On solution sets of multivalued differential equations

Let X be a Banach space, 2^x\? the nonempty subsets of X,J = [o,a]?R and F:J×X→2^x\? a multivalued map. We consider U′ ? F(t,u) a.e. on J, u(o) = X_p ? X. A solution of (1) is understood to be a.e. differentiable with u′ Bochner integrable over J such that u(t) =X₀ + ∫^{0 t} u′(s)ds on J and u′(t)?F(t,u(t)) a.e. Under appropriate conditions on F the set S of solutions to (1) is compact ≠ ? in C_X (J), the space of continuous v : J → X with ∣v∣₀ = max∣v(t)∣. We concentrate on maps F with F(t,.) upper semicontinuous andshow that S is connected or even a compact R_δ in the sense of Borsuk. This is interesting in itself, but also in connection with the multivalued Poincare map in case F is periodic in time.     

An implicit function theorem

Suppose thatF:DR ⁿ×R^mRⁿ, withF(x ⁰,y ⁰)=0. The classical implicit function theorem requires thatF is differentiable with respect tox and moreover that ₁ F(x ⁰,y ⁰) is nonsingular. We strengthen this theorem by removing the nonsingularity and differentiability requirements and by replacing them with a one-to-one condition onF as a function ofx.     

On cubic polynomials I. Hua's estimate of exponential sums

Hua andChen gave estimates of sums wheree(z)=e ^2iz and is a polynomial of the typef(x)/q wheref(x)=a _k x ^k +...+a ₁ x with integer coefficients having gcd (q, a _k ,...,a ₁)=1 But no good estimates hold for these sums whenq is small in comparison tok. We therefore consider here a related but different class of polynomials. Special emphasis is given to the cubic case.In subsequent papers of this series we shall deal with cubic exponential sums in many variables and withp-adic and rational zeros of systems of cubic forms.Partially supported by NSF contract NSF-MCS-8015356.     

Connexité des sous-niveaux des fonctionnelles intégrales

Let (T, ℐ, μ) be a σ-finite atomless measure space,p∈[1,∞),E a real Banach space andf a measurable function:E xT→ℝ. We denote byF the functionalF: and byDom(F) its domain, it is the set {uεL ^p(T,E):ū(t)=f(u),t)εL ¹(T)}, and we prove that the sublevelsS(λ)={u:F(u)≤λ} are all connected in the subspaceDom(F) of the Banach spaceL ^p(T, E).     

On the existence of solutions of a class of Monge-Ampére equations

In this paper, the Dirichlet problem for the Monge-Ampére equation det(u _ij)=F(x,u,Δu) on a convex bounded domain ΩσR″ is considered. The author establishes a newC ^o-estimate and gives some new existence results. He also presents a new proof for theC ^3,α-estimates of solutions, which not only weakens the smooth assumptions forF but also applies to more general nonlinear elliptic systems.     

Random walks in quenched i.i.d. space-time random environment are always a.s. diffusive

We consider a general model of discrete-time random walk X_t on the lattice , = 1,..., in a random environment ={(t,x):(t,x)⁺¹} with i.i.d. components (t,x). Previous results on the a.s. validity of the Central Limit Theorem for the quenched model required a small stochasticity condition. In this paper we show that the result holds provided only that an obvious non-degeneracy condition is met. The proof is based on the analysis of a suitable generating function, which allows to estimate L₂ norms by contour integrals.Partially supported by C.N.R. (G.N.F.M.) and M.U.R.S.T. research funds.Partially supported by C.N.R. (G.N.F.M.) and M.U.R.S.T. research funds, by R.F.F.I. grants n. 99-01-00284, 97-01-00714, and CRDF research funds N RM1-2085.Partially supported by C.N.R. (G.N.F.M.) and M.U.R.S.T. research funds.Mathematics Subject Classification (2000): 60J15, 60F05, 60G60, 82B41     

The primitive ideal space of solvable Lie groups

LetG be a connected and simply connected solvable Lie group. In a previous paper (cf.[22]) we associated withG a familyM of geometrical objects (generalized orbits), and with each elementO ofM a unitary equivalence classF(O) of factor representations. IfG is nilpotent,M coincides with the orbit space of the coadjoint representation, and the mapOF(O) reproduces essentially the Kirillov isomorphism betweenM and the dual ofG. As a fargoing extension of this, the principal result of the present paper asserts, that upon assigning to 0M the kernel of the representation, associated with some element ofF(O), of the groupC ^* algebraC ^*(G), we obtain a bijection betweenM and the primitive ideal space ofC ^*(G).This work was supported by a grant from the National Science Foundation     

A new continuation method for complementarity problems with uniformP-functions

The complementarity problem with a nonlinear continuous mappingf from the nonnegative orthantR ₊ ⁿ ofR ⁿ intoR ⁿ can be written as the system of equationsF(x, y) = 0 and(x, y) R ₊ ²ⁿ , whereF denotes the mapping from the nonnegative orthantR ₊ ²ⁿ ofR ²ⁿ intoR ₊ ⁿ × Rⁿ defined byF(x, y) = (x ₁y₁,,x_ny_n, f₁(x) – y₁,, f_n(x) – y_n) for every(x, y) R ₊ ²ⁿ . Under the assumption thatf is a uniformP-function, this paper establishes that the mappingF is a homeomorphism ofR ₊ ²ⁿ ontoR ₊ ⁿ × Rⁿ. This result provides a theoretical basis for a new continuation method of tracing the solution curve of the one parameter family of systems of equationsF(x, y) = tF(x ⁰, y⁰) and(x, y) R ₊ ²ⁿ from an arbitrary initial point(x ⁰, y⁰) R ₊ ²ⁿ witht = 1 until the parametert attains 0. This approach is an extension of the one used in the polynomially bounded algorithm recently given by Kojima, Mizuno and Yoshise for solving linear complementarity problems with positive semi-definite matrices.     

A note on the Liouville method applied to elliptic eventually degenerate fully nonlinear equations governed by the Pucci operators and the Keller–Osserman condition

This paper is devoted to the study of the viscosity solutions of

Cyclic codes over $${{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}$$

In this work, we focus on cyclic codes over the ring \mathbbF₂+u\mathbbF₂+v\mathbbF₂+uv\mathbbF₂{{{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}} , which is not a finite chain ring. We use ideas from group rings and works of AbuAlrub et.al. in (Des Codes Crypt 42:273–287, 2007) to characterize the ring (\mathbbF₂+u\mathbbF₂+v\mathbbF₂+uv\mathbbF₂)/(xⁿ-1){({{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2})/(x^n-1)} and cyclic codes of odd length. Some good binary codes are obtained as the images of cyclic codes over \mathbbF₂+u\mathbbF₂+v\mathbbF₂+uv\mathbbF₂{{{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}} under two Gray maps that are defined. We also characterize the binary images of cyclic codes over \mathbbF₂+u\mathbbF₂+v\mathbbF₂+uv\mathbbF₂{{{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}} in general.     

On the regularity of the minima of quasiconvex integrals

We prove in this paper theC ^∞ regularity for a “very strict” local minimum of classC _loc ^ρ , ρ>3, of functionals with genuine degenerate quasiconvex integrand depending on a vector-valued function u. Such a minimum satisfies the condition: for all x∈Ω, there exists a neighbourhoodK(x) ofx in Ω andC ₁ (x)>0,C ₂ (x)>0,1≥ε(x)>0, such that for all real ϕ∈c ₀ ^∞ (K). This work is supported by the National Natural Science Foundation of China and the Fok Ying Tung Education Foundation.     

Local mappings on spaces of differentiable functions

We study conditions under which a functionalF(u, B) defined for everyu∈C ^k(Ω;R ^m ) and every Borel subsetB of Ω admits the integral representation

Strikte Konvexität für Variationsprobleme auf demn-dimensionalen Torus

We consider a variational problem with an integrandF:R ⁿ ×R×R ⁿ →R that isZ-periodic in the firstn+1 variables and satisfies certain growth-conditions. By a recent result of Moser, there exist for every α∈R ⁿ minimal solutionsu:R ⁿ →R minimising ƒF(x, u(x), u _x (x)) dx with respect to compactly supported variations ofu and such that sup |u(x)-αx|<∞. Given such a minimal solutionu we define the average action (whereB _r is ther-ball around 0∈R ⁿ ) and show thatM(α) is indeed independent of the minimal solutionu satisfying sup |u(x)-αx|<∞. We prove that this average actionM(α) is strictly convex in α.      

Regularity of Weak Solutions of Nonlinear Equations with Discontinuous Coefficients

In this paper, we prove that the weak solutions u∈Wloc^1, p （Ω） （1 〈p〈∞） of the following equation with vanishing mean oscillation coefficients A（x）： -div[（A（x）△↓u·△↓u）p-2/2 A（x）△↓u＋│F（x）│^p-2 F（x）]=B（x, u, △↓u）, belong to Wloc^1, q （Ω）（A↓q∈（p, ∞）, provided F ∈ Lloc^q（Ω） and B（x, u, h） satisfies proper growth conditions where Ω ∪→R^N（N≥2） is a bounded open set, A（x）=（A^ij（x）） N×N is a symmetric matrix function.     

The existence of probability measures with specified projections

Let X and Y be locally compact-compact topological spaces, F X×Y is closed, and P(F) is the set of all Borel probability measures on F. For us to find, for the pair of probability measures (_x, _y P (X)×P(Y), a probability measure P(F) such that _X = _X ^–1 , _Y = _Y ^–1 it is necessary and sufficient that, for any pair of Borel sets A X, B Y for which (A× B) F=Ø, the condition _XA+ _YB 1 holds.Translated from Matematicheskie Zametki, Vol. 14, No. 4, pp. 573–576, October, 1973.     

Algebraic cycles on products of elliptic curves over <Emphasis Type="Italic">p</Emphasis>-adic fields

We show that for any odd prime p there is a smooth projective threefold W defined over a p-adic field such that the Chow group CH²(W)/ and the Griffiths group Griff²(W)/ are infinite for suitable primes . We further give examples of smooth projective fourfolds over these p-adic fields for which the -torsion subgroup CH³ is infinite.     

Equivariant K-Homology and Restriction to Finite Cyclic Subgroups

For a discrete group G, we prove that a G-map between proper G–CW-complexes induces an isomorphism in G-equivariant K-homology if it induces an isomorphism in C-equivariant K-homology for every finite cyclic subgroup C of G. As an application, we show that the source of the Baum–Connes assembly map, namely K _* ^G (E(G, in)), is isomorphic to K _* ^G (E(G, )), where E(G, ) denotes the classifying space for the family of finite cyclic subgroups of G. Letting be the family of virtually cyclic subgroups of G, we also establish that and related results.     

A note on uniqueness of entropy solutions to degenerate parabolic equations in {\mathbb{R}^N}

We study the Cauchy problem in \mathbbR^N{\mathbb{R}^N} for the parabolic equation

Existence of the Vejvoda-Shtedry spaces

We investigate the linear periodic problem u _tt −u _xx =F(x, t), u(x+2π, t)=u(x, t+T)=u(x, t), ∈ ℝ², and establish conditions for the existence of its classical solution in spaces that are subspaces of the Vejvoda-Shtedry spaces. Ternopol’ Pedagogical Institute, Ternopol’. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 2, pp. 302–308, February, 1997.     

Complexity bound for the absolute factorization of parametric polynomials

An algorithm is constructed for the absolute factorization of polynomials with algebraically independent parametric coefficients. It divides the parameter space into pairwise disjoint pieces such that the absolute factorization of polynomials with coefficients in each piece is given uniformly. Namely, for each piece there exist a positive integer l ≤ d, l variables C₁, ..., C_l algebraically independent over the ground field F, and rational functions b_J,j of the parameters and of the variables C₁, ..., C_l such that for any parametric polynomial f with coefficients in this piece, there exist c₁, ..., with f = Π _j G _j where G _j = Σ_|J| B _J,j Z ^J is absolutely irreducible. Here Z = (Z₀, ..., Z_n) are the variables of f, each B_J,j is the value of b_J,j at the coefficients of f and c₁, ..., c_l, and denotes the algebraic closure of F. The number of pieces does not exceed (2d²+1)^2n+3d+5, and the algorithm performs arithmetic operations in F (thus the number of operations is exponential in the number r = ( _n+1 ^n+1+d ) of coefficients of f), and its binary complexity is bounded by if F = ℚ and by if , where d is an upper bound on the degrees of polynomials. The techniques used include the Hensel lemma and the quantifier elimination in the theory of algebraically closed fields. Bibliography: 20 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 316, 2004, pp. 5–29.