On the local existence of solutions to a class of nonconvex evolution inclusions

We prove the local existence of solutions to the Cauchy problemx'∈-?_F V(x)+F(x+f(t,x),x(0)=x ₀, where? _FV is the Fréchet subdifferential of a functionV with aψ-monotone subdifferential of order 2,F is an upper semicontinuous set-valued map contained in the Fréchet subdifferential of aφ- convex function of order two andf is a Carathéodory mapping.     

On the dimension of the space of ℝ-places of certain rational function fields

We prove that for every n ∈ ? the space M(K(x ₁, …, x _n ) of ?-places of the field K(x ₁, …, x _n ) of rational functions of n variables with coefficients in a totally Archimedean field K has the topological covering dimension dimM(K(x ₁, …, x _n )) ≤ n. For n = 2 the space M(K(x ₁, x ₂)) has covering and integral dimensions dimM(K(x ₁, x ₂)) = dim_? M(K(x ₁, x ₂)) = 2 and the cohomological dimension dim _G M(K(x ₁, x ₂)) = 1 for any Abelian 2-divisible coefficient group G.     

Additive operators preserving rank-additivity on symmetry matrix spaces

We characterize the additive operators preserving rank-additivity on symmetry matrix spaces. LetS _n(F) be the space of alln×n symmetry matrices over a fieldF with 2,3 ∈F ^*, thenT is an additive injective operator preserving rank-additivity onS _n(F) if and only if there exists an invertible matrixU∈M _n(F) and an injective field homomorphism ? ofF to itself such thatT(X)=cUX ^?U^T, ?X=(x_ij)∈S_n(F) wherec∈F ^*,X ^?=(?(x _ij)). As applications, we determine the additive operators preserving minus-order onS _n(F) over the fieldF.     

The hilbert series of the polynomial identities for the tensor square of the grassmann algebra

LetE be the Grassmann (or exterior) algebra of an infinite-dimensional vector space over a fieldK of characteristic 0. We compute the Hilbert series of the relatively free algebraF _m(varE?_K E)=K(x _l,...,x _m) /(T(E?_K E)∩K〈x _l,...,x _m〉) of the variety of algebrasvarE?_KE, whereT(E?_K E) is the set of all polynomial identities forE?_K EE from the free associative algebraK〈x ₁,x₂…〉.     

A nonlinear parabolic-Sobolev equation

Let M and L be (nonlinear) operators in a reflexive Banach space B for which Rg(M + L) = B and $\text{¦(Mx ? My) + α(Lx ? Ly)¦ ? | mx ? My |}$ for all α > 0 and pairs x, y in D(M) ∩ D(L). Then there is a unique solution of the Cauchy problem (Mu(t))′ + Lu(t) = 0, Mu(0) = v₀. When M and L are realizations of elliptic partial differential operators in space variables, this gives existence and uniqueness of generalized solutions of boundary value problems for nonlinear partial differential equations of mixed parabolic-Sobolev type.     

Singularities ofJ-holomorphic curves in almost complex 4-manifolds

This note concerns the structure of singularities of mapsf from a neighborhood of {0} in the complex plane ? to an almost complex manifold (V, J), which areJ-holomorphic in the sense thatdf oi =J odf and are singular (i.e.,df = 0) at {0}. The main result is that whenV has dimension 4, the topology of these singularities is the same as in the case whenJ is integrable. Thus, if the image Imf =C is not multiply-covered, there is a neighborhoodU of the pointx = f(0), such that the pair (U, U ∩C) is homeomorphic to the cone overS ³,K _x whereK _x is an algebraic knot in S³ that depends only on the germC atx.     

On the small sieve. II. sifting by composite numbers

For any set A of natural numbers let F(x, A) denote the number of natural numbers up to x that are divisible by no element of A and let H(x, K) be the maximum of F(x, A) when A runs over the sets not containing 1 and having a sum of reciprocals not greater than K. A logarithmic asymptotic formula is given for H(x, K)—in particular it shows H(x, K) < x^ε for K > K₀(ε)—and some related problems are discussed.     

A limiting absorption principle for Schrödinger operators with spherically symmetric exploding potentials

LetH=?Δ+V(r) be a Schrödinger operator with a spherically symmetric exploding potential, namely,V(r)=V _S(r)+V _L(r), whereV _S(r) is short-range and the exploding partV _L(r) satisfies the following assumptions: (a) Λ=lim sup _r→∞ V _L(r)<∞ (but Λ=?∞ is possible). Denote Λ⁺= max(Λ,0). (b)V _L(r)∈C ^2k (r ₀, ∞) and, with someδ>0 such that 2kδ>1: (d/dr) ^j V _L(r) · (Λ⁺?V _L(r))^?1=O(r ^jδ) asr → ∞,j=1, ..., 2k. (c) ∫ _r0 ^∞ dr|V _L(r|^1/2 dr|V _L(r)|^1/2=∞. (d) (d/dr)V _L(r)≦0. Under these assumptions a limiting absorption principle forR(z)=(H?z)^?1 is established. More specifically, ifK ?C ⁺={zImz≧0} is compact andK ∩ (?∞, Λ]=Ø thenR (z) can be extended as a continuous map ofK intoB (Y, Y*) (with the uniform operator topology), whereY ?L ²(R ⁿ) is a weighted-L ² space. To ensure uniqueness of solutions of (H?z)u=f, z ∈K, a suitable radiation condition is introduced.     

An existence-convergence theorem for a class of iterative methods

LetM ₀, characterized byx _k+1=G ₀(x _k),k?0,x ₀ prescribed, be an iterative method for the solution of the operator equationF(x)=0, whereF:X → X is a given operator andX is a Banach space. Let ω:X → X be a given operator, and let the methodM _mbe characterized byx _x+1,m =G _m(x _k,m),k?0,x _0,m prescribed, where $$G_i (x) = G_0 (x) - \sum\limits_{j = 0}^{i - 1} { F'(\omega (x))^{ - 1} F(G_j (x)), i = 1, . . . ,m,} $$ in whichG ₀:X → X is a given operator andF′:X → L(X) is the Fréchet derivative ofF. Sufficient conditions for the existence of a solutionx* _m ofF(x)=0 to which the sequence (x _k,m) generated from methodM _mconverges are given, together with a rate-of-convergence estimate.     

Elliptic equation with a singular potential in a domain with a conic point

This paper deals with the behavior of the nonnegative solutions of the problem $$- \Delta u = V(x)u, \left. u \right|\partial \Omega = \varphi (x)$$ in a conical domain Ω ? ? ⁿ , n ≥ 3, where 0 ≤ V (x) ∈ L₁(Ω), 0 ≤ ?(x) ∈ L₁(?Ω) and ?(x) is continuous on the boundary ?Ω. It is proved that there exists a constant C _*(n) = (n ? 2)²/4 such that if V ₀(x) = (c + λ ₁)|x|^?2, then, for 0 ≤ c ≤ C _*(n) and V(x) ≤ V ₀(x) in the domain Ω, this problem has a nonnegative solution for any nonnegative boundary function ?(x) ∈ L ₁(?Ω); for c > C _*(n) and V(x) ≥ V ₀(x) in Ω, this problem has no nonnegative solutions if ?(x) > 0.     

To what extent is a large space of matrices not closed under product?

Let (K) be a field. Given an arbitrary linear subspace V of M_n(K) of codimension less than n-1, a classical result states that V generates the (K)-algebra M_n(K). Here, we strengthen this statement in three ways: we show that M_n(K) is spanned by the products of the form AB with (A,B)∈V²; we prove that every matrix in M_n(K) can be decomposed into a product of matrices of V; finally, when V is a linear perplane of M_n(K) and n>2, we show that every matrix in M_n(K) is a product of two elements of V.     

A note on the numerical evaluation of finite integrals of oscillatory functions

A method is described for the numerical evaluation of integrals of the form ∫ _?1 ¹ f(x)K(m,x)dx, wheref(x) is smooth in [?1,1], whileK(m,x) is highly oscillatory for large values ofm.     

Odd cycles and matrices with integrality properties

A cycle of a bipartite graphG(V⁺, V^?; E) is odd if its length is 2 (mod 4), even otherwise. An odd cycleC is node minimal if there is no odd cycleC′ of cardinality less than that ofC′ such that one of the following holds:C′ ∩V ⁺ ?C ∩V ⁺ orC′ ∩V ^? ?C ∩V ^?. In this paper we prove the following theorem for bipartite graphs: For a bipartite graphG, one of the following alternatives holds:

-All the cycles ofG are even.

-G has an odd chordless cycle.

-For every node minimal odd cycleC, there exist four nodes inC inducing a cycle of length four.

-An edge (u, v) ofG has the property that the removal ofu, v and their adjacent nodes disconnects the graphG.

To every (0, 1) matrixA we can associate a bipartite graphG(V⁺, V^?; E), whereV ⁺ andV ^? represent respectively the row set and the column set ofA and an edge (i,j) belongs toE if and only ifa _ij = 1. The above theorem, applied to the graphG(V⁺, V^?; E) can be used to show several properties of some classes of balanced and perfect matrices. In particular it implies a decomposition theorem for balanced matrices containing a node minimal odd cycleC, having the property that no four nodes ofC induce a cycle of length 4. The above theorem also yields a proof of the validity of the Strong Perfect Graph Conjecture for graphs that do not containK ₄?e as an induced subgraph.     

Kernel of a map of a shift along the orbits of continuous flows

Let F:M × \mathbbR \mathbb{R} → M be a continuous flow on a topological manifold M. For every subset V ì M V \subset M , we denote by P(V) the set of all continuous functions x:V ? \mathbbR \xi :V \to \mathbb{R} such that \textF( x,x(x) ) = x {\text{F}}\left( {x,\xi (x)} \right) = x for all x ? V x \in V . These functions vanish at nonperiodic points of the flow, while their values at periodic points are integer multiples of the corresponding periods (in general, not minimal). In this paper, the structure of P(V) is described for an arbitrary connected open subset V ì M V \subset M .     

A criterion for generating full matrix algebras

Let M_n(F) be the algebra of n×n matrices over a field F, and let A∈M_n(F) have characteristic polynomial c(x)=p₁(x)p₂(x)?p_r(x) where p₁(x),…,p_r(x) are distinct and irreducible in F[x]. Let X be a subalgebra of M_n(F) containing A. Under a mild hypothesis on the p_i(x), we find a necessary and sufficient condition for X to be M_n(F).     

Period functions for $ {\mathcal{C}^0} $- and $ {\mathcal{C}^1} $-flows

Let F:M ×\mathbbR ? M {\mathbf{F}}:M \times \mathbb{R} \to M be a continuous flow on a manifold M, let V ⊂ M be an open subset, and let x:V ? \mathbbR \xi :V \to \mathbb{R} be a continuous function. We say that ξ is a period function if F(x, ξ(x)) = x for all x ∈ V. Recently, for any open connected subset V ⊂ M; the author has described the structure of the set P(V) of all period functions on V. Assume that F is topologically conjugate to some C¹ {\mathcal{C}^1} -flow. It is shown in this paper that, in this case, the period functions of F satisfy some additional conditions that, generally speaking, are not satisfied for general continuous flows.     

Commutativity of set-valued cosine families

Let K be a closed convex cone with nonempty interior in a real Banach space and let cc(K) denote the family of all nonempty convex compact subsets of K. If {F _t : t ≥ 0} is a regular cosine family of continuous additive set-valued functions F _t : K → cc(K) such that x ∈ F _t (x) for t ≥ 0 and x ∈ K, then $F_t \circ F_s (x) = F_s \circ F_t (x)fors,t \geqslant 0andx \in K$ .     

Weak solution of (r, (x, u(x)) u′(x))′ = (Fu)′(x) with r(0, u(0))u′(0) = ku(0), r(L, u(L))u′ (L) = hu(L) and k, h are suitable elements of [0, ∞]

We consider weak solutions to the nonlinear boundary value problem (r, (x, u(x)) u′(x))′ = (Fu)′(x) with r(0, u(0)) u′(0) = ku(0), r(L, u(L)) u′(L) = hu(L) and k, h are suitable elements of [0, ∞]. In addition to studying some new boundary conditions, we also relax the constraints on r(x, u) and (Fu)(x). r(x, u) > 0 may have a countable set of jump discontinuities in u and r(x, u)^?1?L_q((0, L) × (0, p)). F is an operator from a suitable set of functions to a subset of L_p(0, L) which have nonnegative values. F includes, among others, examples of the form (Fu)(x) = (1 ? H(x ? x₀)) u(x₀), (Fu)(x) = ∫_x^Lf(y, u(y)) dy where f(y, u) may have a countable set of jump discontinuities in u or F may be chosen so that (Fu)′(x) = ? g(x, u(x)) u′(x) ? q(x) u(x) ? f(x, u(x)) where q is a distributional derivative of an L₂(0, L) function.     

Disjoint matchings of graphs

In this paper, we prove a generalization of the familiar marriage theorem. One way of stating the marriage theorem is: Let G be a bipartite graph, with parts S₁ and S₂. If A ? S₁ and F(A) ? S₂ is the set of neighbors of points in A, then a matching of G exists if and only if Σ_x∈S2 min(1, | F^?1(x) ∩ A |) ≥ | A | for each A ? S₁. Our theorem is that k disjoint matchings of G exist if and only Σ_x∈S2 min (k, | F^?1(x) ∩ A |) ≥ k | A | for each A ? S₁.