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A classical result of K. L. Chung and W. Feller deals with the partial sums S k S_k arising in a fair coin-tossing game. If N n N_n is the number of “positive” terms among S 1 S_1 , S 2 S_2 , …, S n S_n then the quantity P ( N 2 n = 2 r ) P(N_{2n} = 2r) takes an elegant form. We lift the restriction on an even number of tosses and give a simple expression for P ( N 2 n + 1 = r ) P(N_{2n+1} = r) , r = 0 r = 0 , 1 1 , 2 2 , …, 2 n + 1 2n+1 . We get to this ansatz by adaptating the Feynman–Kac methodology.

This is a corrigendum to our paper, “On a class of prime entire functions”, published in Acta Math. Sin., Engl. Ser., 25, 1647–1652 (2009).

We introduce a graph Ramsey game called Ramsey, Paper, Scissors. This game has two players, Proposer and Decider. Starting from an empty graph on $n$ vertices, on each turn Proposer proposes a potential edge and Decider simultaneously decides (without knowing Proposer's choice) whether to add it to the graph. Proposer cannot propose an edge which would create a triangle in the graph. The game ends when Proposer has no legal moves remaining, and Proposer wins if the final graph has independence number at least $s$. We prove a threshold phenomenon exists for this game by exhibiting randomized strategies for both players that are optimal up to constants. Namely, there exist constants $0B\sqrt{n}\log{n}$. This is a factor of $\Theta(\sqrt{\log{n}})$ larger than the lower bound coming from the off-diagonal Ramsey number $r(3,s)$.

This work is a contribution to the Project UIDB/04035/2020, funded by FCT - Fundacao para a Ciencia e a Tecnologia, Portugal.

Connectivity is the safety network for biodiversity conservation because connected habitats are more effective for saving the species and ecological functions. The nature of coupling for connectivity also plays an important role in the co-existence of species in cyclic-dominance. The rock-paper-scissors game is one of the paradigmatic mathematical model in evolutionary game theory to understand the mechanism of biodiversity in cyclic-dominance. In this paper, the metapopulation model for rock-paper-scissors with mutations is presented in which the total population is divided into patches and the patches form a network of complete graph. The migration among patches is allowed through simple random walk. The replicator-mutator equations are used with the migration term. When migration is allowed then the population of the patches will synchronized and attain stable state through Hopf bifurcation. Apart form this, two phases are observed when the strategies of one of the species mutate to other two species: co-existence of all the species phase and existence of one kind of species phase. The transition from one phase to another phase is taking place due to transcritical bifurcation. The dynamics of the population of species of rock, paper, scissors is studied in the environment of homogeneous and heterogeneous mutation. Numerical simulations have been performed when mutation is allowed in all the patches (homogeneous mutation) and some of the patches (heterogeneous mutation). It has been observed that when the number of patches is increased in the case of heterogeneous mutation then the population of any of the species will not extinct and all the species will co-exist.

In this note, we make two revisions of the paper [2]. The first one is the asymptotic behavior of the energy functional as $$t\rightarrow T$$ (see [2, Theorem 1.6]), where T is the blow-up time. The second one is the equivalent conditions for the solutions blowing up in finite time or existing globally (see [2, Theorem 1.8]).

We congratulate the authors Bertsimas, Pauphilet and van Parys (hereafter BPvP) and Hastie, Tibshirani and Tibshirani (hereafter HTT) for providing fresh and insightful views on the problem of variable selection and prediction in linear models. Their contributions at the fundamental level provide guidance for more complex models and procedures.

Classic frameworks of rock-paper-scissors game have been assumed in a closed community that a density of each group is only affected by internal factors such as competition interplay among groups and reproduction itself. In real systems in ecological and social sciences, however, the survival and a change of a density of a group can be also affected by various external factors. One of common features in real population systems in ecological and social sciences is population flow that is characterized by population inflow and outflow in a group or a society, which has been usually overlooked in previous works on models of rock-paper-scissors game. In this paper, we suggest the rock-paper-scissors system by implementing population flow and investigate its effect on biodiversity. For two scenarios of either balanced or imbalanced population flow, we found that the population flow can strongly affect group diversity by exhibiting rich phenomena. In particular, while the balanced flow can only lead the persistent coexistence of all groups which accompanies a phase transition through supercritical Hopf bifurcation on different carrying simplices, the imbalanced flow strongly facilitates rich dynamics such as alternative stable survival states by exhibiting various group survival states and multistability of sole group survivals by showing not fully covered but spirally entangled basins of initial densities due to local stabilities of associated fixed points. In addition, we found that, the system can exhibit oscillatory dynamics for coexistence by relativistic interplay of population flows which can capture the robustness of the coexistence state. Applying population flow in the rock-paper-scissors system can ultimately change a community paradigm from closed to open one, and our foundation can eventually reveal that population flow can be also a significant factor on a group density which is independent to fundamental interactions among groups.

We complete a gap in the proof that exceptional polynomials are complete orthogonal systems in the associated Hilbert spaces.

The aim of this paper is to discuss the main result in the paper by D.Y. Gao and X. Lu [On the extrema of a nonconvex functional with double-well potential in 1D, Z. Angew. Math. Phys. (2016) 67:62]. More precisely we provide a detailed study of the problem considered in that paper, pointing out the importance of the norm on the space $C^{1}[a,b]$; because no norm (topology) is mentioned on $C^{1}[a,b]$ we look at it as being a subspace of $W^{1,p}(a,b)$ for $p\in [1,\infty]$ endowed with its usual norm. We show that the objective function has not local extrema with the mentioned constraints for $p\in [1,4)$, and has (up to an additive constant) only a local maximizer for $p=\infty$, unlike the conclusion of the main result of the discussed paper where it is mentioned that there are (up to additive constants) two local minimizers and a local maximizer. We also show that the same conclusions are valid for the similar problem treated in the preprint by X. Lu and D.Y. Gao [On the extrema of a nonconvex functional with double-well potential in higher dimensions, arXiv:1607.03995]., 12 pages; in this version we added the forgotten condition $F(x) \ne 0$ for $x\in (a,b)$ on page 3

In the paper (HA14), unfortunately, a computational error occurred in one estimate. Although the wrong estimate does not affect the main results, we want to present the necessary corrections. Essentially, Lemma 5.2 has to be corrected and, since it is used in the proof of Theorem 5.1, the proof of this theorem also requires an adaptation. (i) The corrected formulation of Lemma 5.2 is as follows. Lemma 5.2 For Lagrange finite elements with a shape-regular family of affine meshes { T n h } h>0 there is a constant C > 0 independent of q and h such that for all w ∈ Wh and q = 2m, m ∈N: CΛq−2 p (∇w,∇Ip h (wq−1))T ∫ T ‖∇w‖l2‖w‖ q−2 0,∞,T dx, ∀T ∈ T n h , (5.1) where Λp = ‖ ∑ndof i=1 |φi|‖0,∞,T is the Lebesgue constant.

We show that a relative entropy condition recently shown by Leger and Vasseur to imply uniqueness and stable $L^2$ dependence on initial data of Lax 1- or $n$-shock solutions of an $n\times n$ system of hyperbolic conservation laws with convex entropy implies Lopatinski stability in the sense of Majda. This means in particular that Leger and Vasseur's relative entropy condition represents a considerable improvement over the standard entropy condition of decreasing shock strength and increasing entropy along forward Hugoniot curves, which, in a recent example exhibited by Barker, Freist\"uhler and Zumbrun, was shown to fail to imply Lopatinski stability, even for systems with convex entropy. This observation bears also on the parallel question of existence, at least for small $BV$ or $H^s$ perturbations, Comment: to appear in Proceedings of the AMS

We indicate how recent results of Grafakos and Teschl (J Fourier Anal Appl 19:167–179, 2013), and Estrada (J Fourier Anal Appl 20:301–320, 2014), relating the Fourier transform of a radial function in $$\mathbb R^n$$ and the Fourier transform of the same function in $$\mathbb R^{n+2}$$ and $$\mathbb R^{n+1}$$ , respectively, are located within known results on transplantation for Hankel transforms.

In this paper, we discuss the validity of the result of Khuangsatung and Kangtunyakarn [Existence and convergence theorem for fixed point problem of various nonlinear mappings and variational inequality problems without some assumptions, Filomat 32(1) (2018) 305–309].

The abstract results and applications presented in “Controllability of fractional neutral stochastic functional differential systems, Z. Angew. Math. Phys. 65 (2014), no. 5, 941–959, are not correct. Moreover, the class of differential control problems studied in [1] is not H-controllable.

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

The Award Committee – Peter Kritzer, Johannes Kepler University Linz, Austria and Jan Vybiral, Charles University, Czech Republic – determined that the following paper exhibits exceptional merit and therefore awarded the prize to: Bernd Carl, Aicke Hinrichs, and Philipp Rudolph for their paper ‘‘Entropy numbers of convex hulls in Banach spaces and applications’’, which appeared in October, 2014. Vol. 30, pp. 555–587. The $3000 prize will be divided between the winners. Each author will also receive a plaque.

The Award Committee – Michael Gnewuch, University of Kaiserslautern, Germany and Friedrich Pillichshammer, Johannes Kepler University, Austria – determined that the following two papers exhibited exceptional merit and therefore awarded the prize to: ShuTezuka for his paper ‘‘On the discrepancy of generalizedNiederreiter sequences’’, which appeared in June–August, 2013. Vol. 29, pp. 240–247. Joos Heintz, Bart Kuijpers, and Andres Rojas Paredes for their paper ‘‘Software engineering and complexity in effective Algebraic Geometry’’, which appeared in February, 2013. Vol. 29, pp. 92–138. The $3000 prize will be divided between the winners. Each author will also receive a plaque.

The preeminent target of present study is to reveal the speed characteristic of ongoing outbreak COVID-19 due to novel coronavirus. On January 2020, the novel coronavirus infection (COVID-19) detected in India, and the total statistic of cases continuously increased to 7 128 268 cases including 109 285 deceases to October 2020, where 860 601 cases are active in India. In this study, we use the Hermite wavelets basis in order to solve the COVID-19 model with time- arbitrary Caputo derivative. The discussed framework is based upon Hermite wavelets. The operational matrix incorporated with the collocation scheme is used in order to transform arbitrary-order problem into algebraic equations. The corrector scheme is also used for solving the COVID-19 model for distinct value of arbitrary order. Also, authors have investigated the various behaviors of the arbitrary-order COVID-19 system and procured developments are matched with exiting developments by various techniques. The various illustrations of susceptible, exposed, infected, and recovered individuals are given for its behaviors at the various value of fractional order. In addition, the proposed model has been also supported by some numerical simulations and wavelet-based results.

Funder: John Simon Guggenheim Memorial Foundation; doi: https://doi.org/10.13039/100005851, Funder: Einstein Visiting Fellow Program, The Rayleigh–Bénard system with stress-free boundary conditions is shown to have a global attractor in each affine space where velocity has fixed spatial average. The physical problem is shown to be equivalent to one with periodic boundary conditions and certain symmetries. This enables a Gronwall estimate on enstrophy. That estimate is then used to bound the L 2 norm of the temperature gradient on the global attractor, which, in turn, is used to find a bounding region for the attractor in the enstrophy–palinstrophy plane. All final bounds are algebraic in the viscosity and thermal diffusivity, a significant improvement over previously established estimates. The sharpness of the bounds are tested with numerical simulations.

Novel coronavirus (COVID-19), a global threat whose source is not correctly yet known, was firstly recognised in the city of Wuhan, China, in December 2019. Now, this disease has been spread out to many countries in all over the world. In this paper, we solved a time delay fractional COVID-19 SEIR epidemic model via Caputo fractional derivatives using a predictor-corrector method. We provided numerical simulations to show the nature of the diseases for different classes. We derived existence of unique global solutions to the given time delay fractional differential equations (DFDEs) under a mild Lipschitz condition using properties of a weighted norm, Mittag-Leffler functions and the Banach fixed point theorem. For the graphical simulations, we used real numerical data based on a case study of Wuhan, China, to show the nature of the projected model with respect to time variable. We performed various plots for different values of time delay and fractional order. We observed that the proposed scheme is highly emphatic and easy to implementation for the system of DFDEs.

The purpose of this short paper is to present a new form of the so called early exercise premium for the American type derivatives. The decomposition we derived consists of the corresponding European derivative and a derivative with a stochastic maturity. In different particular cases we reach to the well known form for the American put option where the underlying asset is driven by a Brownian motion or a Levy process.

The first symptomatic infected individuals of coronavirus (Covid-19) was confirmed in December 2020 in the city of Wuhan, China. In India, the first reported case of Covid-19 was confirmed on 30 January 2020. Today, coronavirus has been spread out all over the world. In this manuscript, we studied the coronavirus epidemic model with a true data of India by using Predictor-Corrector scheme. For the proposed model of Covid-19, the numerical and graphical simulations are performed in a framework of the new generalised Caputo sense non-integer order derivative. We analysed the existence and uniqueness of solution of the given fractional model by the definition of Chebyshev norm, Banach space, Schauder's second fixed point theorem, Arzel's-Ascoli theorem, uniform boundedness, equicontinuity and Weissinger's fixed point theorem. A new analysis of the given model with the true data is given to analyse the dynamics of the model in fractional sense. Graphical simulations show the structure of the given classes of the non-linear model with respect to the time variable. We investigated that the mentioned method is copiously strong and smooth to implement on the systems of non-linear fractional differential equation systems. The stability results for the projected algorithm is also performed with the applications of some important lemmas. The present study gives the applicability of this new generalised version of Caputo type non-integer operator in mathematical epidemiology. We compared that the fractional order results are more credible to the integer order results.

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.

In this short paper, we show that there does not exist a noncompact Type-I [Formula: see text]-solution of the Ricci flow with positive curvature in dimension 3.

This paper is devoted to the study of the qualitative properties of solutions to boundary-value problems for strongly elliptic differential-difference equations. Some results for these equations such as existence and smoothness of generalized solutions in certain subdomains of Q were obtained earlier. Nevertheless, the smoothness of generalized solutions of such problems can be violated near the boundary of these subdomains even for infinitely differentiable right-hand side. The subdomains are defined as connected components of the set that is obtained from the domain Q by throwing out all possible shifts of the boundary Q by vectors of a certain group generated by shifts occurring in the difference operators. For the one dimensional Neumann problem for differential-difference equations there were obtained conditions on the coefficients of difference operators, under which for any continuous right-hand side there is a classical solution of the problem that coincides with the generalized solution. 2 Also there was obtained the smoothness (in Sobolev spaces W k ) of generalized solutions of the second and the third boundary-value problems for strongly elliptic differential-difference equations in subdomains excluding -neighborhoods of certain points. However, the smoothness (in Ho lder spaces) of generalized solutions of the second boundary-value problem for strongly elliptic differential-difference equations on the boundary of adjacent subdomains was not considered. In this paper, we study this question in Ho lder spaces. We establish necessary and sufficient conditions for the coefficients of difference operators that guarantee smoothness of the generalized solution on the boundary of adjacent subdomains for any right-hand side from the Ho lder space.

Part of the work was done while the authors were visiting the South China University of Science and Technology (SUSTech), Shenzhen, in 2018, and we are grateful (in particular to Professor Cai Heng Li) for the hospitality that we received.The authors would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme Groups, representations and applications: new perspectives (supported by EPSRC grant no.EP/R014604/1), where further work on this paper was undertaken. In particular we acknowledge a Simons Fellowship (Cameron) and a Kirk Distinguished Visiting Fellowship (Praeger) during this programme. Schneider thanks the Centre for the Mathematics of Symmetry and Computation of The University of Western Australia and Australian Research Council Discovery Grant DP160102323 for hosting his visit in 2017 and acknowledges the support of the CNPq projects Produtividade em Pesquisa (project no.: 308212/2019-3) and Universal (project no.:421624/2018-3). Diagonal groups are one of the classes of finite primitive permutation groups occurring in the conclusion of the O'Nan-Scott theorem. Several of the other classes have been described as the automorphism groups of geometric or combinatorial structures such as affine spaces or Cartesian decompositions, but such structures for diagonal groups have not been studied in general. The main purpose of this paper is to describe and characterise such structures, which we call diagonal semilattices. Unlike the diagonal groups in the O'Nan-Scott theorem, which are defined over finite characteristically simple groups, our construction works over arbitrary groups, finite or infinite. A diagonal semilattice depends on a dimension m and a group T. For m=2, it is a Latin square, the Cayley table of T, though in fact any Latin square satisfies our combinatorial axioms. However, for m≥3, the group T emerges naturally and uniquely from the axioms. (The situation somewhat resembles projective geometry, where projective planes exist in great profusion but higher-dimensional structures are coordinatised by an algebraic object, a division ring.) A diagonal semilattice is contained in the partition lattice on a set Ω, and we provide an introduction to the calculus of partitions. Many of the concepts and constructions come from experimental design in statistics. We also determine when a diagonal group can be primitive, or quasiprimitive (these conditions turn out to be equivalent for diagonal groups). Associated with the diagonal semilattice is a graph, the diagonal graph, which has the same automorphism group as the diagonal semilattice except in four small cases with m

The paper is devoted to the study of Gromov-Hausdorff convergence and stability of irreversible metric-measure spaces, both in the compact and noncompact cases. While the compact setting is mostly similar to the reversible case developed by J. Lott, K.-T. Sturm and C. Villani, the noncompact case provides various surprising phenomena. Since the reversibility of noncompact irreversible spaces might be infinite, it is motivated to introduce a suitable nondecreasing function that bounds the reversibility of larger and larger balls. By this approach, we are able to prove satisfactory convergence/stability results in a suitable – reversibility depending – Gromov-Hausdorff topology. A wide class of irreversible spaces is provided by Finsler manifolds, which serve to construct various model examples by pointing out genuine differences between the reversible and irreversible settings. We conclude the paper by proving various geometric and functional inequalities (as Brunn-Minkowski, Bishop-Gromov, log-Sobolev and Lichnerowicz inequalities) on irreversible structures.

The inconsistent or consistent general fuzzy matrix equation are studied in this paper. The aim of this paper is threefold. Firstly, general strong fuzzy matrix solutions of consistent general fuzzy matrix equation are derived, and an algorithm for obtaining general strong fuzzy solutions of general fuzzy matrix equation by Core-EP inverse is also established. Secondly, if inconsistent or consistent general fuzzy matrix equation satisfies $ X\in R(S^{k}) $, the unique solution or unique least squares solution of consistent or inconsistent general fuzzy matrix equation are given by Core-EP inverse. Thirdly, we present an algorithm for obtaining Core-EP inverse. Finally, we present some examples to illustrate the main results.

Differential-algebraic equations (DAEs) are widely used for modeling of dynamical systems. In numerical analysis of DAEs, consistent initialization and index reduction are important preprocessing prior to numerical integration. Existing DAE solvers commonly adopt structural preprocessing methods based on combinatorial optimization. Unfortunately, the structural methods fail if the DAE has numerical or symbolic cancellations. For such DAEs, methods have been proposed to modify them to other DAEs to which the structural methods are applicable, based on the combinatorial relaxation technique. Existing modification methods, however, work only for a class of DAEs that are linear or close to linear. This paper presents two new modification methods for nonlinear DAEs: the substitution method and the augmentation method. Both methods are based on the combinatorial relaxation approach and are applicable to a large class of nonlinear DAEs. The substitution method symbolically solves equations for some derivatives based on the implicit function theorem and substitutes the solution back into the system. Instead of solving equations, the augmentation method modifies DAEs by appending new variables and equations. The augmentation method has advantages that the equation solving is not needed and the sparsity of DAEs is retained. It is shown in numerical experiments that both methods, especially the augmentation method, successfully modify high-index DAEs that the DAE solver in MATLAB cannot handle., Comment: A preliminary version of this paper is to appear in Proceedings of the 44th International Symposium on Symbolic and Algebraic Computation (ISSAC 2019), Beijing, China, July 2019

The paper is devoted to coverings by translative homothets and illuminations of convex bodies. For a given positive number $\alpha$ and a convex body $B$, $g_{\alpha}(B)$ is the infimum of $\alpha$-powers of finitely many homothety coefficients less than 1 such that there is a covering of $B$ by translative homothets with these coefficients. $h_{\alpha}(B)$ is the minimal number of directions such that the boundary of $B$ can be illuminated by this number of directions except for a subset whose Hausdorff dimension is less than $\alpha$. In this paper, we prove that $g_{\alpha}(B)\leq h_{\alpha}(B)$, find upper and lower bounds for both numbers, and discuss several general conjectures. In particular, we show that $h_{\alpha} (B) > 2^{d-\alpha}$ for almost all $\alpha$ and $d$ when $B$ is the $d$-dimensional cube, thus disproving the conjecture from Research Problems in Discrete Geometry by Brass, Moser, and Pach.

Let $\Omega\subset\mathbb R^n$ be a $C^1$ domain, or more generally, a Lipschitz domain with small local Lipschitz constant. In this paper it is shown that if $u$ is a function harmonic in $\Omega$ and continuous in $\overline \Omega$ which vanishes in a relatively open subset $\Sigma\subset\partial\Omega$ and moreover the normal derivative $\partial_\nu u$ vanishes in a subset of $\Sigma$ with positive surface measure, then $u$ is identically $0$., Comment: More detailed explanation in some argument involving integration by parts and in Remark 3.3. An additional appendix with a self-contained proof of Lemma 4.3, whose proof was not included in the paper previously

The purpose of this paper is to recommend an iterative scheme to approximate a common element of the solution sets of the split problem of variational inclusions, split generalized equilibrium problem and fixed point problem for non-expansive mappings. We prove that the sequences generated by the recommended iterative scheme strongly converge to a common element of solution sets of stated split problems. In the end, we provide a numerical example to support and justify our main result. The result studied in this paper generalizes and extends some widely recognized results in this direction.

In this paper, the author focuses on the joint effects of diffusion and advection on the dynamics of a classical two species Lotka-Volterra competition-diffusion-advection system, where the ratio of diffusion and advection rates are supposed to be a positive constant. For comparison purposes, the two species are assumed to have identical competition abilities throughout this paper. The results explore the condition on the diffusion and advection rates for the stability of former species. Meanwhile, an asymptotic behavior of the stable coexistence steady states is obtained.

This paper is dedicated to L p bounds on eigenfunctions of a Schrodinger-type operator ( − Δ g ) α / 2 + V on closed Riemannian manifolds for critically singular potentials V. The operator ( − Δ g ) α / 2 is defined spectrally in terms of the eigenfunctions of − Δ g . We obtain also quasimodes and spectral clusters estimates. As an application, we derive Strichartz estimates for the fractional wave equation ( ∂ t 2 + ( − Δ g ) α / 2 + V ) u = 0 . The wave kernel techniques recently developed by Bourgain-Shao-Sogge-Yao [4] and Shao-Yao [27] play a key role in this paper. We construct a new reproducing operator with several local operators and some good error terms. Moreover, we shall prove that these local operators satisfy certain variable coefficient versions of the “uniform Sobolev estimates” by Kenig-Ruiz-Sogge [18] . These enable us to handle the critically singular potentials V and prove the quasimode estimates.

The study of the Dirichlet problem with arbitrary measurable data for harmonic functions in the unit disk 𝔻 is due to the dissertation of Luzin. Later on, the known monograph of Vekua was devoted to boundary-value problems only with Holder continuous data for generalized analytic functions, i.e., continuous complex-valued functions f(z) of the complex variable z = x + iy with generalized first partial derivatives by Sobolev satisfying equations of the form $$ {\partial}_{\overline{z}}f+ af+b\overline{f}=c, $$ where the complexvalued functions a; b, and c are assumed to belong to the class Lp with some p > 2 in smooth enough domains D in ℂ. Our last paper [12] contained theorems on the existence of nonclassical solutions of the Hilbert boundaryvalue problem with arbitrary measurable data (with respect to logarithmic capacity) for generalized analytic functions f : D → ℂ such that $$ {\partial}_{\overline{z}}f=g $$ with the real-valued sources. On this basis, the corresponding existence theorems were established for the Poincare problem on directional derivatives and, in particular, for the Neumann problem to the Poisson equations △U = G ∈ Lp; p > 2, with arbitrary measurable boundary data over logarithmic capacity. The present paper is a natural continuation of the article [12] and includes, in particular, theorems on the existence of solutions for the Hilbert boundary-value problem with arbitrary measurable data for the corresponding nonlinear equations of the Vekua type $$ {\partial}_{\overline{z}}f(z)=h(z)q\left(f(z)\right). $$ On this basis, existence theorems were also established for the Poincar´e boundary-value problem and, in particular, for the Neumann problem for the nonlinear Poisson equations of the form △U(z) = H(z)Q(U(z)) with arbitrary measurable boundary data over logarithmic capacity. The Dirichlet problem was investigated by us for the given equations, too. Our approach is based on the interpretation of boundary values in the sense of angular (along nontangential paths) limits that are a conventional tool of the geometric function theory. As consequences, we give applications to some concrete semi-linear equations of mathematical physics arising from modelling various physical processes. Those results can also be applied to semi-linear equations of mathematical physics in anisotropic and inhomogeneous media.

In their 2012 paper, Bobadilla and Koll\'ar studied topological conditions which guarantee that a proper map of complex algebraic varieties is a topological or differentiable fibration. They also asked whether a certain finiteness property on the relative covering space can imply that a proper map is a fibration. In this paper, we answer positively the integral homology version of their question in the case of abelian varieties, and the rational homology version in the case of compact ball quotients. We also propose several conjectures in relation to the Singer-Hopf conjecture in the complex projective setting., Comment: published/final version

In this paper we propose a new approach to the construction of quadrature formulas of interpolation rational type on an interval. In the introduction, a brief analysis of the results on the topic of the research is carried out. Most attention is paid to the works of mathematicians of the Belarusian school on approximation theory – Gauss, Lobatto, and Radau quadrature formulas with nodes at the zeros of the rational Chebyshev – Markov fractions. Rational fractions on the segment, generalizing the classical orthogonal Jacobi polynomials with one weight, are defined, and some of their properties are described. One of the main results of this paper consists in constructing quadrature formulas with nodes at zeros of the introduced rational fractions, calculating their coefficients in an explicit form, and estimating the remainder. This result is preceded by some auxiliary statements describing the properties of special rational functions. Classical methods of mathematical analysis, approximation theory, and the theory of functions of a complex variable are used for proof. In the conclusion a numerical analysis of the efficiency of the constructed quadrature formulas is carried out. Meanwhile, the choice of the parameters on which the nodes of the quadrature formulas depend is made in several standard ways. The obtained results can be applied for further research of rational quadrature formulas, as well as in numerical analysis.

This paper is devoted to exact (in the sense of the order of smallness) estimates of the best rational approximations of functions with derivative of generalized finite variation on a finite segment of a straight line in uniform and integral metrics. The obtained results were announced in the authors' paper in 2014. They are analogous to the results of the first author, where A. Khatamov establishes exact (in the sense of the order of smallness) estimates of the best spline approximations of functions with derivative of generalized finite variation on a finite segment of a straight line in uniform and integral metrics. Results announced by the authors in 2014 generalize those obtained by N.Sh. Zagirov in 1982, namely, exact (in the sense of the order of smallness) estimates of rational approximations of functions with generalized finite variation in the integral metric, to the best rational approximations of functions with derivative of generalized finite variation on a finite segment in uniform and integral metrics. Generally speaking, the calculation of exact (in the sense of the order of smallness) estimates for the best approximations for any class of functions in any metric is a difficult problem.

The renormalization group method has been developed to investigate p-adic q-state Potts models on the Cayley tree of order k. This method is closely related to the examination of dynamical behavior of the p-adic Potts–Bethe mapping which depends on the parameters q, k. In Mukhamedov and Khakimov [Chaotic behavior of the p-adic Potts–Behte mapping. Discrete Contin. Dyn. Syst.38 (2018), 231–245], we have considered the case when q is not divisible by p and, under some conditions, it was established that the mapping is conjugate to the full shift on $\kappa _p$ symbols (here $\kappa _p$ is the greatest common factor of k and $p-1$ ). The present paper is a continuation of the forementioned paper, but here we investigate the case when q is divisible by p and k is arbitrary. We are able to fully describe the dynamical behavior of the p-adic Potts–Bethe mapping by means of a Markov partition. Moreover, the existence of a Julia set is established, over which the mapping exhibits a chaotic behavior. We point out that a similar result is not known in the case of real numbers (with rigorous proofs).

This paper presents sufficient conditions for a substitution tiling dynamical system of $\mathbb {R}^2$ , generated by a generalized substitution on three letters, to be topologically mixing. These conditions are shown to hold on a large class of tiling substitutions originally presented by Kenyon in 1996. This problem was suggested by Boris Solomyak, and many of the techniques that are used in this paper are based on the work by Kenyon, Sadun, and Solomyak [Topological mixing for substitutions on two letters. Ergod. Th. & Dynam. Sys.25(6) (2005), 1919–1934]. They studied one-dimensional tiling dynamical systems generated by substitutions on two letters and provided similar conditions sufficient to ensure that one-dimensional substitution tiling dynamical systems are topologically mixing. If a tiling dynamical system of $\mathbb {R}^2$ satisfies our conditions (and thus is topologically mixing), we can construct additional topologically mixing tiling dynamical systems of $\mathbb {R}^2$ . By considering the stepped surface constructed from a tiling $T_\sigma $ , we can get a new tiling of $\mathbb {R}^2$ by projecting the surface orthogonally onto an irrational plane through the origin.

In this paper, uniformly most powerful unbiased test for testing the stress-strength model has been presented for the first time. The end of the paper is recommending a method which is appropriate for no large data where a normal asymptotic distribution is not applicable. The previous methods for inference on stress-strength models use almost all the asymptotic properties of maximum likelihood estimators. The distribution of components is considered exponential and generalized logistic. A corresponding unbiased confidence interval is constructed, too. We compare presented methodology with previous methods and show the method of this paper is logically better than other methods. Interesting result is that our recommended method not only uses from small sample size but also has better result than other ones.

In 1993, Shub and Smale posed the problem of finding a sequence of univariate polynomials of degree $N$ with condition number bounded above by $N$. In a previous paper by C. Belt\'an, U. Etayo, J. Marzo and J. Ortega-Cerd\`a, it was proved that the optimal value of the condition number is of the form $O(\sqrt{N})$, and the sequence demanded by Shub and Smale was described by a closed formula (for large enough $N\geqslant N_0$ with $N_0$ unknown) and by a search algorithm for the rest of the cases. In this paper we find concrete estimates for the constant hidden in the $O(\sqrt{N})$ term and we describe a simple formula for a sequence of polynomials whose condition number is at most $N$, valid for all $N=4M^2$, with $M$ a positive integer., Comment: 21 pages