On the inverse problem relative to dynamics of the <Emphasis Type="Italic">w</Emphasis> function

Let P be the set of prime numbers and P(n) denote the largest prime factor of integer n > 1. Write

On the <Emphasis Type="Italic">K</Emphasis>-Theory of Graph <Emphasis Type="Italic">C</Emphasis><Superscript>*</Superscript>-Algebras

We classify graph C ^*-algebras, namely, Cuntz-Krieger algebras associated to the Bass-Hashimoto edge incidence operator of a finite graph, up to strict isomorphism. This is done by a purely graph theoretical calculation of the K-theory of the C ^*-algebras and the method also provides an independent proof of the classification up to Morita equivalence and stable equivalence of such algebras, without using the boundary operator algebra. A direct relation is given between the K ₁-group of the algebra and the cycle space of the graph. We thank Jakub Byszewski for his input in Sect. 2.8. The position of the unit in K ₀( _Ч) was guessed based on some example calculations by Jannis Visser in his SCI 291 Science Laboratory at Utrecht University College.     

Riesz <Emphasis Type="Italic">s</Emphasis>-Equilibrium Measures on <Emphasis Type="Italic">d</Emphasis>-Rectifiable Sets as <Emphasis Type="Italic">s</Emphasis> Approaches <Emphasis Type="Italic">d</Emphasis>

Let A be a compact set in of Hausdorff dimension d. For s ∈ (0,d) the Riesz s-equilibrium measure μ ^s is the unique Borel probability measure with support in A that minimizes

<Emphasis Type="Italic">W</Emphasis><Superscript>1,<Emphasis Type="Italic">p</Emphasis></Superscript>(R<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)-boundedness for maximal functions

In this paper, we get W ^1,p (R ⁿ )-boundedness for tangential maximal function and nontangential maximal function, which improves J.Kinnunen, P.Lindqvist and Tananka’s results. Supported by the key Academic Discipline of Zhejiang Province of China under Grant No.2005 and the Zhejiang Provincial Natural Science Foundation of China.     

Point sets with low <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-discrepancy

In this paper we study the L _p -discrepancy of digitally shifted Hammersley point sets. While it is known that the (unshifted) Hammersley point set (which is also known as Roth net) with N points has L _p -discrepancy (p an integer) of order (log N)/N, we show that there always exists a shift such that the digitally shifted Hammersley point set has L _p -discrepancy (p an even integer) of order which is best possible by a result of W. Schmidt. Further we concentrate on the case p = 2. We give very tight lower and upper bounds for the L ₂-discrepancy of digitally shifted Hammersley point sets which show that the value of the L ₂-discrepancy of such a point set mostly depends on the number of zero coordinates of the shift and not so much on the position of these. This work is supported by the Austrian Research Fund (FWF), Project P17022-N12 and Project S8305.     

On hamiltonicity of <Emphasis Type="Italic">P</Emphasis><Subscript>3</Subscript>-dominated graphs

We introduce a new class of graphs which we call P ₃-dominated graphs. This class properly contains all quasi-claw-free graphs, and hence all claw-free graphs. Let G be a 2-connected P ₃-dominated graph. We prove that G is hamiltonian if α(G ²) ≤ κ(G), with two exceptions: K _2,3 and K _1,1,3. We also prove that G is hamiltonian, if G is 3-connected and |V(G)| ≤ 5δ(G) − 5. These results extend known results on (quasi-)claw-free graphs. This paper was completed when both authors visited the Center for Combinatorics, Nankai University, Tianjin. They gratefully acknowledge the hospitality and support of the Center for Combinatorics and Nankai University. The work of E.Vumar is sponsored by SRF for ROCS, REM.     

On the Chromatic Number of the Square of the Kneser Graph <Emphasis Type="Italic">K</Emphasis>(2<Emphasis Type="Italic">k</Emphasis>+1, <Emphasis Type="Italic">k</Emphasis>)

The Kneser graph K(n, k) is the graph whose vertices are the k-element subsets of an n-element set, with two vertices adjacent if the sets are disjoint. The chromatic number of the Kneser graph K(n, k) is n–2k+2. Zoltán Füredi raised the question of determining the chromatic number of the square of the Kneser graph, where the square of a graph is the graph obtained by adding edges joining vertices at distance at most 2. We prove that (K²(2k+1, k))4k when k is odd and (K²(2k+1, k))4k+2 when k is even. Also, we use intersecting families of sets to prove lower bounds on (K²(2k+1, k)), and we find the exact maximum size of an intersecting family of 4-sets in a 9-element set such that no two members of the family share three elements.This work was partially supported by NSF grant DMS-0099608Final version received: April 23, 2003     

The strong <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-<Emphasis Type="Italic">greedy</Emphasis> property of the Walsh system

For any 0 < ? < 1 one can find a measurable set E ? [0, 1] with the measure |E| > 1 ? ? such that for each function f(x) ε L ¹ (0, 1) a function g(x) ε L ¹ (0, 1) exists such that it coincides with f (x) on E, its Fourier—Walsh series converges to it in the metric of L ¹ (0, 1), and all nonzero terms of the sequence of Fourier coefficients of the new function obtained by the Walsh system have the modulo decreasing order; consequently, the greedy algorithm for this function converges to it in the L ¹ (0, 1)-norm.     

On the sequential order continuity of the <Emphasis Type="Italic">C</Emphasis>(<Emphasis Type="Italic">K</Emphasis>)-space

As shown in [1], for each compact Hausdorff space K without isolated points, there exists a compact Hausdorff P′-space X but not an F-space such that C(K) is isometrically Riesz isomorphic to a Riesz subspace of C(X). The proof is technical and depends heavily on some representation theorems. In this paper we give a simple and direct proof without any assumptions on isolated points. Some generalizations of these results are mentioned.     

Results on <Emphasis Type="Italic">F</Emphasis>-continuous graphs

For any nontrivial connected graph F and any graph G, the F-degree of a vertex v in G is the number of copies of F in G containing v. G is called F-continuous if and only if the F-degrees of any two adjacent vertices in G differ by at most 1; G is F-regular if the F-degrees of all vertices in G are the same. This paper classifies all P ₄-continuous graphs with girth greater than 3. We show that for any nontrivial connected graph F other than the star K _1,k , k ⩾ 1, there exists a regular graph that is not F-continuous. If F is 2-connected, then there exists a regular F-continuous graph that is not F-regular.      

On <Emphasis Type="Italic">U</Emphasis>-orthodox semigroups

As a generalization of an orthodox semigroup in the class of regular semigroups, a type W semigroup was first investigated by El-Qallali and Fountain. As an analogy of the type W semigroups in the class of abundant semigroups, we introduce the U-orthodox semigroups. It is shown that the maximum congruence μ contained in on U-orthodox semigroups can be determined. As a consequence, we show that a U-orthodox semigroup S can be expressed by the spined product of a Hall semigroup W _U and a V-ample semigroup (T,V). This theorem not only generalizes a known result of Hall-Yamada for orthodox semigroups but also generalizes another known result of El-Qallali and Fountain for type W semigroups. This work was supported by National Natural Science Foundation of China (Grant No. 10671151) and Natural Science Foundation of Shaanxi Province (Grant No. SJ08A06), and partially by UGC (HK) (Grant No. 2160123)     

On <Emphasis Type="Italic">k</Emphasis>-th Power of Upper Bound Graphs

We consider k-th power of upper bound graphs. According to the characterization of upper bound graphs, we obtain a characterization of k-th power of upper bound graphs. That is, for a connected upper bound graph G, G^k is an upper bound graph if and only if for any pair of A^k -simplicial vertices s₁, s₂ such that , there exists a G^k -simplicial vertex s satisfying the conditions: and . Furthermore we also get some properties on squares of upper bound graphs.AMS Subject Classification: 05C62.     

On <Emphasis Type="Italic">S</Emphasis>-quasinormal and <Emphasis Type="Italic">c</Emphasis>-normal subgroups of a finite group

Let be a saturated formation containing the class of supersolvable groups and let G be a finite group. The following theorems are presented: (1) G ∈ if and only if there is a normal subgroup H such that G/H ∈ and every maximal subgroup of all Sylow subgroups of H is either c-normal or S-quasinormally embedded in G. (2) G ∈ if and only if there is a normal subgroup H such that G/H ∈ and every maximal subgroup of all Sylow subgroups of F*(H), the generalized Fitting subgroup of H, is either c-normal or S-quasinormally embedded in G. (3) G ∈ if and only if there is a normal subgroup H such that G/H ∈ and every cyclic subgroup of F*(H) of prime order or order 4 is either c-normal or S-quasinormally embedded in G. Supported by the Natural Science Foundation of China and the Natural Science Foundation of Guangxi Autonomous Region (No. 0249001). Corresponding author. Supported in part by the Natural Science Foundation of China (10571181), NSF of Guangdong Province (06023728) and ARF(GDEI).     

<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> discrepancy of generalized two-dimensional Hammersley point sets

We determine the L _p discrepancy of the two-dimensional Hammersley point set in base b. These formulas show that the L _p discrepancy of the Hammersley point set is not of best possible order with respect to the general (best possible) lower bound on L _p discrepancies due to Roth and Schmidt. To overcome this disadvantage we introduce permutations in the construction of the Hammersley point set and show that there always exist permutations such that the L _p discrepancy of the generalized Hammersley point set is of best possible order. For the L ₂ discrepancy such permutations are given explicitly. F.P. is supported by the Austrian Science Foundation (FWF), Project S9609, that is part of the Austrian National Research Network “Analytic Combinatorics and Probabilistic Number Theory”.     

On modules with <Emphasis Type="Italic">d</Emphasis>-Koszul-type submodules

The so-called weakly d-Koszul-type module is introduced and it turns out that each weakly d-Koszul-type module contains a d-Koszul-type submodule. It is proved that, M ∈ W H J^d（A） if and only if M admits a filtration of submodules： 0 belong to U0 belong to U1 belong to ... belong to Up = M such that all Ui/Ui-1 are d-Koszul-type modules, from which we obtain that the finitistic dimension conjecture holds in W H J^d（A） in a special case. Let M ∈ W H J^d（A）. It is proved that the Koszul dual E（M） is Noetherian, Hopfian, of finite dimension in special cases, and E（M） ∈ gr0（E（A））. In particular, we show that M ∈ W H J^d（A） if and only if E（G（M）） ∈ gr0（E（A））, where G is the associated graded functor.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> approximation capability of RBF neural networks

L ^p approximation capability of radial basis function (RBF) neural networks is investigated. If g: R ₊¹ → R ¹ and ∈ L _loc ^p (R ⁿ ) with 1 ≤ p < ∞, then the RBF neural networks with g as the activation function can approximate any given function in L ^p (K) with any accuracy for any compact set K in R ⁿ , if and only if g(x) is not an even polynomial. Partly supported by the National Natural Science Foundation of China (10471017)     

<Emphasis Type="Italic">LJ</Emphasis>-spaces

In this paper LJ-spaces are introduced and studied. They are a common generalization of Lindelöf spaces and J-spaces researched by E. Michael. A space X is called an LJ-space if, whenever {A, B} is a closed cover of X with A ∩ B compact, then A or B is Lindelöf. Semi-strong LJ-spaces and strong LJ-spaces are also defined and investigated. It is demonstrated that the three spaces are different and have interesting properties and behaviors.     

<Emphasis Type="Italic">k</Emphasis>-Free-like groups

The following results are proved. In Theorem 1, it is stated that there exist both finitely presented and not finitely presented 2-generated nonfree groups which are k-free-like for any k ⩾ 2. In Theorem 2, it is claimed that every nonvirtually cyclic (resp., noncyclic and torsion-free) hyperbolic m-generated group is k-free-like for every k ⩾ m + 1 (resp., k ⩾ m). Finally, Theorem 3 asserts that there exists a 2-generated periodic group G which is k-free-like for every k ⩾ 3. Supported by NSF (grant Nos. DMS 0455881 and DMS-0700811). (A. Yu. Olshanskii, M. V. Sapir) Supported by RFBR project No. 08-01-00573. (A. Yu. Olshanskii) Supported by BSF grant (USA–Israel). (M. V. Sapir) Translated from Algebra i Logika, Vol. 48, No. 2, pp. 245–257, March–April, 2009.     

On representation theorem of <Emphasis Type="Italic">G</Emphasis>-expectations and paths of <Emphasis Type="Italic">G</Emphasis>-Brownian motion

We give a very simple and elementary proof of the existence of a weakly compact family of probability measures {Pθ ： θ∈θ} representing an important sublinear expectation- G-expectation E[·]. We also give a concrete approximation of a bounded continuous function X（ω） by an increasing sequence of cylinder functions Lip（Ω） in order to prove that Cb（Ω） belongs to the completion of Lip（Ω） under the natural norm E[｜·｜].     

The construction of infinite families of any κ-tight optimal and singular κ-tight optimal directed double loop networks

The double loop network (DLN) is a circulant digraph with n nodes and outdegree 2. It is an important topological structure of computer interconnection networks and has been widely used in the designing of local area networks and distributed systems. Given the number n of nodes, how to construct a DLN which has minimum diameter? This problem has attracted great attention. A related and longtime unsolved problem is for any given non-negative integer k, is there an infinite family of k-tight optimal DLN? In this paper, two main results are obtained (1) for any k ≥ 0, the infinite families of k-tight optimal DLN can be constructed, where the number n(k,e,c) of their nodes is a polynomial of degree 2 in e with integral coefficients containing a parameter c. (2) for any k ≥ 0,an infinite family of singular k-tight optimal DLN can be constructed.