Stability and bifurcations on a finite time interval in variational inequalities

In this paper, using the Brouwer topological degree, we prove an existence result for finite variational inequalities. This approach is also applied to obtain the existence of periodic solutions for a class of evolution variational inequalities.

In this paper, using the Brouwer topological degree, the authors prove an existence result for finite variational inequalities. This approach is also used to obtain the existence of periodic solutions for a class of evolution variational inequalities.

1. Integral transforms on a sphere.- 1. The generalized kernels (xy)?+-, (+-xy + iO)?.- 2. The operator E(?), its relation with the Fourier and Mellin transform.- 3. The action of E(?) on spherical functions.- 4. Operators related with the transform E(?).- 5. The spaces Hs(?, Sn-1). The operator E(?) on the spaces Hs(?, Sn-1).- 6. An analog of the Paley-Wiener theorem for the operator E(?).- 2. The Fourier transform and convolution operators on spaces with weighted norms.- 1. The spaces H?s(?n).- 2. The Fourier transform on the spaces Hs?(?n).- 3. Convolution operator on the spaces Hs?(?n).- 4. The spaces Hs?(?m,?m-n).- 5. Transversal operators and special representations.- 6. Estimates for the convolution operator on the spaces H?s(?m, ?m-n).- 3. Meromorphic pseudodifferential operators.- 1. Canonical meromorphic pseudodifferential operators.- 2. Operations on canonical meromorphic pseudodifferential operators.- 3. General meromorphic pseudodifferential operators.- 4. Traces of meromorphic pseudodifferential operators.- 5. Meromorphic pseudodifferential operators on strongly oscillating functions.- 6. Estimates for meromorphic pseudodifferential operators.- 7. Periodic meromorphic pseudodifferential operators.- 8. Change of variables in meromorphic pseudodifferential operators.- 4. Pseudodifferential operators with discontinuous symbols on manifolds with conical singularities.- 1. Pseudodifferential operators on ?n.- 2. Pseudodifferential operators on a conic manifold.- 3. Pseudodifferential operators on manifolds with conical points.- 4. Algebras generated by pseudodifferential operators of order zero.- 5. The spectrum of a C* -algebra of pseudodifferential operators with discontinuous symbols on a closed manifold.- 1. Results from the theory* of C* -algebras.- 2. The spectrum of a C* -algebra of pseudodifferential operators with discontinuities of the first kind in the symbols on a smooth closed manifold (statement of the main theorem).- 3. Representations of the algebra $$ \mathfrak{G} $$(?) generated by the operators E(?)-1F(o, ?)E(?).- 4. Representations of an algebra $$ \mathfrak{G} $$(lx).- 5. Proof of theorem 2.1.- 6. Ideals in the algebra of pseudodifferential operators with discontinuous symbols.- 7. Spectra of C* -algebras of pseudodifferential operators on a manifold with conical points.- 8. The spectrum of a C* -algebra of pseudodifferential operators with oscillating symbols.- 6. The spectrum of a C* -algebra of pseudodifferential operators on a manifold with boundary.- 1. The algebras $$ \mathfrak{G} $$c(?).- 2. The algebras $$ \mathfrak{G} $$(?).- 3. The algebras $$ \mathfrak{G} $$c(l?).- 4. The algebras $$ \mathfrak{G} $$(l?).- 5. The spectrum of an algebra of pseudodifferential operators on a manifold with boundary.- Bibliographical sketch.- References.

An effective numerical technique for solving a special class of ordinary difference equations

Point symmetries are obtained for all equations in the KP hierarchy. The Lie algebra for each equation is infinite dimensional and involves several arbitrary functions of the corresponding time tN. The symmetry algebra is a semidirect sum of a Virasoro algebra and a Kac–Moody one. The “positive” part of this algebra is embedded into the known W∞ algebra of KP symmetries and into the free fermion algebra ĝl(∞). The corresponding action on the tau-function is presented. The negative part of the point symmetries does not fit into the free fermion algebra, but is embedded into a P∞ algebra, based on the algebra of pseudodifferential operators.

The Quantization of Edge Symbols.- On Rays of Minimal Growth for Elliptic Cone Operators.- Symbolic Calculus of Pseudo-differential Operators and Curvature of Manifolds.- Weyl Transforms, Heat Kernels, Green Functions and Riemann Zeta Functions on Compact Lie Groups.- On the Fourier Analysis of Operators on the Torus.- Wave Kernels of the Twisted Laplacian.- Super-exponential Decay of Solutions to Differential Equations in ?d.- Gevrey Local Solvability for Degenerate Parabolic Operators of Higher Order.- A New Aspect of the L p-extension Problem for Inhomogeneous Differential Equations.- Continuity in Quasi-homogeneous Sobolev Spaces for Pseudo-differential Operators with Besov Symbols.- Continuity and Schatten Properties for Pseudo-differential Operators on Modulation Spaces.- Algebras of Pseudo-differential Operators with Discontinuous Symbols.- A Class of Quadratic Time-frequency Representations Based on the Short-time Fourier Transform.- A Characterization of Stockwell Spectra.- Exact and Numerical Inversion of Pseudo-differential Operators and Applications to Signal Processing.- On the Product of Localization Operators.- Gelfand-Shilov Spaces, Pseudo-differential Operators and Localization Operators.- Continuity and Schatten Properties for Toeplitz Operators on Modulation Spaces.- Microlocalization within Some Classes of Fourier Hyperfunctions.

This paper deals with the asymptotic behaviour of the solutions of linear initial boundary value problems with constant coefficients on the half-line and on finite intervals. We assume that the boundary data are periodic in time and we investigate whether the solution becomes time-periodic after sufficiently long time. Using Fokas’ transformation method, we show that, for the linear Schrödinger equation, the linear heat equation and the linearized KdV equation on the half-line, the solutions indeed become periodic for large time. However, for the same linear Schrödinger equation on a finite interval, we show that the solution, in general, is not asymptotically periodic; actually, the asymptotic behaviour of the solution depends on the commensurability of the time period T of the boundary data with the square of the length of the interval over π .

We consider stability and bifurcation on a finite time interval in a thermoviscoelastoplastic contact problem posed in the form of a variational system depending on a parameter. To describe stability and bifurcation, the latter being understood as stability loss on a finite time interval, we use the frequency-domain approach. We present frequency-domain conditions for the existence of determining observations and the absolute dichotomy of the variational equation.

The set ℓ Pol of complex numbers ζ , which can be approximated in a polynomial time cζ ·n k depending from the precision 2−n, is a field. The operations of the field can be approximated in a time<c · n 2 and better relative to the time one needs to approximate the operands ζ 1 ,ζ 2 with precision 2−n. In the case of addition and multiplication it can be done for |ζ 1 |, |ζ 2 |<a with a c depending from a and the constants in the running time of the polynomial time approximation of ζ 1,ζ 2, but in the case of division the constant depends additionally from |ζ | . The same holds for the ring of the polynomials ℓ Pol [x ] and the field of the rational functions f ∈ ℓ Pol (x ). The constant factor in the polynomial time bound of the approximation of f (ζ ) depend additionally from the distance of ζ from the singularities of f . The field ℓ Pol is algebraically closed. In the case that the polynomials have constant coefficients there exist algorithms to approximate the roots of the polynomials in a linear time with the Newton method if the roots are simple and an isolation of the roots is precomputed [Sch82]. But this precomputation needs much time [ESY06]. The best known algorithms to solve this problem need a time O (m 5(τ +logm )2), m the degree of the polynomial and τ the length of the binary representation of the coefficients. The problem with not constant coefficients has been attacked on base of bitstreams in [Eig08], [ESY06]. In [Eig08] on can find an excellent overview of the state of art. In the case the coefficients are not fixed constants but given by approximations the situation is much more complicated because the precomputation depends from the precision of the approximation. It is clear that in general the approximation of a root of a polynomial with coefficients, which cannot be approximated in a time O (n k ), cannot be done in this time. Ker-I Ko discusses the approximation problems on base of the unary representation of the numbers [Ko91]. Polynomial time algorithms relative to dyadic and unary representation are obviously not the same. The motivation of this paper is the question how far the mentioned problems can be solved by global polynomial time algorithms. This are algorithms with a running time <c·n k with the constant c only depending from the constants of the polynomial time approximations of the arguments and the domain of the function to be computed. We prove that there exist analytic functions, which are computable on all finite domains in this sense and that analytic functions for which this is true can be represented by a power series in z.

Pseudo differential operators and integrable systems in AXIOM

We discuss a sub-algebra ¸ of Pseudo Differential Operators of which is Fredholm-closed and commutative modulo the ideal of compact operators. The algebra ¸ contains in particular all LP - Fourier-multipliers of Mihlin type. is shown to be an algebra with symbol, and its symbol space is determined. As an application we get simple criteria for an operator in ¸to be Fredholm. Furthermore, we regard algebras ¸ on Sobolev spaces , which are isometrically isomorphic to ¸ and have equivalent properties. Finally we conclude Fredholm criteria for mroe general Differential and Pseudo Differential Operators.

Given an algebra of pseudo-differential operators on a manifold, an elliptic element is said to be a reduction of orders, if it induces isomorphisms of Sobolev spaces with a corresponding shift of smoothness. Reductions of orders on a manifold with boundary refer to boundary value problems. We employ specific smooth symbols of arbitrary real orders and with parameters, and we show that the associated operators induce isomorphisms between Sobolev spaces on a given manifold with boundary. Such operators for integer orders have the transmission property and belong to the calculus of Boutet de Monvel [1], cf. also [9]. In general, they fit to the algebra of boundary value problems without the transmission property in the sense of [17] and [24]. Order reducing elements of the present kind are useful for constructing parametrices of mixed elliptic problems. We show that order reducing symbols have the Volterra property and are parabolic of anisotropy 1; analogous relations are formulated for arbitrary anisotropies. We then investigate parameter-dependent operators, apply a kernel cut-off construction with respect to the parameter and show that corresponding holomorphic operator-valued Mellin symbols reduce orders in weighted Sobolev spaces on a cone with boundary. We finally construct order reducing operators on a compact manifold with conical singularities and boundary.

An algebra of pseudo‐differential operators

Algebras of pseudodifferential operators, volume 43 : B.A. Plamenevskii, Mathematics and its Applications (Soviet Series). Kluwer, Dordrecht, 1989. 291 pp., Dfl. 210, US $99, £69, ISBN 0-7923-0231-1

The trace class ideal of a $$H^*$$-algebra of pseudo-differential operators on $$(0,\infty )$$