Degree 2k Research Articles

The Truncated Moment Problem on Reducible Cubic Curves I: Parabolic and Circular Type Relations

In this article we study the bivariate truncated moment problem (TMP) of degree 2k on reducible cubic curves. First we show that every such TMP is equivalent after applying an affine linear transformation to one of 8 canonical forms of the curve. The case of the union of three parallel lines was solved in Zalar (Linear Algebra Appl 649:186–239, 2022. https://doi.org/10.1016/j.laa.2022.05.008), while the degree 6 cases in Yoo (Integral Equ Oper Theory 88:45–63, 2017). Second we characterize in terms of concrete numerical conditions the existence of the solution to the TMP on two of the remaining cases concretely, i.e., a union of a line and a circle y(ay+x2+y2)=0,a∈R\\{0}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$y(ay+x^2+y^2)=0, a\\in {\\mathbb {R}}{\\setminus } \\{0\\}$$\\end{document}, and a union of a line and a parabola y(x-y2)=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$y(x-y^2)=0$$\\end{document}. In both cases we also determine the number of atoms in a minimal representing measure.

The strong truncated Hamburger moment problem with and without gaps

The strong truncated Hamburger moment problem (STHMP) of degree (−2k1,2k2) asks to find necessary and sufficient conditions for the existence of a positive Borel measure, supported on R, such that βi=∫xidμ(−2k1≤i≤2k2). The first solution of the STHMP, covering also its matrix generalization, was established by Simonov [60], who used the operator approach and described all solutions in terms of self-adjoint extensions of a certain symmetric operator. Using the solution of the truncated Hamburger moment problem and the properties of Hankel matrices we give an alternative solution of the STHMP and describe concretely all minimal solutions, i.e., solutions having the smallest support. Then, using the equivalence with the STHMP of degree (−2k,2k), we obtain the solution of the 2–dimensional truncated moment problem (TMP) of degree 2k with variety xy=1, first solved by Curto and Fialkow [22]. Our addition to their result is the fact previously known only for k=2, that the existence of a measure is equivalent to the existence of a flat extension of the moment matrix. Further on, we solve the STHMP of degree (−2k1,2k2) with one missing moment in the sequence, i.e., β−2k1+1 or β2k2−1, which also gives the solution of the TMP with variety x2y=1 as a special case, first studied by Fialkow in [33].

The truncated moment problem on the union of parallel lines

In this article we study the bivariate truncated moment problem (TMP) of degree 2k on the union of parallel lines. First we present an alternative proof of Fialkow's solution [28] to the TMP on the union of two parallel lines (TMP–2pl) using the solution of the truncated Hamburger moment problem (THMP). We add a new equivalent solvability condition, which is then used together with the THMP, to solve the TMP on the union of three parallel lines (TMP–3pl), our second main result of the article. Finally, we establish a sufficient condition for the existence of a solution to the TMP on the union of n parallel lines in the pure case, i.e. when the moment matrix Mk is of the highest possible rank, or equivalently the only column relations come from the union of n lines. The condition is based on the feasibility of a certain linear matrix inequality, corresponding to the extension of Mk by adding rows and columns indexed by some monomials of degree k+1. The proof is by induction on n, where n≥2 and for the base of induction n=2 we use the solution of the TMP–2pl.

Explicit computation of some families of Hurwitz numbers, II

Abstract We continue our computation, using a combinatorial method based on Gronthendieck’s dessins d’enfant, of the number of (weak) equivalence classes of surface branched covers matching certain specific branch data. In this note we concentrate on data with the surface of genus g as source surface, the sphere as target surface, 3 branching points, degree 2k, and local degrees over the branching points of the form [2, …, 2], [2h + 1, 3, 2, …, 2], π = [ d i ] i = 1 ℓ . $\begin{array}{} \displaystyle \pi=[d_i]_{i=1}^\ell. \end{array}$ We compute the corresponding (weak) Hurwitz numbers for several values of g and h, getting explicit arithmetic formulae in terms of the di ’s.

On coloring numbers of graph powers

The weak r-coloring numbers wcolr(G) of a graph G were introduced by the first two authors as a generalization of the usual coloring number col(G), and have since found interesting theoretical and algorithmic applications. This has motivated researchers to establish strong bounds on these parameters for various classes of graphs.Let Gp denote the pth power of G. We show that, all integers p>0 and Δ≥3 and graphs G with Δ(G)≤Δ satisfy col(Gp)∈O(p⋅wcol⌈p∕2⌉(G)(Δ−1)⌊p∕2⌋); for fixed tree width or fixed genus the ratio between this upper bound and worst case lower bounds is polynomial in p. For the square of graphs G, we also show that, if the maximum average degree 2k−2<mad(G)≤2k, then col(G2)≤(2k−1)Δ(G)+2k+1.

Explicit computation of some families of Hurwitz numbers

We compute the number of (weak) equivalence classes of branched covers from a surface of genus g to the sphere, with 3 branching points, degree 2k, and local degrees over the branching points of the form (2,…,2), (2h+1,1,2,…,2), π=dii=1ℓ, for several values of g and h. We obtain explicit formulae of arithmetic nature in terms of the local degrees di. Our proofs employ a combinatorial method based on Grothendieck’s dessins d’enfant.

On Eulerian orientations of even-degree hypercubes

It is well known that every Eulerian orientation of an Eulerian 2k-edge connected (undirected) graph is strongly k-edge connected. A long-standing goal in the area is to obtain analogous results for other types of connectivity, such as node connectivity. We show that every Eulerian orientation of the hypercube of degree 2k is strongly k-node connected.

The correlahedron

We introduce a new geometric object, the correlahedron, which we conjecture to be equivalent to stress-energy correlators in planar mathcal{N}=4 super Yang-Mills. Re-expressing the Grassmann dependence of correlation functions of n chiral stress-energy multiplets with Grassmann degree 4k in terms of 4(n + k)-linear bosonic variables, the resulting expressions have an interpretation as volume forms on a Gr(n+k, 4+n+k) Grassmannian, analogous to the expressions for planar amplitudes via the amplituhedron. The resulting volume forms are to be naturally associated with the correlahedron geometry. We construct such expressions in this bosonised space both directly, in general, from Feynman diagrams in twistor space, and then more invariantly from specific known correlator expressions in analytic superspace. We give a geometric interpretation of the action of the consecutive lightlike limit and show that under this the correlahedron reduces to the squared amplituhedron both as a geometric object as well as directly on the corresponding volume forms. We give an explicit easily implementable algorithm via cylindrical decompositions for extracting the squared amplituhedron volume form from the squared amplituhedron geometry with explicit examples and discuss the analogous procedure for the correlators.

Runge-Kutta Central Discontinuous Galerkin Methods for the Special Relativistic Hydrodynamics

AbstractThis paper develops Runge-Kutta PK-based central discontinuous Galerkin (CDG) methods with WENO limiter for the one- and two-dimensional special relativistic hydrodynamical (RHD) equations, K = 1,2,3. Different from the non-central DG methods, the Runge-Kutta CDG methods have to find two approximate solutions defined on mutually dual meshes. For each mesh, the CDG approximate solutions on its dual mesh are used to calculate the flux values in the cell and on the cell boundary so that the approximate solutions on mutually dual meshes are coupled with each other, and the use of numerical flux will be avoided. The WENO limiter is adaptively implemented via two steps: the “troubled” cells are first identified by using a modified TVB minmod function, and then the WENO technique is used to locally reconstruct new polynomials of degree (2K+1) replacing the CDG solutions inside the “troubled” cells by the cell average values of the CDG solutions in the neighboring cells as well as the original cell averages of the “troubled” cells. Because the WENO limiter is only employed for finite “troubled” cells, the computational cost can be as little as possible. The accuracy of the CDG without the numerical dissipation is analyzed and calculation of the flux integrals over the cells is also addressed. Several test problems in one and two dimensions are solved by using our Runge-Kutta CDG methods with WENO limiter. The computations demonstrate that our methods are stable, accurate, and robust in solving complex RHD problems.

The Cutter Trajectory Avoiding Intersections of Cuts

The technological support of cutting process often deals with different restrictions on cutter trajectory, particularly, (1) the part cut off a sheet does not require additional cuts; (2) the trajectory avoids intersections of the cuts. Let the homeomorphic image of a cutting plan be a plane Eulerian graph. Then a trail in this graph can be interpreted as the tool trajectory. The polynomial time algorithm for constructing of special type trails for a plane Eulerian graph being is considered in this paper. The considered type of trails allow an instrument path correspond the mentioned conditions. Algorithm works by transforming the initial plane Eulerian graph to a plane 4-regular graph for which the problem can be solved by polynomial time by constructing an AOE-trail, i.e. trail where adjacent edges are a neighbours in the cyclic order for their incident vertex, and a cycle of passed edges does not enclose not passed ones. This transformation needs splitting of all vertices of degree 2k, k ≥ 3 to k fictive ones and adding a cycle containing k fictive edges incident to these vertices. Then AOE-trail is being constructed for obtained 4-regular graph. This trail is to be the non-intersected OE-trail for the initial graph after absorbing all the fictive edges and vertices obtained while splitting vertices of initial graph.

Runge–Kutta discontinuous Galerkin methods for the special relativistic magnetohydrodynamics

This paper develops PK-based non-central and central Runge–Kutta discontinuous Galerkin (DG) methods with WENO limiter for the one- and two-dimensional special relativistic magnetohydrodynamical (RMHD) equations, K=1,2,3. The non-central DG methods are locally divergence-free, while the central DG are “exactly” divergence-free but have to find two approximate solutions defined on mutually dual meshes. The adaptive WENO limiter first identifies the “troubled” cells by using a modified TVB minmod function, and then uses the WENO technique and the cell average values of the DG solutions in the neighboring cells as well as the original cell averages of the “troubled” cells to locally reconstruct a new polynomial of degree (2K+1) inside the “troubled” cells instead of the DG solution. The WENO limiting procedure does not destroy the locally or “exactly” divergence-free property of magnetic field and is only employed for finite “troubled” cells so that the computational cost can be as little as possible. Several test problems in one and two dimensions are solved by using our non-central and central Runge–Kutta DG methods with WENO limiter. The numerical results demonstrate that our methods are stable, accurate, and robust in resolving complex wave structures.

Some Direct Estimates for Linear Combination of Linear Positive Convolution Operators

In this paper we have estimated some direct results for the even positive convolution integrals on C_2π, Banach space of 2 &pi; - periodic functions. Here, positive kernels are of finite oscillations of degree 2<i>k</i>. Technique of linear combination is used for improving order of approximation. Property of Central factorial numbers, inverse formulas, mixed algebraic –trigonometric formula is used throughout the paper.

Locally injective [formula omitted]-colourings of planar graphs

A colouring of the vertices of a graph is called injective if every two distinct vertices connected by a path of length 2 receive different colours, and it is called locally injective if it is an injective proper colouring. We show that for k≥4, deciding the existence of a locally injective k-colouring, and of an injective k-colouring, are NP-complete problems even when restricted to planar graphs. It is known that every planar graph of maximum degree ≤35k−52 allows a locally injective k-colouring. To compare the behaviour of planar and general graphs we show that for general graphs, deciding the existence of a locally injective k-colouring becomes NP-complete for graphs of maximum degree 2k (when k≥7).

On properties of integrals of the Legendre polynomial

Properties of the integrals $$P_{n0} (x) = P_n (x),P_{nk} (x) = \int\limits_{ - 1}^x {P_{n,k - 1} (y)dy} $$ of the Legendre polynomials Pn(x) on the base interval −1 ≤ x ≤ 1 are systematically considered. The generating function $$(1 - 2xz + z^2 )^{k - 1/2} = Q_k (x,z) + ( - 1)^k (2k - 1)!!\sum\limits_{n = k}^\infty {P_{nk} (x)z^{n + k} } $$ is defined; here, Q0 = 0 and Qk with k > 0 is a polynomial of degree 2k − 1 in each of the variables x and z. A second-order differential equation is derived, an analogue of Rodrigues’ formula is obtained, and the asymptotic behavior as n → ∞ is determined. It is proved that the representation $$P_{nk} (x) = (x^2 - 1)^k f_{nk} (x)$$ holds if and only if n ≥ k, where fnk is a polynomial divisible by neither x − 1 nor x + 1. The main result is the sharp bound $$|P_{nk} (\cos \theta )| < \frac{{A_k }} {{\nu ^{k + 1/2} }}\sin ^{k - 1/2} \theta ,n \geqslant k.$$ Here, $$\nu ^2 = \left( {n + \frac{1} {2}} \right)^2 - \left( {k^2 - \frac{1} {4}} \right)\left( {1 - \frac{4} {{\pi ^2 }}} \right),A_k = \sqrt t _k J_k (t_k ) \sim \mu _1 k^{1/6} ,\mu _1 = 0.674885, $$ where tk is the maximum of the function \(\sqrt t J_k (t)\) on the half-axis t > 0 and Jk(t) is the Bessel function. The first values Ak and differences Ak − μ1k1/6 are tabulated below as follows: Open image in new window

Geometry of the inversion in a finite field and partitions of PG(2k − 1, q) in normal rational curves

Let \({L = \mathbb{F}_{q^n}}\) be a finite field and let \({F=\mathbb{F}_q}\) be a subfield of L. Consider L as a vector space over F and the associated projective space that is isomorphic to PG(n − 1, q). The properties of the projective mapping induced by \({x \mapsto x^{-1}}\) have been studied in Csajbok (Finite Fields Appl. 19:55–66, 2013), Faina et al. (Eur. J. Comb. 23:31–35, 2002), Havlicek (Abh. Math. Sem. Univ. Hamburg 53:266–275, 1983), Herzer (Abh. Math. Sem. Univ. Hamburg 55:211–228 1985, Handbook of Incidence Geometry. Buildings and Foundations. Elsevier, Amsterdam, 1995). The image of any line is a normal rational curve in some subspace. In this note a more detailed geometric description is achieved. Consequences are found related to mixed partitions of the projective spaces; in particular, it is proved that for any positive integer k, if q ≥ 2k − 1, then there are partitions of PG(2k − 1, q) in normal rational curves of degree 2k − 1. For smaller q the same construction gives partitions in (q + 1)-tuples of independent points.

Cones of multipowers and combinatorial optimization problems

The cone of multipowers is dual to the cone of nonnegative polynomials. The relation of the former cone to combinatorial optimization problems is examined. Tensor extensions of polyhedra of combinatorial optimization problems are used for this purpose. The polyhedron of the MAX-2-CSP problem (optimization version of the two-variable constraint satisfaction problem) of tensor degree 4k is shown to be the intersection of the cone of 4k-multipowers and a suitable affine space. Thus, in contrast to SDP relaxations, the relaxation to a cone of multipowers becomes tight even for an extension of degree 4.

Hybridizable discontinuous Galerkin method (HDG) for Stokes interface flow

In this paper, we present a hybridizable discontinuous Galerkin (HDG) method for solving the Stokes interface problems with discontinuous viscosity and variable surface tension. The jump condition of the stress tensor across the interface is naturally incorporated into the HDG formulation through a constraint on the numerical flux. The most important feature of HDG method compared to other DG methods is that it reduces the number of globally coupled unknowns significantly when high order approximate polynomials are used. For problems with polygonal interfaces, it provides optimal convergence rates of order k+1 in L2-norm for the velocity, pressure and as well as the gradient of velocity. Furthermore, a new approximate velocity can be obtained by an element-by-element postprocessing which converges with order k+2 in the L2-norm. For Stokes interface problems with curved interfaces, we use general curvilinear element to ensure the optimal convergence rates. An error estimate is given for the approximation of the interface. It indicates that curvilinear elements of degree 2k+1 should be used for optimal convergence rate of order k+1.

Counting and computing the Rand and block distances of pairs of set partitions

The Rand distance of two set partitions is the number of pairs {x,y} such that there is a block in one partition containing both x and y, but x and y are in different blocks in the other partition. Let R(n,k) denote the number of distinct (unordered) pairs of partitions of n that have Rand distance k. For fixed k we prove that R(n,k) can be expressed as ∑jC(j,k)(nj)Bn−j where C(j,k) is a non-negative integer, Bn is the nth Bell number, and the summation range is of size less than 2k. If n⩾2k+2 then R(n,(n2)−k) can be expressed as a polynomial of degree 2k in n. This polynomial is explicitly determined for 0⩽k⩽3.We explain how to compute R(n,k) for k=0,1,…,(n2) in time O(Bn2) using the idea of a refinement of two partitions; we also present an O(n) algorithms for computing the refinement and coarsening when the partitions are represented as restricted growth strings.The block distance of two set partitions is the number of elements that are not in common blocks. We give formulae and asymptotics for B(n,k), the number of pairs of partitions of {1,2,…,n} with block distance k; a formula that is based on N(n), the number of pairs of partitions with no blocks in common. We develop an O(n) algorithm for computing the block distance.

THE HASSE PRINCIPLE FOR CERTAIN HYPERELLIPTIC CURVES AND FORMS

We prove that, for each positive integer n≥2, there is an infinite arithmetic family of hyperelliptic curves of genus n violating the Hasse principle explained by the Brauer–Manin obstruction. Using these families of curves, we show that, for any positive integer k≥1, there are infinitely many algebraic and arithmetic families of forms in three variables of degree 4k+2 such that they are counterexamples to the Hasse principle explained by the Brauer–Manin obstruction.

Hitting time results for Maker‐Breaker games

AbstractWe study Maker‐Breaker games played on the edge set of a random graph. Specifically, we analyze the moment a typical random graph process first becomes a Maker's win in a game in which Maker's goal is to build a graph which admits some monotone increasing property \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \begin{align*}\mathcal{P}\end{align*} \end{document}. We focus on three natural target properties for Maker's graph, namely being k ‐vertex‐connected, admitting a perfect matching, and being Hamiltonian. We prove the following optimal hitting time results: with high probability Maker wins the k ‐vertex connectivity game exactly at the time the random graph process first reaches minimum degree 2k; with high probability Maker wins the perfect matching game exactly at the time the random graph process first reaches minimum degree 2; with high probability Maker wins the Hamiltonicity game exactly at the time the random graph process first reaches minimum degree 4. The latter two statements settle conjectures of Stojaković and Szabó. We also prove generalizations of the latter two results; these generalizations partially strengthen some known results in the theory of random graphs. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011