we study the existence of positive solutions of the above problem. These results enrich the ones in the existing literatures. Finally, some examples are included to demonstrate our main results in this paper and we give out an open problem.

In this paper, a new approach to the definition of the interpolating rational process of Fejer – Hermite with first-type Chebyshev – Markov nodes on a segment is studied and some of its approximating properties are described. In the introduction a brief analysis of the results on the topic of the research is carried out. Herein, the methods of the construction of interpolating processes, in particular, Fejer – Hermite processes, in the polynomial and rational approximation are analysed. A new method to determine the interpolating rational Fejer – Hermite process is proposed. One of the main results of this paper is the proof of the uniform convergence of this process for an arbitrary function, which is continuous on the segment, under some restrictions for the poles of approximating functions. This result is preceded by some auxiliary statements describing the properties of special rational functions. The classic methods of mathematical analysis, approximation theory, and theory of functions of a complex variable are used to prove the results of the work. Moreover, we present the numerical analysis of the effectiveness of the application of the constructed interpolating Fejer – Hermite process for the approximation of a continuous function with singularities. The choice of parameters, on which the nodes of interpolation depend, is made in several standard ways. The obtained results can be applied to further study the approximating properties of interpolating processes.

A goal of the paper is to prove new inequalities connecting some functionals of probability distribution functions. These inequalities are based on the strict convexity of functions used in the definition of the functionals. The starting point is the paper “Cramer–von Mises distance: probabilistic interpretation, confidence intervals and neighborhood of model validation” by Ludwig Baringhaus and Norbert Henze. The present paper provides a generalization of inequality obtained in probabilistic interpretation of the Cramer–von Mises distance. If the equality holds there, then a chance to give characterization of some probability distribution functions appears. Considering this fact and a special character of the functional, it is possible to create a class of free-of-distribution two sample tests.

In this paper, we prove the integrability of certain classes of dynamical systems on the tangent bundles of four-dimensional manifolds (systems with four degrees of freedom). The force field considered possessed so-called variable dissipation; they are generalizations of fields studied earlier. This paper continues earlier works of the author devoted to systems on the tangent bundles of two- and three-dimensional manifolds.

Since the second half of the twentieth century, wide studies of Sobolev-type equations are undertaken. These equations contain items that are derivatives with respect to time of the second order derivatives of the unknown function with respect to space variables. They can describe nonstationary processes in semiconductors, in plasm, phenomena in hydrodinamics and other ones. Notice that wide studies of qualitative properties of solutions of Sobolev-type equations exist. Namely, results about existence and uniqueness of solutions, their asymptotics and blow-up are known. But there are few results about exact solutions of Sobolev-type equations. There are books and papers about exact solutions of partial equations, but they are devoted mainly to classical equations, where the first or second order derivative with respect to time or the derivative with respect to time of the first order derivative of the unknown function with respect to the space variable is equal to a stationary expression. Therefore it is interesting to study exact solutions of Sobolev-type equations. In the present paper, a fourth order nonlinear partial equation is studied. Three classes of its exact solutions are built. They are expressed in terms of special functions (solutions of some ordinary differential equations). For two of these classes subsets that can be expressed in elementary functions are built, for the third one subsets that can be described in elementary functions and an implicit function (without a quadrature) are built.

Typically, a researcher faces computational difficulties in assessing the parameters of a certain model when modeling the relationship between different indicators of analysis. A suitable model is generally obtained by a sequential refinement of the features included in its composition and, therefore, by performing multiple repetitions of computational algorithms. At the same time, the computational complexity of these algorithms begins to play a significant role in modeling. A certain set of indicators is used to reduce the number of iterations. These indicators are responsible for the quality of the constructed model and capable of “signaling" about the need to adjust the model. The model parameters and the quality functional value are such indicators in regression modeling. They are able to answer the question of the appropriateness of building a particular model and are indicators of the quality of the resulting functional dependence. In this paper, we study methods and algorithms for constructing and evaluating the main indicators of L∞-regression — a quality indicator and model parameters. The first part of the paper describes the most efficient computational procedures for determining parameters in the case of a three-dimensional uniform regression model, indicates the complexity of these algorithms, and gives a geometric interpretation. In the second part of the paper, a series of theorems on estimating the values of the parameters of threedimensional L∞-regression is presented, and a formula for calculation of an indicator of sample uniformity is provided.

We show that the phase tropical pair-of-pants is (ambient) isotopic to the complex pair-of-pants. This paper can serve as an addendum to the author's joint paper with Ruddat arXiv:2001.08267 where an isotopy between complex and ober-tropical pairs-of-pants was shown. Thus all three versions are isotopic., 10 pages, 8 figures. arXiv admin note: text overlap with arXiv:2001.08267

The work is devoted to the fundamental problem of studying the solvability of the initial-boundary value problem for a pseudo-parabolic equation (also called Sobolev type equations) with a fairly smooth boundary. In this paper, the initial-boundary value problem for a quasilinear equation of a pseudoparabolic type with a nonlinear Neumann–Dirichlet boundary condition is studied. From a physical point of view, the initial-boundary-value problem we are considering is a mathematical model of quasi-stationary processes in semiconductors and magnetics, taking into account the most diverse physical factors. Many approximate methods are suitable for finding eigenvalues and eigenfunctions of tasks boundary conditions of which are linear with respect to the function and its derivatives. Among these methods, Galerkin’s method leads to the simplest calculations. In the paper, by means of the Galerkin method the existence of a weak solution of a pseudoparabolic equation in a bounded domain is proved. The use of the Galerkin approximations allows us to get an estimate above the time of the solution existence. Using Sobolev ’s attachment theorem, a priori solution estimates are obtained. The local theorem of the existence of the solution has been proved. The uniqueness of the weak generalized solution of the initial-boundary value problem of quasi-linear equations of pseudoparabolic type is proved on the basis of a priori estimates.A special place in the theory of nonlinear equations is taken by the range of studies of unlimited solutions, or, as they are otherwise called, modes with exacerbation. Nonlinear evolutionary problems that allow unlimited solutions are globally intractable: solutions increase indefinitely over a finite period of time. Sufficient conditions have been obtained for the destruction of its solution over finite time in a limited area with a nonlinear Neumann–Dirichle boundary condition.

In Enochs' relative homological dimension theory occur the so called (co)resolvent and (co)proper dimensions which are defined using proper and coproper resolutions constructed by precovers and preenvelopes, respectively. Recently, some authors have been interested in relative homological dimensions defined by just exact sequences. In this paper, we contribute to the investigation of these relative homological dimensions. We first study the relation between these two kinds of relative homological dimensions and establish some "transfer results" under adjoint pairs. Then, relative global dimensions are studied which lead to nice characterizations of some properties of particular cases of self-orthogonal subcategories. At the end of the paper, relative derived functors are studied and generalizations of some known results of balance for relative homology are established., To appear in a Special Issue of Algebra Colloquium; proceedings of The International Conference on Algebra and Related Topics (ICART 2018)

Purpose The purpose of this paper is to propose an efficient computational technique, which uses Haar wavelets collocation approach coupled with the Newton-Raphson method and solves the following class of system of Lane–Emden equations: −(tk1y′(t))′=t−ω1f1(t,y(t),z(t)), −(tk2z′(t))′=t−ω2f2(t,y(t),z(t)),where t > 0, subject to the following initial values, boundary values and four-point boundary values: y(0)=γ1, y′(0)=0, z(0)=γ2, z′(0)=0, y′(0)=0, y(1)=δ1, z′(0)=0, z(1)=δ2, y(0)=0, y(1)=n1z(v1), z(0)=0, z(1)=n2y(v2),where n1,n2,v1,v2∈(0,1) and k1≥0, k2≥0, ω1<1, ω2<1, γ1, γ2, δ1, δ2 are real constants. Design/methodology/approach To deal with singularity, Haar wavelets are used, and to deal with the nonlinear system of equations that arise during computation, the Newton-Raphson method is used. The convergence of these methods is also established and the results are compared with existing techniques. Findings The authors propose three methods based on uniform Haar wavelets approximation coupled with the Newton-Raphson method. The authors obtain quadratic convergence for the Haar wavelets collocation method. Test problems are solved to validate various computational aspects of the Haar wavelets approach. The authors observe that with only a few spatial divisions the authors can obtain highly accurate solutions for both initial value problems and boundary value problems. Originality/value The results presented in this paper do not exist in the literature. The system of nonlinear singular differential equations is not easy to handle as they are singular, as well as nonlinear. To the best of the knowledge, these are the first results for a system of nonlinear singular differential equations, by using the Haar wavelets collocation approach coupled with the Newton-Raphson method. The results developed in this paper can be used to solve problems arising in different branches of science and engineering.

Let H2(ⅅ2) be the Hardy space over the bidisk ⅅ2, and let Mψ,ϕ = [(ψ(z) − ϕ(w))2] be the submodule generated by (ψ(z) − ϕ(w))2, where ψ(z) and ϕ(w) are nonconstant inner functions. The related quotient module is denoted by Nψ,ϕ = H2(ⅅ2) ⊖ Mψ,ϕ. In this paper, we give a complete characterization for the essential normality of Nψ,ϕ. In particular, if ψ(z)= z, we simply write Mψ,ϕ and Nψ,ϕ as Mϕ and Nϕ respectively. This paper also studies compactness of evaluation operators L(0)∣nϕ and R(0)ϕnϕ, essential spectrum of compression operator Sz on Nϕ, essential normality of compression operators Sz and Sw on Nϕ.

This paper presents a novel and generalised mathematical formulation for mathematical and theoretical chemistry. The formulation presented employs network theoretic methods and standard Conformal Geometric Algebra (Conformal Clifford Algebra) with a unique algebraic extension that introduces special non-integer basis vectors, known as ‘shades’. The methodology developed lays preliminary foundations for both generic and more specialised characterisations and modelling of chemical systems by computing a multi-vector function, termed the ‘hyperfield’ (an algebra-geometric characterisation of a many-agent chemical system). In this paper, the formulation has been applied to inorganic compounds, in the context of their emergent properties (such as boiling and melting points) and intra-molecular characteristics. Case studies have been presented for the calculation of melting point, boiling point, maximum solubility in water and equilibrium bond length of 21 diatomic molecules. This work finds that a network theoretic model of many-agent systems modelled in $${\mathbf{R}}^{\mathrm{4,1}}$$ space, using an algebraic extension of non-integer basis vectors, presents a promising and unifying method for theoretical mathematical chemistry.

A very short proof of an unpublished result due to O. Gabber is given. More exactly, let R be a regular local ring containing a finite field k. Let G be a simply-connected reductive group scheme over k. It is proved that a principal G-bundle over R is trivial if it is trivial over the fraction field of R. This is the mentioned unpublished result due to O. Gabber. In this paper, this result is derived from a purely geometric one, proved in another paper of the author and stated in the Introduction.

In a joint paper of the author with Alexei Stepanov, it was established that for any two comaximal ideals A and B of a commutative ring R, A + B = R, and any n ≥ 3 one has [E(n,R,A),E(n,R,B)] = E(n,R,AB). Alec Mason and Wilson Stothers constructed counterexamples demonstrating that the above equality may fail when A and B are not comaximal, even for such nice rings as ℤ [i]. The present note proves a rather striking result that the above equality and, consequently, also the stronger equality [GL(n,R,A), GL(n,R,B)] = E(n,R,AB) hold whenever R is a Dedekind ring of arithmetic type with infinite multiplicative group. The proof is based on elementary calculations in the spirit of the previous papers by Wilberd van der Kallen, Roozbeh Hazrat, Zuhong Zhang, Alexei Stepanov, and the author, and also on an explicit computation of the multirelative SK1 from the author’s paper of 1982, which, in its turn, relied on very deep arithmetical results by Jean-Pierre Serre and Leonid Vaserstein (as corrected by Armin Leutbecher and Bernhard Liehl). Bibliography: 50 titles.

The classical Morse–Sard theorem claims that for a mapping v : R n → R m + 1 of class C k the measure of critical values v ( Z v , m ) is zero under condition k ≥ n − m . Here the critical set, or m-critical set is defined as Z v , m = { x ∈ R n : rank ∇ v ( x ) ≤ m } . Further Dubovitskiĭ in 1957 and independently Federer and Dubovitskiĭ in 1967 found some elegant extensions of this theorem to the case of other (e.g., lower) smoothness assumptions. They also established the sharpness of their results within the C k category. Here we formulate and prove a bridge theorem that includes all the above results as particular cases: namely, if a function v : R n → R d belongs to the Holder class C k , α , 0 ≤ α ≤ 1 , then for every q > m the identity H μ ( Z v , m ∩ v − 1 ( y ) ) = 0 holds for H q -almost all y ∈ R d , where μ = n − m − ( k + α ) ( q − m ) . Intuitively, the sense of this bridge theorem is very close to Heisenberg's uncertainty principle in theoretical physics: the more precise is the information we receive on measure of the image of the critical set, the less precisely the preimages are described, and vice versa. The result is new even for the classical C k -case (when α = 0 ); similar result is established for the Sobolev classes of mappings W p k ( R n , R d ) with minimal integrability assumptions p = max ⁡ ( 1 , n / k ) , i.e., it guarantees in general only the continuity (not everywhere differentiability) of a mapping. However, using some N-properties for Sobolev mappings, established in our previous paper, we obtained that the sets of nondifferentiability points of Sobolev mappings are fortunately negligible in the above bridge theorem. We cover also the case of fractional Sobolev spaces. The proofs of the most results are based on our previous joint papers with J. Bourgain and J. Kristensen (2013, 2015). We also crucially use very deep Y. Yomdin's entropy estimates of near critical values for polynomials (based on algebraic geometry tools).

A regular system is a Delone set in Euclidean space with a transitive group of symmetries or, in other words, the orbit of a crystallographic group. The local theory for regular systems, created by the geometric school of B. N. Delone, was aimed, in particular, to rigorously establish the “local-global-order” link, i.e., the link between the arrangement of a set around each of its points and symmetry/regularity of the set as a whole. The main result of this paper is a proof of the so-called 10R-theorem. This theorem asserts that identity of neighborhoods within a radius 10R of all points of a Delone set (in other words, an (r, R)-system) in 3D Euclidean space implies regularity of this set. The result was obtained and announced long ago independently by M. Shtogrin and the author of this paper. However, a detailed proof remains unpublished for many years. In this paper, we give a proof of the 10R-theorem. In the proof, we use some recent results of the author, which simplify the proof.

The aim of this (mostly expository) article is to show a connection between the finiteness conditions arising in twisted K-theory. There are two different conditions arising naturally in two main approaches to the problem of computing the index of the appropriate family of elliptic operators (the approach of Nistor and Troitsky and the approach of Mathai, Melrose, and Singer). These conditions are formulated absolutely differently, but in some sense they should be close to each other. In this paper, we find this connection and prove the corresponding formal statement. Thereby it is shown that these conditions map to each other. This opens a possibility to synthesize these approaches. It is also shown that the finiteness condition arising in the paper of Nistor and Troitsky is a special case of the finiteness condition that appears in the paper of Emerson and Meyer, where the theorem of Nistor and Troitsky is proved not only for the case of a bundle of Lie groups, but also for the case of a general groupoid.

In the classical theory of solid mechanics, the behavior of solids is described by partial differential equations (PDE) through Newton’s second law of motion. However, when spontaneous cracks and fractures exist, such PDE models are inadequate to characterize the discontinuities of physical quantities such as the displacement field. Recently, a peridynamic continuum model was proposed which only involves the integration over the differences of the displacement field. A linearized peridynamic model can be described by the integro-differential equation with initial values. In this paper, we study the well-posedness and regularity of a linearized peridynamic model with singular kernel. The novelty of the paper is that the singular kernel is represented as the Laplacian of a regular function. This let to convert the model to an operator valued Volterra integral equation. Then the existence and regularity of the solution of the peridynamics problem are established through the study of the Volterra integral equation.

In this paper, we suggest a new type of tasks for control theory for stochastic dynamical systems — programmed control with Probability 1 (PCP1). PCP1 is an application of an invariant theory. We use the PCP1 concept for dynamical processes described by a system of Ito differential equations with jump-diffusion (GSDES). The considered equations include the drift, the diffusion, and the jumps, together or not. Features of our approach are both a wide set of dynamical systems and investigation of such systems for their unique trajectories. Our method is based on the concept of a stochastic first integral (SFI) for GSDES and its equations which author studied before. The purpose of the present paper is to construct a differential equation system (both stochastic and deterministic) using a known set of FIs for the investigating process. Several examples are given.

This paper is concerned with a reaction-diffusion heroin model in a bound domain. The objective of this paper is to explore the threshold dynamics based on threshold parameter and basic reproduction number (BRN) ℜ0, and it is proved that if ℜ0<1, heroin spread will be extinct, while if ℜ0>1, heroin spread is uniformly persistent and there exists a positive heroin-spread steady state. We also obtain that the explicit formula of ℜ0 and global attractiveness of constant positive steady state (PSS) when all parameters are positive constants. Our simulation results reveal that compared to the homogeneous setting, the spatial heterogeneity has essential impacts on increasing the risk of heroin spread.

This paper is a contribution to frame theory. Frames in a Hilbert space are generalizations of orthonormal bases. In particular, Gabor frames of $L^2(\mathbb{R})$, which are made of translations and modulations of one or more windows, are often used in applications. More precisely, the paper deals with a question posed in the last years by Christensen and Hasannasab about the existence of overcomplete Gabor frames, with some ordering over $\mathbb{Z}$, which are orbits of bounded operators on $L^2(\mathbb{R})$. Two classes of overcomplete Gabor frames which cannot be ordered over $\mathbb{Z}$ and represented by orbits of operators in $GL(L^2(\mathbb{R}))$ are given. Some results about operator representation are stated in a general context for arbitrary frames, covering also certain wavelet frames., Comment: 12 pages

Let $\unicode[STIX]{x1D703}$ be an irrational real number. The map $T_{\unicode[STIX]{x1D703}}:y\mapsto (y+\unicode[STIX]{x1D703})\!\hspace{0.6em}{\rm mod}\hspace{0.2em}1$ from the unit interval $\mathbf{I}= [\!0,1\![$ (endowed with the Lebesgue measure) to itself is ergodic. In a short paper [Parry, Automorphisms of the Bernoulli endomorphism and a class of skew-products. Ergod. Th. & Dynam. Sys.16 (1996), 519–529] published in 1996, Parry provided an explicit isomorphism between the measure-preserving map $[T_{\unicode[STIX]{x1D703}},\text{Id}]$ and the unilateral dyadic Bernoulli shift when $\unicode[STIX]{x1D703}$ is extremely well approximated by the rational numbers, namely, if $$\begin{eqnarray}\inf _{q\geq 1}q^{4}4^{q^{2}}~\text{dist}(\unicode[STIX]{x1D703},q^{-1}\mathbb{Z})=0.\end{eqnarray}$$ A few years later, Hoffman and Rudolph [Uniform endomorphisms which are isomorphic to a Bernoulli shift. Ann. of Math. (2)156 (2002), 79–101] showed that for every irrational number, the measure-preserving map $[T_{\unicode[STIX]{x1D703}},\text{Id}]$ is isomorphic to the unilateral dyadic Bernoulli shift. Their proof is not constructive. In the present paper, we relax notably Parry’s condition on $\unicode[STIX]{x1D703}$: the explicit map provided by Parry’s method is an isomorphism between the map $[T_{\unicode[STIX]{x1D703}},\text{Id}]$ and the unilateral dyadic Bernoulli shift whenever $$\begin{eqnarray}\inf _{q\geq 1}q^{4}~\text{dist}(\unicode[STIX]{x1D703},q^{-1}\mathbb{Z})=0.\end{eqnarray}$$ This condition can be relaxed again into $$\begin{eqnarray}\inf _{n\geq 1}q_{n}^{3}~(a_{1}+\cdots +a_{n})~|q_{n}\unicode[STIX]{x1D703}-p_{n}| where $[0;a_{1},a_{2},\ldots ]$ is the continued fraction expansion and $(p_{n}/q_{n})_{n\geq 0}$ the sequence of convergents of $\Vert \unicode[STIX]{x1D703}\Vert :=\text{dist}(\unicode[STIX]{x1D703},\mathbb{Z})$. Whether Parry’s map is an isomorphism for every $\unicode[STIX]{x1D703}$ or not is still an open question, although we expect a positive answer.