Applications of Riemannian and Einstein–Weyl Geometry in the Theory of Second-Order Ordinary Differential Equations

We consider some properties of four-dimensional Riemannian spaces whose metric coefficients are associated with the coefficients of second-order nonlinear differential equations, and we study the properties of three-dimensional Einstein–Weyl spaces related to the dual equations b = g(a,b,b), where the function g(a,b,b) satisfies a special partial differential equation.     

The Problem of Finding the Kernels in the System of Integro-Differential Maxwell’s Equations

Journal of Applied and Industrial Mathematics - We pose the direct and inverse problems of finding the electromagnetic field and the diagonal memory matrix for the reduced canonical system of...     

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...     

Solutions to the Definite Solution Problem of Differential Equations in Space W_2~l

In this PaPer,we diseuss the definite solution Problem of differential equations in Hilbert funetion sPaeeW_2~l.Aeeording to different definite solutionProblems,we ehoose the eorresPonding     

Faraday’s Form and Maxwell’s Equations in the Heisenberg Group

In this note we present a geometric formulation of Maxwell’s equations in Carnot groups (connected simply connected nilpotent Lie groups with stratified Lie algebra) in the setting of the intrinsic complex of differential forms defined by M. Rumin. Restricting ourselves to the first Heisenberg group \mathbbH¹{\mathbb{H}^{1}}, we show that these equations are invariant under the action of suitably defined Lorentz transformations, and we prove the equivalence of these equations with differential equations “in coordinates”. Moreover, we analyze the notion of “vector potential”, and we show that it satisfies a new class of 4th order evolution differential equations.     

Green’s Functions of Partial Differential Equations with Involutions

In this paper we develop a way of obtaining Green''s functions of partial differential equations with linear involutions by reducing the equation to a higher-order PDE without involutions. The developed theory is applied to a model of heat transfer in a conducting plate which is bent in half.     

Resonance Problem for a Class of Duffing’s Equations

Consider the Duffing's equation+g(x)=f(t),(1)where g∈C(R,R)and f∈P≡{f∈C(R,R);f is ω-periodic for some ω>0}.The functiong is said to be resonant if there exists f∈P such that eq.(1) has no bounded solutions on[0,∞).Using a generalized version of the Poincare-Birkhoff fixed point theorem,theauthors establish conditions on g which guarantee the following result holds:for any f∈Pwith period ω,there exists K≥0 such that eq.(1) has infinitely many kω-periodic solutionsfor every integer k≥K.In such a case,g is clearly non-resonant.     

An Inverse Problem for Maxwell’s Equations in Anisotropic Media

The authors consider Maxwell's equations for an isomagnctic anisotropic and inhomogeneous medium in two dimensions, and discuss an inverse problem of determining the permittivity tensor (ε1ε2ε2ε3) and the permeabilityμin the constitutive relations from a finite number of lateral boundary measurements. Applying a Carlcman estimate, the authors prove an estimate of the Lipschitz type for stability, provided thatε1,ε2,ε3,μsatisfy some a priori conditions.     

On Schwarzian Triangle Functions,Automorphic Forms and a Generalization of Ramanujan’s Triple Differential Equations

Let G be the group of the fractional linear transformations generated by

$$T(\tau ) = \tau + \lambda ,S(\tau ) = \frac{{\tau \cos \frac{\pi }{n} + \sin \frac{\pi }{n}}}{{ - \tau \sin \frac{\pi }{n} + \cos \frac{\pi }{n}}};$$

where

$$\lambda = 2\frac{{\cos \frac{\pi }{m} + \cos \frac{\pi }{n}}}{{\sin \frac{\pi }{n}}};$$

m, n is a pair of integers with either n ≥ 2,m ≥ 3 or n ≥ 3,m ≥ 2; τ lies in the upper half plane H.

A fundamental set of functions f₀, f_i and f_∞ automorphic with respect to G will be constructed from the conformal mapping of the fundamental domain of G. We derive an analogue of Ramanujan’s triple differential equations associated with the group G and establish the connection of f₀, f_i and f_∞ with a family of hypergeometric functions.     

Estimates of the Green’s Functions for the Elasto-static Equations and Stokes Equations in a Three Dimensional Lipschitz Domain

In this paper, we study a Green’s functions G ^E , G ^S for an elasto-static equations and Stokes equations in a three-dimensional bounded Lipschitz domain Ω. We prove that there is a positive constant c > 0 depending on the Lipschitz constant such that for all . Furthermore, we show that there is a positive constant η ∈ (0,1) depending on the Lipschitz constant such that for all . The second author is partially supported by Korea Research Foundation Grant KRF C-00005.     

On the Exceptional Set of Goldbach’s Problem in Short Intervals

Denote by E[X,X+H] the set of even integers in [X,X+H] that are not a sum of two primes (i.e. that are not Goldbach numbers). Here we prove that there exists a (small) positive constant such that for we have .     

Convergence of Newton‘’s Method and Uniqueness of the Solution of Equations in Banach SpacesⅡ

Some results on convergence of Newton‘s method in Banach spaces are established under the assumption that the derivative of the opderators satisfies the radius or center Lipschitz condition with a weak L average.     

Chebyshev’s Problem of the Moments of Nonnegative Polynomials

Mathematical Notes - We study the problem of P. L. Chebyshev (proposed in 1883) concerning the extreme values of moments of nonnegative polynomials with weight on the interval $$[-1,1]$$ at a fixed...     

On the Geometrical and Physical Meaning of Newton’s Solution to Kepler’s Problem

In the short treatise De Motu (1684),which serves as a precursor to the Principia Mathematica (1687),Newton essentially deals with the following two problems.     

Global Differential Geometry of 1-Resolvable C∞ Curves in the Plane

A C nonimmersed curve in the Euclidean plane c: S1 E² is 1-resolvable if its lift to the orthonormal frame bundle c: S¹ F extends to an immersion. The objective of this paper is to relate the shape of the planar curve with the differential topology of its lift. Specifically, we derive inequalities relating geometric invariants of c with topological invariants of c. The corresponding equalities will identify the simplest 1-resolvable curves.