Stable blow-up dynamics in the -critical and -supercritical generalized Hartree equation

We study stable blow-up dynamics in the generalized Hartree equation with radial symmetry, which is a Schrödinger-type equation with a nonlocal, convolution-type nonlinearity: First, we consider the -critical case in dimensions and obtain that a generic blow-up has a self-similar structure and exhibits not only the square root blowup rate , but also the log-log correction (via asymptotic analysis and functional fitting), thus, behaving similarly to the stable blow-up regime in the -critical nonlinear Schrödinger equation. In this setting, we also study blow-up profiles and show that generic blow-up solutions converge to the rescaled , a ground state solution of the elliptic equation . We also consider the -supercritical case in dimensions . We derive the profile equation for the self-similar blow-up and establish the existence and local uniqueness of its solutions. As in the NLS -supercritical regime, the profile equation exhibits branches of nonoscillating, polynomially decaying (multi-bump) solutions. A numerical scheme of putting constraints into solving the corresponding ordinary differential equation is applied during the process of finding the multi-bump solutions. Direct numerical simulation of solutions to the generalized Hartree equation by the dynamic rescaling method indicates that the is the profile for the stable blow-up. In this supercritical case, we obtain the blow-up rate without any correction. This blow-up happens at the focusing level , and thus, numerically observable (unlike the -critical case). In summary, we find that the results are similar to the behavior of stable self-similar blowup solutions in the corresponding settings for the nonlinear Schrödinger equation. Consequently, one may expect that the form of the nonlinearity in the Schrödinger-type equations is not essential in the stable formation of singularities.     

Uniform asymptotics and zeros of the associated Pollaczek polynomials

Using a difference-equation method in a previous paper, we study the associated Pollaczek polynomials defined by a three-term recurrence relation. Two asymptotic approximations are derived for these polynomials; one holds for with and , and the other holds for with t in a neighborhood of . An asymptotic formula is also provided for their largest zeros.     

Development and analysis of a malaria transmission mathematical model with seasonal mosquito life-history traits

In this paper, we develop and analyze a malaria model with seasonality of mosquito life-history traits: periodic-mosquitoes per capita birth rate, -mosquitoes death rate, -probability of mosquito to human disease transmission, -probability of human to mosquito disease transmission, and -mosquitoes biting rate. All these parameters are assumed to be time dependent leading to a nonautonomous differential equation system. We provide a global analysis of the model depending on two threshold parameters and (with ). When , then the disease-free stationary state is locally asymptotically stable. In the presence of the human disease-induced mortality, the global stability of the disease-free stationary state is guarantied when . On the contrary, if , the disease persists in the host population in the long term and the model admits at least one positive periodic solution. Moreover, by a numerical simulation, we show that a sub-critical (backward) bifurcation is possible at . Finally, the simulation results are in accordance with the seasonal variation of the reported cases of a malaria-epidemic region in Mpumalanga province in South Africa.     

A difference equation approach to Plancherel-Rotach asymptotics for -orthogonal polynomials

In this paper, we employ a difference equation approach to study the Plancherel-Rotach asymptotics of -orthogonal polynomials about their largest zeros. Our method for -difference equations is an analogue to the turning point problem for Hermite differential equations. It works well in the toy problems of Stieltjes-Wigert polynomials and -Hermite polynomials.     

Finite‐size corrections at the hard edge for the Laguerre β ensemble

A fundamental question in random matrix theory is to quantify the optimal rate of convergence to universal laws. We take up this problem for the Laguerre β ensemble, characterized by the Dyson parameter β, and the Laguerre weight , in the hard edge limit. The latter relates to the eigenvalues in the vicinity of the origin in the scaled variable . Previous work has established the corresponding functional form of various statistical quantities—for example, the distribution of the smallest eigenvalue, provided that . We show, using the theory of multidimensional hypergeometric functions based on Jack polynomials, that with the modified hard edge scaling , the rate of convergence to the limiting distribution is , which is optimal. In the case , general the explicit functional form of the distribution of the smallest eigenvalue at this order can be computed, as it can for and general . An iterative scheme is presented to numerically approximate the functional form for general .     

The Riemann–Hilbert analysis to the Pollaczek–Jacobi type orthogonal polynomials

In this paper, we study polynomials orthogonal with respect to a Pollaczek–Jacobi type weight The uniform asymptotic expansions for the monic orthogonal polynomials on the interval (0,1) and outside this interval are obtained. Moreover, near , the uniform asymptotic expansion involves Airy function as , and Bessel function of order α as in the neighborhood of , the uniform asymptotic expansion is associated with Bessel function of order β as . The recurrence coefficients and leading coefficient of the orthogonal polynomials are expressed in terms of a particular Painlevé III transcendent. We also obtain the limit of the kernel in the bulk of the spectrum. The double scaled logarithmic derivative of the Hankel determinant satisfies a σ‐form Painlevé III equation. The asymptotic analysis is based on the Deift and Zhou's steepest descent method.     

Using Symmetries à Rebours

We consider nonclassical symmetries of partial differential equations (PDEs) in dimensions. Given a th‐order ordinary differential equation in the unknown we are able to find the most general scalar PDE of a given order which can be reduced via a nonclassical symmetry to .     

Connection formulae for asymptotics of the fifth Painlevé transcendent on the imaginary axis: I

Leading terms of asymptotic expansions for the general complex solutions of the fifth Painlevé equation as are found. These asymptotics are parameterized by monodromy data of the associated linear ODE, The parameterization allows one to derive connection formulae for the asymptotics. We provide numerical verification of the results. Important special cases of the connection formulae are also considered.     

Inverse scattering transform for the defocusing Ablowitz–Ladik system with arbitrarily large nonzero background

In this work we develop the inverse scattering transform (IST) for the defocusing Ablowitz–Ladik (AL) equation with arbitrarily a large nonzero background at space infinity. The IST was developed in previous works under the assumption that the amplitude of the background satisfies a “small norm” condition . On the other hand, Ohta and Yang recently showed that the defocusing AL system, which is modulationally stable for , becomes unstable if , and exhibits discrete rogue wave solutions, some of which are regular for all times. Here, we construct the IST for the defocusing AL with , analyze the spectrum, and characterize the soliton and rational solutions from a spectral point of view. We formulate the direct and inverse problems by using a suitable uniformization variable, and pose the inverse problem as an RHP across a simple contour in the complex plane of the uniform variable. As a by‐product of the IST, we also obtain explicit soliton solutions, which are the discrete analog of the celebrated Kuznetsov–Ma, Akhmediev, Peregrine solutions, and which mimic the corresponding solutions for the focusing AL equation. Soliton solutions that are the analog of the dark soliton solutions of the defocusing AL equation in the case are also presented.     

Filling the complexity gaps for colouring planar and bounded degree graphs

A colouring of a graph is a function such that for every . A -regular list assignment of is a function with domain such that for every , is a subset of of size . A colouring of respects a -regular list assignment of if for every . A graph is -choosable if for every -regular list assignment of , there exists a colouring of that respects . We may also ask if for a given -regular list assignment of a given graph , there exists a colouring of that respects . This yields the -Regular List Colouring problem. For , we determine a family of classes of planar graphs, such that either -Regular List Colouring is -complete for instances with , or every is -choosable. By using known examples of non--choosable and non--choosable graphs, this enables us to classify the complexity of -Regular List Colouring restricted to planar graphs, planar bipartite graphs, planar triangle-free graphs, and planar graphs with no -cycles and no -cycles. We also classify the complexity of -Regular List Colouring and a number of related colouring problems for graphs with bounded maximum degree.     

A dichotomy for the kernel by H-walks problem in digraphs

Let be a digraph which may contain loops, and let be a loopless digraph with a coloring of its arcs . An -walk of is a walk of such that is an arc of , for every . For , we say that reaches by -walks if there exists an -walk from to in . A subset is a kernel by -walks of if every vertex in reaches by -walks some vertex in , and no vertex in can reach another vertex in by -walks. A panchromatic pattern is a digraph such that every -arc-colored digraph has a kernel by -walks. In this study, we prove that every digraph is either a panchromatic pattern, or the problem of determining whether an arc-colored digraph has a kernel by -walks is -complete.     

On perfect matchings in matching covered graphs

A graph is matching-covered if every edge of is contained in a perfect matching. A matching-covered graph is strongly coverable if, for any edge of , the subgraph is still matching-covered. An edge subset of a matching-covered graph is feasible if there exist two perfect matchings and such that , and an edge subset with at least two edges is an equivalent set if a perfect matching of contains either all edges in or none of them. A strongly matchable graph does not have an equivalent set, and any two independent edges of form a feasible set. In this paper, we show that for every integer , there exist infinitely many -regular graphs of class 1 with an arbitrarily large equivalent set that is not switching-equivalent to either or , which provides a negative answer to a problem of Lukot’ka and Rollová. For a matching-covered bipartite graph , we show that has an equivalent set if and only if it has a 2-edge-cut that separates into two balanced subgraphs, and is strongly coverable if and only if every edge-cut separating into two balanced subgraphs and satisfies and .     

Reconstruction from the deck of -vertex induced subgraphs

The - deck of a graph is its multiset of subgraphs induced by vertices; we study what can be deduced about a graph from its -deck. We strengthen a result of Manvel by proving for that when is large enough ( suffices), the -deck determines whether an -vertex graph is connected ( suffices when , and cannot suffice). The reconstructibility of a graph with vertices is the largest such that is determined by its -deck. We generalize a result of Bollobás by showing for almost all graphs. As an upper bound on , we have . More generally, we compute whenever , which involves extending a result of Stanley. Finally, we show that a complete -partite graph is reconstructible from its -deck.     

Upper bounds on the uniquely restricted chromatic index

Golumbic, Hirst, and Lewenstein define a matching in a simple, finite, and undirected graph to be uniquely restricted if no other matching covers exactly the same set of vertices. We consider uniquely restricted edge-colorings of , defined as partitions of its edge set into uniquely restricted matchings, and study the uniquely restricted chromatic index of , defined as the minimum number of uniquely restricted matchings required for such a partition. For every graph , where is the classical chromatic index, the acyclic chromatic index, and the strong chromatic index of . While Vizing's famous theorem states that is either the maximum degree of or , two famous open conjectures due to Alon, Sudakov, and Zaks, and to Erdős and Nešetřil concern upper bounds on and in terms of . Since is sandwiched between these two parameters, studying upper bounds in terms of is a natural problem. We show that with equality if and only if some component of is . If is connected, bipartite, and distinct from , and is at least , then, adapting Lovász's elegant proof of Brooks’ theorem, we show that . Our proofs are constructive and yield efficient algorithms to determine the corresponding edge-colorings.     

Solutions with concentration for conservation laws with discontinuous flux and its applications to numerical schemes for hyperbolic systems

Measure-valued weak solutions for conservation laws with discontinuous flux are proposed and explicit formulae have been derived. We propose convergent discontinuous flux-based numerical schemes for the class of hyperbolic systems that admit nonclassical -shocks, by extending the theory of discontinuous flux for nonlinear conservation laws to scalar transport equation with a discontinuous coefficient. The article also discusses the concentration phenomenon of solutions along the line of discontinuity, for scalar transport equations with a discontinuous coefficient. The existence of the solutions for transport equation is shown using the vanishing viscosity approach and the asymptotic behavior of the solutions is also established. The performance of the numerical schemes for both scalar conservation laws and systems to capture the -shocks effectively is displayed through various numerical experiments.     

Extending edge-colorings of complete hypergraphs into regular colorings

Let be the collection of all -subsets of an -set . Given a coloring (partition) of a set , we are interested in finding conditions under which this coloring is extendible to a coloring of so that the number of times each element of appears in each color class (all sets of the same color) is the same number . The case was studied by Sylvester in the 18th century and remained open until the 1970s. The case is extensively studied in the literature and is closely related to completing partial symmetric Latin squares. For , we settle the cases , and completely. Moreover, we make partial progress toward solving the case where . These results can be seen as extensions of the famous Baranyai’s theorem, and make progress toward settling a 40-year-old problem posed by Cameron.     

Linearly χ-bounding (P6, C4)-free graphs*

Given two graphs and , a graph is -free if it contains no induced subgraph isomorphic to or . Let and be the path on vertices and the cycle on vertices, respectively. In this paper we show that for any -free graph it holds that , where and are the chromatic number and clique number of , respectively. Our bound is attained by several graphs, for instance, the 5-cycle, the Petersen graph, the Petersen graph with an additional universal vertex, and all -critical -free graphs other than (see Hell and Huang [Discrete Appl. Math. 216 (2017), pp. 211–232]). The new result unifies previously known results on the existence of linear -binding functions for several graph classes. Our proof is based on a novel structure theorem on -free graphs that do not contain clique cutsets. Using this structure theorem we also design a polynomial time -approximation algorithm for coloring -free graphs. Our algorithm computes a coloring with colors for any -free graph in time.     

Strong chromatic index of unit distance graphs

The strong chromatic index of a graph , denoted by , is defined as the least number of colors in a coloring of edges of , such that each color class is an induced matching (or: if edges and have the same color, then both vertices of are not adjacent to any vertex of ). A graph is a unit distance graph in if vertices of can be uniquely identified with points in , so that is an edge of if and only if the Euclidean distance between the points identified with and is 1. We would like to find the largest possible value of , where is a unit distance graph (in and ) of maximum degree . We show that , where is a unit distance graph in of maximum degree . We also show that the maximum possible size of a strong clique in unit distance graph in is linear in and give a tighter result for unit distance graphs in the plane.     

A scaling limit for the length of the longest cycle in a sparse random digraph

We discuss the length of the longest directed cycle in the sparse random digraph , constant. We show that for large there exists a function such that a.s. The function where is a polynomial in . We are only able to explicitly give the values , although we could in principle compute any .     

Minimum codegree condition for perfect matchings in k-partite k-graphs

Let be a -partite -graph with vertices in each partition class, and let denote the minimum codegree of . We characterize those with and with no perfect matching. As a consequence, we give an affirmative answer to the following question of Rödl and Ruciński: if is even or , does imply that has a perfect matching? We also give an example indicating that it is not sufficient to impose this degree bound on only two types of -sets.