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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 this study, we investigate a discrete-time version of logit dynamics, as applied to the rock–paper–scissors (RPS) game. First, we show that around the Nash equilibrium point, an attracting closed invariant curve appears due to the Neimark–Sacker bifurcation. Next, near the resonance point, we find a period-three attracting cycle, which can be thought of as a counterpart to the cyclically stable set in the RPS game with best response dynamics. Moreover, we show that the cycle can coexist with an attracting closed invariant curve, a period-three saddle cycle, and the attracting or repelling Nash equilibrium point. Finally, we use the codimension-two bifurcation theory to specify the set of heteroclinic bifurcations that destroy the coexistence of the attractors.

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

In 1903, Epstein published his proof of meromorphic continuation and a functional equation for Dirichlet series associated with quadratic forms, now called Epstein zeta-functions. However, already in 1889 (or even earlier) Hurwitz was aware of these results as his mathematical diaries and some unpublished notes (in an almost final form) found in his estate at the ETH Zurich show. In this article we present and analyze Hurwitz's notes and compare his reasoning with Epstein's paper in detail.

Two limit cycles for the continuous model of the rockscissorspaper (RSP) game is constructed.The Hopf bifurcation method is used for the construction of limit cycles.The results give a partial answer to an open question posed by Neumann and Schuster. In this paper we construct two limit cycles with a heteroclinic polycycle for the three-dimensional continuous model of the rockscissorspaper (RSP) game between bacteriocin producing bacteria. Our construction gives a partial answer to an open question posed by Neumann and Schuster (2007) concerning how many limit cycles can coexist for the RSP game.

The objective of this contribution is to improve recent stability results for a system coupling ordinary differential equations to a vectorial transport partial differential equation by proposing a new structure of Lyapunov functional. Following the same process of most of the investigations in literature, that are based on an a priori selection of Lyapunov functionals and use the usual integral inequalities (Jensen, Wirtinger, Bessel...), we will present an efficient method to estimate the exponential decay rate of this coupled system leading to a tractable test expressed in terms of linear matrix inequalities. These LMI conditions stem from the new design of a candidate Lyapunov functional, but also the inherent properties of the Legendre polynomials, that are used to build a projection of the infinite dimensional part of the state of the system. Based on these polynomials and using the appropriate Bessel-Legendre inequality, we will prove an exponential stability result and in the end, we will show the efficiency of our approach on academic example.

For a rational f : C → C with a conformal measure μ we show that if there is a subset of the Julia set J(f) of positive μ -measure whose points are not eventual preimages of critical or parabolic points and have limit sets not contained in the union of the limit sets of recurrent critical points, then μ is non-atomic, μ(J(f))=1 , ω(x)=J(f) for μ -a.e. point x∈J(f) and f is conservative, ergodic and exact. The proof uses a version of the Lebesgue Density Theorem valid for Borel measures and conformal balls.

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

In this short paper, we consider the formation of singularities for the solutions of the Euler–Poisson equations with damping with attractive forces in R n ( n ≥ 1 ) and the repulsive forces in R . Under the appropriate assumptions on initial data, we can obtain the divergence of the velocity will blowup in the finite time.

The paper discusses the existence of a linear manifold in a vicinity of a steady state for stabilization of the Hamiltonian systems arising in optimal control problems for economic growth models. It is shown that such stable manifold exists for almost all possible values of model parameters guaranteeing the existence of a steady state. Research is based on the qualitative analysis of the Hamiltonian dynamics, which plays a key role for investigating the asymptotic behavior of optimal trajectories. A procedure is proposed for stabilization of the Hamiltonian system, whose trajectories converge to equilibrium and approximate the optimal solution with the quadratic accuracy at a vicinity of the steady state. Basing on properties of the Hamiltonian matrices, the classification of steady states is provided and the sensitivity analysis for identification of their character is implemented with respect to model parameters. The proposed approach is applied to the model dealing with dynamic optimization of the resource productivity. © 2018 18–01–00221a The paper is supported by Russin Foundation for Basic Research (Project No. 18–01–00221a).

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

Although planar quadratic differential systems and their applications have been studied in more than one thousand papers, we still have no complete understanding of these systems. In this paper we have two objectives. First we provide a brief bibliographical survey on the main results about quadratic systems. Here we do not consider the applications of these systems to many areas as in Physics, Chemist, Economics, Biology, … Second we characterize the new class of planar separable quadratic polynomial differential systems. For such class of systems we provide the normal forms which contain one parameter, and using the Poincare compactification and the blow up technique, we prove that there exist 10 non-equivalent topological phase portraits in the Poincare disc for the separable quadratic polynomial differential systems.

Carsten Thomassen and I proved that in any graph G , the number of cycles containing a specified edge as well as all the vertices of odd degree is odd if and only if G is eulerian. Where all vertices have even degree this is a theorem of Shunichi Toida and where all vertices have odd degree it is Andrew Thomason’s extension of Smith’s Theorem. Our proof was not algorithmic. In this paper, I extend Thomason’s algorithm to one which, in a non-eulerian graph, finds a second cycle containing a specified edge and all the odd-degree vertices. Kenneth Berman extended Thomason’s Theorem to spanning trees: he proved that if T is a spanning tree such that the degree of every vertex in G − E ( T ) is odd, then there is an even number of spanning trees of G with the same degree as T at each vertex. In this paper, I give a common generalization of all of these theorems to a parity theorem about “good” trees in general graphs. My proof provides an algorithm for finding a second “good” tree.

In this paper, we study various aspects of the ODE's flow $X$ solution to the equation $\partial_t X(t,x)=b(X(t,x))$, $X(0,x)=x$ in the $d$-dimensional torus $Y_d$, where $b$ is a regular $Z^d$-periodic vector field from $R^d$ in $R^d$. We present an original and complete picture in any dimension of all logical connections between the following seven conditions involving the field $b$: - the everywhere asymptotics of the flow $X$, - the almost-everywhere asymptotics of the flow $X$, - the global rectification of the vector field $b$ in $Y_d$, - the ergodicity of the flow related to an invariant probability measure which is absolutely continuous with respect to Lebesgue's measure, - the unit set condition for Herman's rotation set $C_b$ composed of the means of $b$ related to the invariant probability measures, - the unit set condition for the subset $D_b$ of $C_b$ composed of the means of $b$ related to the invariant probability measures which are absolutely continuous with respect to Lebesgue's measure, - the homogenization of the linear transport equation with oscillating data and the oscillating velocity $b(x/\varepsilon)$ when $b$ is divergence free. The main and surprising result of the paper is that the almost-everywhere asymptotics of the flow $X$ and the unit set condition for $D_b$ are equivalent when $D_b$ is assumed to be non empty, and that the two conditions turn to be equivalent to the homogenization of the transport equation when $b$ is divergence free. In contrast, using an elementary approach based on classical tools of PDE's analysis, we extend the two-dimensional results of Oxtoby and Marchetto to any $d$-dimensional Stepanoff flow: this shows that the ergodicity of the flow may hold without satisfying the everywhere asymptotics of the flow.

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

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

The post-buckling of a clamped column made of nonlinear elastic material and subject to axial compression is investigated in this paper. The Ramberg−Osgood type constitutive relation is adopted and it is expanded by using Taylor series. Based on Euler−Bernoulli beam theory, the exact governing equations are expressed in terms of rotation angle. Because the equations contain strongly nonlinear terms that are functions of the rotation angle, analytical solutions are virtually impossible. In this respect, this paper is focused on presenting an alternative method to construct concise yet accurate analytical approximate solutions for post-buckling of the Ramberg−Osgood column that is related to large rotation amplitude. The improved harmonic balance method is used to solve the nonlinear governing equations which are simplified via the Maclaurin series expansion and orthogonal Chebyshev polynomials. In addition, numerical solutions by applying the shooting method on the governing equations are obtained for comparison. The second-order analytical approximate solutions presented in this paper show excellent accuracy by comparing with numerical solutions. The analytical approximate method and numerical result presented in this paper can be applied as design guidelines for designing engineering structures that sustain large deformation, such as slender nonlinear compression of aluminum alloy columns, rods or braces.

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

We are concerned with two kinds of singular limits of the Cauchy problem to the two-layer rotating shallow water equations as the Rossby number and the Froude number tend to zero. First we construct the uniform estimates for the strong solutions to the system under the condition that the Froude number is small enough. Different from the previously studied cases, the large operator of this model is not skew-symmetric. One of the key new ideas in this paper is to obtain the uniform estimates using the special structure of the system rather than the antisymmetry of the large operator. After that the convergence of the equations with ill-prepared data to a two-layer incompressible Navier-Stokes system is proved with the help of Strichartz estimates constructed in this paper.

The paper deals with the Caldirola–Kanai Hamiltonian in which the potential energy is defined as a linear function of the position variable. The exact solution of the Schrodinger equation using this Hamiltonian is presented. It is shown that this solution can be utilized in the theory of electric conduction of metals and semiconductors. The main idea of this paper is that if the Caldirola–Kanai Hamiltonian is applied to conduction electrons , it is possible to calculate successfully the drift velocity of the electrons in a constant electric field. The alternative calculation of the drift velocity in the framework of the density-matrix theory which is also presented leads to the same result. A useful lesson is drawn: if the friction coefficient , the parameter defining the Caldirola–Kanai Hamiltonian, concerns the conduction electrons, it is precisely equal to the reciprocal value of the relaxation time used in the kinetic theory of the electron transport in solids.

The problem of partitioning a large complex network equitably into sparse modules with given rules can be modeled by the equitable list d-degenerate coloring of graphs. This paper establishes theoretical results on such a coloring based on a newly proposed conjecture which states that every graph G is equitably d-degenerate k-colorable and equitably d-degenerate k-choosable for every integer k ≥ ( Δ ( G ) + 1 ) / ( d + 1 ) . This conjecture is strong as it implies the Hajnal-Szemeredi theorem on equitable coloring, the equitable list coloring conjecture (Kostochka, Pelsmajer, and West, 2003), the equitable vertex arboricity conjecture (Wu, Zhang, and Li, 2013), and the equitable list vertex arboricity conjecture (Zhang, 2016). In this paper, we confirm this unified conjecture for globally coupled networks, ( d + 1 ) -degenerate graphs, graphs with bounded maximum average degree, and planar graphs with large maximum degree. The equitable d-degenerate k-colorability part of this conjecture is also verified for interval graphs, generalizing a result of Niu, Li, and Zhang (2021).

The classical global criteria for the existence of Hamilton cycles only apply to graphs with large edge density and small diameter. In a series of papers Asratian and Khachatryan developed local criteria for the existence of Hamilton cycles in finite connected graphs, which are analogues of the classical global criteria due to Dirac (1952), Ore (1960), Jung (1978), and Nash-Williams (1971). The idea was to show that the global concept of Hamiltonicity can, under rather general conditions, be captured by local phenomena, using the structure of balls of small radii. (The ball of radius $r$ centered at a vertex $u$ is a subgraph of $G$ induced by the set of vertices whose distances from $u$ do not exceed $r$.) Such results are called localization theorems and present a possibility to extend known classes of finite Hamiltonian graphs. In this paper we formulate a general approach for finding localization theorems and use this approach to formulate local analogues of well-known results of Bauer et al. (1989), Bondy (1980), H\"aggkvist and Nicoghossian (1981), and Moon and Moser (1963). Finally we extend two of our results to infinite locally finite graphs and show that they guarantee the existence of Hamiltonian curves, introduced by K\"undgen, Li and Thomassen (2017)., Comment: 25 pages, 3 figures

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

In this paper we consider the problem of stochastic H 2 / H ∞ control for Poisson jump-diffusion system driven by Brownian motion and Poisson process. Firstly, a mean-field stochastic bounded real lemma(SBRL) is derived in this paper. Secondly, a sufficient condition for the solvability of Poisson jump-diffusion linear quadratic (LQ) optimal control of discrete-time mean-field type is presented. Thirdly, based on the results of SBRL and LQ control, the sufficient conditions for the existence of mean-field stochastic H 2 / H ∞ control of Poisson jump-diffusion system are established by the solvability of the coupled matrix value equation. Finally, an example of recursive algorithm is presented to demonstrate the effectiveness of the proposed theory.

In this paper we study the well-posedness for the inhomogeneous nonlinear Schrodinger equation i ∂ t u + Δ u = λ | x | − α | u | β u in Sobolev spaces H s , s ≥ 0 . The well-posedness theory for this model has been intensively studied in recent years, but much less is understood compared to the classical NLS model where α = 0 . The conventional approach does not work particularly for the critical case β = 4 − 2 α d − 2 s . It is still an open problem. The main contribution of this paper is to develop the well-posedness theory in this critical case (as well as non-critical cases). To this end, we approach to the matter in a new way based on a weighted L p setting which seems to be more suitable to perform a finer analysis for this model. This is because it makes it possible to handle the spatially decaying factor | x | − α in the nonlinearity more efficiently. This observation is a core of our approach that covers the critical case successfully.

In this paper we give the necessary and sufficient conditions for the Discrete Maximum Principle (DMP) to hold. We prove the convergence of the nonlinear finite element method applied to the logistic equation by using that the Jacobian matrix evaluated in the supersolution, provided by the a priori bound, is a non-singular M-matrix, which is proved in a fast way using both, the positiveness of its principal eigenvalue and the DMP. Meanwhile a positive subsolution provides the coercivity constant. The numerical simulations show that the nonlinear finite element approximate solutions do not oscillate if the DMP is fulfilled. The characterization of the DMP and the mesh sizes guaranteeing the existence of positive sub- and supersolutions of the nonlinear finite element approximate problem, in the case of variable coefficients and all types of boundary conditions are some of the novelties of this paper. The excellent performance of the method is tested in two examples with boundary layers caused by very small diffusion.

In this paper, the one-side physical vacuum problem for the one dimensional compressible Euler equations with time-dependent damping is considered. Near the physical vacuum boundary, the sound speed is C 1 / 2 -Holder continuous. The coefficient of the time-dependent damping is given by μ ( 1 + t ) λ , ( 0 λ , 0 μ ) which decays by order −λ in time. First we give an one-side physical vacuum background solution whose density and velocity have a growing order with respect to time. Then the main purpose of this paper is to prove the stability of this background solution under the assumption that 0 λ 1 , 0 μ or λ = 1 , 2 μ . The pointwise convergence rate of the density, velocity and the expanding rate of the physical vacuum boundary are also given. The proof is based on the space-time weighted energy estimates, elliptic estimates and the Hardy inequality in the Lagrangian coordinates.

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

In the article, we prove the large data scattering for two models, i.e. the defocusing quintic nonlinear Schrodinger equation on R 2 × T and the defocusing cubic nonlinear Schrodinger equation on R 3 × T . Both of the two equations are mass supercritical and energy critical. The main ingredients of the proofs contain global Stricharz estimate, profile decomposition and energy induction method. This paper is the second project of our series work (two papers, together with [31] ) on large data scattering for the defocusing critical NLS with integer index nonlinearity on low dimensional waveguides. At this point, this category of problems are almost solved except for two remaining resonant system conjectures and the quintic NLS problem on R × T .

In this paper, we consider the following non-linear equations in unbounded domains Ω with exterior Dirichlet condition: { ( − Δ ) p s u ( x ) = f ( u ( x ) ) , x ∈ Ω , u ( x ) > 0 , x ∈ Ω , u ( x ) = 0 , x ∈ R n ∖ Ω , where ( − Δ ) p s is the fractional p-Laplacian defined as (0.1) ( − Δ ) p s u ( x ) = C n , s , p P . V . ∫ R n | u ( x ) − u ( y ) | p − 2 [ u ( x ) − u ( y ) ] | x − y | n + s p d y with 0 s 1 and p ≥ 2 . We first establish a maximum principle in unbounded domains involving the fractional p-Laplacian by estimating the singular integral in (0.1) along a sequence of approximate maximum points. Then, we obtain the asymptotic behavior of solutions far away from the boundary. Finally, we develop a sliding method for the fractional p-Laplacians and apply it to derive the monotonicity and uniqueness of solutions. There have been similar results for the classical Laplacian [3] and for the fractional Laplacian [39] , which are linear operators. Unfortunately, many approaches there no longer work for the fully non-linear fractional p-Laplacian here. To circumvent these difficulties, we introduce several new ideas, which enable us not only to deal with non-linear non-local equations, but also to remarkably weaken the conditions on f ( ⋅ ) and on the domain Ω. We believe that the new methods developed in our paper can be widely applied to many problems in unbounded domains involving non-linear non-local operators.

In this paper, we investigate steady incompressible Euler flows with nonvanishing vorticity in a planar bounded domain. Let q be a harmonic function that corresponds to an irrotational flow. This paper proves that if q has k isolated local extremum points on the boundary, then there exist two kinds of steady Euler flows with small vorticity supported near these k points. For the first kind, near each maximum point the vorticity is positive and near each minimum point the vorticity is negative. For the second kind, near each minimum point the vorticity is positive and near each maximum point the vorticity is negative. Moreover, near these k points, the flow is characterized by a semilinear elliptic equation with a given profile function in terms of the stream function. The results are achieved by solving a certain variational problem for the vorticity and studying the limiting behavior of the extremizers.

The paper illustrate the development and validation of two regression models i.e., poisson and artificial neural network employed to predict the significance of different climatic variables for an institutional building in composite climate. For prediction of outer wall temperature using five input parameters (ambient temperature, relative humidity, global radiation, wind speed and wind direction), two modeling approaches: poisson and artificial neural network were used. The results of both the model were validated through test data analysis in terms of three statistical measures namely, R-squared, root mean square error and mean absolute error. The performances of both the models are comparable in terms of train and test data. Comparative statistical analysis of both the model shows a better fit for the artificial neural network model as it was capable of adjusting themselves to the unpredicted changes in the input data. For the current study where weather parameters are continuously fluctuating poisson model was preferred as it generates the estimation coefficient values for the variables while neural model is not mathematically defined. The study demonstrates that neural model should be used as an alternative method for temperature prediction. The content and format of the paper has been widely used for designing and retrofitting of buildings by considering the most significant climatic variables for future work. Also, it helps the architects and engineers to find the relevant insulating material for the building envelope as per the requirement and subsequently improves its thermal behavior.

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

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

The initial boundary value problems for compressible Navier-Stokes-Poisson and Euler-Poisson equations of plasma are considered on exterior domains in this paper. With the radial symmetry assumption, the global existence of solutions to compressible Navier-Stokes-Poisson equations with the large initial data on a domain exterior to a ball in R n ( n ≥ 1 ) is proved. Moreover, without any symmetry assumption, the global existence of smooth solutions near a given constant steady state for both compressible Navier-Stokes-Poisson and Euler-Poisson equations on an exterior domain in R 3 with physical boundary conditions is also established with the exponential stability. A key issue addressed in this paper is on the global-in-time regularity of solutions near physical boundaries. This is in particular so for the 3-D compressible Navier-Stokes-Poisson equations to which global smooth solutions of initial boundary value problems are seldom found in literature to the best of knowledge.

In this paper, we derive sharp bounds on the semigroup of the linearized incompressible Navier-Stokes equations near a stationary shear layer in the half plane and in the half space ($\mathbb{R}_+^2$ or $\mathbb{R}_+^3$), with Dirichlet boundary conditions, assuming that this shear layer in spectrally unstable for Euler equations. In the inviscid limit, due to the prescribed no-slip boundary conditions, vorticity becomes unbounded near the boundary. The novelty of this paper is to introduce boundary layer norms that capture the unbounded vorticity and to derive sharp estimates on this vorticity that are uniform in the inviscid limit., Comment: this greatly revised and shortened the previous version

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

Let $B=\{ B_{t}\} _{t\ge 0}$ be a one-dimensional standard Brownian motion, to which we associate the exponential additive functional $A_{t}=\int _{0}^{t}e^{2B_{s}}ds,\,t\ge 0$. Starting from a simple observation of generalized inverse Gaussian distributions with particular sets of parameters, we show, with the help of a result by Matsumoto--Yor (2000), that for every $x\in \mathbb{R}$ and for every finite stopping time $\tau $ of the process $\{ e^{-B_{t}}A_{t}\} _{t\ge 0}$, there holds the identity in law \begin{align*} \left( e^{B_{\tau}}\!\sinh x+\beta (A_{\tau }), \, Ce^{B_{\tau}}\!\cosh x+\hat{\beta}(A_{\tau }), \, e^{-B_{\tau }}\!A_{\tau } \right) \stackrel{(d)}{=} \left( \sinh (x+B_{\tau }), \, C\cosh (x+B_{\tau }), \, e^{-B_{\tau }}\!A_{\tau } \right) , \end{align*} which extends an identity due to Bougerol (1983) in several aspects. Here $\beta =\{ \beta (t)\} _{t\ge 0}$ and $\hat{\beta}=\{ \hat{\beta}(t)\} _{t\ge 0}$ are one-dimensional standard Brownian motions, $C$ is a standard Cauchy variable, and $B$, $\beta $, $\hat{\beta}$ and $C$ are independent. Using an argument relevant to derivation of the above identity, we also present some invariance formulae for Cauchy variable involving an independent Rademacher variable., Comment: 43 pages. Changes from the first version are: positivity condition imposed on the stopping time $\tau $ is removed from Abstract, on which a remark is inserted in Remark 1.1; the assertion of Theorem 1.2 is fairly extended; two papers by Barndorff-Nielsen and two books are added for descriptions of GIG and related laws; a paper by Matsumoto--Yor (2003) is referred to in the newly added Remark A.1

The main goal of this paper is to discover some new analytical solutions of a fractional form of the Bogoyavlensky–Konopelchenko equation via two new analytical schemes. This model is considered as a particular case of (2 + 1)–dimensional version of the well–known KdV equation where it describes the interaction between the Riemann wave propagating and the long-wave propagation along the x , y –axises. An efficient fractional derivative called Atangana–Baleanu-Riemann derivative is utilized to convert the standard form of the model into a nonlinear fractional PDE with an–integer order. The basic idea in these methods is to use a new variable to transform the form of the equation into a nonlinear equation with ordinary derivatives. The novelty of the present paper is that the new solutions determined by applying these two powerful analytical methods can not be found in previous articles. Several two and three-dimensional figures have been depicted to illustrate the dynamic behavior of the acquired solutions. Another advantage of these two methods is their applicability in solving similar models using this fractional derivative operator.

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

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

Deviance information criterion (DIC) has been widely used for Bayesian model comparison, especially after Markov chain Monte Carlo (MCMC) is used to estimate candidate models. This paper first studies the problem of using DIC to compare latent variable models when DIC is calculated from the conditional likelihood. In particular, it is shown that the conditional likelihood approach undermines theoretical underpinnings of DIC. A new version of DIC, namely DIC L , is proposed to compare latent variable models. The large sample properties of DIC L are studied. A frequentist justification of DIC L is provided. Like AIC, DIC L provides an asymptotically unbiased estimator to the expected Kullback–Leibler (KL) divergence between the DGP and a predictive distribution. Some popular algorithms, such as the EM, Kalman and particle filtering algorithms, are introduced to compute DIC L for latent variable models. Moreover, this paper studies the problem of using DIC to compare misspecified models. A new version of DIC, namely DIC M , is proposed and it can be regarded as a Bayesian version of TIC. A frequentist justification of DIC M is provided under misspecification. DIC L and DIC M are illustrated using asset pricing models and stochastic volatility models.

In the Graph bipartization (or Odd Cycle Transversal) problem, the objective is to decide whether a given graph G can be made bipartite by the deletion of k vertices for some given k. The parameterized complexity of Odd Cycle Transversal was resolved in the breakthrough paper of Reed, Smith and Vetta [Operations Research Letters, 2004], who developed an algorithm running in time O ( 3 k k m n ) . The question of improving the dependence on the input size to linear, which was another long standing open problem in the area, was resolved by Iwata et al. [SICOMP 2016] and Ramanujan and Saurabh [TALG 2017], who presented O ( 4 k ( m + n ) ) and 4 k k O ( 1 ) ( m + n ) algorithms respectively. In this paper, we obtain a faster algorithm that runs in time 3 k k O ( 1 ) ( m + n ) and hence preserves the linear dependence on the input size while nearly matching the dependence on k incurred by the algorithm of Reed, Smith and Vetta.

A correction is noted to the previously published paper (Zou and Lv, 2017). Regularity, positive recurrence and stationary distribution are considered in this paper. A complete threshold analysis of strong stochastic persistence and extinction is investigated. Stochastic Hopf bifurcation is considered from the viewpoint of numerical simulations.

International audience; In this paper, we consider a semi-linear fourth order parabolic equation in a bounded smooth domain Ω with homogeneous Dirichlet and Neumann boundary conditions. The main result of this paper is the null controllability and the exact controllability to the trajectories at any time T > 0 for the associated control system with a control function acting at the interior.; Dans ce papier, on considère uneéquation parabolique semi-linéaire de quatrième ordre dans un domaine borné régulier Ω avec des conditions aux limites de type Dirichlet et Neumann homogènes. Le résultat principal de ce papier concerne la contrôlabilitéà zéro et la contrôlabilité exacte pour tout T > 0 du système de contrôle associé avec un contrôle agissantà l'interieur.

This paper is concerned with the large time behaviors of the entropy solutions to one-dimensional scalar convex conservation laws, of which the initial data are assumed to approach two arbitrary L ∞ periodic functions as x → − ∞ and x → + ∞ , respectively. We show that the solutions approach the Riemann solutions at algebraic rates as time increases. Moreover, a new discovery in this paper is that the difference between the two periodic perturbations at two infinities may generate a constant shift on the background shock wave, which is different from the result in [11] , where the two periodic perturbations are the same.

This paper studies a new gradient regularity in Lorentz spaces for solutions to a class of quasilinear divergence form elliptic equations with nonhomogeneous Dirichlet boundary conditions: { div ( A ( x , ∇ u ) ) = div ( | F | p − 2 F ) in Ω , u = σ on ∂ Ω , where Ω ⊂ R n ( n ≥ 2 ), the nonlinearity A is a monotone Caratheodory vector valued function defined on W 0 1 , p ( Ω ) for p > 1 and the p-capacity uniform thickness condition is imposed on the complement of our bounded domain Ω. Moreover, for given data F ∈ L p ( Ω ; R n ) , the problem is set up with general Dirichlet boundary data σ ∈ W 1 , p ( Ω ) . In this paper, the optimal good-λ type bounds technique is applied to prove some results of fractional maximal estimates for gradient of solutions. And the main ingredients are the action of the cut-off fractional maximal functions and some local interior and boundary comparison estimates developed in previous works [45] , [52] , [53] and references therein.

Local regressions have been widely employed for decomposing atmospheric data series. However, the use of local quadratic regressions is less extended. The current paper is grounded on the hypothesis that local linear regressions are able to capture CO2 and CH4 temporal evolution equally as well as quadratic linear regressions. Thus, the current paper pursues the following goals: (1) to quantify the temporal variability of both gases by using the local linear and local quadratic regression method; (2) to analyse the main statistics of the detrended series over time; (3) to compare results between the local linear and local quadratic regression method. Minimum mixing ratios in late summer and maximum in winter were detected for both gases. Atmospheric increases of an average of 1.98 ppm year−1 for CO2 and 11 ppb year−1 for CH4 were found when applying the local linear regression method. Alternatively, an increase of 2.24 ppm year−1 for CO2 and around 10.34 ppb year−1 for CH4 was obtained when the local quadratic regression method was applied. The Pearson correlation coefficients (0.21–0.40) showed acceptable values due to the large amount of available data. Statistically significant differences for the initial and the smoothed trend, as well as statistically significant differences for the seasonal component, were reported when comparing the local linear with the local quadratic method. Overall, both methods proved easy to apply and both provided acceptable data accuracy.

To our knowledge the article in hand is first ever study for existence of Si’lnikov type of chaos for Nuclear Spin Generator (NSG) system. This paper deals with the existence of heteroclinic Si’lnikov-type orbits within an NSG chaotic system involving three parameters and having three equilibrium points. Method of undetermined coefficient is applied for analytical analysis of heteroclinic orbits. As an outcome the Si’lnikov criteria guarantees that the NSG system has a Smale horseshoe type chaos. Uniform convergence of series expansion for heteroclinic orbit is verified in this paper. Lyapunov exponents spectrum and sensitivity dependence are discussed. Numerical simulations are compiled for the verification of analytical results and bifurcation diagrams are displayed. Though examination of Si’lnikov criterion for NSG system is an exhaustive study but we have succeeded to explore an other aspect of the system by this criterion.

For the exterior Dirichlet problems (EDP) of the Helmholtz equation in 2D, when the Sommerfeld radiation condition is satisfied, there always exists the unique solution. Denote the unbounded domain S ∞ outside of a bounded simply-connected domain, where the interior boundary Γ i n is non-circular. The standard MFS is first studied by using the complex Hankel function H 0 ( 1 ) ( k r ) as the fundamental solutions (FS). However, some solutions can not be obtained correctly (e.g., the spurious eigenvalues called). This paper is devoted to explore the spurious eigenvalues and their removals. For the non-circular interior boundary Γ i n , the error analysis is briefly made for the standard MFS, and some guidance is provided to bypass the spurious eigenvalues. Denote two nodes: P = ( ρ , θ ) and Q = ( R , ϕ ) , and r = | P Q ‾ | = R 2 + ρ 2 − 2 R ρ cos ⁡ ( θ − ϕ ) . To completely eliminate all spurious eigenvalues, we solicit the new combined FS: ( ∂ ∂ R ± i k ) H 0 ( 1 ) ( k r ) with i = − 1 , and propose the new modified MFS. The new algorithms are simple, and the strict analysis may be made. The bounds of errors are derived, and the polynomial convergence rates can be achieved. The bounds of condition numbers are derived for circular Γ ∘ i n only, to display the exponential growth via the number of the new FS used. Numerical experiments are carried to support the analysis made. This paper is the first time for the MFS to explore the spurious eigenvalues and their removals, accompanied with strict analysis of errors and stability.