On a permutablity problem for groups

Letm, n be positive integers. We denote byR(m, n) (respectivelyP(m, n)) the class of all groupsG such that, for everyn subsetsX ₁, X₂, . . .,X _n of sizem ofG there exits a non-identity permutation σ such that $X_1 X_2 ...X_n \cap X_{\sigma (1)} X_{\sigma (2)} ...X_{\sigma (n)} \ne \not 0$ (respectively X₁X₂ . . .X _n = X_σ(1)X{σ(2)} . . . X{gs(n)}). Let G be a non-abelian group. In this paper we prove that

1. G ∈ P(2,3) if and only ifG isomorphic to S₃, whereS _n is the symmetric group onn letters.
2. G ∈ R(2, 2) if and only if¦G¦ ≤ 8.
3. IfG is finite, thenG ∈ R(3, 2) if and only if¦G¦ ≤ 14 orG is isomorphic to one of the following: SmallGroup(16,i), i ∈ {3, 4, 6, 11, 12, 13}, SmallGroup(32,49), SmallGroup(32, 50), where SmallGroup(m, n) is the nth group of orderm in the GAP [13] library.

     

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.     

Modification of a Finsler space by a normalized semi-parallel vector field

LetF _n be a Finsler space with metric functionF(x, y). M. Matsumoto [6] has defined a modified Finsler spaceF _n ^* whose metric functionF ^*(x, y) is given byF ^*2 = = F² + (X_i(x)yⁱ)², whereX _i are the components of a covariant vector which is a function of coordintae only. Since a concurrent vector is a function of coordinate only, Matsumoto and Eguchi [9] have studied various properties of the modified Finsler spaceF _n ^* under the assumption thatX _i are the components of a concurrent vector field inF _n. In this paper we shall introduce the concept of semi-parallel vector field inF _n and study the properties of modified Finsler spaceF _n ^* .     

A note on pseudocompactifications

There exists a countable spaceV _ω such that:

1. V _ω has a single non-isolated point,
2. V _ω has no pseudocompactificationX witht(X)=ω.

     

On HCO spaces. An uncountable compactT 2 space,different from ℵ1+1, which is homeomorphic to each of its uncountable closed subspaces,which is homeomorphic to each of its uncountable closed subspaces

LetX be an Hausdorff space. We say thatX is a CO space, ifX is compact and every closed subspace ofX is homeomorphic to a clopen subspace ofX, andX is a hereditarily CO space (HCO space), if every closed subspace is a CO space. It is well-known that every well-ordered chain with a last element, endowed with the interval topology, is an HCO space, and every HCO space is scattered. In this paper, we show the following theorems: Theorem (R. Bonnet):

1. Every HCO space which is a continuous image of a compact totally disconnected interval space is homeomorphic to β+1 for some ordinal β.
2. Every HCO space of countable Cantor-Bendixson rank is homeomorphic to α+1 for some countable ordinal α.

Theorem (S. Shelah):Assume \(\diamondsuit _{\aleph _1 } \) . Then there is a HCO compact space X of Cantor-Bendixson rankω ₁} and of cardinality ?₁ such that:

1. X has only countably many isolated points,
2. Every closed subset of X is countable or co-countable,
3. Every countable closed subspace of X is homeomorphic to a clopen subspace, and every uncountable closed subspace of X is homeomorphic to X, and
4. X is retractive.

In particularX is a thin-tall compact space of countable spread, and is not a continuous image of a compact totally disconnected interval space. The question whether it is consistent with ZFC, that every HCO space is homeomorphic to an ordinal, is open.     

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*.     

Complex Banach spaces whose duals areL 1-spaces

We prove that for a complex Banach spaceA the following properties are equivalent:

1. A ^* is isometric to anL ₁(μ)-space;
2. every family of 4 balls inA with the weak intersection property has a non-empty intersection;
3. every family of 4 balls inA such that any 3 of them have a non-empty intersection, has a non-empty intersection.

     

Existence of best approximation elements inC(Q,X)

Generalizing the result of A. L. Garkavi (the caseX = ?) and his own previous result concerningX = ?), the author characterizes the existence subspaces of finite codimension in the spaceC(Q, X) of continuous functions on a bicompact spaceQ with values in a Banach spaceX, under some assumptions concerningX. Under the same assumptions, it is proved that in the space of uniform limits of simple functions, each subspace of the form $$\left\{ {g \in B:\smallint _Q \left\langle {g(t),d\mu _i } \right\rangle = 0,i = 1,...,n} \right\},$$ whereμ _i ∈ C(Q, X)^* are vector measures of regular bounded variation, is an existence subspace (the integral is understood in the sense of Gavurin).     

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.     

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.     

Analytische Fortsetzbarkeit von zufälligen Potenzreihen bezüglich abhängiger Zufallsvariabler

Рассматриваются слу чайная величина \(\mathfrak{X} = (X_n (\omega ))\) , удовлетворяющая усл овиюE(X _n ⁴ )≦M, и соответствующ ий случайный степенн ой ряд \(f_x (z;\omega ) = \mathop \sum \limits_{n = 0}^\infty a_n X_n (\omega )z^n\) . Устанавливаются тео ремы непродолжимост и почти наверное:

1. дляf _x при условиях с лабой мультипликати вности на \(\mathfrak{X}\) ,
2. для \(f_{\tilde x}\) , где \(\mathop \mathfrak{X}\limits^ \sim = (\mathop X\limits^ \sim _n )\) есть подп оследовательность в \(\mathfrak{X}\) ,
3. для по крайней мере од ного из рядовf _x′ илиf _x″ , где \(\mathfrak{X}'\) и \(\mathfrak{X}''\) — некоторые п ерестановки \(\mathfrak{X}\) , выбираемые универс ально, т. е. независимо от коэффициентовa _n .

     

Cores,cutsets and the fixed point property

The purpose of this paper is the analysis and application of the concepts of a core (a pair of chains) and cutset in the fixed point theory for posets. The main results are:

1. (Theorem 3) If P is chain-complete and (*), it contains a cutset S such that every nonempty subset of S has a join or a meet in P, then P has the fixed point property (FPP),
2. (Theorem 5) If P or Q is chain-complete, Q satisfies (*) and both P and Q have the FPP, then P x Q has the FPP.
3. (Theorem 6) Let P or Q be chain-complete and there exist p∈P and a finite sequence f ₁, f ₂, ..., f _n of order-preserving mappings of P into P such that $$\left( {\forall x\varepsilon P} \right)x \leqslant f_1 \left( x \right) \geqslant f_2 \left( x \right) \leqslant \cdots \geqslant f_n \left( x \right) \leqslant p$$ If P and Q have the FPP then P x Q has the FPP.
4. (Theorem 7) If T is an ordered set with the FPP and {P _t :t∈T} is a disjoint family of ordered sets with the FPP then its ordered sum ∪{P _t :t∈T} has the FPP.

     

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 .

     

On the exactness of certain inequalities in approximation theory

We prove the following: for every sequence {F_v}, F_v ? 0, F_v > 0 there exists a functionf such that

1. E_n(f)?F_n (n=0, 1, 2, ...) and
2. A_kn^?k? _v=1 ⁿ v^k?1 F_v?1?Ω_k (f, n^?1) (n=1, 2, ...).

     

Degenerate complementary cones induced by aK 0-matrix

In this paper we prove two results concerning the unionC of all the degenerate complementary cones associated with the linear complementarity problem (M, q) whereM is aK ₀-matrix.

1. C is the same as the set of allq ∈R ⁿ for which (M, q) has infinitely many solutions.
2. C is the same as the boundary of the set of allq ∈ R ⁿ for which (M, q) has a solution, an easily observable geometric result for a 2 × 2K ₀-matrix.

     

w* sequential convergence

LetX be an infinite dimensional Banach space, andX* its dual space. Sequences {χ _n ^* } _n=1 ^∞ ?X* which arew* converging to 0 while inf _n ‖x* _n ‖>0, are constructed.     

On an Ulam's problem

The existence and the uniqueness (with respect to a filtration-equivalence) of a vector flowX on ? ⁿ ,n≥3, such that:

1. X has not any stationary points on ? ⁿ ;
2. all orbits ofX are bounded;
3. there exists a filtration forX are proved in the present note.

     

Convergence of representation averages and of convolution powers

LetS be a locally compact (σ-compact) group or semi-group, and letT(t) be a continuous representation ofS by contractions in a Banach spaceX. For a regular probability μ onS, we study the convergence of the powers of the μ-averageUx=∫T(t)xdμ(t). Our main results for random walks on a groupG are:

1. if μ is adapted and strictly aperiodic, and generates a recurrent random walk, thenU ⁿ (U-I) converges strongly to 0. In particular, the random walk is completely mixing.
2. If μ×μ is ergodic onG×G, then for every unitary representationT(.) in a Hilbert space,U ⁿ converges strongly to the orthogonal projection on the space of common fixed points. These results are proved for semigroup representations, along with some other results (previously known only for groups) which do not assume ergodicity.
3. If μ is spread-out with supportS, then $\left\| {\mu ^{n + K} - \mu ^n } \right\| \to 0$ if and only if e $ \in \overline { \cup _{j = 0}^\infty S^{ - j} S^{j + K} } .$ .

     

Platitude d'une adherence schematique et lemme de Hironaka generalise

The main aim of this article is to prove the following:Theorem (Generalized Hironaka's lemma). Let X→Y be a morphism of schemes, locally of finite presentation, x a point of X and y=f(x). Assume that the following conditions are satisfied:

1. O _Y,y is reduced.
2. f is universally open at the generic points of the components of X_y which contain x.
3. For every maximal generisation y′ of y in Y and every maximal generisation x′ of x in X which belongs to X_y, we have dim_x, (X_y')=dim_x(X_y)=d.
4. X_y is reduced at the generic points of the components of X_y which contain x and (X_y)_red is geometrically normal over K(y) in x.

Then there exist an open neighbourhood U of x in X and a subscheme U₀ of U which have the same underlying space as U such that f₀:U₀\arY is normal (i.e. f₀ is a flat morphism whose geometric fibers are normal).