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This note is related to a nice short paper of M. Ochiai. We prove in a very fast way that the two-parameter family of Heegaard diagrams, constructed by Ochiai, encodes the genuine three-sphere. The result is obtained, up to isotopy, by using a sequence of only three moves in this order: a Whitehead–Zieschang reduction, a band sum and a cancellation of a handle.

The object of this note is to point out and correct an error in the paper [21 of Fox, purporting to prove Fenchel's conjecture that a finitely generated. infinite Fuchsian group has a torsion-free normal subgroup of finite index. Fox divided the proof of the main result in his paper into four cases and an error occurs in the proofs of Cases (III) and (IV). A direct proof of Case (III) was given. While a direct proof of Case (IV) can be given, the author shows that it also follows indirectly from the result in the paper [1] of Burns and Solitar. Given any three integers a > 1, b > 1 and c > 1, there exist permutations A of order a and B of order b, such that AB has order c. Fox divided the proof of this lemma into the four cases: (I) ab 0 (mod 2) and max{a, b} ? c < a + b, (II) ab 0 (mod 2) and c : a + b, (III) a b c 1 (mod 2), (IV) a b 1 and c 0 (mod 2), and subcases. In the subcase m(a + b 2) + 2 ? c < (m + l)(b + 1) 1 of Case (III) where m is the largest integer not exceeding (c l)/(b + 1), and it is assumed that a ? b < c, the number of distinct symbols in the permutations [2, p. 64] nl~~~W B ( Wv-sI .. WI2 I7 (vWb B = (vl * *t*v W_ ***W P2) * (V W2 .. * ]Pi+]1)'

Submitted by Vitor Silverio Rodrigues (vitorsrodrigues@reitoria.unesp.br) on 2014-05-27T11:25:28Z No. of bitstreams: 0Bitstream added on 2014-05-27T14:41:41Z : No. of bitstreams: 1 2-s2.0-79951846250.pdf: 494002 bytes, checksum: 56b6ee8beddda3e7dae971355d44a19f (MD5) Made available in DSpace on 2014-05-27T11:25:28Z (GMT). No. of bitstreams: 0 Previous issue date: 2011-03-01 Item merged in doublecheck by Felipe Arakaki (arakaki@reitoria.unesp.br) on 2015-12-11T17:28:11Z Item was identical to item(s): 71803, 21370 at handle(s): http://hdl.handle.net/11449/72321, http://hdl.handle.net/11449/21804 Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) The simplest necessary conditions for an entire function ψ(x) =∞ ∑ k=0 γk xk/k! to be in the Laguerre-Pólya class are the Turán inequalities γ2 k- γk+1γk-1 ≥ 0. These are in fact necessary and sufficient conditions for the second degree generalized Jensen polynomials associated with ψ to be hyperbolic. The higher order Turán inequalities 4(γ2 n - γn-1γn+1)(γ2n +1 - γnγn+2) - (γnγn+1 - γn-1γn+2) 2 ≥ 0 are also necessary conditions for a function of the above form to belong to the Laguerre-Pólya class. In fact, these two sets of inequalities guarantee that the third degree generalized Jensen polynomials are hyperbolic. Pólya conjectured in 1927 and Csordas, Norfolk and Varga proved in 1986 that the Turán inequalities hold for the coefficients of the Riemann ψ-function. In this short paper, we prove that the higher order Turán inequalities also hold for the ψ-function, establishing the hyperbolicity of the associated generalized Jensen polynomials of degree three. © 2010 American Mathematical Society. Departamento de Ciências de Computação e Estatística IBILCE, Universidade Estadual Paulista, 15054-000 São José do Rio Preto, SP Departamento de matemática Aplicada IMECC UNICAMP, 13083-859 Campinas, SP Departamento de Ciências de Computação e Estatística IBILCE, Universidade Estadual Paulista, 15054-000 São José do Rio Preto, SP FAPESP: 03/01874-2 FAPESP: 06/60420-0 CNPq: 305622/2009-9 CAPES: DGU-160

Let p p be an odd prime. h 1 ( p ) {h_1}(p) is the first factor of the class number of field Q ( ζ p ) Q({\zeta _p}) . We proved that \[ h 1 ( p ) ⩽ { 2 p ( p − 1 8 ( 2 l / 2 + 1 ) 4 / l ) ( p − 1 ) / 4 , if l is even, 2 p ( p − 1 8 ( 2 l − 1 ) 2 / l ) ( p − 1 ) / 4 , if l is odd . {h_1}(p) \leqslant \left \{ \begin {gathered} 2p{\left ( {\frac {{p - 1}} {{8{{({2^{l/2}} + 1)}^{4/l}}}}} \right )^{(p - 1)/4}},\quad {\text {if }}l\;{\text {is even,}} \hfill \\ 2p{\left ( {\frac {{p - 1}} {{8{{({2^l} - 1)}^{2/l}}}}} \right )^{(p - 1)/4}},\quad {\text {if }}l\;{\text {is odd}}{\text {.}} \hfill \\ \end {gathered} \right . \] From that we obtain h 1 ( p ) ⩽ 2 p ( ( p − 1 ) / 31.997158 … ) ( p − 1 ) / 4 {h_1}(p) \leqslant 2p{((p - 1)/31.997158 \ldots )^{(p - 1)/4}} which is better than Carlitz’s and Metsänkyla’s results. For the fields Q ( ζ 2 n ) Q({\zeta _{{2^n}}}) and Q ( ζ p n ) ( n ⩾ 2 ) Q({\zeta _{{p^n}}})(n \geqslant 2) , we get the similar results.

In this paper, we carry out in an abstract order context some real subset combinatorial problems. Specifically, let [Formula: see text] be a finite poset, where [Formula: see text] is an order-reversing and involutive map such that [Formula: see text] for each [Formula: see text]. Let [Formula: see text] be the Boolean lattice with two elements and [Formula: see text] the family of all the order-preserving 2-valued maps [Formula: see text] such that [Formula: see text] if [Formula: see text] for all [Formula: see text]. In this paper, we build a family [Formula: see text] of particular subsets of [Formula: see text], that we call [Formula: see text]-bases on [Formula: see text], and we determine a bijection between the family [Formula: see text] and the family [Formula: see text]. In such a bijection, a [Formula: see text]-basis [Formula: see text] on [Formula: see text] corresponds to a map [Formula: see text] whose restriction of [Formula: see text] to [Formula: see text] is the smallest 2-valued partial map on [Formula: see text] which has [Formula: see text] as its unique extension in [Formula: see text]. Next we show how each [Formula: see text]-basis on [Formula: see text] becomes, in a particular context, a sub-system of a larger system of linear inequalities, whose compatibility implies the compatibility of the whole system.

The concept of graph coloring has become a very active field of research that enhances many practical applications and theoretical challenges. Various methods have been applied in carrying out this study. Let G be a finite group. In this paper, we introduce a new graph of groups, which is a commuting order product prime graph of finite groups as a graph having the elements of G as its vertices and two vertices are adjacent if and only if they commute and the product of their order is a prime power. This is an extension of the study for order product prime graph of finite groups. The graph's general presentations on dihedral groups, generalized quaternion groups, quasi-dihedral groups, and cyclic groups have been obtained in this paper. Moreover, the commuting order product prime graph on these groups has been classified as connected, complete, regular, or planar. These results are used in studying various and recently introduced chromatic numbers of graphs.

All groups in the family of Baumslag-Solitar groups (i.e., groups of the form G(m,n) = 〈a, b; a−1bma = bn〉, where m and n are nonzero integers) for which the residual nilpotence condition holds if and only if the residual p-finiteness condition holds for some prime number p are described. It has turned out, in particular, that the group G(pr, −pr), where p is an odd prime and r ≥ 1, is residually nilpotent, but it is residually q-finite for no prime q. Thus, an answer to the existence problem for noncyclic one-relator groups possessing such a property (formulated by McCarron in his 1996 paper) is obtained. A simple proof of the statement that an arbitrary residually nilpotent noncyclic one-relator group which has elements of finite order is residual p-finite for some prime p, which was announced in the same paper of McCarron, is also given.

Conjectures on permanents are well-known unsettled conjectures in linear algebra. Let A be an n×n matrix and Sn be the symmetric group on n element set. The permanent of A is defined as perA=∑σ∈Sn∏i=1naiσ(i). The Merris conjectured that for all n×n doubly stochastic matrices (denoted by Ωn), nperA≥min1≤i≤n∑j=1nperA(j|i), where A(j|i) denotes the matrix obtained from A by deleting the jth row and ith column. Foregger raised a question whether per(tJn+(1−t)A)≤perA for 0≤t≤nn−1 and for all A∈Ωn, where Jn is a doubly stochastic matrix with each entry 1n. The Merris conjecture is one of the well-known conjectures on permanents. This conjecture is still open for n≥4. In this paper, we prove the Merris inequality for some classes of matrices. We use the sub permanent inequalities to prove our results. Foregger’s inequality is also one of the well-known inequalities on permanents, and it is not yet proved for n≥5. Using the concepts of elementary symmetric function and subpermanents, we prove the Foregger’s inequality for n=5 in [0.25, 0.6248]. Let σk(A) be the sum of all subpermanents of order k. Holens and Dokovic proposed a conjecture (Holen–Dokovic conjecture), which states that if A∈Ωn,A≠Jn and k is an integer, 1≤k≤n, then σk(A)≥(n−k+1)2nkσk−1(A). In this paper, we disprove the conjecture for n=k=4.

In this paper, the compositional structure of a finite group G is investigated, which has the Sylow 2-subgroup that is permutable with some non p-nilpotent biprimary subgroups, which contain the Sylow р-subgroup of G for all odd simple divisors of the р order of the group G, and such biprimary subgroups are taken one by one for each odd р, and mark the set SB(G). In this work, the existence of the subset SB(G)* in SB(G) is proved, which consists of р-closed subgroups. The main result of this paper is as follows: if the Sylow 2-subgroup of the group G is permutable with all subgroups SB(G)*, then G may have simple non-abelian compositional factors only of L2(7) type, if p > 3, and additionally of L2(3f) type, f = 3a, a ≥ 1, if p = 3.

We present a theorem in the present paper on an approximation to functions of a Holder class on a countable union of segments lying on a positive ray by entire functions of order 1/2 bounded on this ray. Problems related to the approximation of entire functions on subsets of the semiaxis by using entire functions of order 1/2 are closely related to problems of approximating functions on subsets of the whole axis using entire functions of exponential type but have their own specifics. We consider segments In in this paper with lengths of order n such that the distance between In and In + 1 is also of order n. Cases of the whole semiaxis or the union of finitely many segments and a ray were considered in previous papers. As for the problem of approximating functions of the Holder class on the union of a countable set of segments on the whole axis, it turns out that the approximation rate at neighborhoods of the segment endpoints as the type of the functions increases is higher than that in a neighborhood of their midpoints.

In 1998, R. Downey in his review paper outlined a research agenda for the study and description of sufficient conditions for computable representability of linear orderings, namely, the problem of describing order properties P such that, for any low linear ordering Limplies that, the feasibility of the condition P(L) implies that L has a computable presentation. This paper is a part of the research program initiated by R. Downey. It is shown in the paper that any low linear ordering whose factor order (in other words, condensation) is η (the order type of naturally ordered natural numbers) has a computable presentation via a 0—computable isomorphism if this ordering does not contain a strongly η-like infinite interval. A countable linear ordering is said to be strongly η-like if there exists some natural number k such that each maximal block of the ordering is of cardinality no more than k. It is also proved that the above result does not hold for a 0—computable isomorphism instead of the 0—computable one. Namely, we construct a low linear ordering L with condensation η and without strongly η-like infinite interval so that L has no computable presentation via a 0— computable isomorphism.

In this paper, we study Carlitz&rsquo, s type higher-order (p,q)-Genocchi polynomials. To be specific, we define Carlitz&rsquo, s type higher-order (p,q)-Genocchi polynomials and Carlitz&rsquo, s type higher-order (h,p,q)-Genocchi polynomials. This paper also explores properties including distribution relation and symmetric identities. In addition, we find alternating (p,q)-power sums. We identify symmetric identities using Carlitz&rsquo, s type higher-order (h,p,q)-Genocchi polynomials and alternating (p,q)-power sums.

In the paper, we extend known results concerning crossing numbers for join of graphs of order six. We give the crossing number of the join product G+Dn, where the graph G consists of two leaves incident with two opposite vertices of one 4-cycle, and Dn consists on n isolated vertices. The proof is done with the help of software that generates all cyclic permutations for a given number k, and creates a new graph COG for a calculating the distances between all (k − 1)! vertices of the graph. Finally, by adding new edges to the graph G, we are able to obtain the crossing number of the join product with the discrete graph Dn for two other graphs. The methods used in the paper are new, and they are based on combinatorial properties of cyclic permutations.

The Farey sequence of order n consists of all reduced fractions a / b between 0 and 1 with positive denominator b less or equal to n. In a series of former papers we obtained a limit function which describes a scaling property of the Farey sequence of order n for $$n \rightarrow \infty $$ in the vicinity of any fixed fraction a / b and which is independent of a / b. In this paper we derive a representation formula for a sequence of functions used in the Franel–Landau theorem to determine the number of Farey-fractions in prescribed intervals, and we establish a corresponding limit function for this sequence as well. This is combined with a former result to derive a remarkable representation formula for arithmetic means of related functions.

In a paper in 1962, Guido Zappa asked whether a non-trivial coset of a Sylow p-subgroup of a finite group could contain only elements whose orders are powers of p, and also in that case, at least one element of order p. The first question was raised again recently in a 2014 paper by Daniel Goldstein and Robert Guralnick, when generalising an answer by John Thompson in 1967 to a similar question by L.J. Paige. In this paper we give a positive answer to both questions of Zappa, showing somewhat surprisingly that in a number of non-abelian finite simple groups (including PSL ( 3 , 4 ) , PSU ( 5 , 2 ) and the Janko group J 3 ), some non-trivial coset of a Sylow 5-subgroup (of order 5) contains only elements of order 5. Also Zappa's first question is studied in more detail. Various possibilities for the group and its Sylow p-subgroup P are eliminated, and it then follows that | P | ≥ 5 and | P | ≠ 8 . It is an open question as to whether the order of the Sylow p-subgroup can be 7 or 9 or more.

Recently a lot of papers have been devoted to partial infinite reciprocal sums of a higher-order linear recursive sequence. In this paper, we continue this program by finding a sequence which is asymptotically equivalent to partial infinite sums, including a reciprocal of polynomial applied to linear higher order recurrences.

Chen et al. (2004) strongly conjectured that R(Tn,Wm)=2n-1 if the maximum degree of Tn is small and m is even. Related to the conjecture, it is interesting to know for which tree Tn, we have R(Tn,Wm) > 2n-1 for even m. In this paper, we find the Ramsey number R(Tn,W8) for tree Tn with the maximum degree of Tn is at least n-3, namely R(Tn,W8) > 2n-1 for almost all such tree Tn. We also prove that if the maximum degree of Tn is large, then R(Tn,Wm) is a function of both m and n. In the end, we refine the conjecture of Chen et al. by giving the condition of small maximum degree. For a tree Tn with large maximum degree and even m, the R(Tn,Wm) is also unknown in general. In this paper, we shall determine the Ramsey number R(Tn,W8) for all trees Tn of order n with the maximum degree of Tn is at least n-3., 7 pages

Given a digital image (or digital object) (X,k),X⊂Zn, this paper initially establishes a group structure of the set of self-k-isomorphisms of (X,k) with the function composition, denoted by Isok(X) or Autk(X). In particular, let Ckn,l be a simple closed k-curve with l elements in Zn. Then, the group Isok(Ckn,l) is proved to be isomorphic to the standard dihedral group Dl with order l. The calculation of this quantity Isok(Ckn,l) is a key step for obtaining many new results. Indeed, it is essential for exploring many features of Isok(X). Furthermore, this quantity is proved to be a digital topological invariant. After proceeding with an Isok(X)-action on (X,k), we investigate some properties of fixed point sets by this action. Finally, we explore various features of fixed point sets by this action from the viewpoint of digital k-curve theory. This paper only deals with k-connected digital images (X,k) whose cardinality is equal to or greater than 2. Besides, we discuss some errors that have appeared in the lilterature.

In this paper, we introduce a new operator in order to derive some new symmetric properties of [Formula: see text]-Fibonacci and [Formula: see text]-Lucas numbers and Fibonacci polynomials. By making use of the new operator defined in this paper, we give some new generating functions for [Formula: see text]-Fibonacci and Pell numbers and Fibonacci polynomials.

The purpose of the paper is to study new algebraic operations and fundamental properties of totally dependent-neutrosophic sets and totally dependent-neutrosophic soft sets. First, the in-coordination relationships among the original inclusion relations of totally dependent-neutrosophic sets (called type-1 and typ-2 inclusion relations in this paper) and union (intersection) operations are analyzed, and then type-3 inclusion relation of totally dependent-neutrosophic sets and corresponding type-3 union, type-3 intersection, and complement operations are introduced. Second, the following theorem is proved: all totally dependent-neutrosophic sets (based on a certain universe) determined a generalized De Morgan algebra with respect to type-3 union, type-3 intersection, and complement operations. Third, the relationships among the type-3 order relation, score function, and accuracy function of totally dependent-neutrosophic sets are discussed. Finally, some new operations and properties of totally dependent-neutrosophic soft sets are investigated, and another generalized De Morgan algebra induced by totally dependent-neutrosophic soft sets is obtained.

In this paper we give a combinatorial characterization of tight fusion frame (TFF) sequences using Littlewood-Richardson skew tableaux. The equal rank case has been solved recently by Casazza, Fickus, Mixon, Wang, and Zhou. Our characterization does not have this limitation. We also develop some methods for generating TFF sequences. The basic technique is a majorization principle for TFF sequences combined with spatial and Naimark dualities. We use these methods and our characterization to give necessary and sufficient conditions which are satisfied by the first three highest ranks. We also give a combinatorial interpretation of spatial and Naimark dualities in terms of Littlewood-Richardson coefficients. We exhibit four classes of TFF sequences which have unique maximal elements with respect to majorization partial order. Finally, we give several examples illustrating our techniques including an example of tight fusion frame which can not be constructed by the existing spectral tetris techniques. We end the paper by giving a complete list of maximal TFF sequences in dimensions less than ten., 31 pages

Let f be a transcendental meromorphic function of nite order in the plane such that f (m) has nitely many zeros for some positive integer m ≥ 2: Suppose that f (k) and f share a CM, where k ≥ 1 is a positive integer, a 0 is a nite complex value. Then f is an entire function such that f (k) − a = c(f − a); where c 0 is a nonzero constant. The results in this paper are concerning a conjecture of Bruck (5). An example is provided to show that the results in this paper, in a sense, are the best possible.

In the paper [4, Theorem 1] Gat, Goginava and the author proved that the maximal operator σκ,∗ of Marcinkiewicz-Fejer means of Walsh-Kaczmarz-Fourier series, is bounded from the dyadic Hardy space Hp into the space Lp for p > 2/3 . Moreover, Goginava and the author showed that σκ,∗ is not bounded from the Hardy space H2/3 to the space L2/3 [6, Theorem 1]. The main aim of this paper is to show that the maximal operator σκ,∗ f := supn∈P |σκ n f | log3/2(n+1) , is bounded from the Hardy space H2/3 into the space L2/3. Moreover, we prove that the order of deviant behavior of the n th Walsh-Kacmarz-Marcinkiewicz-Fejer mean is exactly log3/2(n+1) in the endpoint p = 2/3 . Mathematics subject classification (2010): 42C10.

In this paper, we establish some new Ostrowski type inequalities for s logarithmically convex functions. Some applications of our results to P.D.F.’s and in numerical integration are given. 1. INTRODUCTION Let f : I [0;1] ! R be a di¤erentiable mapping on I , the interior of the interval I, such that f 0 2 L [a; b] where a; b 2 I with a < b. If jf 0 (x)j M , then the following inequality holds (see [2]): (1.1) f(x) 1 b a Z b a f(u)du M b a " (x a) + (b x) 2 # : This inequality is well known in the literature as the Ostrowski inequality. For some results which generalize, improve and extend the inequality (1.1) see ([2]-[6]) and the references therein. Let us recall some known de…nitions and results which we will use in this paper. The function f : I R ! [0;1) is said to be log convex or multiplicatively convex if log t is convex, or, equivalenly, if for all x; y 2 I and t 2 [0; 1] ; one has the inequality (See [7], p.7): f (tx+ (1 t) y) [f (x)] + [f (y)] t) : In [1], Akdemir and Tunc were introduced the class of s logarithmically convex functions in the …rst sense as the following: De…nition 1. A function f : I R0 ! R+ is said to be s logarithmically convex in the …rst sense if (1.2) f ( x+ y) [f (x)] s [f (y)] s for some s 2 (0; 1], where x; y 2 I and s + s = 1: In [8], authors introduced the class of s logarithmically convex functions in the second sense as the following: De…nition 2. A function f : I R0 ! R+ is said to be s logarithmically convex in the second sense if (1.3) f (tx+ (1 t) y) [f (x)] s [f (y)] (1 t) 2000 Mathematics Subject Classi…cation. 26D10, 26A15, 26A16, 26A51. Key words and phrases. Hadamard’s inequality, s geometrically convex functions . 1 2 MEVLUT TUNC AND AHMET OCAK AKDEMIR for some s 2 (0; 1], where x; y 2 I and t 2 [0; 1]. Clearly, when taking s = 1 in De…nition 1.2 or De…nition 1.3, then f becomes the standard logarithmically convex function on I. The main purpose of this paper is to establish some new Ostrowski’s type inequalities for s logarithmically convex functions. We also give some applications to P.D.F.’s and to midpoint formula. 2. THE NEW RESULTS In order to prove our main results, we will use following Lemma which was used by Alomari and Darus (see [3]): Lemma 1. Let f : I R ! R, be a di¤erentiable mapping on I where a; b 2 I, with a < b. Let f 0 2 L[a; b]; then the following equality holds

This paper is about the classification of frieze groups. A frieze is a decorative strip of paper (or wood, stone,…) on which a pattern is produced by the periodic repetition of a picture along the strip. A symmetry of a frieze is an isometry of the plane that leaves the pattern unchanged, and a frieze group is the group of symmetries of some frieze. A popular exercise is to start with a given frieze and then try to identify its frieze group. However, in order to do this we need to know that there are only seven possible frieze groups, and what these groups are. Seven friezes, with different frieze groups, are illustrated below, in such a way that each pattern is invariant under the same translation, namely x → x + 1. Our task is to show that (up to a change in the motif, and a simple change of coordinates in the plane) these are the only frieze patterns.

In this paper, we define the diamond Marcel Riesz operator of order (α, β) on the function f by M (f) = Kα,β ∗ f, where Kα,β is diamond kernel of Marcel Riesz, α, β ∈ C, the symbol ∗ designates the convolution, and f ∈ S, S is the Schwartz space of functions. Our purpose of this paper is to obtain the operator N (α,β) = [ M (α,β) ]−1 such that if M (α,β)(f) = φ, then N (α,β)φ = f. Our results generalize formulae appearing in A. Kananthai [On the convolutions of the diamond kernel of Marcel Riesz, Applied Mathematics and Computation, 114(2000), 95 − 101]. Mathematics Subject Classification: 46F10, 46F12

As known, the method to obtain a sequence space by using convergence field of an infinite matrix is an old method in the theory of sequence spaces. However, the study of convergence field of an infinite matrix in the space of almost convergent sequences is so new (see [15]). The purpose of this paper is to introduce the new spaces and consisting of all sequences whose Cesaro transforms of order one are in the spaces f and f0, respectively. Also, in this paper, we show that and are linearly isomorphic to the spaces f and f0, respectively. The β- and γ-duals of the spaces and are computed. Furthermore, the classes ( : μ) and (μ: ) of infinite matrices are characterized for any given sequence space μ, and determined the necessary and sufficient conditions on a matrix A to satisfy BC-core(Ax) ⊆ K-core(x), K-core(Ax) ⊆ BC-core(x), BC-core(Ax) ⊆ BC-core(x), BC-core(Ax) ⊆ st-core(x) for all x ∈ l∈.

A selfadjoint involutive matrix J endows C n with the indefinite Krein space struc- ture endowed by the inner product (·,·) given by (x,y) := � Jx,y� , x,y ∈ C n . For a pair of J−selfadjoint matrices A,B with positive eigenvalues, LogA J LogB is called the J−chaotic order or the indefinite chaotic order .S ano (8), proved as an application of Furuta inequality of indefinite type that LogA J LogB if and only if A r J A r 2 B p A r 2 r p+r for all p > 0a ndr > 0. In this paper, we prove Sano's result using a different approach. In the process, some other re- sults due to Bebiano, Lemos, Providencia and Soares (4, 9) are reobtained. The techniques in this paper are inspired by (5).

There has been an emerging interest in developing an approximation theory for manifold-valued functions. In this paper, we address the following fundamental problem: Let M be a manifold with a metric d. For each smoothness factor r > 0 and approximation order R > 0, is there an approximation operator Ah = Ah;r,R that maps samples of any f : R→ M on a grid of size h to an approximant fh = Ah(f |hZ) : R→ M with the properties that a) supx d(fh(x), f(x)) = O(h) whenever f is a bounded C function, and b) fh is C smooth ? The case of M = R is of course well-studied. In the recent paper [14], the authors show that subdivision methods can be used to create arbitrarily smooth interpolants for M -valued data, addressing b) above. In this paper we further show that interpolatory subdivision schemes can be used to solve a) above. So, altogether, we establish the fact that if a linear interpolatory subdivision scheme possesses a smoothness order r and an approximation order R, then a nonlinear interpolatroy subdivision scheme for M -valued data constructed based on this linear scheme has the same smoothness and approximation orders. In other words, subdivision schemes furnish a constructive approximation method for solving the open problem posted above. We discuss the construction of quasi-interpolants of manifold-valued data based on general (not necessarily interpolatory) subdivision schemes. Acknowledgments. The work of this research was partially supported by the National Science Foundation grant DMS 0542237. The empirical observation on approximation equivalence (Figure 1) was first presented in the Sixth International Conference on Curves and Surfaces (Avignon, France, June 29-July 5, 2006.) and further discussed in a mini-symposium in the 12th International Conference in Approximation Theory (San Antonio, Texas, March 4-8, 2007) and the MAIA 2007 conference (Alesund, Norway, August 22-26, 2007.) A preliminary effort on this problem was made in [5]. The second named author thanks Nira Dyn for inviting him to a workshop on “Subdivision and Refinability” (Pontignano, Siena, Italy, May 1-4, 2008), at which the main result of this paper was first presented.

A subgroup H of finite group G is called pronormal in G if for every element x of G, H is conjugate to Hx in 〈H, Hx〉. A finite group G is called PRN-group if every cyclic subgroup of G of prime order or order 4 is pronormal in G. In this paper, we find all PRN-groups and classify minimal non-PRN-groups (non-PRN-group all of whose proper subgroups are PRN-groups). At the end of the paper, we also classify the finite group G, all of whose second maximal subgroups are PRN-groups.

In his paper "Finite groups have many conjugacy classes" (J. London Math. Soc (2) 46 (1992), 239-249), L. Pyber proved the to date best general lower bounds for the number of conjugacy classes of a finite group in terms of the order of the group. In this paper we strengthen the main results in Pyber's paper., 13 pages

In the paper, it is proved that, for any odd n ≥ 1039, there are words u(x, y) and υ(x, y) over the group alphabet {x, y} such that, if a and b are any two noncommuting elements of the free Burnside group B(m, n), then, for some k, the elements u(ak, b) and υ(ak, b) freely generate a free Burnside subgroup of the group B(m, n). In particular, the facts proved in the paper imply the uniform nonamenability of the group B(m, n) for odd n, n ≥ 1039.

In this paper, sufficient conditions are obtained, so that the second order neutral delay differential equation $$ (r(t)(y(t) - p(t)y(t - \tau ))')' + q(t)G(y(h(t)) = f(t) $$ has a positive and bounded solution, where q, h, f ∈ C ([0, ∞), ℝ) such that q(t) ≥ 0, but ≢ 0, h(t) ≤ t, h(t) → ∞ as t → ∞, r ∈ C (1) ([0, ∞), (0, ∞)), p ∈ C (2) [0, ∞), ℝ), G ∈ C(ℝ, ℝ) and τ ∈ ℝ+. In our work r(t) ≡ 1 is admissible and neither we assume G is non-decreasing, xG(x) > 0 for x ≠ 0, nor we take G is Lipschitzian. Hence the results of this paper improve many recent results.

In §1 we study the p-groups G containing exactly p + 1 subgroups of order p p and exponent p. A number of counting theorems and results on subgroups of maximal class and p-groups with few subgroups of given type are also proved. Counting theorems play crucial role in the whole paper. This paper is a continuation of (Ber1, Ber3, Ber4, BJ2). We use the same notation, however, for the sake of convenience, we recall it in the following paragraph. In what follows, p is a prime, n, m, k, s, t are natural numbers, G is a finite p-group of order |G|, o(x) is the order of x ∈ G, n(G) = h x ∈ G | o(x) ≤ p n i , ∗(G) = h x ∈ G | o(x) = p n i and ℧n(G) = h x p n | x ∈ Gi . A p-group G is said to be absolutely regular if |G/℧1(G)| 2 and cl(G) = m − 1. As usually, G ' , �(G), Z(G) denote the derived subgroup, Frattini subgroup and center of G, respectively. Let i = {H 2 and n > 3 for p = 2, let Mpn = h a, b | a p n 1 = b p = 1, a b = a 1+p n 2 i . Let D2m, Q2m and SD2m be dihedral, generalized quaternion and semidihedral groups of order 2 m , and let Cpn, Epn be cyclic and elementary abelian groups

Let w be a complex symmetric matrix of order r, and Δ1(w), . . . , Δr(w) the principal minors of w. If w belongs to the Siegel right half-space, then it is known that Re (Δk(w)/Δk-1(w)) > 0 for k = 1, . . . , r. In this paper we study this property in three directions. First we show that this holds for general symmetric right half-spaces. Second we present a series of non-symmetric right half-spaces with this property. We note that case-by-case verifications up to dimension 10 tell us that there is only one such irreducible non-symmetric tube domain. The proof of the property reduces to two lemmas. One is entirely generalized to non-symmetric cases as we prove in this paper. This is the third direction. As a byproduct of our study, we show that the basic relative invariants associated to a homogeneous regular open convex cone Ω studied earlier by the first author are characterized as the irreducible factors of the determinant of right multiplication operators in the complexification of the clan associated to Ω.

In this paper, we investigate the existence of solutions of second order im-pulsive neutral functional differential inclusions which the nonlinearity F admits convexand non-convex values. Some results under weaker conditions are presented. Our resultsextend previous ones. The methods rely on a fixed point theorem for condensing mul-tivalued maps and Schaefer’s fixed point theorem combined with lower semi-continuousmultivalued operators with decomposable values. 1. IntroductionIn this paper, we consider the existence results of solutions for the followingsecond-order neutral functional differential inclusions with the form(1.1) p(t)u 0 (t)−Z tt−τ q(s)u(s)ds 0 ∈ F(t,u t ), a.e.t∈ [0,T]\{t 1 ,t 2 ,··· ,t m },(1.2) ∆u| t=t k = I k (u(t −k )), ∆u 0 | t=t k = J k (u(t −k )), k= 1,2,··· ,m,(1.3) u(t) = a(t),t∈ [−τ,0],u 0 (0) = η,where F: [0,T] × C→ P(R n ) is a multi-valued map, C= {ϕ : [−τ,0] → R ; ϕis continuous everywhere except for a finite number of points ˜tat which ϕ(˜t − ) andϕ(˜t

This paper considers some random processes of the form X n+1=T X n +B n (mod p) where B n and X n are random variables over (ℤ/pℤ) d and T is a fixed d×d integer matrix which is invertible over the complex numbers. For a particular distribution for B n , this paper improves results of Asci to show that if T has no complex eigenvalues of length 1, then for integers p relatively prime to det (T), order (log p)2 steps suffice to make X n close to uniformly distributed where X 0 is the zero vector. This paper also shows that if T has a complex eigenvalue which is a root of unity, then order p b steps are needed for X n to get close to uniformly distributed for some positive value b≤2 which may depend on T and X 0 is the zero vector.

We prove that if G is a locally finite group admitting an automorphism φ of order four such that CG(φ) is Chernikov, then G has a soluble subgroup of finite index. 1. Introduction. The study of centralizers in locally finite groups is an interesting direction of group theory. In particular, the situation that a locally finite group G admits an automorphism φ of finite order (or, equivalenly, contains an element φ) such that CG(φ) is Chernikov has received some attention in the past. Recall that a group C is Chernikov if it has a subgroup of finite index that is a direct product of finitely many groups of type Cp∞ for various primes p (quasicyclic p-groups). By a deep result obtained independently by Shunkov (22) and also by Kegel and Wehrfritz (7) Chernikov groups are precisely the locally finite groups satisfying the minimal condition on subgroups; that is, any non-empty set of subgroups possesses a minimal subgroup. Hartley proved in 1988 that if φ is of prime-power order, then G has a locally soluble subgroup of finite index (see (4, Theorem 1)). The case of the theorem where φ has order two was handled by Asar in (1) and for automorphisms of arbitrary prime order the theorem was proved by Turau in (25). Both Hartley's and Turau's works depend on the classification of finite simple groups. We now quote a paragraph from the introduction in Hartley's paper. "It seems likely that much more remains to be said. Possibly Theorem 1 remains true even if 'locally soluble' is replaced by 'soluble' and 'prime power order' is replaced by 'finite order', and if that is too much to hope for, then at least some progress in that direction might be feasible". In 1992 Hartley did manage to show that if the automorphism φ has arbitrary finite order and CG(φ) is finite, then indeed G has a locally soluble subgroup of finite index (5). Hartley's question if this is true with φ of arbitrary finite order and CG(φ) Chernikov remains unanswered, the main difficulty being that the problem does not reduce to finite groups. In the present paper we are concerned with the part of Hartley's implicit question that deals with possible replacement of 'locally soluble' by 'soluble'. Trying to figure out a way to handle the question one quickly finds themself faced with the following long- standing problem. Suppose a finite group G admits a fixed-point-free automorphism

In this paper, necessary and sufficient conditions for the oscillation and asymptotic behaviour of solutions of the second order neutral delay differential equation (NDDE) $$\left[ {r(t)(y(t) - p(t)y(t - \tau ))'} \right]^\prime + q(t)G(y(h(t))) = 0$$ are obtained, where q, h ∈ C([0, ∞), ℝ) such that q(t) ≥ 0, r ∈ C (1) ([0, ∞), (0, ∞)), p ∈ C ([0, ∞), ℝ), G ∈ C (ℝ, ℝ) and τ ∈ ℝ+. Since the results of this paper hold when r(t) ≡ 1 and G(u) ≡ u, therefore it extends, generalizes and improves some known results.

In this paper, we determine the bound of the valency of Cayley graphs of Frobenius groups with respect to normal Cayley subsets which guarantees to be Ramanujan. We see that if the ratio between the orders of the Frobenius kernel and complement is not so small, then this bound coincides with the trivial one coming from the trivial estimate of the largest non-trivial eigenvalue of the graphs. Moreover, in the cases of the dihedral groups of order twice odd primes, which are special cases of the Frobenius groups, we determine the same bound for the Cayley graphs of the groups with respect to not only normal but also all Cayley subsets. As is the case of abelian groups which we have treated in the previous papers, such a bound is equal to the trivial one in the above sense or, as exceptional cases, exceeds one from it. We then clarify that the latter occurs if and only if the corresponding prime is represented by a quadratic polynomial in a finite family., Comment: 12 pages

where the least upper bound is taken over all possible sigma-perturbations Q corresponding to a given σ > 0 is called [1] the higher sigma-exponent (the Izobov exponent) of system (1). It is well known that the sets of characteristic exponents of the original and perturbed systems (1) and (2) coincide for all σ > 2M ; therefore, the relation ∇σ(A) = λn(A) is valid for such σ. It follows from the results in [2, 3] providing a complete description of the higher sigma-exponent treated as a function of the parameter σ that there exists a unique critical value σ0(A) ∈ [0, 2M ] such that ∇σ(A) = λn(A) for all σ > σ0(A) and ∇σ(A) > λn(A) for 0 < σ < σ0(A). By using the Lyapunov, Grobman, and Perron irregularity coefficients [4, pp. 67, 73; 5, p. 80] σL(A), σG(A), and σP(A), respectively, one can obtain more accurate upper bounds for σ0(A). In particular, it was shown in [6] that the inequality σ0(A) ≤ σL(A) holds, which, by [7], can be strengthened to the estimate σ0(A) ≤ σG(A). If n = 2, then, by [8], we have the even stronger inequality σ0(A) ≤ σP(A), which fails [9, 10] for systems of higher dimension. In the paper [9], the irregularity coefficient σλ(A) was constructed, which is also an upper bound for σ0(A) but, in addition, does not exceed σG(A) and coincides with σP(A) for two-dimensional systems. In the present paper, on the basis of the algorithm constructed in [1] for the computation of the higher sigma-exponent ∇σ(A), we obtain explicit formulas expressing the critical value σ0(A) in terms of the Cauchy matrix XA(t, s) of the unperturbed system (1). Let us introduce related notation. For each m ∈ N, by D0(m) we denote the set of all subsets d of the set {1, . . . ,m− 1} ⊂ N. If d ∈ D (m) is nonempty, then its elements di, i = 1, . . . , s, where s := |d| is the number of elements in d, are assumed to be numbered in ascending order, so that d = {d1, . . . , ds} and 0 < d1 < · · · < ds < m. The class of nonempty sets d ∈ D0(m) is denoted by D (m). We also set ‖d‖ := d1 + · · · + ds for d = ∅ and ‖d‖ := 0 for d = ∅. Furthermore, for convenience of computations, we assume that d0 = 0 and ds+1 = m for all d ∈ D (m). Under these assumptions, we define Ξ(m,d) := ∑s i=0 ln ‖XA (di+1, di)‖, where m ∈ N, d ∈ D (m), and s := |d|.

Let p be a prime number. This paper solves the question of the structure of the group D(P) of endo-permutation modules over an arbitrary finite p-group P, that was open after Dade’s original papers in 1978 ([19], [20]), and it gives a proof of the conjectures proposed in [4] and [10]. This leads to a presentation of D(P) by explicit generators and relations, generalizing the presentation obtained by Dade when P is abelian. A key result of independent interest is the explicit description of the kernel of the natural map from the Burnside group to the group of rational characters, in terms of the extraspecial group of order p 3 and exponent p if p≠2, or of all dihedral groups of order at least 8 if p=2.

By the classification theorem by F. Oort and J. Tate [6], any group scheme of prime order is isomorphic to a group scheme Ga,b under the suitable choice of a and b. We computed the torsors for some kinds of group schemes Ga,b in [8], which is a joint work with T. Sekiguchi, as in the following way: denote by p a prime number and by m = φ(p−1) the value of the Euler function φ. Suppose p is a prime ideal lying over p (which splits completely in Z[ζ]), where ζ is a primitive (p − 1)-st root of the unity. In case p is principal, the sequence 0 → μp,B → G m m,B p −→ Gm,B → 0 is exact, and the Galois descent of μp,B is isomorphic to Ga,b under the suitable choice of a and b, thus one can compute the torsors for this kinds of group schemes. The non-principal case is solved by Y. Koide [3] by using our method. The aim of this paper is to study some group schemes of order a power of a prime number. In section from 1 to 3, we would like to review the main result of the papers [6] by F. Oort and J. Tate, [4] by Y. Koide and T. Sekiguchi, and [8] by T. Sekiguchi and Y. Toda. In section 4, we give our main result, namely, the torsor for the Galois descent of μpn,B.

Let G = ((A,B),E) be an instance of the stable marriage problem where every vertex ranks its neighbors in a strict order of preference. A matching M in G is popular if M does not lose a head-to-head election against any matching. Popular matchings are a well-studied generalization of stable matchings, introduced with the goal of enlarging the set of admissible solutions, while maintaining a certain level of fairness. Every stable matching is a min-size popular matching. Unfortunately, when there are edge costs, it is NP-hard to find a popular matching of minimum cost -- even worse, the min-cost popular matching problem is hard to approximate up to any factor. Let opt be the cost of a min-cost popular matching. Our goal is to efficiently compute a matching of cost at most opt by paying the price of mildly relaxing popularity. Our main positive results are two bi-criteria algorithms that find in polynomial time a near-popular or quasi-popular matching of cost at most opt. Moreover, one of the algorithms finds a quasi-popular matching of cost at most that of a min-cost popular fractional matching, which could be much smaller than opt. Key to the other algorithm is a polynomial-size extended formulation for an integral polytope sandwiched between the popular and quasi-popular matching polytopes. We complement these results by showing that it is NP-hard to find a quasi-popular matching of minimum cost, and that both the popular and quasi-popular matching polytopes have near-exponential extension complexity. This version of the paper goes beyond the conference version [12] in the following two points: (i) the algorithm for finding a quasi-popular matching of cost at most that of a min-cost popular fractional matching is new; (ii) the proofs from Section 6.1 and Section 7.3 are now self-contained (the conference version used constructions from [10] to show these lower bounds).

This paper is concerned with the well-known and long-standingk(G V)-problem: If the finite groupGacts faithfully and irreducibly on the finiteGF(p)-moduleVandpdoes not divide the order ofG, is the numberk(GV) of conjugacy classes of the semidirect productGVbounded above by the order ofV? Over the past two decades, through the work of numerous people, by using deep character theoretic arguments this question has been answered in the affirmative except for ρ = 5 for which it is still open. In this paper we suggest a new approach to thek(G V)-problem which is independent of most of the previous work on the problem and which is mainly group theoretical. To demonstrate the potential of the new line of attack we use it to solve thek(G V)-problem for solvableGand large ρ.

Let K \mathbf {K} be a field of characteristic 0 0 and let ( G n ( x ) ) n = 0 ∞ (G_{n}(x))_{n=0}^{\infty } be a linear recurring sequence of degree d d in K [ x ] \mathbf {K}[x] defined by the initial terms G 0 , … , G d − 1 ∈ K [ x ] G_0,\ldots ,G_{d-1}\in \mathbf {K}[x] and by the difference equation \[ G n + d ( x ) = A d − 1 ( x ) G n + d − 1 ( x ) + ⋯ + A 0 ( x ) G n ( x ) , for n ≥ 0 , G_{n+d}(x)=A_{d-1}(x)G_{n+d-1}(x)+\cdots +A_0(x)G_{n}(x), \quad \mbox {for} \,\, n\geq 0, \] with A 0 , … , A d − 1 ∈ K [ x ] A_0,\ldots ,A_{d-1}\in \mathbf {K}[x] . Finally, let P ( x ) P(x) be an element of K [ x ] \mathbf {K}[x] . In this paper we are giving fairly general conditions depending only on G 0 , … , G d − 1 , G_0,\ldots ,G_{d-1}, on P P , and on A 0 , … , A d − 1 A_0,\ldots ,A_{d-1} under which the Diophantine equation \[ G n ( x ) = G m ( P ( x ) ) G_{n}(x)=G_{m}(P(x)) \] has only finitely many solutions ( n , m ) ∈ Z 2 , n , m ≥ 0 (n,m)\in \mathbb {Z}^{2},n,m\geq 0 . Moreover, we are giving an upper bound for the number of solutions, which depends only on d d . This paper is a continuation of the work of the authors on this equation in the case of second-order linear recurring sequences.

We continue our study of the different representations of an integer n as a sum of two squares initiated in our final paper written with Paul Erdős [1]. We let r(n) denote the number of representations n = a 2 + b 2 counted in the usual way, that is, with regard to both the order and sign of a and b . We have where x is the non-principal Dirichlet character (mod 4); moreover, r ( n )≥0 if and only if n has no prime factor p ≡ 3 (mod 4) with odd exponent. We define the function b ( n ) on the sequence of representable numbers as the least possible value of | b |—for example, we have b (13) = 2, b (25) = 0, b (65) = 1—and we write here and throughout the paper the star denotes that the sum is restricted to the representable integers. The problem considered here is to find an asymptotic formula for this sum, or, less ambitiously, to determine the order of magnitude of the function B(x) .

In the present paper we study $$G_2$$ -periodic $$p$$ -adic quasi Gibbs measures for $$p$$ -adic Potts model on a Cayley tree of order two. In the case $$q=3$$ , we prove the occurrence of a phase transition and construct ART quasi Gibbs measures for $$p$$ -adic Potts model on a Cayley tree of order $$k\geq3$$ .

This paper considers “lazy” random walks supported on a random subset of k elements of a finite group G with order n. If k=⌈a log2 n⌉ where a>1 is constant, then most such walks take no more than a multiple of log2 n steps to get close to uniformly distributed on G. If k=log2 n+f(n) where f(n)→∞ and f(n)/log2 n→0 as n→∞, then most such walks take no more than a multiple of (log2 n) ln(log2 n) steps to get close to uniformly distributed. To get these results, this paper extends techniques of Erdos and Renyi and of Pak.

In a recent paper Edmunds, Gurka, and Opic [5] showed that Sobolev spaces of order k, based on the Zygmund spaces L n/k (log L)α (R n ), are continuously embedded into L ∞ (R n ) if α > 1/p′, p n/k. In this paper we replace L n/k (log L)α (R n ) by the Lebesgue space L n/k (R n ) and increase the smoothness of the functions involved by a "logarithmic" order α > 1/p′ to obtain the continuous embedding into L ∞ (R n ). Both approaches turn out to be equivalent. We also derive results of Trudinger-type [16] on embeddings into Orlicz spaces in the limit case k = n/p as well as results of Brezis-Wainger-type [2] on almost Lipschitz continuity in the superlimiting case k = n/p + 1.