The vertex isoperimetric problem for the powers of the diamond graph

We introduce a new graph for all whose cartesian powers the vertex isoperimetric problem has nested solutions. This is the fourth kind of graphs with this property besides the well-studied graphs like hypercubes, grids, and tori. In contrast to the mentioned graphs, our graph is not bipartite. We present an exact solution to the vertex isoperimetric problem on our graph by introducing a new class of orders that unifies all known isoperimetric orders defined on the cartesian powers of graphs.     

A counterexample to a conjecture related to the Jacobian problem

Translated fromMatematicheskie Zametki, Vol. 58, No. 3, pp. 452–455, September, 1995.     

A problem related to Foulkes’s conjecture

In this paper we study a class of symmetric matricesT indexed by positive integers m≥ n≥2 and defined as follows: for any positive integersp andq let ?_p,q be the set of partitions ofU = {1,2,3, ...,pq} into p blocks each of sizeq. Letm ≥n ≥ 2 be positive integers. By atransversal of α = A₁/A₂/.../A_n ∈ ?_n,m we mean a partitionß = B₁/B₂/.../B_m ∈ ? _m,n such that ‖A _i ∩B _j = 1 for every i= 1,2, ...,n and everyj = 1,2, ...,m. LetM be the zero-one matrix with rows indexed by the elements of ?_n,m and columns indexed by the elements of ?_m,n such that M_αß = 1 iffß is a transversal of α. We are interested in finding the eigenvalues and eigenspaces of the symmetric matrixT = MM^t. The nonsingularity ofT implies Foulkes’s Conjecture (for these values of m andn). In the casen = 2 we completely determine the eigenvalues and eigenspaces of T and in so doing demonstrate the non-singularity ofT. Forn = 3 we develop a fast algorithm for computing the eigenvalues ofT, and give numerical results in the cases m = 3,4, 5, 6.     

From Grushin to Heisenberg via an isoperimetric problem

The Grushin plane is a right quotient of the Heisenberg group. Heisenberg geodesics' projections are solutions of an isoperimetric problem in the Grushin plane.     

Boundary behaviour of the unique solution to a singular Dirichlet problem with a convection term

By Karamata regular variation theory and constructing comparison functions, we derive that the boundary behaviour of the unique solution to a singular Dirichlet problem −Δu=b(x)g(u)+λq|∇u|, u>0, x∈Ω, u_|∂Ω=0, which is independent of λq|∇uλ|, where Ω is a bounded domain with smooth boundary in RN, λ∈R, q∈(0,2], lims→⁰⁺g(s)=+∞, and b is non-negative on Ω, which may be vanishing on the boundary.     

On the isoperimetric problem in Euclidean space with density

We study the isoperimetric problem for Euclidean space endowed with a continuous density. In dimension one, we characterize isoperimetric regions for a unimodal density. In higher dimensions, we prove existence results and we derive stability conditions, which lead to the conjecture that for a radial log-convex density, balls about the origin are isoperimetric regions. Finally, we prove this conjecture and the uniqueness of minimizers for the density exp by using symmetrization techniques. First and second authors are partially supported by MCyT-Feder research project MTM2004-01387, fourth author by the National Science Foundation.     

A computer-assisted proof of Saari's conjecture for the planar three-body problem

The five relative equilibria of the three-body problem give rise to solutions where the bodies rotate rigidly around their center of mass. For these solutions, the moment of inertia of the bodies with respect to the center of mass is clearly constant. Saari conjectured that these rigid motions are the only solutions with constant moment of inertia. This result will be proved here for the planar problem with three nonzero masses with the help of some computational algebra and geometry.

     

On the isoperimetric problem with respect to a mixed Euclidean–Gaussian density

The isoperimetric problem with respect to the product-type density on the Euclidean space Rh×Rk is studied. In particular, existence, symmetry and regularity of minimizers is proved. In the special case k=1, also the shape of all the minimizers is derived. Finally, a conjecture about the minimality of large cylinders in the case k>1 is formulated.     

The exact asymptotic behaviour of the unique solution to a singular nonlinear Dirichlet problem

By Karamata regular varying theory, a perturbed argument and constructing comparison functions, we show the exact asymptotic behaviour of the unique solution near the boundary to a singular Dirichlet problem −Δu=b(x)g(u)+λf(u), u>0, x∈Ω, u_|∂Ω=0, which is independent on λf(u), and we also show the existence and uniqueness of solutions to the problem, where Ω is a bounded domain with smooth boundary in RN, λ>0, g∈C¹((0,∞),(0,∞)) and there exists γ>1 such that , ∀ξ>0, , the function is decreasing on (0,∞) for some s₀>0, and b is nonnegative nontrivial on Ω, which may be vanishing on the boundary.     

Demailly's Conjecture and the containment problem

We investigate Demailly's Conjecture for a general set of sufficiently many points. Demailly's Conjecture generalizes Chudnovsky's Conjecture in providing a lower bound for the Waldschmidt constant of a set of points in projective space. We also study a containment between symbolic and ordinary powers conjectured by Harbourne and Huneke that in particular implies Demailly's bound, and prove that a general version of this containment holds for generic determinantal ideals and defining ideals of star configurations.     

A geometric programming approach to the Van der Waerden conjecture on doubly stochastic matrices

LetD _p be the set of all doubly stochastic square matrices of orderp i.e. the set of allp × p matrices with non-negative entries with row and column sums equal to unity. The permanent of ap × p matrixA = (a _ij ) is defined byP(A) = _Sp II _i=1 ^p a _i(i) whereS _p is the symmetric group of orderp. Van der Waerden conjectured thatP(A) p !/p ^p for all A AD _p with equality occurring if and only ifA = J _p , whereJ _p is the matrix all of whose entries are equal to 1/p.The validity of this conjecture has been shown for a few values ofp and for generalp under certain assumptions. In this paper the problem of finding the minimum of the permanent of a doubly stochastic matrix has been formulated as a reversed geometric program with a single constraint and an equivalent dual program is given. A related problem of reversed homogeneous posynomial programming problem is also studied.     

An application of Newton–Puiseux charts to the Jacobian problem

We study 2-dimensional Jacobian maps using so-called Newton–Puiseux charts. These are multi-valued coordinates near divisors of resolutions of indeterminacies at infinity of the Jacobian map in the source space as well as in the target space. The map expressed in these charts takes a very simple form, which allows us to detect a series of new analytical and topological properties. We prove that the Jacobian Conjecture holds true for maps (f,g) whose topological degree is ≤5, for maps with gcd(degf,degg)≤16 and for maps with. gcd(degf,degg) equal to 2 times a prime.     

The Alon–Tarsi conjecture: A perspective on the main results

The Alon–Tarsi conjecture states that if $n$ is even, then the sum of the signs of the Latin squares of order $n$ is non-zero (Alon and Tarsi, 1992). The conjecture has been proven in the cases $n=p+1$ (Drisko, 1997), and $n=p?1$ (Glynn, 2010), where $p$ is an odd prime. This paper is intended to be a concise and largely self-contained account of these results, along with streamlined, and in some cases, original proofs that should be readily accessible to a mathematician with a background in combinatorics. We also discuss the relation between the Alon–Tarsi conjecture and Rota’s basis conjecture (Huang and Rota, 1994), and present some related problems, such as Zappa’s extension of the Alon–Tarsi conjecture (Zappa, 1997), and Drisko’s proof of the extended conjecture for $n=p$ (Drisko, 1998).     

A counterexample to the ``composition conjecture"

In this note we construct a class of counterexamples to the ``composition conjecture" concerning an infinitesimal version of the center problem for the polynomial Abel equation in the complex domain.

     

A proof of a conjecture on the Randić index of graphs with given girth

The Randić index R(G) of a graph G is defined by , where is the degree of a vertex u in G and the summation extends over all edges uv of G. Aouchiche, Hansen and Zheng proposed the following conjecture: For any connected graph on n≥3 vertices with Randić index R and girth g,

with equalities if and only if . This paper is devoted to giving a confirmative proof to this conjecture.