Arithmetic Properties of Summands of Partitions

Let d d, d2 2. We prove that for almost all partitions of an integer the parts are well distributed in residue classes mod d. The limitations of the uniformity of this distribution are also studied.     

Hahn-Banach extension property for Banach spaces over non-archimedean fields

LetK be a locally compact non-archimedean non-trivially valued field. It is proved the theorem: For a Banach space overK containing a dense subspace with the Hahn-Banach extension property one of the following two mutually exclusive conditions holds:E is a non-archimedean Banach space or the space {xE:f(x)=0 for allfE ^*} has no non-trivial continuous linear functionals. Two corollaries are also obtained.     

k-regular factors and semi-k-regular factors in bipartite graphs

LetG be a graph, andk1 an integer. LetU be a subset ofV(G), and letF be a spanning subgraph ofG such that deg _F (x)=k for allx V(G)–U. If deg _F (x)k for allxU, thenF is called an upper semi-k-regular factor with defect setU, and if deg _F (x)k for allxU, thenF is called a lower semi-k-regular factor with defect setU. Now letG=(X, Y;E(G)) be a bipartite graph with bipartition (X,Y) such that X=Yk+2. We prove the following two results.(1) Suppose that for each subsetU ₁X such that U ₁=max{k+1, X+1/2},G has an upper semi-k-regular factor with defect setU ₁Y, and for each subsetU ₂Y such that U ₂=max{k+1, X+1/2},G has an upper semi-k-regular factor with defect setXU ₂. ThenG has ak-factor.(2) Suppose that for each subsetU ₁X such that U ₁=X–1/k+1,G has a lower semi-k-regular factor with defect setU ₁Y, and for each subsetU ₂Y such that U ₂=X–1/k+1,G has a lower semi-k-regular factor with defect setXU ₂. ThenG has ak-factor.     

Analytic Approximations for a Singularly Perturbed Convection—Diffusion Problem with Discontinuous Data in a Half-Infinite Strip

We consider a singularly perturbed convection—diffusion equation, –u+v u=0, defined on a half-infinite strip, (x,y)(0,)×(0,1) with a discontinuous Dirichlet boundary condition: u(x,0)=1, u(x,1)=u(0,y)=0. Asymptotic expansions of the solution are obtained from an integral representation in two limits: (a) as the singular parameter 0⁺ (with fixed distance r to the discontinuity point of the boundary condition) and (b) as that distance r0⁺ (with fixed ). It is shown that the first term of the expansion at =0 contains an error function or a combination of error functions. This term characterizes the effect of discontinuities on the -behavior of the solution and its derivatives in the boundary or internal layers. On the other hand, near the point of discontinuity of the boundary condition, the solution u(x,y) is approximated by a linear function of the polar angle at the point of discontinuity (0,0).     

A Note on the Recursive Sequence x n + 1 = p k x n + p k − 1 x n − 1 +...+ p 1 x n − k + 1

We present some comments on the behavior of solutions of the difference equation where p _i 0, i = 1,..., k, k N, and x _–k ,..., x _–1 R.     

On signpost systems and connected graphs

By a signpost system we mean an ordered pair (W, P), where W is a finite nonempty set, P W × W × W and the following statements hold: if (u, v, w) P, then (v, u, u) P and (v, u, w) P, for all u, v, w W; if u v; then there exists r W such that (u, r, v) P, for all u, v W. We say that a signpost system (W, P) is smooth if the folowing statement holds for all u, v, x, y, z W: if (u, v, x), (u, v, z), (x, y, z) P, then (u, v, y) P. We say thay a signpost system (W, P) is simple if the following statement holds for all u, v, x, y W: if (u, v, x), (x, y, v) P, then (u, v, y), (x, y, u) P.By the underlying graph of a signpost system (W, P) we mean the graph G with V(G) = W and such that the following statement holds for all distinct u, v W: u and v are adjacent in G if and only if (u, v, v) P. The main result of this paper is as follows: If G is a graph, then the following three statements are equivalent: G is connected; G is the underlying graph of a simple smooth signpost system; G is the underlying graph of a smooth signpost system.Research was supported by Grant Agency of the Czech Republic, grant No. 401/01/0218.     

Rectangular convexity of convex domains of constant width

An E R ² is r-convex if for every x, y E there exists a closed rectangle R such that x, y R and R E. Several results about r-convexity appeared in [1]. Its authors formulated a conjecture about conditions for a compact, convex set in R ² to be r-convex. We prove this conjecture in the case of convex domains of constant width.     

Linear control theory and quasi-differential equations

Summary LetI be an interval of the real line, and letA andB ben×n complex-valued, Lebesgue measurable, matrix functions defined onI such thatAL _loc ¹(I) andBL _loc (I). Ifx=[x₁x₂x _n ] ^t andu=[u ₁ u ₂u _n ] ^t are column vectors defined onI such thatxAC _loc ¹anduL _loc ¹(I) then the linear control problem considered isx(t)=A(t) x(t)+B(t)u(t) (tI) wherex is the response, andu is the control. This paper is concerned with the problem of determining necessary and sufficient conditions onA andB to make (^*) fully controllable onI, without departing from the basic requirementsAL _loc ¹ (I) andBL _loc (I)Dedicated to Professor H.-W. Knobloch on the occasion of his sixtieth birthday     

Projectivities in Moufang-Klingenberg planes

Let = = (,,) be a Moufang-Klingenberg plane coordinatized by a local alternative ring R. We define the projectivities of a line g in geometrically as products of perspectivities. It is shown that under certain conditions the group of projectivities of g is generated by the algebraically defined permutations xx+t (tR), xcx (cR a unit), xx ^–.     

Inequalities of Different Metrics for Differentiable Periodic Functions, Polynomials, and Splines

We obtain new inequalities of different metrics for differentiable periodic functions. In particular, for p, q (0, ], q > p, and s [p, q], we prove that functions satisfy the unimprovable inequality

A Class of Iterative Algorithms and Solvability of Nonlinear Variational Inequalities Involving Multivalued Mappings

The solvability of the following class of nonlinear variational inequality (NVI) problems based on a class of iterative procedures, which possess an equivalence to a class of projection formulas, is presented.Determine an element x ^* K and u ^* T(x ^*) such that u ^*, x – x ^* 0 for all x K where T: K P(H) is a multivalued mapping from a real Hilbert space H into P(H), the power set of H, and K is a nonempty closed convex subset of H. The iterative procedure adopted here is represented by a nonlinear variational inequality: for arbitrarily chosen initial points x ⁰, y ⁰ K, u ⁰ T(y ⁰) and v ⁰ T(x ⁰), we have u ^k + x ^k+1 – y ^k , x – x ^k+1 0, x K, for u ^k T(y ^k ) and for k 0where v ^k + y ^k – x ^k , x – y ^k 0, x K and for v ^k T(x ^k ).     

Strengthening of the Kolmogorov Comparison Theorem and Kolmogorov Inequality and Their Applications

We obtain a strengthened version of the Kolmogorov comparison theorem. In particular, this enables us to obtain a strengthened Kolmogorov inequality for functions x L ^x (r), namely,

Nonexistence of certain symmetric spherical codes

A spherical 1-codeW is any finite subset of the unit sphere inn dimensionsS ^n–1, for whichd(u, v)1 for everyu, v fromW, uv. A spherical 1-code is symmetric ifuW implies –uW. The best upper bounds in the size of symmetric spherical codes onS ^n–1 were obtained in [1]. Here we obtain the same bounds by a similar method and improve these bounds forn=5, 10, 14 and 22.     

Two results concerning symmetric bi-derivations on prime rings

Summary LetR be a ring. A bi-additive symmetric mappingD(.,.): R × R R is called a symmetric bi-derivation if, for any fixedy R, the mappingx D(x, y) is a derivation. The purpose of this paper is to prove two results concerning symmetric bi-derivations on prime rings. The first result states that, ifD ₁ andD ₂ are symmetric bi-derivations on a prime ring of characteristic different from two and three such thatD ₁(x, x)D ₂(x,x) = 0 holds for allx R, then eitherD ₁ = 0 orD ₂ = 0. The second result proves that the existence of a nonzero symmetric bi-derivation on a prime ring of characteristic different from two and three, such that [[D(x, x),x],x] Z(R) holds for allx R, whereZ(R) denotes the center ofR, forcesR to be commutative.     

On a classical Nicoletti boundary value problem

We shall derive existence, uniqueness and comparison results for the functional differential equationx(t)=f(t, x), a. e.tI, with classical Nicoletti boundary conditionsx _i(t_i)=y _iX, iA, whereI is a real interval,A is a nonempty set andX is a Banach space.     

Addition of sequences in general fields

IfX, Y are two finite subsets of a fieldL and the characteristic ofL is either 0 or is sufficiently large compared to the cardinalities ofX andY then there exists az L uniquely representable asx+y,xX,yY.Dedicated to Prof. Dr. E. Hlawka on the occasion of his 60th birthday     

A Remark on Bernstein,Prokhorov, Bennett,Hoeffding, and Talagrand Inequalities

Let X ₁,..., X_n be independent random variables such that {X_j 1}=1 and E X _j=0 for all j. We prove an upper bound for the tail probabilities of the sum M _n=X₁+...+ X_n. Namely, we prove the inequality {M _nx} 3.7 {S_n x}, where S _n=₁+...+ _n is a sum of centered independent identically distributed Bernoulli random variables such that E S _n ² =ME M _n ² and {_k=1}=E S _n ² /(n+E S _n ² ) for all k (we call a random variable Bernoulli if it assumes at most two values). The inequality holds for x at which the survival function x{S _nx} has a jump down. For remaining x, the inequality still holds provided that we interpolate the function between the adjacent jump points linearly or log-linearly. If necessary, in order to estimate {S _nx} one can use special bounds for binomial probabilities. Up to the factor at most 2.375, the inequality is final. The inequality improves the classical Bernstein, Prokhorov, Bennett, Hoeffding, Talagrand, and other bounds.     

Covering sets by subsets

Let to every elementx of a finite setM be associated some nonempty subsetM (x) ofM in such a way that the implicationyM(x)xM(y) is fulfilled. We prove two upper estimations for the least number of setsM(x) which are necessary to coverM. Several applications to number theory are presented.     

A White Noise Approach to Stochastic Neumann Boundary-Value Problems

We illustrate the use of white noise analysis in the solution of stochastic partial differential equations by explicitly solving the stochastic Neumann boundary-value problem LU(x)–c(x)U(x)=0, xDR ^d ,(x)U(x)=–W(x), xD, where L is a uniformly elliptic linear partial differential operator and W(x), xR ^d , is d-parameter white noise.     

Affine modifications and affine hypersurfaces with a very transitive automorphism group

We study the modificationAA of an affine domainA which produces another affine domainA=A[I/f] whereI is a nontrivial ideal ofA andf is a nonzero element ofI. First appeared in passing in the basic paper of O. Zariski [Zar], it was further considered by E. D. Davis [Da]. In [Ka1] its geometric counterpart was applied to construct contractible smooth affine varieties non-isomorphic to Euclidean spaces. Here we provide certain conditions (more general than those in [Ka1]) which guarantee preservation of the topology under a modification.As an application, we show that the group of biregular automorphisms of the affine hypersurfaceXC ^k+2, given by the equationuv=(p(x ₁,...,x_k) wherepC[x ₁,...,x _k ],k2, actsm-transitively on the smooth part regX ofX for anymN. We present examples of such hypersurfaces diffeomorphic to Euclidean spaces.Partially supported by the NSA grant MDA904-96-01-0012.