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.     

Partial integrability of Hamiltonian systems with homogeneous potential

In this paper we consider systems with n degrees of freedom given by the natural Hamiltonian function of the form $$ H = \frac{1} {2}p^T Mp + V(q), $$ where q = (q ₁, …, q _n ) ∈ ? ⁿ , p = (p ₁, …, p _n ) ∈ ? ⁿ , are the canonical coordinates and momenta, M is a symmetric non-singular matrix, and V (q) is a homogeneous function of degree k ∈ ?*. We assume that the system admits 1 ? m < n independent and commuting first integrals F ₁, … F _m . Our main results give easily computable and effective necessary conditions for the existence of one more additional first integral F _m+1 such that all integrals F ₁, … F _m+1 are independent and pairwise commute. These conditions are derived from an analysis of the differential Galois group of variational equations along a particular solution of the system. We apply our result analysing the partial integrability of a certain n body problem on a line and the planar three body problem.     

The maximal Fejér operator on the spacesH p (R×···×R)

It is shown that the maximal operator of the Fejér means of a tempered distribution is bounded from thed-dimensional Hardy spaceH _p (R×···×R) toL _p (R ^d ) (1/2<p<∞) and is of weak type (H ₁ ^?i ,L ₁) (i=1,…,d), where the Hardy spaceH ₁ ^?i is defined by a hybrid maximal function. As a consequence, we obtain that the Fejér means of a functionf ∈H ₁ ^?i ?L(logL) ^d?1 converge a.e. to the function in question. Moreover, we prove that the Fejér means are uniformly bounded onH _p (R×···×R) whenever 1/2<p<∞. Thus, in casef ∈H _p (R×···×R) the Fejér means converge tof inH _p (R×···×R) norm. The same results are proved for the conjugate Fejér means, too.     

Word maps and word maps with constants of simple algebraic groups

In the present paper, we consider word maps w: G ^m → G and word maps with constants w _Σ: G ^m → G of a simple algebraic group G, where w is a nontrivial word in the free group F _m of rank m, w _Σ = w ₁ σ ₁ w ₂ ··· w _r σ _r w _{r + 1}, w ₁, …, w _{r + 1} ∈ F _m , w ₂, …, w _r ≠ 1, Σ = {σ ₁, …, σ _r | σ _i ∈ G Z(G)}. We present results on the images of such maps, in particular, we prove a theorem on the dominance of “general” word maps with constants, which can be viewed as an analogue of a well-known theorem of Borel on the dominance of genuine word maps. Besides, we establish a relationship between the existence of unipotents in the image of a word map and the structure of the representation variety R(Γ_w, G) of the group Γ_w = F _m /<w>.     

On the condensation points of the lagrange spectrum

For a positive integerN, L(N) denotes the set of Lagrange values of all sequences (a _k:k=0, ±1, ±2,…) of positive integers with lim sup _k ^ak=N. It is shown that for anyN≥3L(N) has infinitely many condensation points. Such points can be realized as Markov values of symmetric doubly periodic sequences whose period consists of a semi-symmetric tuple.     

Linear maps preserving M–P inverses of matrices between matrix spaces over fields

Suppose F is a field of characteristic not 2. Let n and m be two arbitrary positive integers with n≥2. We denote by M _n (F) and S _n (F) the space of n×n full matrices and the space of n×n symmetric matrices over F, respectively. All linear maps from S _n (F) to M _m (F) preserving M–P inverses of matrices are characterized first, and thereby all linear maps from S _n (F) (M _n (F)) to S _m (F) (M _m (F)) preserving M–P inverses of matrices are characterized, respectively.     

Chebyshev's inequalities for functions monotone in the mean

LetR be the reals ≥ 0. LetF be the set of mapsf: {1, 2, ?,n} →R. Choosew ∈ F withw _i = w(i) > 0. PutW _i = w₁ + ? + w_i. Givenf ∈ F, define \(\bar f\) ∈F by $$\bar f\left( i \right) = \frac{{\left\{ {w_i f\left( 1 \right) + \ldots + w_i f\left( i \right)} \right\}}}{{W_i }}.$$ Callf mean increasing if \(\bar f\) is increasing. Letf ₁, ?, f_t be mean decreasing andf _t+1,?: f_t+u be mean increasing. Put $$k = W_n^u \min \left\{ {w_i^{u - 1} W_i^{t - u} } \right\}.$$ Then $$k\mathop \sum \limits_{i = 1}^n w_i f_1 \left( i \right) \ldots f_{t + u} \left( i \right) \leqslant \mathop \prod \limits_{j = 1}^{t + u} (\mathop \sum \limits_{i = 1}^n w_i f_1 (i)).$$      

Separability of wild automorphisms of a polynomial ring

For a large class (including the Nagata automorphism) of wild automorphisms F of k[x, y, z] (where k is a field of characteristic zero), we prove that we can find a weight w such that there exists no tame automorphism with the same w-weight multidegree.     

Absolutely continuous cocycles over irrational rotations

For homeomorphisms $$\left( {z,w} \right)\mathop \to \limits^{T\varphi } \left( {z . e^{2xi\alpha } ,\varphi \left( z \right)w} \right)$$ (z, w ∈S ¹,α is irrational,?:S ¹→S ¹) of the torusS ¹×S ¹ it is proved thatT? has countable Lebesgue spectrum in the orthocomplement of the eigenfunctions whenever? is absolutely continuous with nonzero topological degree and the derivative of? is of bounded variation. Some other cocycles with bounded variation are studied and generalizations of the above result to certain distal homeomorphisms on finite dimensional tori are presented.     

An embedding theorem for partial algebras and the free completion of a free partial algebra within a primitive class

Let U be any nontrivial primitive class of partial algebras, i.e. there existsA ∈ U with |A|≥2, and U is closed with respect to homomorphic images (in the weak sense), subalgebras (on closed subsets) and cartesian products of U-algebras, and let U _f denote the—also nontrivial and primitive—class of all full U-algebras. Then every U-algebra with at least two elements is a relative algebra of some U _f -algebra. For any U-algebraAsetU _A =U _i εI({i}×(A ^K _i—domf _i ^A )), where (K _i) _i εI is the type under consideration. Furthermore let F(N, U) denote any U-algebra U-freely generated by some setN (and let F (M, U _f ) be similarly defined). Then for every nonempty setM there exists a setN satisfyingM ?N such that there exists a bijective mapping σ:U _{F(N, U)} →N ?M satisfying σ((i, α)) ? α(K _i ) for all (i, α) ∈U _{F (N, U)}, and, for the structureg=(g _i)_iεI defined by ,g _i : =f _i ^{F(N, U)} ∪ {(α, σ((i, α))) | (i, α ∈U _{F(N, U)}} id _M induces an isomorphism betweenF(M, U _f ), and (F(N, U)g).     

On rings whose number of centralizers of ideals is finite

LetR be a ring. For the setF of all nonzero ideals ofR, we introduce an equivalence relation inF as follows: For idealsI andJ, I～J if and only ifV _R (I)=V _R(J), whereV _R() is the centralizer inR. LetI _R=F/～. Then we can see thatn(I _R), the cardinality ofI _R, is 1 if and only ifR is either a prime ring or a commutative ring (Theorem 1.1). An idealI ofR is said to be a commutator ideal ifI is generated by{st?ts; s∈S, t∈T} for subsetS andT ofR, andR is said to be a ring with (N) if any commutator ideal contains no nonzero nilpotent ideals. Then we have the following main theorem: LetR be a ring with (N). Thenn(I _R) is finite if and only ifR is isomorphic to an irredundant subdirect sum ofS⊕Z whereS is a finite direct sum of non commutative prime rings andZ is a commutative ring (Theorem 2.1). Finally, we show that the existence of a ringR such thatn(I _R)=m for any given natural numberm.     

On-Line Dimension for Posets Excluding Two Long Incomparable Chains

For a positive integer k, let k?+?k denote the poset consisting of two disjoint k-element chains, with all points of one chain incomparable with all points of the other. Bosek, Krawczyk and Szczypka showed that for each k?≥?1, there exists a constant c _k so that First Fit will use at most $c_kw^2$ chains in partitioning a poset P of width at most w, provided the poset excludes k?+?k as a subposet. This result played a key role in the recent proof by Bosek and Krawczyk that O(w ^16logw ) chains are sufficient to partition on-line a poset of width w into chains. This result was the first improvement in Kierstead’s exponential bound: (5 ^w ???1)/4 in nearly 30 years. Subsequently, Joret and Milans improved the Bosek–Krawczyk–Szczypka bound for the performance of First Fit to 8(k???1)² w, which in turn yields the modest improvement to O(w ^14logw ) for the general on-line chain partitioning result. In this paper, we show that this class of posets admits a notion of on-line dimension. Specifically, we show that when k and w are positive integers, there exists an integer t?=?t(k,w) and an on-line algorithm that will construct an on-line realizer of size t for any poset P having width at most w, provided that the poset excludes k?+?k as a subposet.     

Measurable events indexed by products of trees

A tree T is said to be homogeneous if it is uniquely rooted and there exists an integer b ≥ 2, called the branching number of T, such that every t ∈ T has exactly b immediate successors. A vector homogeneous tree T is a finite sequence (T ₁,...,T _d ) of homogeneous trees and its level product ?T is the subset of the Cartesian product T ₁×...×T _d consisting of all finite sequences (t ₁,...,t _d ) of nodes having common length. We study the behavior of measurable events in probability spaces indexed by the level product ?T of a vector homogeneous tree T. We show that, by refining the index set to the level product ?S of a vector strong subtree S of T, such families of events become highly correlated. An analogue of Lebesgue’s density Theorem is also established which can be considered as the “probabilistic” version of the density Halpern-Läuchli Theorem.     

Torsion and Abelianization in Equivariant Cohomology

Let X be a topological space upon which a compact connected Lie group G acts. It is well known that the equivariant cohomology H ^* _G (X; Q) is isomorphic to the subalgebra of Weyl group invariants of the equivariant cohomology H ^* _T (X; Q), where T is a maximal torus of G. This relationship breaks down for coefficient rings k other than Q. Instead, we prove that under a mild condition on k the algebra H ^* _G (X; k) is isomorphic to the subalgebra of H ^* _T (X; k) annihilated by the divided difference operators.     

Variations of Landau’s theorem for p-regular and p-singular conjugacy classes

The well-known Landau’s theorem states that, for any positive integer k, there are finitely many isomorphism classes of finite groups with exactly k (conjugacy) classes. We study variations of this theorem for p-regular classes as well as p-singular classes. We prove several results showing that the structure of a finite group is strongly restricted by the number of p-regular classes or the number of p-singular classes of the group. In particular, if G is a finite group with O_p(G) = 1 then |G/F(G)|_p' is bounded in terms of the number of p-regular classes of G. However, it is not possible to prove that there are finitely many groups with no nontrivial normal p-subgroup and kp-regular classes without solving some extremely difficult number-theoretic problems (for instance, we would need to show that the number of Fermat primes is finite).     

MV-closures of Wajsberg hoops and applications

In this paper we construct, given a Wajsberg hoop A, an MV-algebra MV(A) such that the underlying set A of A is a maximal filter of MV(A) and the quotient MV(A)/A is the two element chain. As an application we provide a topological duality for locally finite Wajsberg hoops based on a previously known duality for locally finite MV-algebras. We also give another duality for k-valued Wajsberg hoops based on a different representation of k-valued MV-algebras and show the relation to the first duality. We also apply this construction to give a topological representation for free k-valued Wajsberg hoops.     

Range of cube-indexed random walk

For given finite, connected, bipartite graphG=(V,E) with distinguishedν ₀ ∈V, set {fx189-1} Our main result says there is a fixedb so that whenG is a Hamming cube ({0, 1} ⁿ with the usual adjacency), andf is chosen uniformly fromF, the probability thatf takes more thanb values is at most e^?Ω(n). this settles in a very strong way a conjecture of I. Benjamini, O. Häggström and E. Mossel [2]. The proof is based on entropy considerations and a new log-concavity result.     

Regular principal factors in free objects in varieties in the interval [B 2 ,NB 2 ∨G n ]

The author has shown previously how to associate a completely 0-simple semigroup with a connected bipartite graph containing labelled edges and how to describe the regular principal factors in the free objects in the Rees-Sushkevich varieties RS _n generated by all completely 0-simple semigroups over groups from the Burnside variety G _n of groups of exponent dividing a positive integer n by employing this graphical construction. Here we consider the analogous problem for varieties containing the variety B ₂ , generated by the five element Brandt semigroup B ₂, and contained in the variety NB ₂ ∨G _n where NB ₂ is the variety generated by all left and right zero semigroups together with B ₂. The interval [NB ₂ ,NB ₂ ∨G _n ] is of particular interest as it is an important interval, consisting entirely of varieties generated by completely 0-simple semigroups, in the lattice of subvarieties of RS _n .     

On diversity and stability of unit bases for the Euclidean metric

We prove the existence of a family Ω(n) of 2 ^c (where c is the cardinality of the continuum) subgraphs of the unit distance graph (E ⁿ , 1) of the Euclidean space E ⁿ , n ≥ 2, such that (a) for each graph G ? Ω(n), any homomorphism of G to (E ⁿ , 1) is an isometry of E ⁿ ; moreover, for each subgraph G ₀ of the graph G obtained from G by deleting less than c vertices, less than c stars, and less than c edges (we call such a subgraph reduced), any homomorphism of G ₀ to (E ⁿ , 1) is an isometry (of the set of the vertices of G ₀); (b) each graph G ? Ω(n) cannot be homomorphically mapped to any other graph of the family Ω(n), and the same is true for each reduced subgraph of G.     

Sets of vectors with many orthogonal paris:

What is the most number of vectors inR ^d such that anyk+1 contain an orthogonal pair? The 24 positive roots of the root systemF ₄ inR ⁴ show that this number could exceeddk.