An existence theorem for the complementarity problem

LetC be a pointed, solid, closed and convex cone in then-dimensional Euclidean spaceE ⁿ ,C* its polar cone,M:C→E ⁿ a map, andq a vector inE ⁿ . The complementarity problem (q|M) overC is that of finding a solution to the system $$(q|M) x \varepsilon C, M(x) + q \varepsilon C{^*} , \left\langle {x, M(x) + q} \right\rangle = 0.$$ It is shown that, ifM is continuous and positively homogeneous of some degree onC, and if (q|M) has a unique solution (namely,x=0) forq=0 and for someq=q ₀ ∈ intC*, then it has a solution for allq ∈E ⁿ .     

Fixed point theory and product,sub-, super-spaces

In this paper, we definen-segmentwise metric spaces and then we prove the following results:

1. (i)|Let (X, d) be ann-segmentwise metric space. ThenX ⁿ has the fixed point property with respect to uniformly continuous bounded functions if and only if, for any continuous functionF: C ^*(X) → C^*(X) and for anyn-tuple of distinct points x₁, x₂, ?, x_n ∈X, there exists anh ∈C ^*(X) such that $$F(h)(x_1 ) = h(x_1 ),i = 1,2,...,n;$$ whereC ^*(X) has either the uniform topology or the subspace product (Tychonoff) topology \((C^ * (X) \subseteq X^X )\) .
2. LetX _i (i = 1, 2, ?) be countably compact Hausdorff spaces such thatX ₁ × ? × X_n has the fixed point property for alln ∈N Then the product spaceX ₁ × X₂ × ? has the fixed point property. We shall also discuss several problems in the Fixed Point Theory and give examples if necessary. Among these examples, we have:
3. There exists a connected metric spaceX which can be decomposed as a disjoint union of a closed setA and an open setB such thatA andB have the fixed point property andX does not have.
4. There exists a locally compact metrizable spaceX which has the fixed point property but its one-point compactificationX ⁺ does not have the fixed point property.

Other relevant results and examples will be presented in this paper.     

Uppers to zero in R[x] and almost principal ideals

Let R be an integral domain with quotient field K and f(x) a polynomial of positive degree in K[x]. In this paper we develop a method for studying almost principal uppers to zero ideals. More precisely, we prove that uppers to zero divisorial ideals of the form I = f(x)K[x] ∩ R[x] are almost principal in the following two cases:

• J, the ideal generated by the leading coefficients of I, satisfies J ^?1 = R.
• I ^?1 as the R[x]-submodule of K(x) is of finite type.

Furthermore we prove that for I = f(x)K[x] ∩ R[x] we have:

• I ^?1 ∩ K[x] = (I: K(x) I).
• If there exists p/q ∈ I ^?1 ? K[x], then (q, f) ≠ 1 in K[x]. If in addition q is irreducible and I is almost principal, then I′ = q(x)K[x] ∩ R[x] is an almost principal upper to zero.

Finally we show that a Schreier domain R is a greatest common divisor domain if and only if every upper to zero in R[x] contains a primitive polynomial.     

Commuting and centralizing generalized derivations on lie ideals in prime rings

Let R be a noncommutative prime ring of characteristic different from 2, U the Utumi quotient ring of R, C the extended centroid of R, and L a noncentral Lie ideal of R. If F and G are generalized derivations of R and k ≥1 a fixed integer such that [F(x), x] _k x ? x[G(x), x] _k = 0 for any x ∈ L, then one of the following holds:

1. either there exists an a ∈ U and an α ∈ C such that F(x) = xa and G(x) = (a + α)x for all x ∈ R
2. or R satisfies the standard identity s ₄(x ₁, …, x ₄) and one of the following conclusions occurs

1. there exist a, b, c, q ∈ U, such that a ?b + c ?q ∈ C and F(x) = ax + xb, G(x) = cx + xq for all x ∈ R
2. there exist a, b, c ∈ U and a derivation d of U such that F(x) = ax+d(x) andG(x) = bx+xc?d(x) for all x ∈ R, with a + b ? c ∈ C.

     

A generalization of the katona theorem for crosst-intersecting families

LetX be ann-element set and letA and? be families of subsets ofX. We say thatA and? are crosst-intersecting if |A ∩ B| ≥ t holds for all A ∈A and for allB ∈ ?. Suppose thatA and ? are crosst-intersecting. This paper first proves a crosst-intersecting version of Harper's Theorem:

1. There are two crosst-intersecting Hamming spheresA ₀,? ₀ with centerX such that |A| ≤ |A ₀| and|?| ≤ |? ₀| hold.
2. Suppose thatt ≥ 2 and that the pair of integers (|A) is maximal with respect to direct product ordering among pairs of crosst-intersecting families. Then,A and? are Hamming spheres with centerX.

Using these claims, the following conjecture of Frankl is proven:

1. Ifn + t = 2k ? 1 then |A| |?| ≤ max \(\left\{ {\left( {K_k^n + \left( {_{k - 1}^{n - 1} } \right)} \right)^2 ,K_k^n K_{k - 1}^n } \right\}\) holds, whereK _l ⁿ is defined as \(\left( {_n^n } \right)\left( {_{n - 1}^n } \right) + \cdots + \left( {_l^n } \right).\)
2. Ifn + t = 2k then |A| |? ≤ (K _k ⁿ )² holds.

The extremal configurations are also determined.     

Right inverses for linear, constant coefficient partial differential operators on distributions over open half spaces

Results of Hörmander on evolution operators together with a characterization of the present authors [Ann. Inst. Fourier, Grenoble 40, 619–655 (1990)] are used to prove the following: Let P ∈ ?[z₁,...,z _n ] and denote by P _m its principal part. If P ? P_m is dominated by P _m then the following assertions for the partial differential operators P(D) and P _m(D) are equivalent for N ∈ S ^n?1:

1. P(D) and/or P_m D)admit a continuous linear right inverse on C ^∞(H ₊(N)).
2. P(D) admits a continuous linear right inverse on C ^∞ (? ⁿ ) and a fundamental solution E ∈ C ^∞(?ⁿ) satisfying Supp $E \subset \overline {H - (N)} $

where H ₊(N) := {x ∈ ? ⁿ :±(x,N) τ; 0}.     

Group inverse for a class of 2×2 anti-triangular block matrices over skew fields

Suppose K is a skew field. Let K ^m×n denote the set of all m×n matrices over K. In this paper, we give necessary and sufficient conditions for the existence and explicit representations of the group inverses of the block matrices in the following three cases, respectively:

1. $\mathrm{rank}(S)=\mathrm{rank}(B^{\pi}A)$ ;
2. $\mathrm{rank}(S)=\mathrm{rank}(AB^{\pi})$ ;
3. $\mathrm{rank}(S)=\mathrm{rank}(B^{\pi}A)=\mathrm{rank}(AB^{\pi})$ ,

where A,B,C??K ^n×n , B ^# exists, R(B)=R(C), N(B)=N(C) and S=B ^?? AB ^?? . The paper??s conclusions generalized some related results of Zhao and Bu (Electron. J. Linear Algebra 21:63?C75, 2010).     

Effective analysis of integral points on algebraic curves

LetK be an algebraic number field,S?S _\t8 a finite set of valuations andC a non-singular algebraic curve overK. Letx∈K(C) be non-constant. A pointP∈C(K) isS-integral if it is not a pole ofx and |x(P)| _v >1 impliesv∈S. It is proved that allS-integral points can be effectively determined if the pair (C, x) satisfies certain conditions. In particular, this is the case if

1. x:C→P ¹ is a Galois covering andg(C)≥1;
2. the integral closure of $\bar Q$ [x] in $\bar Q$ (C) has at least two units multiplicatively independent mod $\bar Q$ *.

This generalizes famous results of A. Baker and other authors on the effective solution of Diophantine equations.     

n-almost prime submodules

Let R be a commutative ring with identity. A proper submodule N of an R-module M will be called prime [resp. n-almost prime], if for r ∈ R and a ∈ M with ra ∈ N [resp. ra ∈ N \ (N: M) ^n?1 N], either a ∈ N or r ∈ (N: M). In this note we will study the relations between prime, primary and n-almost prime submodules. Among other results it is proved that:

1. If N is an n-almost prime submodule of an R-module M, then N is prime or N = (N: M)N, in case M is finitely generated semisimple, or M is torsion-free with dim R = 1.
2. Every n-almost prime submodule of a torsion-free Noetherian module is primary.
3. Every n-almost prime submodule of a finitely generated torsion-free module over a Dedekind domain is prime.
4. There exists a finitely generated faithful R-module M such that every proper submodule of M is n-almost prime, if and only if R is Von Neumann regular or R is a local ring with the maximal ideal m such that m ² = 0.
5. If I is an n-almost prime ideal of R and F is a flat R-module with IF ≠ F, then IF is an n-almost prime submodule of F.

     

An embedding theorem for a weighted space of Sobolev type and correct solvability of the Sturm-Liouville equation

We consider the weighted space W ₁ ⁽²⁾ (?,q) of Sobolev type $$W_1^{(2)} (\mathbb{R},q) = \left\{ {y \in A_{loc}^{(1)} (\mathbb{R}):\left\| {y''} \right\|_{L_1 (\mathbb{R})} + \left\| {qy} \right\|_{L_1 (\mathbb{R})} < \infty } \right\} $$ and the equation $$ - y''(x) + q(x)y(x) = f(x),x \in \mathbb{R} $$ Here f ε L ₁(?) and 0 ? q ∈ L ₁ ^loc (?). We prove the following:

1. The problems of embedding W ₁ ⁽²⁾ (?q) ? L ₁(?) and of correct solvability of (1) in L ₁(?) are equivalent
2. an embedding W ₁ ⁽²⁾ (?,q) ? L ₁(?) exists if and only if $$\exists a > 0:\mathop {\inf }\limits_{x \in R} \int_{x - a}^{x + a} {q(t)dt > 0} $$

     

Sur les Suites Presque Echangeables DansL q , 1≦q<2

Let 1≦q<p<2. We construct a bounded sequence (X _n ) _n∈N inL _q which defines a typeσ onL _q , such that:

1. (X _n ) _n∈N is equivalent to the unit vector basis ofl _p .
2. The l-conic classK ₁(σ) generated byσ is not relatively compact for the topology of uniform convergence on bounded sets ofL _q .
3. (X _n ) _n∈N has no almost exchangeable subsequence after any change of density.

This sequence does not verify the two natural conditions inL _q -spaces that ensure the existence of an almost symmetric subsequence.     

Approximation by antiderivatives

Let Ω ?C be an open set with simply connected components and suppose that the functionφ is holomorphic on Ω. We prove the existence of a sequence {φ ^(?n)} ofn-fold antiderivatives (i.e., we haveφ ⁽⁰⁾(z)∶=φ(z) andφ ^(?n)(z)=dφ ^(?n?1)(z)/dz for alln ∈ N₀ and z ∈ Ω) such that the following properties hold:

1. For any compact setB ?Ω with connected complement and any functionf that is continuous onB and holomorphic in its interior, there exists a sequence {n _k} such that {φ^?nk} converges tof uniformly onB.
2. For any open setU ?Ω with simply connected components and any functionf that is holomorphic onU, there exists a sequence {m _k} such that {φ^?mk} converges tof compactly onU.
3. For any measurable setE ?Ω and any functionf that is measurable onE, there exists a sequence {p _k} such that {φ ^(-Pk)} converges tof almost everywhere onE.

     

Restricted greedy clique decompositions and greedy clique decompositions ofK 4-free graphs

A greedy clique decomposition of a graph is obtained by removing maximal cliques from a graph one by one until the graph is empty. It has recently been shown that any greedy clique decomposition of a graph of ordern has at mostn ²/4 cliques. In this paper, we extend this result by showing that for any positive integerp, 3≤p any clique decomposisitioof a graph of ordern obtained by removing maximal cliques of order at leastp one by one until none remain, in which case the remaining edges are removed one by one, has at mostt _p-1( ⁿ ) cliques. Heret _p-1( ⁿ ) is the number of edges in the Turán graph of ordern, which has no complete subgraphs of orderp. In connection with greedy clique decompositions, P. Winkler conjectured that for any greedy clique decompositionC of a graphG of ordern the sum over the number of vertices in each clique ofC is at mostn ²/2. We prove this conjecture forK ₄-free graphs and show that in the case of equality forC andG there are only two possibilities:

1. G?K _n/2,n/2
2. G is complete 3-partite, where each part hasn/3 vertices.

We show that in either caseC is completely determined.     

Некоторые точные инт егральные оценки про изводных рациональных и алгеб раических функций. Пр иложения

Exact estimates are obtained for integrals of absolute values of derivatives and gradients, for integral moduli of continuity and for major variations of piecewise algebraic functions (in particular, for polynomials, rational functions, splines, etc.). These results are applied to the problems of approximation theory and to the estimates of Laurent and Fourier coefficients. Typical results:

1. IfE is a measurable subset of the circle or of a line in thez-plane andR(z) is a rational function of degree ≦n, ¦R(z)¦≦ (z∈E), then ∝_E ¦R′(z)¦dz¦≦ 2πn; the latter estimate is exact forn=0, 1, ... and everyE with positive measure;
2. Iff(x ₁,x ₂, ...,x _m) is a real valued piecewise algebraic function of order (n, k) on the unit ballD?R ^m (in particular, a real valued rational function of order ≦n), and ¦f¦≦1 onD, then ∝_D¦gradf¦dx≦2π ^m/2n/Π(m/2); herem≧1, n≧0, 1≦k<∞;
3. LetE=Π={z∶¦z¦=1}, and letc _m(R) be the mth Laurent coefficient ofR onΠ,C _m(n)=sup{¦cm(R)¦}, where sup is taken over allR from 1), then 1/7 min {n/¦m¦, 1} ≦C _m(n) ≦ min {n/¦m¦, 1}.

     

On the completion of excellent rings in characteristicp>0

Si considera il seguente problema posto da Grothendieck (E.G.A.): SeA è un anello eccellente edm un ideale diA, (A, m) ^{^}=m-adico completamento diA è eccellente? Si mostra che la risposta è positiva nei seguenti casi:

1. A=algebra di tipo finito su un DVR completo di caratteristicap>0;
2. A=algebra di tipo finito su un DVRC contenente un corpok di caratteristicap>0 e finito suk [C ^p ] oppure tale che:

1. per ogni sottocampok′ dik contenentek ^p tale che [k:k′]<∞, il modulo universale finito dei differenzialiD _{k′ (C)} esiste;
2. il corpo residuoK diC soddisfa rank _KK ? _K/k <∞
3. C ha una Der-base.

     

Positive Lyapunov exponent for generic one-parameter families of unimodal maps

Letf _{a a∈A} be a C² one-parameter family of non-flat unimodal maps of an interval into itself anda* a parameter value such that

1. f_a* satisfies the Misiurewicz Condition,
2. f_a* satisfies a backward Collet-Eckmann-like condition,
3. the partial derivatives with respect tox anda of f _a ⁿ (x), respectively at the critical value and ata*, are comparable for largen.

Thena* is a Lebesgue density point of the set of parameter valuesa such that the Lyapunov exponent of f_a at the critical value is positive, and f_a admits an invariant probability measure absolutely continuous with respect to the Lebesgue measure. We also show that given f_a* satisfying (a) and (b), condition (c) is satisfied for an open dense set of one-parameter families passing through f_a*.     

Analysis of bounded sets in a topological tensor product

The aim of this paper is to investigate the nature of bounded sets in a topological ∈-tensor product EX_∈* F of any two locally convex topological vector spaces E and F over the same scalar field K. Next, we apply the results of this investigation to the study of each of the following:

1. Totally summable families in EX_∈*F;
2. ∈-tensor product of DF-spaces;
3. Topological nature of the dual of E X_∈*F, where E and F are strong duals of Banach spaces;
4. Properties of bounded sets in an ∈-tensor product of metrizable spaces.

Forπ-tensor product, the result corresponding to (b) is well known (see Grothendieck¹) that if E and F are DF-spaces then EX_π* F and EX_π* F are DF-spaces and that the strong topology on the topological dual (EX_π*F)′, which equals the space of continuous bilinear forms on EXF, coincides with the bibounded topology. We study each of the problems from (a) to (d) for ∈-tensor products. For terminology, notations and the well-known results in the theory of topological vector spaces and the topological tensor products we refer to [1–11]. However, for convenience in presentation of the results of our investigation we give a brief survey of notations and fundamental theorems which are needed throughout this paper.     

On approximation of a function and its derivatives by a polynomial and its derivatives

Let g∈C~q[-1, 1] be such that g~((k))(±1)=0 for k=0,…,q. Let P_n be an algebraic polynomialof degree at most n, such that P_n~((k))(±1)=0 for k=0,…,[_2~ (q+1)]. Then P_n and its derivativesP_n~((k)) for k≤q well approximate g and its respective derivatives, provided only that P_n well approxi-mates g itself in the weighted norm ‖g(x)-P_n(x) (1-x~2)~(1/2)~q‖This result is easily extended to an arbitrary f∈C~q[-1, 1], by subtracting from f the polynomial ofminnimal degree which interpolates f~((0))…,f~((q)) at±1. As well as providing easy criteria for judging the simultaneous approximation properties of a givenPolynomial to a given function, our results further explain the similarities and differences betweenalgebraic polynomial approximation in C~q[-1, 1] and trigonometric polynomial approximation in thespace of q times differentiable 2π-periodic functions. Our proofs are elementary and basic in character,permitting the construction of actual error estimates for simultaneous approximation proedures for smallvalues of q.     

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.     

Singular integrals along Lipschitz curves with holomorphic kernels

If γ(x)=x+iA(x),tan ^?1‖A′‖_∞<ω<π/2,S _ω ⁰ ={z∈C}| |argz|<ω, or, |arg(-z)|<ω} We have proved that if φ is a holomorphic function in S _ω ⁰ and \(\left| {\varphi (z)} \right| \leqslant \frac{C}{{\left| z \right|}}\) , denotingT f (z)= ∫?(z-ζ)f(ζ)dζ, ?f∈C ₀(γ), ?z∈suppf, where C_c(γ) denotes the class of continuous functions with compact supports, then the following two conditions are equivalent:

1. T can be extended to be a bounded operator on L²(γ);
2. there exists a function ?₁ ∈H ^∞(S _ω ⁰ ) such that ?′₁(z)=?(z)+?(-z), ?z∈S _ω ⁰ ?z∈S _w ⁰ .