Bott functions and the euler characteristic

In terms of the Euler characteristic, we obtain the condition of existence of Bott functions on differentiable manifolds that have a set of critical points formed by connected homeomorphic submanifolds. State Navigation Academy, Kirovograd. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 10, pp. 1431–1432, October, 1999.     

Asymptotics of functions that are a generalization of the euler gamma function

The asymptotic behavior of two classes of functions expressed by integrals is considered in the paper. For special values of the parameters the functions of these classes are 1/(z) or (z+1). The necessity of their study is occasioned by the connection problem for an ordinary linear differential equation with two singular points. By means of a number of lemmas and the method of descent, a theorem is proved which gives the asymptotics of these functions as ¦z¦ in some sector.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 104, pp. 123–129, 1981.     

Compositions of convex functions and fully linear models

Derivative-free optimization (DFO) is the mathematical study of the optimization algorithms that do not use derivatives. One branch of DFO focuses on model-based DFO methods, where an approximation of the objective function is used to guide the optimization algorithm. Proving convergence of such methods often applies an assumption that the approximations form fully linear models—an assumption that requires the true objective function to be smooth. However, some recent methods have loosened this assumption and instead worked with functions that are compositions of smooth functions with simple convex functions (the max-function or the \(\ell _1\) norm). In this paper, we examine the error bounds resulting from the composition of a convex lower semi-continuous function with a smooth vector-valued function when it is possible to provide fully linear models for each component of the vector-valued function. We derive error bounds for the resulting function values and subgradient vectors.     

Note on the compressible euler equations with zero temperature

This note presents the behavior of solution of the compressible Euler equations as the temperature drops to zero by the simple riemann problem.     

Further results on the euler and Genocchi numbers

Summary We characterize the ordinary generating functions of the Genocchi and median Genocchi numbers as unique solutions of some functional equations and give a direct algebraic proof of several continued fraction expansions for these functions. New relations between these numbers are also obtained.     

Linear algebra of convex sets and the euler characteristic

It is shown that surprisingly many operations over convex sets can be extended to multilinear mappings on appropriate chosen linear spaces if one replaces convex sets by their characteristic functions. Some generalizations of results known in combinatorial geometry of convex sets are given too.     

Linear algebra of convex sets and the euler characteristic

It is shown that surprisingly many operations over convex sets can be extended to multilinear mappings on appropriate chosen linear spaces if one replaces convex sets by their characteristic functions. Some generalizations of results known in combinatorial geometry of convex sets are given too.     

Incompressible navier-stokes and euler limits of the boltzmann equation

We consider solutions of the Boltzmann equation, in a d-dimensional torus, d = 2, 3, For macroscopic times τ = t/?^N, ? « 1, t ≧ 0, when the space variations are on a macroscopic scale x = ?^N?1r, N ≧ 2, x in the unit torus. Let u(x, t) be, for t ≦ t₀, a smooth solution of the incompressible Navier Stokes equations (INS) for N = 2 and of the Incompressible Euler equation (IE) for N > 2. We prove that (*) has solutions for t ≦ t₀ which are close, to O(?²) in a suitable norm, to the local Maxwellian [p/(2πT)^d/2]exp{?[v ? ?u(x,t)]²/2T } with constant density p and temperature T . This is a particular case, defined by the choice of initial values of the macroscopic variables, of a class of such solutions in which the macroscopic variables satisfy more general hydrodynamical equations. For N ≧ 3 these equations correspond to variable density IE while for N = 2 they involve higher-order derivatives of the density.     

Gevrey regularity with weight for incompressible euler equation in the half plane

In this work we prove the weighted Gevrey regularity of solutions to the incompressible Euler equation with initial data decaying polynomially at infinity. This is motivated by the well-posedness problem of vertical boundary layer equation for fast rotating fluid. The method presented here is based on the basic weighted L²-estimate, and the main difficulty arises from the estimate on the pressure term due to the appearance of weight function.     

Existence of global smooth solutions for euler equations with symmetry

Eventhough existence of global smooth solutions for one dimensional quasilinear hyperbolic systems has been well established, much less is known about the corresponding results for higher dimensional cases. In this paper, we study the existence of global smoothe solutions for the initial-boundary value problem ofo Euler equtions satisfying γ law with damping and exisymmetry, or spherical symmetry. When the damping is strong enough, we give some sufficient conditions for existence of global smooth solutions as 1<γ< 5 3 and 5 3 <γ<3 . The proof is based on technical estimation of the C ¹ norm of the solutions.     

Vassiliev invariants and finite-dimensional approximations of the euler equation in magnetohydrodynamics

We consider Hamiltonian systems that correspond to Vassiliev invariants defined by Chen’s iterated integrals of logarithmic differential forms. We show that Hamiltonian systems generated by first-order Vassiliev invariants are related to the classical problem of motion of vortices on the plane. Using second-order Vassiliev invariants, we construct perturbations of Hamiltonian systems for the classical problem of n vortices on the plane. We study some dynamical properties of these systems.     

Hyperbolic euler formula in the n-dimensional Minkowski space

In this paper, the properties ofn-dimensional Minkowski space discussed by using Clifford algebra. The hyperbolic Euler formula is given inn-dimensional Minkowski space.