Onsager’s Conjecture on the Energy Conservation for Solutions of Euler Equations in Bounded Domains

Journal of Nonlinear Science - The Onsager’s conjecture has two parts: conservation of energy, if the exponent is larger than 1 / 3, and the possibility of dissipative Euler...     

Compatible path-cycle-decompositions of plane graphs

Sabidussi's Conjecture states that given an Euler trail T in a connected Euler graph, there always exists a cycle decomposition S such that consecutive edges of T belong to different cycles in S. This conjecture has been solved in a generalized form for planar Euler graphs. A similar result for arbitrary planar graphs is the content of this paper.     

Almost integrable evolution equations

We present a 2-component equation with exactly two nontrivial generalized symmetries, a counterexample to Fokas' conjecture that equations with as many symmetries as components are integrable. Furthermore we prove the existence of infinitely many evolution equations with finitely many symmetries. We introduce the concept of almost integrability to describe the situation where one has a finite number of symmetries. The symbolic calculus of Gel'fand-Dikiî andp-adic analysis are used to prove our results.     

Singularity analysis of a variant of the Painlevé–Ince equation

We examine by singularity analysis an equation derived by reduction using Lie point symmetries from the Euler–Bernoulli Beam equation which is the Painlevé–Ince Equation with additional terms. The equation possesses the same leading-order behaviour and resonances as the Painlevé–Ince Equation and has a Right Painlevé Series. However, it has no Left Painlevé Series. A conjecture for the existence of Left Painlevé Series for ordinary differential equations is given.     

Sui principi variazionali di Lagrange-D'Alembert e di Hamilton nella dinamica dei fluidi ideali classici

We show that the Euler equation of classical ideal fluid dynamics can be deduced either from the Lagrange-D' Alembert principle or from the Hamilton principle by means of only one method, which is valid both to ideal compressible and to incompressible fluids. Futhermore, making use of some thermodynamical relations we determine the conditions under which the motion of the medium must occur in order to be possible to introduce the Hamilton's action A and the Lagrangian L of the motion.     

A numerical method for the first-order wave equation with discontinuous initial data

We introduce a method, constructed such that numerical solutions of the wave equation are well behaved when the solutions also contain discontinuities. The wave equation serves as a model problem for the Euler equations when the solution contains a contact discontinuity. Numerical computations of linear equations and the Euler equations in one and two dimensions are presented. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 353–365, 1998     

The loop expansion of the Kontsevich integral, the null-move and S-equivalence

The Kontsevich integral of a knot is a graph-valued invariant which (when graded by the Vassiliev degree of graphs) is characterized by a universal property; namely it is a universal Vassiliev invariant of knots. We introduce a second grading of the Kontsevich integral, the Euler degree, and a geometric null-move on the set of knots. We explain the relation of the null-move to S-equivalence, and the relation to the Euler grading of the Kontsevich integral. The null-move leads in a natural way to the introduction of trivalent graphs with beads, and to a conjecture on a rational version of the Kontsevich integral, formulated by the second author and proven in Geom. Top 8 (2004) 115 (see also Kricker, preprint 2000, math/GT.0005284).     

The Arc‐Weighted Version of the Second Neighborhood Conjecture

Seymour conjectured that every oriented simple graph contains a vertex whose second neighborhood is at least as large as its first. Seymour's conjecture has been verified in several special cases, most notably for tournaments by Fisher  6 . One extension of the conjecture that has been used by several researchers is to consider vertex‐weighted digraphs. In this article we introduce a version of the conjecture for arc‐weighted digraphs. We prove the conjecture in the special case of arc‐weighted tournaments, strengthening Fisher's theorem. Our proof does not rely on Fisher's result, and thus can be seen as an alternate proof of said theorem.     

Do 3n − 5 edges force a subdivision of K5?

A conjecture of Dirac states that every simple graph with n vertices and 3n ? 5 edges must contain a subdivision of K₅. We prove that a topologically minimal counterexample is 5-connected, and that no minor-minimal counterexample contains K₄ – e. Consequently, Dirac's conjecture holds for all graphs that can be embedded in a surface with Euler characteristic at least ? 2.     

On the Extension of Onsager’s Conjecture for General Conservation Laws

Journal of Nonlinear Science - The aim of this work is to extend and prove the Onsager conjecture for a class of conservation laws that possess generalized entropy. One of the main findings of this...     

Sequential decision problems on isolated time domains

In this paper, we shall study the deterministic dynamic sequence problem on isolated time domains. After introducing the Euler equations and the transversality condition, we shall prove that the Euler equations and transversality condition are sufficient for the existence of the optimal solution. We shall also introduce the Bellman equation on isolated time scales. This equation will generalize the well-known Bellman equation in the theory of dynamic programming. As an application in financial economics, we shall optimize a sequence problem of growth model on isolated time domains.     

Circular Distributions and Euler Systems, II

The purpose of this paper is to investigate a conjecture about the universality of the circular distribution made by Robert Coleman. The algebraic property of the universal distribution is the main ingredient in studying Euler system of Kolyvagin and Rubin. We study the universality of the circular distribution by using the Iwasawa theory and the theory of the Euler systems. The conjecture is a characterization of Euler systems in the case of number field. The results here assert that Euler systems are essentially made out of cyclotomic units.     

On the Boltzmann equation for long‐range interactions

We study the Boltzmann equation without Grad's angular cutoff assumption. We introduce a suitable renormalized formulation that allows the cross section to be singular in both the angular and the relative velocity variables. Angular singularities occur as soon as one is interested in long‐range interactions, while singularities in the relative velocity variable occur in the study of soft potentials, in particular, Coulomb interaction. Together with several new estimates, this new formulation enables us to prove existence of weak solutions and to give a proof of a conjecture by Lions (appearance of strong compactness) under general, fully realistic assumptions. © 2001 John Wiley & Sons, Inc.     

A nonlinear stochastic growth model on discrete time domains

In this paper, we introduce nonlinear stochastic dynamic problems on discrete time domains where events may occur at unevenly spaced time points. We define Euler equation and transversality condition for the problem. We prove that the Euler equation and the transversality condition are sufficient for the existence of the optimal solution. Next we generalize discrete time Cagan type rational expectation model to multivariate case. As an application of the main results, we obtain an explicit solution to a log-linearized nonlinear stochastic growth model.     

关于Euler数一个猜想的证明

研究Euler数的猜想:对于任何素数p≡1 mod4,E(p-1)／20 mod p．根据Euler数本身的性质和Fourier级数对于Gauss和的应用,证明了这个猜想是成立的．     

Fair reception and Vizing's conjecture

In this paper we introduce the concept of fair reception of a graph which is related to its domination number. We prove that all graphs G with a fair reception of size γ(G) satisfy Vizing's conjecture on the domination number of Cartesian product graphs, by which we extend the well‐known result of Barcalkin and German concerning decomposable graphs. Combining our concept with a result of Aharoni, Berger and Ziv, we obtain an alternative proof of the theorem of Aharoni and Szabó that chordal graphs satisfy Vizing's conjecture. A new infinite family of graphs that satisfy Vizing's conjecture is also presented. © 2009 Wiley Periodicals, Inc. J Graph Theory 61: 45‐54, 2009     

An adaptive scheme for the approximation of dissipative systems

We propose a new scheme for the long time approximation of a diffusion when the drift vector field is not globally Lipschitz. Under this assumption, a regular explicit Euler scheme–with constant or decreasing step–may explode and implicit Euler schemes are CPU-time expensive. The algorithm we introduce is explicit and we prove that any weak limit of the weighted empirical measures of this scheme is a stationary distribution of the stochastic differential equation. Several examples are presented including gradient dissipative systems and Hamiltonian dissipative systems.     

Around a theorem of F. Dyson and A. Lenard: Energy equilibria for point charge distributions in classical electrostatics

We discuss several results in electrostatics: Onsager’s inequality, an extension of Earnshaw’s theorem, and a result stemming from the celebrated conjecture of Maxwell on the number of points of electrostatic equilibrium. Whenever possible, we try to provide a brief historical context and references.     

Counterexamples to a conjecture by Gross,Mansour and Tucker on partial-dual genus polynomials of ribbon graphs

Gross, Mansour and Tucker introduced the partial-dual orientable genus polynomial and the partial-dual Euler genus polynomial. They computed these two partial-dual genus polynomials of four families of ribbon graphs, posed some research problems and made some conjectures. In this paper, we introduce the notion of signed interlace sequences of bouquets and obtain the partial-dual Euler genus polynomials for all ribbon graphs with the number of edges less than 4 and the partial-dual orientable genus polynomials for all orientable ribbon graphs with the number of edges less than 5 in terms of signed interlace sequences. We check all the conjectures and find a counterexample to the Conjecture 3.1 in their paper: There is no orientable ribbon graph having a non-constant partial-dual genus polynomial with only one non-zero coefficient. Motivated by this counterexample, we further find an infinite family of counterexamples to the conjecture.     

On the Blaschke conjecture for 3-webs

We find relative differential invariants of orders eight and nine for a planar nonparallelizable 3-web such that their vanishing is necessary and sufficient for a 3-web to be linearizable. This resolves the Blaschke conjecture for 3-webs. We also give the algorithm for determining whether a given 3-web is linearizable, find the linearity condition for 3-webs and establish its relationships to the condition that a plane curve consists of flexes and to the Euler equation in gas-dynamics.