A resultant system as the set of coefficients of a single resultant

Explicit expressions for polynomials forming a homogeneous resultant system of a set of m+1 homogeneous polynomial equations in n+1<m+1 variables are given. These polynomials are obtained as coefficients of a homogeneous resultant for an appropriate system of n+1 equations in n+1 variables, which is explicitly constructed from the initial system. Similar results are obtained for mixed resultant systems of sets of n + 1 sections of line bundles on a projective variety of dimension n < m. As an application, an algorithm determining whether one of the orbits under an action of an affine irreducible algebraic group on a quasi-affine variety is contained in the closure of another orbit is described.     

Generating differential equations satisfied by products of functions

Based on the coefficients of two homogeneous linear differential equations, a method is proposed to construct a third homogeneous linear differential equations which is satisfied by all products of the form uv, where u and v satisfy, respectively, the first and the second given differential equation. The method was used recently in the computation of rapidly oscillatory integrals with kernels which are products of Bessel functions and their variants.     

The Fredholm alternative for parabolic evolution equations with inhomogeneous boundary conditions

We study the Fredholm properties of parabolic evolution equations on R with inhomogeneous boundary values. These problems are transformed into evolution equations with inhomogeneities taking values in certain extrapolation spaces. Assuming that the underlying homogeneous problem is asymptotically hyperbolic, we show the Fredholm alternative for these equations. The results are applied to parabolic partial differential equations.     

Global nontrivial bifurcation of homogeneous operators with an application to the p-Laplacian

We prove the existence of a global solution branch of nontrivial solutions for a class of equations by a blow-up method. In particular, positively homogeneous problems and equations with the p-Laplace operator are considered.     

Periodic solutions for functional-differential equations of mixed type

In this paper we study the existence of ω-periodic solutions for some functional-differential equations of mixed type. Among the main results are the averaging principle and existence theorems for some equations with homogeneous nonlinearities. We use here the coincidence degree theory of Mawhin.     

Sturm theorems for second order linear nonhomogenous differential equations and localization of zeros of the solution

It is shown that Sturm theorems, formulated in the 1830??s ([1], [2], [3] and [4]) and valid for second order linear homogeneous differential equation L(y)??y??+a(x)y??+b(x)y=0, could as well be formulated for the class of nonhomogeneous linear differential equations L(y)=f(x). Criteria for the existence of oscillatory solutions of nonhomogeneous equations, as well as more exact locations of the zeros are given.     

Weighted and regularity estimates for nonlinear equations on Reifenberg flat domains

Global weighted Lp estimates are obtained for the gradient of solutions to nonlinear elliptic Dirichlet boundary value problems over a bounded nonsmooth domain. Morrey and Hölder regularity of solutions are also established, as a consequence. These results generalize various existing estimates for nonlinear equations. The nonlinearities are of at most linear growth and assumed to have a uniform small mean oscillation. The boundary of the domain, on the other hand, may exhibit roughness but assumed to be sufficiently flat in the sense of Reifenberg. Our approach uses maximal function estimates and Vitali covering lemma, and also known regularity results of solutions to nonlinear homogeneous equations.     

Three-webs with covariantly constant curvature and torsion tensors

In this paper, we study a special class of multidimensional 3-webs with covariantly constant curvature and torsion tensors. In the first part, we prove that 3-webs of the class belong to G-webs, i.e., there is a subfamily of adapted frames whose components of curvature and torsion tensors are constant. The structure of the homogeneous space G/H carrying the 3-web is described. Structure equations of the G-group are found. In the second part, we find structure equations of the W ^??-web and finite equations of some special web classes.     

High-order accurate equations describing vibrations of thin bars

A method for deriving one-dimensional wave propagation equations in thin inhomogeneous anisotropic bars based on the mathematical homogenization theory for periodic media is used to obtain equations governing the longitudinal and transverse vibrations of a homogeneous circular bar. The equations are derived up to O(ε⁸) terms and take into account variable body forces and surface loads. Here, ε is the ratio of the bar’s typical thickness to the typical wavelength.     

Free vibration of non-uniform Euler–Bernoulli beam under various supporting conditions using Chebyshev wavelet collocation method

This paper proposes operational matrix of rth integration of Chebyshev wavelets. A general procedure of this matrix is given. Operational matrix of rth integration is taken as rth power of operational matrix of first integration in literature. But, this study removes this disadvantage of Chebyshev wavelets method. Free vibration problems of non-uniform Euler–Bernoulli beam under various supporting conditions are investigated by using Chebyshev Wavelet Collocation Method. The proposed method is based on the approximation by the truncated Chebyshev wavelet series. A homogeneous system of linear algebraic equations has been obtained by using the Chebyshev collocation points. The determinant of coefficients matrix is equated to the zero for nontrivial solution of homogeneous system of linear algebraic equations. Hence, we can obtain ith natural frequencies of the beam and the coefficients of the approximate solution of Chebyshev wavelet series that satisfied differential equation and boundary conditions. Mode shapes functions corresponding to the natural frequencies can be obtained by normalizing of approximate solutions. The computed results well fit with the analytical and numerical results as in the literature. These calculations demonstrate that the accuracy of the Chebyshev wavelet collocation method is quite good even for small number of grid points.     

Nonlinear differential equations as invariants under group action on coset bundles: Burgers and Korteweg-de Vries equation families

Given a group, its coset spaces provide all homogeneous spaces for its action. A subgroup chain allows for the construction of a bundle of sections over a coset space of independent variables, where the fiber coordinates are dependent variables and all their partial derivatives up to some order, (i.e., the kth order jet). In this coset bundle, group invariants take the form of differential equations. We present two families of group-subgroup chains, one leading to various tensor Burgers-type differential equations, and the other to Korteweg-de Vries equations with an nth space derivative. Maps of the Hopf-Cole type appear in both families as transformations which intertwine the original group action to a multiplier realization of a normally extended group, yielding a new differential equation with greater symmetry.     

Stabilization of statistics in wave and Klein-Gordon equations with mixing. Scattering theory for infinite energy solutions

We consider wave and Klein-Gordon equations in the whole space ?ⁿ with arbitraryn≥2. We assume initial data to be homogeneous random functions in ?ⁿ with zero expectation and finite mean density of energy. Moreover, we assume initial data fit mixing condition of Ibragimov-Linnik type. We consider the distributions of the random solution at the moment of timet. The main results mean the convergence of this distribution to some Gaussian measure ast→∞. This is a central limit theorem for wave and Klein-Gordon equations. The limit Gaussian measures are invariant measures for equations considered. Corresponding stationary random solutions are ergodic and mixing in time. The results are inspired by mathematical problems of statistical physics.     

On C 1+α regularity of solutions of Isaacs parabolic equations with VMO coefficients

We prove that boundary value problems for fully nonlinear second-order parabolic equations admit L _p -viscosity solutions, which are in C ^1+α for an ${\alpha \in (0, 1)}$ . The equations have a special structure that the “main” part containing only second-order derivatives is given by a positive homogeneous function of second-order derivatives and as a function of independent variables it is measurable in the time variable and, so to speak, VMO in spatial variables.     

Non-convex self-dual Lagrangians and variational principles for certain PDEʼs

We study the concept and the calculus of non-convex self-dual (Nc-SD) Lagrangians and their derived vector fields which are associated to many partial differential equations and evolution systems. They yield new variational resolutions for large class of partial differential equations with variety of linear and non-linear boundary conditions including many of the standard ones. This approach seems to offer several useful advantages: It associates to a boundary value problem several potential functions which can often be used with relative ease compared to other methods such as the use of Euler–Lagrange functions. These potential functions are quite flexible, and can be adapted to easily deal with both non-linear and homogeneous boundary value problems.     

On the number of solutions of a type of one-dimensional pseudodifferential equations in Sobolev-Slobodetsky space

The paper considers homogeneous, one-dimensional pseudodifferential equations of nonnegative order with symbols of the form Σ _i=1 ^N th(k _i x + ω _i )A _i (ξ). Using a relationship between such equations and the systems of singular equations, some estimates for the number of solutions of pseudodifferential equations in the Sobolev-Slobodetsky space are obtained.     

Sub-Finsler geometry in dimension three

We define the notion of sub-Finsler geometry as a natural generalization of sub-Riemannian geometry with applications to optimal control theory. We compute a complete set of local invariants, geodesic equations, and the Jacobi operator for the three-dimensional case and investigate homogeneous examples.     

Strengthened Cauchy-Schwarz inequality for biorthogonal wavelets in Sobolev spaces

We prove a strengthened Cauchy-Schwarz inequality for one-dimensional biorthogonal wavelets. The functional frame is given by a class of Hilbert spaces, defined in terms of weighted Fourier transforms, which contain as relevant examples the standard Sobolev spaces H(s) as well as their homogeneous version. Intended applications concern multilevel and hierarchical methods for numerical approximation of partial differential equations.     

Efficient spectral ultraspherical-Galerkin algorithms for the direct solution of 2nth-order linear differential equations

Some efficient and accurate algorithms based on the ultraspherical-Galerkin method are developed and implemented for solving 2nth-order linear differential equations in one variable subject to homogeneous and nonhomogeneous boundary conditions using a spectral discretization. We extend the proposed algorithms to solve the two-dimensional 2nth-order differential equations. The key to the efficiency of these algorithms is to construct appropriate base functions, which lead to linear systems with specially structured matrices that can be efficiently inverted, hence greatly reducing the cost and roundoff errors.     

Decay estimates for homogeneous Boussinesq equations in a semi-infinite pipe

In this paper, we establish the spatial decay bounds for homogeneous Boussinesq equations in a semi-infinite pipe flow. Assuming that the entrance velocity and magnetic field data are restricted appropriately, and it converges to laminar flow as the distance down the pipe tends to infinity, we derive a second order differential inequality that leads to an exponential decay estimate for the energy E(z,t) defined in (27). We also indicate how to establish the explicit bound for the total energy.