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Let S and S ˜ be naturally valenced association schemes on sets X and X ˜ , respectively, and let ϕ be a (combinatorial) morphism from ( X , S ) to ( X ˜ , S ˜ ) . In Xu (2009) [X2] , a necessary and sufficient condition was given for ϕ to induce an algebra homomorphism from the scheme ring C S to the scheme ring C S ˜ . The present paper provides new techniques with which this result can be proved without assuming ker ( ϕ ) to be finite. To do this, we will first need to prove that for any normal closed subset T of S, whether T is finite or infinite, the quotient S / / T is a naturally valenced association scheme on the set X / T . We will also need to discuss scheme ring homomorphisms of naturally valenced association schemes, and prove some isomorphism theorems without assuming the kernels of the scheme ring homomorphisms to be finite. As a direct consequence, for a naturally valenced commutative association scheme S on a set X and any closed subset T of S, the quotient S / / T is a naturally valenced commutative association scheme on the set X / T . The approach in this paper is different from Xu (2009) [X2] , and quasi-algebraic morphisms of naturally valenced association schemes are also studied.

A subsemigroup S of a semigroup Q is a local left order in Q if, for every group H -class H of Q , S ∩ H is a left order in H in the sense of group theory. That is, every q ∈ H can be written as a ♯ b for some a , b ∈ S ∩ H , where a ♯ denotes the group inverse of a in H . On the other hand, S is a left order in Q and Q is a semigroup of left quotients of S if every element of Q can be written as c ♯ d where c , d ∈ S and if, in addition, every element of S that is square cancellable lies in a subgroup of Q . If one also insists that c and d can be chosen such that c R d in Q , then S is said to be a straight left order in Q . This paper investigates the close relation between local left orders and straight left orders in a semigroup Q and gives some quite general conditions for a left order S in Q to be straight. In the light of the connection between locality and straightness we give a complete description of straight left orders that improves upon that in our earlier paper.

In previous papers (J. Combin Theory Ser. A 103 (2003) 387) and (J. Combin. Theory Ser. A 105 (2004) 359) Saharon Shelah and I formulated a conditional chromatic number theorem, which described a setting in which the chromatic number of the plane takes on two different values depending upon the axioms for set theory. We also constructed examples of a distance graph on the real line R and difference graphs on the real plane R2 whose chromatic numbers depend upon the system of axioms we choose for set theory. Ideas developed there are extended in the present paper to construct difference graphs on the real space Rn, whose chromatic number is a positive integer in the Zermelo–Fraenkel-choice system of axioms, and is not countable (if it exists) in a consistent system of axioms with limited choice, studied by Solovay (Ann. Math. Ser. 2 (1970) 1). These examples illuminate how heavily combinatorial results can depend upon the underlying set theory, help appreciate the potential complexity of the chromatic number of n-space problem, and suggest that the chromatic number of n-space may depend upon the system of axioms chosen for set theory.

The concept of weak contraction from the case of single-valued mappings is extended to multi-valued mappings and then corresponding convergence theorems for the Picard iteration associated to a multi-valued weak contraction are obtained. The main results in this paper extend, improve and unify a multitude of classical results in the fixed point theory of single and multi-valued contractive mappings and also improve recent results from the paper [P.Z. Daffer, H. Kaneko, Fixed points of generalized contractive multi-valued mappings, J. Math. Anal. Appl. 192 (1995), 655–666].

In this paper, the equivalence of the strong convergence between the modified Mann and Ishikawa iterations with errors in two different schemes by Xu [Y.G. Xu, Ishikawa and Mann iteration process with errors for nonlinear strongly accretive operator equations, J. Math. Anal. Appl. 224 (1998) 91–101] and Liu [L.S. Liu, Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces, J. Math. Anal. Appl. 194 (1995) 114–125] respectively is proven for the generalized strongly successively Φ-pseudocontractive mappings without Lipschitzian assumption. Our results generalize the recent results of the papers [Zhenyu Huang, F. Bu, The equivalence between the convergence of Ishikawa and Mann iterations with errors for strongly successively pseudocontractive mappings without Lipschitzian assumption, J. Math. Anal. Appl. 325 (1) (2007) 586–594; B.E. Rhoades, S.M. Soltuz, The equivalence between the convergences of Ishikawa and Mann iterations for an asymptotically nonexpansive in the intermediate sense and strongly successively pseudocontractive maps, J. Math. Anal. Appl. 289 (2004) 266–278; B.E. Rhoades, S.M. Soltuz, The equivalence between Mann–Ishikawa iterations and multi-step iteration, Nonlinear Anal. 58 (2004) 219–228] by extending to the most general class of the generalized strongly successively Φ-pseudocontractive mappings and hence improve the corresponding results of all the references given in this paper by providing the equivalence of convergence between all of these iteration schemes for any initial points u 1 , x 1 in uniformly smooth Banach spaces.

Text The problem of determining the maximum size of coefficients of cyclotomic polynomials has been studied extensively. Let A ( n ) be the maximum absolute value of a coefficient of Φ n ( x ) , the n th cyclotomic polynomial. This paper further investigates a variation of A ( n ) introduced by Pomerance and Ryan. We consider the function B ( n ) , the maximum absolute value of a coefficient of any divisor of x n − 1 . In the first part of this paper we give an analogue of a well-known result for A ( n ) proved independently by Felsch and Schmidt, and by Justin, and give an upper bound for B ( n ) for a general n . In the second part of the paper we resolve a question of Pomerance and Ryan giving an explicit formula for B ( p 2 q ) . We then give upper and lower bounds for B ( p q r ) where p q r are primes. Video For a video summary of this paper, please visit http://www.youtube.com/watch?v=KLAdOHS_Ekw .

In the previous paper [O.J. Brison, J.E. Nogueira, Linear recurring sequence subgroups in finite fields, Finite Fields Appl. 9 (2003) 413–422] the authors investigated when, and how, the elements of a multiplicative subgroup of a finite field can be written, without repetition, as a cyclically-closed second order linear recurring sequence. Here, the earlier results are extended; in particular a general method is suggested for attempting to prove a given subgroup to be “standard,” in the terminology of the previous paper, while a lifting theorem for “nonstandard” subgroups is proved.

This paper is the first of several papers in which we prove, for the case where the fields of coefficients are of characteristic zero, four open problems posed in the work of Melvyn Nathanson (2003) [1] concerning the solutions of a functional equation arising from multiplication of quantum integers [ n ] q = q n − 1 + q n − 2 + ⋯ + q + 1 . In this paper, we prove one of the problems. The next papers, namely [2] , [3] , [4] by Lan Nguyen, contain the solutions to the other 3 problems.

In a Polish group G, a property is said to hold for a generic g∈G if the set on which it does not hold is meager in G; a property holds for almost every (ae) g∈G if the set on which it does not hold is Haar null. In this paper we study the properties of generic and ae elements of the Baer–Specker Group ZZ, the space of all self-maps of Z. We find that with respect to most of the properties under consideration in this paper, the properties of a generic element and ae element are complementary. Our results are similar in flavor to results by Dougherty and Mycielski for the group S∞ of all permutations of N.

In an earlier paper with Whittle, we showed that there is a tree that displays, up to a natural equivalence, all non-trivial 3-separations of a 3-connected matroid M. The purpose of this paper is to give a polynomial-time algorithm for constructing such a tree for M.

Let G be a finite p-solvable group and N be a normal subgroup of G. Suppose that the p-regular elements of N have exactly two G-conjugacy class sizes. In this paper it is shown that, if H is a p-complement of N, then either H is abelian or H is a product of a q-group for some prime q{double barred pipep and a central subgroup of G. © 2011 Elsevier Inc., This paper has been written during the visit of the first and fourth authors to Departamento de Matematicas at the Universidad Jaume I of Castellon, Spain. They Would like to express their deep gratitude for their warm hospitality. The second and the third authors are supported by Proyecto MTM2010-19938-C03-02 and the second author is also supported by grant Fundacio Caixa-Castello P11 B2010-47.

Given a set A ⊂ N let σ A ( n ) denote the number of ordered pairs ( a , a ′ ) ∈ A × A such that a + a ′ = n . The celebrated Erdős–Turan conjecture states that if A ⊂ N such that σ A ( n ) ⩾ 1 for all sufficiently large n, then the representation function σ A ( n ) must be unbounded. For each positive integer m, let R m be the least positive integer r such that there exists a set A ⊆ Z m with A + A = Z m and σ A ( n ) ⩽ r . Ruzsa's method in [I.Z. Ruzsa, A just basis, Monatsh. Math. 109 (1990) 145–151] implies that R m must be bounded. It is pleasure to call R m a Ruzsa's number. In this paper we prove that all Ruzsa's numbers R m ⩽ 288 . This improves the previous bound R m ⩽ 5120 . Several related open problems are proposed. Video abstract For a video summary of this paper, please visit http://www.youtube.com/watch?v=hgDwkwg_LzY .

The classical Eckmann–Hilton argument shows that two monoid structures on a set, such that one is a homomorphism for the other, coincide and, moreover, the resulting monoid is commutative. This argument immediately gives a proof of the commutativity of the higher homotopy groups. A reformulation of this argument in the language of higher categories is: suppose we have a one object, one arrow 2-category, then its Hom-set is a commutative monoid. A similar argument due to A. Joyal and R. Street shows that a one object, one arrow tricategory is ‘the same’ as a braided monoidal category. In this paper we begin to investigate how one can extend this argument to arbitrary dimension. We provide a simple categorical scheme which allows us to formalise the Eckmann–Hilton type argument in terms of the calculation of left Kan extensions in an appropriate 2-category. Then we apply this scheme to the case of n-operads in the author's sense and classical symmetric operads. We demonstrate that there exists a functor of symmetrisation Sym n from a certain subcategory of n-operads to the category of symmetric operads such that the category of one object, one arrow, … , one ( n − 1 ) -arrow algebras of A is isomorphic to the category of algebras of Sym n ( A ) . Under some mild conditions, we present an explicit formula for Sym n ( A ) which involves taking the colimit over a remarkable categorical symmetric operad. We will consider some applications of the methods developed to the theory of n-fold loop spaces in the second paper of this series.

This paper is a continuation of the paper [G. Parmeggiani, Pushing up point stabilizers, I, J. Algebra 319 (9) (2008) 3854–3884]. Under the same hypotheses, we determine those amalgams which involve SL n ( q ) , n ⩾ 3 , as factors.

Let F be a solvable Lie subalgebra of the Lie algebra gl n ( C ) ( = C n × n as a vector space). Let f k ( x 1 , x 2 , … , x p ) , ( k = 1 , 2 , … , r ) , be polynomials in the commuting variables x 1 , x 2 , … , x p with coefficients in C . For n × n matrices M 1 , M 2 , … , M r , let F ( x 1 , x 2 , … , x p ) = ∑ k = 1 r M k f k ( x 1 , x 2 , … , x p ) and let δ F ( x 1 , x 2 , … , x p ) = det F ( x 1 , x 2 , … , x p ) . In this paper, we prove that, for A 1 , A 2 , … , A p , M 1 , M 2 , … , M r ∈ F , if one value of the matrix-valued function F ( A 1 , A 2 , … , A p ) (the value depends on the product order of the variables) is nilpotent, then, (a) all values of F ( A 1 , A 2 , … , A p ) are nilpotent; (b) all values of δ F ( A 1 , A 2 , … , A p ) (again depends on the product order of the variables) are nilpotent, and one value is 0. This generalizes the recent result in [7] and makes his result accurate. The main tool we use in this paper is the representation theory of solvable Lie algebras.

This paper studies the permutation representation of the symplectic group Sp(2m,Fq), where q is odd, on the 1-spaces of its natural module. The complete submodule lattice for the modulo ℓ reduction of this permutation module is known for all odd primes ℓ not dividing q. In this paper we determine the complete submodule lattice for the mod2 reduction. Similar results are then obtained for the orthogonal group O(5,Fq).

Let G be a simple undirected graph with the characteristic polynomial of its Laplacian matrix L ( G ) , P ( G , x ) = ∑ k = 0 n ( - 1 ) k c k x n - k . Aleksandar Ilic [A. Ilic, Trees with minimal Laplacian coefficients, Comput. Math. Appl. 59 (2010) 2776–2783] identified n-vertex trees with given matching number q which simultaneously minimize all Laplacian coefficients. In this paper, we give another proof of this result. Generalizing the approach in the above paper, we determine n-vertex trees with given matching number q which have the second minimal Laplacian coefficients. We also identify the n-vertex trees with a perfect matching having the largest and the second largest Laplacian coefficients, respectively. Extremal values on some indices, such as Wiener index, modified hyper-Wiener index, Laplacian-like energy, incidence energy, of n-vertex trees with matching number q are obtained in this paper.

Learning regular languages from queries was introduced by Angluin in a seminal paper that also provides the L^@? algorithm. This algorithm, as well as other existing inference methods, finds the exact language accepted by the automaton. However, when only finite languages are used, the construction of a deterministic finite cover automaton (DFCA) is sufficient. A DFCA of a finite language U is a finite automaton that accepts all sequences in U and possibly other sequences that are longer than any sequence in U. This paper presents an algorithm, called L^l, that finds a minimal DFCA of an unknown finite language in polynomial time using membership and language queries, a non-trivial adaptation of [email protected]?s L^@? algorithm. As the size of a minimal DFCA of a finite language U may be much smaller than the size of the minimal automaton that accepts exactly U, L^l can provide substantial savings over existing automata inference methods.

In this paper we investigate linear three-term recurrence formulae Zn=T(n)Zn-1+U(n)Zn-2(n⩾2) with sequences of integers (T(n))n⩾0 and (U(n))n⩾0, which are ultimately periodic modulo m, e.g.(T(n)modm)n⩾0=(a0,a1,a2,…,aρ,T1,T2,…,Tw¯)(U(n)modm)n⩾0=(b0,b1,b2,…,bρ,U1,U2,…,Uw¯).In a former paper of this journal the authors computed explicitly the coefficients of a linear three-term recurrence formula for zn=Zrn+i with 0⩽i

The present paper deals with counting the number and orders of regular semisimple elements, together with the number of primary classes of GL(n,q). The approach used in this paper depends essentially on partitions of positive integers ⩽n. We give the numbers of regular semisimple elements and primary classes of GL(n,q) for n∈{1,2,…,6} and see that the number of regular semisimple elements is an integral polynomial in q, while the number of primary classes is a rational polynomial in q.

The concept of the rough set was originally proposed by Pawlak as a formal tool for modeling and processing incomplete information in information systems. Then in 1990, Dubois and Prade first introduced the rough fuzzy sets and fuzzy rough sets as a fuzzy extension of the rough sets. The aim of this paper is to present a new extension of the rough set model on the different universe. i.e., the fuzzy rough sets model between two different universes is presented based on the fuzzy compatible relation R ∼ α , which is defined by a fuzzy relation R ∼ between two different non-empty universes U and V and the threshold α ( α ∈ (0, 1]). Several properties of this rough sets model are given, and the relationships of this model with others rough set model are examined, too. Furthermore, we also discuss two extended models of the fuzzy rough sets model based on the fuzzy rough set model on different universes. i.e., the degree fuzzy rough set and the variable precision fuzzy rough set. Meanwhile, some principal conclusions of this two rough sets model are also established. Finally, a test example is applied to interpret the background of the application and the ideals for the fuzzy rough sets model that is presented in this paper.

In a previous paper (J. Combin. Theory 48 (1990) 255) we showed that for any infinite set of (finite) graphs drawn in a fixed surface, one of the graphs is isomorphic to a minor of another. In this paper we extend that result in two ways: •we generalize from graphs to hypergraphs drawn in a fixed surface, in which each edge has two or three ends, and•the edges of our hypergraphs are labelled from a well-quasi-order, and the minor relation is required to respect this order. This result is another step in the proof of Wagner's conjecture, that for any infinite set of graphs, one is isomorphic to a minor of another.

The classical Dedekind sums were found in transformation formulae of η-functions. It is known that these sums have some properties, especially a reciprocity law s ( a , c ) + s ( c , a ) = a 2 + c 2 − 3 a c + 1 12 a c . Sczech (1984) [5] expanded his theory for analogies of Dedekind sums and log | η ( z ) | in imaginary quadratic fields. And in my paper [6] (Okada, 1989) we studied analogies of Dedekind sums in function fields. In this paper we correct some mistakes, change some notations and rewrite some lines in the previous paper. And also in this paper we consider another analogies of Dedekind sums and a reciprocity law for these Dedekind sums.

One major area of difficulty in developing an algorithm for embedding a graph on a surface is handling bridges which have more than one possible placement. This paper addresses a number of published algorithms where this has not been handled correctly. This problem arises in certain presentations of the Demoucron, Malgrange and Pertuiset planarity testing algorithm. It also occurs in an algorithm of Filotti for embedding 3-regular graphs on the torus. The same error appears in an algorithm for embedding graphs of arbitrary genus by Filotti, Miller and Reif. It is also present in an algorithm for embedding graphs of arbitrary genus by Djidjev and Reif. The omission regarding the Demoucron, Malgrange and Pertuiset planarity testing algorithm is easily remedied. However there appears to be no way of correcting the algorithms of the other papers without making the algorithms take exponential time.

This paper answers the question of Anderson, Frazier, Lauve, and Livingston: for which finite commutative rings R is the zero-divisor graph Γ ( R ) planar? We build upon and extend work of Akbari, Maimani, and Yassemi, who proved that if R is any local ring with more than 32 elements, and R is not a field, then Γ ( R ) is not planar. They left open the question: “Is it true that, for any local ring R of cardinality 32, which is not a field, Γ ( R ) is not planar?” In this paper we answer this question in the affirmative. We prove that if R is any local ring with more than 27 elements, and R is not a field, then Γ ( R ) is not planar. Moreover, we determine all finite commutative local rings whose zero-divisor graph is planar.

Traceability codes are combinatorial objects introduced by Chor, Fiat and Naor in 1994 to be used in traitor tracing schemes to protect digital content. A k-traceability code is used in a scheme to trace the origin of digital content under the assumption that no more than k users collude. It is well known that an error correcting code of high minimum distance is a traceability code. When does this ‘error correcting construction’ produce good traceability codes? The paper explores this question.Let ℓ be a fixed positive integer. When q is a sufficiently large prime power, a suitable Reed–Solomon code may be used to construct a 2-traceability code containing q⌈ℓ/4⌉ codewords. The paper shows that this construction is close to best possible: there exists a constant c, depending only on ℓ, such that a q-ary 2-traceability code of length ℓ contains at most cq⌈ℓ/4⌉ codewords. This answers a question of Kabatiansky from 2005.Barg and Kabatiansky (2004) asked whether there exist families of k-traceability codes of rate bounded away from zero when q and k are constants such that q⩽k2. These parameters are of interest since the error correcting construction cannot be used to construct k-traceability codes of constant rate for these parameters: suitable error correcting codes do not exist when q⩽k2 because of the Plotkin bound. Kabatiansky (2004) answered Barg and Kabatiansky's question (positively) in the case when k=2. This result is generalised to the following: whenever k and q are fixed integers such that k⩾2 and q⩾k2−⌈k/2⌉+1, or such that k=2 and q=3, there exist infinite families of q-ary k-traceability codes of constant rate.

Let f=(f1,…,fn) be an n-tuple of formal power series in noncommutative indeterminates Z1,…,Zn such that f(0)=0 and the Jacobian detJf(0)≠0, and let g=(g1,…,gn) be its inverse with respect to composition. We assume that f and g have nonzero radius of convergence and g is a bounded free holomorphic function on the open unit ball [B(H)n]1, where B(H) is the algebra of bounded linear operators an a Hilbert space H. In this paper, several results concerning the noncommutative multivariable operator theory on the unit ball [B(H)n]1− are extended to the noncommutative domainBf(H):={X∈B(H)n:g(f(X))=X and ‖f(X)‖⩽1} for an appropriate evaluation X↦f(X). We develop an operator model theory and dilation theory for Bf(H), where the associated universal model is an n-tuple (MZ1,…,MZn) of left multiplication operators acting on a Hilbert space H2(f) of formal power series. All the results of this paper have commutative versions.

Text For each positive integer m, let σ A ( n ) = ♯ { ( a , a ′ ) ∈ A 2 : a + a ′ = n } , δ A ( n ) = ♯ { ( a , a ′ ) ∈ A 2 : a − a ′ = n } , where n ∈ Z m and A is a subset of Z m . Recently Chen proved that for each positive integer m, there exists a set A ⊆ Z m such that A + A = Z m and σ A ( n ) ⩽ 288 for any n ∈ Z m . In this paper, the following results are proved: (i) for each positive integer m, there exists a set A ⊆ Z m such that Z m = A − A and δ A ( n ) ⩽ 7 for all n ∈ Z m with at most 3 exceptions; (ii) for each positive integer m, there exists a set A ′ ⊆ Z m with A ′ + A ′ = Z m and A ′ − A ′ = Z m such that σ A ′ ( n ) ⩽ 26 and δ A ′ ( n ) ⩽ 24 for all n ∈ Z m with at most 3 exceptions. Video For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=GfZ07Fg5qXE .

In this paper we consider two von Neumann subalgebras B0 and B of a type II1 factor N. For a map φ on N, we define||φ||∞,2=sup{||φ(x)||2:||x||⩽1},and we measure the distance between B0 and B by the quantity ||EB0−EB||∞,2. Under the hypothesis that the relative commutant in N of each algebra is equal to its center, we prove that close subalgebras have large compressions which are spatially isomorphic by a partial isometry close to 1 in the ||·||2-norm. This hypothesis is satisfied, in particular, by masas and subfactors of trivial relative commutant. A general version with a slightly weaker conclusion is also proved. As a consequence, we show that if A is a masa and u∈N is a unitary such that A and uAu∗ are close, then u must be close to a unitary which normalizes A. These qualitative statements are given quantitative formulations in the paper.

This paper deals with a generating function of the Jacobi polynomials that satisfies the following properties (I) and (II). (I) The generating function is the kernel of an integral operator that is unitary. (II) The image of the unitary operator is a reproducing kernel Hilbert space of analytic functions and the reproducing kernel is given as a special value of the generating function above. A generating function that satisfies (I) is given in Watanabe (1998) [11] . The purpose of this paper is to give a generating function that satisfies (I) and (II). From a group theoretical point of view, a similar construction for zonal spherical functions is given in Watanabe (2006) [12] .

In a previous paper, we studied the cosets of weight 4 of binary extended 3-error-correcting BCH codes of length 2^m (where m is odd). We expressed the number of codewords of weight 4 in such cosets in terms of exponential sums of three types, including the Kloosterman sums K(a), [email protected]?F^*. In this paper, we derive some congruences which link Kloosterman sums and cubic sums. This allows us to study the divisibility of Kloosterman sums modulo 24. More precisely, if we know the traces of a and of a^1^/^3, we are able to evaluate K(a) modulo 24 and to compute the number of those a giving the same value of K(a) modulo 24.

The aim of this paper is to find those pairs of generalized quasi-arithmetic means on an open real interval I for which the arithmetic mean is invariant, i.e., to characterize those continuous strictly monotone functions φ , ψ : I → R and Borel probability measures μ , ν on the interval [ 0 , 1 ] such that φ −1 ( ∫ 0 1 φ ( t x + ( 1 − t ) y ) d μ ( t ) ) + ψ −1 ( ∫ 0 1 ψ ( t x + ( 1 − t ) y ) d ν ( t ) ) = x + y ( x , y ∈ I ) holds. Under at most fourth-order differentiability assumptions and certain nondegeneracy conditions on the measures, the main results of this paper show that there are three classes of the solutions φ , ψ : they are equivalent either to linear, or to exponential or to power functions.

Motivated by the research in reconfigurable memory array structures, this paper studies the complexity and algorithms for the constrained minimum vertex cover problem on bipartite graphs (NIN-CVCB) defined as follows: given a bipartite graph G = (V, E) with vertex bipartition V = U ∪ L and two integers ku and kl, decide whether there is a minimum vertex cover in G with at most ku vertices in U and at most kl vertices in L. It is proved in this paper that the MIN-CVCB problem is NP-complete. This answers a question posed by Hasan and Liu. A parameterized algorithm is developed for the problem, in which classical results in matching theory and recently developed techniques in parameterized computation theory are nicely combined and extended. The algorithm runs in time O(1.26ku + kl + (ku + kl) |G|) and significantly improves previous algorithms for the problem.

We study the Pettis integral for multi-functions F : Ω → cwk ( X ) defined on a complete probability space ( Ω , Σ , μ ) with values into the family cwk ( X ) of all convex weakly compact non-empty subsets of a separable Banach space X. From the notion of Pettis integrability for such an F studied in the literature one readily infers that if we embed cwk ( X ) into l ∞ ( B X ∗ ) by means of the mapping j : cwk ( X ) → l ∞ ( B X ∗ ) defined by j ( C ) ( x ∗ ) = sup ( x ∗ ( C ) ) , then j ○ F is integrable with respect to a norming subset of B l ∞ ( B X ∗ ) ∗ . A natural question arises: When is j ○ F Pettis integrable? In this paper we answer this question by proving that the Pettis integrability of any cwk ( X ) -valued function F is equivalent to the Pettis integrability of j ○ F if and only if X has the Schur property that is shown to be equivalent to the fact that cwk ( X ) is separable when endowed with the Hausdorff distance. We complete the paper with some sufficient conditions (involving stability in Talagrand's sense) that ensure the Pettis integrability of j ○ F for a given Pettis integrable cwk ( X ) -valued function F.

It was proved by Avramov and Buchweitz that if A is a commutative local complete intersection ring with finitely generated modules M and N, then the Ext groups between M and N vanish from some step if and only if the Ext groups between N and M vanish from some step. This paper shows that the same is true under the weaker conditions that A is Gorenstein and that M and N have finite complete intersection dimension. The result is also proved if A is Gorenstein and has finite Cohen-Macaulay type. Similar results are given for two types of non-commutative rings: Frobenius algebras and complete semi-local algebras., Comment: 18 pages. Paper completely rewritten since version 1 contains an error in the proof of Proposition 2.3

Let M be a 3-connected matroid other than a wheel or a whirl. In the next paper in this series, we prove that there is an element whose deletion from M or M⁎ is 3-connected and whose only 3-separations are equivalent to those induced by M. The strategy used to prove this theorem involves showing that we can remove some element from a leaf of the tree of 3-separations of M. The main result of this paper is designed to allow us to do this.

In this paper we continue our study, begun in part I, of the exceptional set of integers, not restricted by elementary congruence conditions, which cannot be represented as sums of three or four squares of primes. We correct a serious oversight in our first paper, but make further progress on the exponential sums estimates needed, together with an embellishment of the previous sieve technique employed. This leads to an improvement in our bounds for the maximal size of the exceptional sets., Comment: 30 pages

This note is a continuation of a paper by the same authors that appeared in 2002 in the same journal. First we extend the method of the previous paper proving an asymptotic formula for the number of permutations for which the associated permutation polynomial has d coefficients in specified fixed positions equal to 0. This also applies to the function Nq,d that counts the number of permutations for which the associated permutation polynomial has degree

For a set A of nonnegative integers the representation functions R 2 ( A , n ) , R 3 ( A , n ) are defined as the number of solutions of the equation n = a + a ′ , a , a ′ ∈ A with a a ′ , a ⩽ a ′ , respectively. Let D ( 0 ) = 0 and let D ( a ) denote the number of ones in the binary representation of a. Let A 0 be the set of all nonnegative integers a with even D ( a ) and A 1 be the set of all nonnegative integers a with odd D ( a ) . In this paper we show that (a) if R 2 ( A , n ) = R 2 ( N ∖ A , n ) for all n ⩾ 2 N − 1 , then R 2 ( A , n ) = R 2 ( N ∖ A , n ) ⩾ 1 for all n ⩾ 12 N 2 − 10 N − 2 except for A = A 0 or A = A 1 ; (b) if R 3 ( A , n ) = R 3 ( N ∖ A , n ) for all n ⩾ 2 N − 1 , then R 3 ( A , n ) = R 3 ( N ∖ A , n ) ⩾ 1 for all n ⩾ 12 N 2 + 2 N . Several problems are posed in this paper.

Quasimodules for vertex algebras are generalizations of modules for vertex algebras. These new objects arise from a generalization of locality for fields. Quasimodules tie together module theory and twisted module theory, and both twisted and untwisted modules feature Poincare–Birkhoff–Witt-like spanning sets. This paper generalizes these spanning set results to quasimodules for certain Mobius vertex algebras. In particular this paper presents two spanning sets, one featuring a difference-zero ordering restriction on modes and another featuring a difference-one ordering restriction.

This paper is composed of two parts. In the first part, an improved algorithm is presented for the problem of finding length-bounded two vertex-disjoint paths in an undirected planar graph. The presented algorithm requires O(n^3b"m"i"n) time and O(n^2b"m"i"n) space, where b"m"i"n is the smaller of the two given length bounds. In the second part of this paper, we consider the minmax k vertex-disjoint paths problem on a directed acyclic graph, where k>=2 is a constant. An improved algorithm and a faster approximation scheme are presented. The presented algorithm requires O(n^k^+^1M^k^-^1) time and O(n^kM^k^-^1) space, and the presented approximation scheme requires O((1/@e)^k^-^1n^2^klog^k^-^1M) time and O((1/@e)^k^-^1n^2^k^-^1log^k^-^1M) space, where @e is the given approximation parameter and M is the length of the longest path in an optimal solution.

In this paper, we focus on the edit distance between two given strings where block-edit operations are allowed and better fitting to the human natural edit behaviors. Previous results showed that this problem is NP-hard when block moves are allowed. Various approximations to this problem have been proposed in recent years. However, this problem can be solved by the polynomial-time optimization algorithms if some reasonable restrictions are applied. The restricted variations which we consider involve character insertions, character deletions, block copies and block deletions. In this paper, three problems are defined with different measuring functions, which are P(EIS,C), P(EI,L) and P(EI,N). Then we show that with some preprocessing, the minimum block edit distances of these three problems can be obtained by dynamic programming in O(nm), O(nmlogm) and O(nm2) time, respectively, where n and m are the lengths of the two input strings.

We extend the calculus of relations to embed a regular category A into a family of pseudo-abelian tensor categories T(A,d) depending on a degree function d. Under the condition that all objects of A have only finitely many subobjects, our main results are as follows: 1. Let N be the maximal proper tensor ideal of T(A,d). We show that T(A,d)/N is semisimple provided that A is exact and Mal'cev. Thereby, we produce many new semisimple, hence abelian, tensor categories. 2. Using lattice theory, we give a simple numerical criterion for the vanishing of N. 3. We determine all degree functions for which T(A,d) is Tannakian. As a result, we are able to interpolate the representation categories of many series of profinite groups such as the symmetric groups S_n, the hyperoctahedral groups S_n\semidir Z_2^n, or the general linear groups GL(n,F_q) over a fixed finite field. This paper generalizes work of Deligne, who first constructed the interpolating category for the symmetric groups S_n. It also extends (and provides proofs for) a previous paper math.CT/0605126 on the special case of abelian categories., Comment: v1: 52 pages; v2: 52 pages, proof of Lemma 7.2 fixed, otherwise minor changes

This paper introduces ♠ (pronounce spades), a stochastic process algebra for discrete-event systems, that extends traditional process algebra with timed actions whose delay is governed by general (a.o. continuous) probability distributions. The operational semantics is defined in terms of stochastic automata, a model that uses clocks--like in timed automata--to symbolically represent randomly timed systems, cf. the accompanying paper [P.R. D'Argenio, J.-P. Katoen, A theory of stochastic systems. Part I: Stochastic automata. Inf. Comput. (2005), to appear]. We show that stochastic automata and ♠ are equally expressive, and prove that the operational semantics of a term up to α-conversion of clocks, is unique (modulo symbolic bisimulation). (Open) probabilistic and structural bisimulation are proven to be congruences for ♠, and are equipped with an equational theory. The equational theory is shown to be complete for structural bisimulation and allows to derive an expansion law.

The motivation for the paper is the geometric approach to learning Bayesian network (BN) structure. The basic idea of our approach is to represent every BN structure by a certain uniquely determined vector so that usual scores for learning BN structure become affine functions of the vector representative. The original proposal from Studený et al. (2010) [26] was to use a special vector having integers as components, called the standard imset, as the representative. In this paper we introduce a new unique vector representative, called the characteristic imset, obtained from the standard imset by an affine transformation.Characteristic imsets are (shown to be) zero-one vectors and have many elegant properties, suitable for intended application of linear/integer programming methods to learning BN structure. They are much closer to the graphical description; we describe a simple transition between the characteristic imset and the essential graph, known as a traditional unique graphical representative of the BN structure. In the end, we relate our proposal to other recent approaches which apply linear programming methods in probabilistic reasoning.

This paper is devoted to the investigation of Lie algebras of local infinitesimal CR automorphisms. Such algebras are naturally associated to germs of homogeneous CR manifolds. The authors introduce a corresponding abstract notion of CR algebra. A CR algebra is a pair $(L,L_1)$(L,L1), consisting of a real Lie algebra $L$L and a subalgebra $L_1$L1 of the complexification $\bold C\otimes_{\bold R} L$C⊗RL, such that the factor space $L/L\cap L_1$L/L∩L1 is finite-dimensional. The authors investigate some formal properties of CR algebras and construct some "fibrations'' (i.e., $L$L-equivariant submersions) of such algebras. They introduce three new notions of nondegeneracy of CR algebras---strict, weak and ideal nondegeneracy. These three concepts are weaker than those used previously by some other authors. The authors intend to extend the application of the E. Cartan method of investigating the equivalence of CR structures to some larger classes of CR manifolds. One of the main ideas of this paper is a decomposition of arbitrary CR algebras into three "parts'': totally real, totally complex and weakly nondegenerate CR algebras (Theorems 5.3 and 5.4). There are some results about these three special classes of CR algebras. Some results about prolongations for transitive CR algebras are also obtained, in particular about maximality of parabolic CR algebras with respect to transitive prolongations.

The variable precision rough set model and graded rough set model are 2 important extended rough set models. This paper aims to make a comparative study of the 2 models. Rough set regions, primitive notions, are proposed first for the 2 models, which classify the universe more precisely. Then, both of their logical meanings related to quantitative indexes and their basic structure are investigated, and their precise descriptions are obtained as well. Furthermore, in the graded rough set model, macroscopic and microscopic algorithms are proposed and analyzed to calculate rough set regions; then, the conclusion is drawn that macroscopic and microscopic algorithms have advantages in time and space complexities, respectively, and a medical example is provided to illustrate the rough set regions and the 2 algorithms. In addition, 3 new properties of the 2 models are investigated, which are the results of extending the classical rough set model, i.e. the relationships between approximations and the basic set, the power actions of approximation operators, and the modifications of approximation operator actions on set operations. Finally, the classical rough set model is used to obtain many corresponding results, and moreover, the relationship and transformation between the 2 models is investigated. The study results of this paper have extended and enriched rough set theory from both operator-oriented and set-oriented points of view.

In this paper we apply the Garland–Lepowsky Theorem to compute the homology of the Lie algebra N=sl n (t C [t]) . By suitably taking Euler characteristics we derive a theorem of Stembridge which gives a combinatorial decomposition of a virtual character of SL n ( C ) . In his theorem, Stembridge indexes the Schur functions that appear in his decomposition by an integer vector of length n and a permutation in Sn. An additional feature of our approach is that we interpret the integer vector and permutation in terms of the affine Weyl group of the Kac–Moody Lie algebra of type A. A second result, given in the Garland–Lepowsky paper (extending an earlier result of Kostant), is an explicit formula for the eigenvalues of the Laplacian in the Koszul complex for computing the homology of N. We give an alternative quite different in nature description of those eigenvalues. The Garland–Lepowsky formula is algebraic, stated in terms of differences of squared norms. Our description is combinatorial, stated in terms of structural features of certain graphs. In order to make this combinatorial interpretation seemingly more natural, we link it directly with the paper of Kostant. We work in this part with the Lie algebra L=T⊕N where T is the span of x⊗1 with x taken from the strictly upper triangular matrices in sl n ( C ) .

Kolmogorov Theorem on the persistence of invariant tori of real analytic Hamiltonian systems is revisited. In this paper we are mainly concerned with the lower bound on the constant of the Diophantine condition required by the theorem. From the existing proofs in the literature, this lower bound turns to be of O ( e 1 / 4 ) , where e is the size of the perturbation. In this paper, by means of careful estimates on Kolmogorov's method, we show that this lower bound can be weakened to be of O ( e 1 / 2 ) . This condition coincides with the optimal one of KAM Theorem. Moreover, we also obtain optimal estimates for the distance between the actions of the perturbed and unperturbed tori. We believe that some ideas contained in this paper may be used for improving several estimates in the general KAM context.

We discuss two issues in using mixtures of polynomials (MOPs) for inference in hybrid Bayesian networks. MOPs were proposed by Shenoy and West for mitigating the problem of integration in inference in hybrid Bayesian networks. First, in defining MOP for multi-dimensional functions, one requirement is that the pieces where the polynomials are defined are hypercubes. In this paper, we discuss relaxing this condition so that each piece is defined on regions called hyper-rhombuses. This relaxation means that MOPs are closed under transformations required for multi-dimensional linear deterministic conditionals, such as Z=X+Y, etc. Also, this relaxation allows us to construct MOP approximations of the probability density functions (PDFs) of the multi-dimensional conditional linear Gaussian distributions using a MOP approximation of the PDF of the univariate standard normal distribution. Second, Shenoy and West suggest using the Taylor series expansion of differentiable functions for finding MOP approximations of PDFs. In this paper, we describe a new method for finding MOP approximations based on Lagrange interpolating polynomials (LIP) with Chebyshev points. We describe how the LIP method can be used to find efficient MOP approximations of PDFs. We illustrate our methods using conditional linear Gaussian PDFs in one, two, and three dimensions, and conditional log-normal PDFs in one and two dimensions. We compare the efficiencies of the hyper-rhombus condition with the hypercube condition. Also, we compare the LIP method with the Taylor series method.