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Abstract: It was shown by Heinrich et al. [The inverse of the star-discrepancy depends linearly on the dimension, Acta Arith. 96 (2001) 279–302] that there exist point sets for which the inverse of the star discrepancy depends linearly on the dimension. In this paper we extend those results by showing that there exist point sets extensible in the modulus and the dimension for which the star discrepancy satisfies a tractability bound for all dimensions and moduli. [Copyright &y& Elsevier]

Abstract: We study the optimal approximation of the solution of an operator equation by certain n-term approximations with respect to specific classes of frames. We consider worst case errors, where f is an element of the unit ball of a Sobolev or Besov space and is a bounded Lipschitz domain; the error is always measured in the -norm. We study the order of convergence of the corresponding nonlinear frame widths and compare it with several other approximation schemes. Our main result is that the approximation order is the same as for the nonlinear widths associated with Riesz bases, the Gelfand widths, and the manifold widths. This order is better than the order of the linear widths iff . The main advantage of frames compared to Riesz bases, which were studied in our earlier papers, is the fact that we can now handle arbitrary bounded Lipschitz domains—also for the upper bounds. [Copyright &y& Elsevier]

Abstract: The gap function of an infinite word over the binary alphabet gives the distances between consecutive 1''s in this word. In this paper we study infinite binary words whose gap function is injective or “almost injective.” A method for computing the subword complexity of such words is given. A necessary and sufficient condition for a function to be the subword complexity function of a binary word whose gap function is increasing is obtained. [Copyright &y& Elsevier]

Abstract: This paper concerns the embedding problem for partial totally symmetric quasigroups. For all , it is shown that any partial totally symmetric quasigroup of order n can be embedded in a totally symmetric quasigroup of order v if v is even and , and this is the best possible such inequality. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to exhibit the main results of A. Masuoka [A. Masuoka, Corings and invertible bimodules, Tsukuba J. Math. 13 (1989) 353–362] and of L. El Kaoutit and J. Gómez-Torrecillas [L. El Kaoutit, J. Gómez-Torrecillas, Comatrix corings and invertible bimodules, Ann. Univ. Ferrara Ser. VII 51 (2005) 263–280] as special cases of a more general result. [Copyright &y& Elsevier]

Abstract: We are interested in spatially extended pattern forming systems close to the threshold of the first instability in case when the so-called degenerated Ginzburg–Landau equation takes the role of the classical Ginzburg–Landau equation as the amplitude equation of the system. This is the case when the relevant nonlinear terms vanish at the bifurcation point. Here we prove that in this situation every small solution of the pattern forming system develops in such a way that after a certain time it can be approximated by the solutions of the degenerated Ginzburg–Landau equation. In this paper we restrict ourselves to a Swift–Hohenberg–Kuramoto–Shivashinsky equation as a model for such a pattern forming system. [Copyright &y& Elsevier]

Abstract: Let D be a simply laced Dynkin diagram of rank r whose affinization has the shape of a star (i.e., ). To such a diagram one can attach a group G whose generators correspond to the legs of the affinization, have orders equal to the leg lengths plus 1, and the product of the generators is 1. The group G is then a 2-dimensional crystallographic group: , where ℓ is 2, 3, 4, and 6, respectively. In this paper, we define a flat deformation of the group algebra of this group, by replacing the relations saying that the generators have prescribed orders by their deformations, saying that the generators satisfy monic polynomial equations of these orders with arbitrary roots (which are deformation parameters). The algebra for is the Cherednik algebra of type , which was studied by Noumi, Sahi, and Stokman, and controls Askey–Wilson polynomials. We prove that is the universal deformation of the twisted group algebra of G, and that this deformation is compatible with certain filtrations on . We also show that if q is a root of unity, then for generic t the algebra is an Azumaya algebra, and its center is the function algebra on an affine del Pezzo surface. For generic q, the spherical subalgebra provides a quantization of such surfaces. We also discuss connections of with preprojective algebras and Painlevé VI. [Copyright &y& Elsevier]

Abstract: Consider a polar space weakly embedded in a projective space . A secant of is the intersection of the point set of with a line of spanned by two non-collinear points of . The geometry consisting of the points of and as lines the secants is a so-called Delta space. In this paper we give a characterization of this and some related geometries. [Copyright &y& Elsevier]

Abstract: Let n be a positive integer. In this paper we estimate the size of the set of linear forms , where and are integers, as . [Copyright &y& Elsevier]

Abstract: This paper presents a systematic investigation of the integrability conditions for nonautonomous quad-graph maps, using the Lax pair approach, the ultra-local singularity confinement criterion and direct construction of conservation laws. We show that the integrability conditions derived from each of the methods are the one and the same, suggesting that there exists a deep connection between these techniques for partial difference equations. [Copyright &y& Elsevier]

Abstract: In this paper, we give a necessary and sufficient condition for the semisimplicity of cyclotomic Brauer algebras of types with . This generalizes [H. Rui, A criterion on the semisimple Brauer algebras, J. Combin. Theory Ser. A 111 (2005) 78–88, 1.2–1.3] and [H. Rui, M. Si, A criterion on the semisimple Brauer algebras, II, J. Combin. Theory Ser. A 113 (2006) 1199–1203, 2.5] on Brauer algebras. [Copyright &y& Elsevier]

Abstract: In this paper we prove that a Lie algebra L is strongly prime if and only if for every nonzero elements . As a consequence, we give an elementary proof, without the classification theorem of strongly prime Jordan algebras, of the fact that a linear Jordan algebra or Jordan pair T is strongly prime if and only if for every . Moreover, we prove that the Jordan algebras at nonzero Jordan elements of strongly prime Lie algebras are strongly prime. [Copyright &y& Elsevier]

Abstract: The general case of the Nagell–Ljunggren equation is and odd primes. In this paper we derive a new bound for which there are no solutions. For small q the problem is harder and we achieve a conditional result: for and some additional condition on must hold. Both functions are quadratic and they leave a small gap for , on which we have no general result. The picture is completed by some explicit, unconditional upper bounds for the absolute values , . [Copyright &y& Elsevier]

Abstract: Let be a graded Artinian K-algebra, where and . (The grading may not necessarily be standard.) Then A has the strong Lefschetz property if there exists an element such that the multiplication is bijective for every . The main results obtained in this paper are as follows: [1.] A has the strong Lefschetz property if and only if there is a linear form such that has the strong Lefschetz property. [2.] If A is Gorenstein, then A has the strong Lefschetz property if and only if there is a linear form such that all central simple modules of have the strong Lefschetz property. [3.] A finite free extension of an Artinian K-algebra with the strong Lefschetz property has the strong Lefschetz property if the fiber does. [4.] The complete intersection defined by power sums of consecutive degrees has the strong Lefschetz property. [Copyright &y& Elsevier]

Abstract: For an algebraic curve over the finite field , we denote by and the number of -rational points and genus of , respectively. Define the asymptotic quantity by . In this paper, we show an improvement on the best known lower bound on . [Copyright &y& Elsevier]

Abstract: This paper deals with the global attraction and global weak attraction of the origin for the generalized Liénard equation. The problem proposed by Jiang is solved by a negative answer and some new criteria for the globally asymptotically stability are obtained by the ways different from the usual Filippov''s conditions. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the second-order equations of Duffing type. Bounds for the derivative of the restoring force are given that ensure the existence and uniqueness of a periodic solution. Furthermore, the stability of the unique periodic solution is analyzed; the sharp rate of exponential decay is determined for a solution that is near to the unique periodic solution. [Copyright &y& Elsevier]

Abstract: In this paper, we show that the stabilizer subgroup of for a without multiple roots can be derived from the stabilizer of . As an application, we construct a family of 3-designs such as , where is a prime power such that and . [Copyright &y& Elsevier]

Abstract: A graph is called circular super-critical if for every vertex of . In this paper, analogous to a result of Dirac on chromatic critical graphs, a sharp lower bound on the vertex degree of circular super-critical graphs is proved. This lower bound provides a partial answer to a question of X. Zhu [The circular chromatic number of induced subgraphs, J. Combin. Theory Ser. B 92 (2004) 177–181]. Some other structural properties of circular super-critical graphs are also presented. [Copyright &y& Elsevier]

Abstract: Let be a convex polytope in the Euclidean space . Consider the group generated by reflections in the facets of . We say that generates a reflection group . The spherical simplices generating simply-laced reflection groups were described by P.J. Cameron, J.J. Seidel and S.V. Tsaranov [Signed graphs, root lattices and Coxeter groups, J. Algebra 164 (1994) 173–209]. The simplices generating the remaining spherical reflection groups were listed in [A. Felikson, Spherical simplices generating discrete reflection groups, Sb. Math. 195 (3–4) (2004) 585–598, arXiv:math.MG/0212244]. In the present paper, we find all Euclidean simplices generating discrete reflection groups. [Copyright &y& Elsevier]

Abstract: The finiteness of the little finitistic dimension of an artin algebra R is known to be equivalent to the existence of a tilting R-module T such that where is the category of all finitely presented R-modules of finite projective dimension. Moreover, T can be taken finitely generated if and only if is contravariantly finite. In this paper, we describe explicitly the structure of T for the IST-algebra, a finite-dimensional algebra with not contravariantly finite. We also characterize the indecomposable modules in , and all tilting classes over this algebra. [Copyright &y& Elsevier]

Abstract: In [J. Du, Q. Fu, J.-P. Wang, Infinitesimal quantum and little q-Schur algebras, J. Algebra 287 (2005) 199–233], little q-Schur algebras were introduced as homomorphic images of infinitesimal quantum in the odd root of unity case. In this paper, we shall introduce little q-Schur algebras at even roots of unity. We shall construct various bases for and give the dimension formula of . [Copyright &y& Elsevier]

Abstract: Let be the cyclotomic q-Schur algebra associated to the Ariki–Koike algebra . We construct a certain subalgebra of , and show that it is a standardly based algebra in the sense of Du and Rui. has a natural quotient , which turns out to be a cellular algebra. In the case where the modified Ariki–Koike algebra is defined, coincides with the cyclotomic q-Schur algebra associated to . In this paper, we discuss a relationship among the decomposition numbers of , and . In particular, we show that some important part of the decomposition matrix of coincides with a part of the decomposition matrix of . [Copyright &y& Elsevier]

Abstract: It is well known that the second homology group of the Steinberg Lie algebra is trivial when . In this paper, we will work out explicitly for which are not necessarily trivial. Consequently, we obtained for . [Copyright &y& Elsevier]

Abstract: In this paper, finite-dimensional solvable Lie algebras with nondegenerate invariant bilinear forms are classified. The classification is based on the investigation of the minimal generating sets, which are proved to be one of the three types: paired type, isolated type and mixed type. [Copyright &y& Elsevier]

Abstract: In this paper we establish the exact blow-up rate of the large solutions of a porous media logistic equation. We consider the carrying capacity function with a general decay rate at the boundary instead of the usual cases when it can be approximated by a distant function. Obtaining the accurate blow-up rate allows us to establish the uniqueness result. Our result covers all previous results on the ball domain and can be further adapted in a more general domain. [Copyright &y& Elsevier]

Abstract: Let A be a finite-dimensional algebra given by quiver and monomial relations. In [E.L. Green, D. Zacharia, Manuscripta Math. 85 (1994) 11–23] we see that the Ext-algebra of A is finitely generated only if all the Ext-algebras of certain cycle algebras overlying A are finitely generated. Here a cycle algebra Λ is a finite-dimensional algebra given by quiver and monomial relations where the quiver is an oriented cycle. The main result of this paper gives necessary and sufficient conditions for the Ext-algebra of such a Λ to be finitely generated; this is achieved by defining a computable invariant of Λ, the smo-tube. We also give necessary and sufficient conditions for the Ext-algebra of Λ to be Noetherian. [Copyright &y& Elsevier]

Abstract: In [B. De Bruyn, P. Vandecasteele, Valuations and hyperplanes of dual polar spaces, J. Combin. Theory Ser. A 112 (2005) 194–211], we introduced the class of the SDPS-valuations of dual polar spaces. We showed that these valuations and all their extensions give rise to hyperplanes of dual polar spaces. We call these hyperplanes SDPS-hyperplanes. In the present paper, we show that a hyperplane of a thick dual polar space is an SDPS-hyperplane if and only if every hex not contained in intersects in either a singular hyperplane or the extension of an ovoid. [Copyright &y& Elsevier]

Abstract: Cryptographic protocols are small programs which involve a high level of concurrency and which are difficult to analyze by hand. The most successful methods to verify such protocols are based on rewriting techniques and automated deduction in order to implement or mimic the process calculus describing the execution of a protocol. We are interested in the intruder deduction problem, that is vulnerability to passive attacks in presence of equational theories which model the protocol specification and properties of the cryptographic operators. In the present paper, we consider the case where the encryption distributes over the operator of an Abelian group or over an exclusive-or operator. We prove decidability of the intruder deduction problem in both cases. We obtain a PTIME decision procedure in a restricted case, the so-called binary case. These decision procedures are based on a careful analysis of the proof system modeling the deductive power of the intruder, taking into account the algebraic properties of the equational theories under consideration. The analysis of the deduction rules interacting with the equational theory relies on the manipulation of -modules in the general case, and on results from prefix rewriting in the binary case. [Copyright &y& Elsevier]

Abstract: Let be a complex, semi-simple Lie algebra, a Cartan subalgebra and D a subdiagram of the Dynkin diagram of . Let be the corresponding semi-simple and Levi subalgebras and consider two invariant solutions and of the pentagon equation for and respectively. Motivated by the theory of quasi-Coxeter quasitriangular quasibialgebras [V. Toledano Laredo, Quasi-Coxeter algebras, Dynkin diagram cohomology and quantum Weyl groups, math.QA/0506529], we study in this paper the existence of a relative twist, that is an element such that the twist of Φ by F is . Adapting the method of Donin and Shnider [J. Donin, S. Shnider, Cohomological construction of quantized universal enveloping algebras, Trans. Amer. Math. Soc. 349 (1997) 1611–1632], who treated the case of an empty D, so that and , we give a cohomological construction of such an F under the assumption that is the image of Φ under the generalised Harish-Chandra homomorphism . We also show that F is unique up to a gauge transformation if is of corank 1 or F satisfies where is an involution acting as −1 on . [Copyright &y& Elsevier]

Abstract: In this paper we study uniqueness properties of solutions of the so-called k-generalized Korteweg–de Vries equations. Our goal is to obtain sufficient conditions on the behavior of the difference of two solutions of (1.1) at two different times and which guarantee that . [Copyright &y& Elsevier]

Abstract: For a canonical threefold X, we know for a sufficiently large n. When , there are few known results about the integer n. This paper introduces an algorithm for computing plurigenera. Furthermore, when is small, especially 1 and 2, plurigenera are computed. This produces for and for when . Also, for and for with 8 possible exceptional cases when . [Copyright &y& Elsevier]

Abstract: Let G be a finite group of characteristic p of one of the types symplectic group , Chevalley group , Suzuki group or Ree group . In this paper we show that if H is a Sylow p-subgroup of G then the restriction of χ to H has at least one linear constituent of multiplicity one for certain irreducible characters χ of G. [Copyright &y& Elsevier]

Abstract: In this paper we consider the existence of nontrivial solutions for an elliptic system, where the nonlinear term is superlinear in one equation and sublinear in the other equation. By constructing two cones and computing the fixed point index in , and , we obtain that the elliptic system has three nontrivial solutions , and . It is remarkable that the third nontrivial solution is established on the Cartesian product of two cones, in which the feature of two equations can be exploited better. [Copyright &y& Elsevier]

Abstract: In this paper, we establish sufficient conditions for the oscillation of the linear non-autonomous systems of difference equations with continuous arguments where , , , , ; τ, γ and , , are non-negative constants. Furthermore, assume , where , , , are continuous, bounded and do not change signs on . [Copyright &y& Elsevier]

Abstract: Let be the polynomial ring in n variables over a field k for some , and its field of fractions. Assume that L is a subfield of containing k over which is algebraic. In spite of being an important issue in Hilbert''s Fourteenth Problem, relations between finite generation of the k-subalgebra of and the extension degree of over L have not been investigated. In the present paper, we give an example of L with such that is not finitely generated for each with for . [Copyright &y& Elsevier]

Abstract: Let K be an infinite field of characteristic , G a locally finite group and KG its group algebra. Let denote the K-linear extension of an involution φ defined on G. In this paper we prove, under some assumptions, that if the set of φ-symmetric units of KG satisfies a group identity then KG satisfies a polynomial identity. Moreover, in case the prime radical of KG is nilpotent we characterize the groups for which the φ-symmetric units satisfy a group identity. [Copyright &y& Elsevier]

Abstract: It is known that, up to isomorphism, there are only one lattice S and one non-lattice semi-lattice in . In [S. Tan, TKK algebras and vertex operator representations, J. Algebra 211 (1999) 298–342] we obtained a vertex operator representation for the baby-TKK algebra . The goal of the present paper is to study the TKK algebra , which includes the baby-TKK algebra as a subalgebra, and its vertex operator representations. As a by-product, we also obtain two representations for the baby-TKK algebra . One of the representations recovers the construction given in [S. Tan, TKK algebras and vertex operator representations, J. Algebra 211 (1999) 298–342]. [Copyright &y& Elsevier]

Abstract: In this paper, we give a new realization of the crystal using modified Nakajima monomials. For the quantum affine algebras of type , we give an explicit characterization of the monomials in , and construct a natural crystal isomorphism between monomial realization and path realization. As a corollary, we obtain a new realization of the crystal for the quantum classical algebras of type . [Copyright &y& Elsevier]

Abstract: Let N be a normal subgroup of a p-solvable group G and let M be a simple FN-module, where F is an algebraically closed field of characteristic p. Let Q be any p-subgroup of G. In this paper, we count the number of simple FG-modules lying over M and having Q as a vertex. [Copyright &y& Elsevier]

Abstract: Let G be a simple simply connected affine algebraic group over an algebraically closed field k of characteristic p for an odd prime p. Let B be a Borel subgroup of G and U be its unipotent radical. In this paper, we determine the second cohomology groups of B and its Frobenius kernels for all simple B-modules. We also consider the standard induced modules obtained by inducing a simple B-module to G and compute all second cohomology groups of the Frobenius kernels of G for these induced modules. Also included is a calculation of the second ordinary Lie algebra cohomology group of Lie(U) with coefficients in k. [Copyright &y& Elsevier]

Abstract: The classical criterion of asymptotic stability of the zero solution of equations is that there exists a function , for some , such that for some . In this paper we prove that if is bounded, is uniformly continuous and bounded, then the condition that can be weakened and replaced by and contains no complete trajectory of , , where , uniformly for . [Copyright &y& Elsevier]

Abstract: This paper discusses a class of second-order derivative nonlinear Schrödinger equations which are used to describe the upper-hybrid oscillation propagation. By establishing a variational problem, applying the potential well argument and the concavity method, we prove that there exists a sharp condition for global existence and blow-up of the solutions to the nonlinear Schrödinger equation. In addition, we also answer the question: how small are the initial data, the global solutions exist? [Copyright &y& Elsevier]

Abstract: The well-known Schiffer conjecture saying that for a smooth bounded domain , if there exists a positive Neumann eigenvalue such that the corresponding Neumann eigenfunction u is constant on the boundary of Ω, then Ω is a ball. In this paper, we shall prove that the Schiffer conjecture holds if and only if the third order interior normal derivative of the corresponding Neumann eigenfunction is constant on the boundary. We also prove a similar result to the Berenstein conjecture for the overdetermined Dirichlet eigenvalue problem. [Copyright &y& Elsevier]

Abstract: Let B be a Douglas algebra and let be the algebra on the disk generated by the harmonic extensions of the functions in B. In this paper we show that is generated by and the complex conjugates of the harmonic extensions of the interpolating Blaschke products invertible in B. Every element S in the Toeplitz algebra generated by Toeplitz operators (on the Bergman space) with symbols in has a canonical decomposition for some R in the commutator ideal ; and S is in iff the Berezin transform vanishes identically on the union of the maximal ideal space of the Douglas algebra B and the set of trivial Gleason parts. [Copyright &y& Elsevier]

Abstract: Let G be a finite group that acts on an abelian monoid A. If is a map so that , for all , then the submonoid of the associated semidirect product is said to be a monoid of IG-type. If A is a finitely generated free abelian monoid of rank n and G is a subgroup of the symmetric group of degree n, then these monoids first appeared in the work of Gateva-Ivanova and Van den Bergh (they are called monoids of I-type) and later in the work of Jespers and Okniński. It turns out that their associated semigroup algebras share many properties with polynomial algebras in finitely many commuting variables. In this paper we first note that finitely generated monoids S of IG-type are epimorphic images of monoids of I-type and their algebras are Noetherian and satisfy a polynomial identity. In case the group of fractions of S is torsion-free abelian then it is characterized when also is a maximal order. It turns out that they often are, and hence these algebras again share arithmetical properties with natural classes of commutative algebras. The characterization is in terms of prime ideals of S, in particular G-orbits of minimal prime ideals in A play a crucial role. Hence, we first describe the prime ideals of S. It also is described when the group is torsion-free. [Copyright &y& Elsevier]

Abstract: In this paper, we deal with the problem of framing 2-variable 2-parameter Bessel functions (2V2PBF) into the context of the representation of the 3-dimensional Lie algebra . Further, we derive generating relations and identities involving 2V2PBF. Furthermore, we deduce some new and known relations involving various other forms of generalized Bessel functions (GBF) and Bessel functions (BF) as applications of these results. [Copyright &y& Elsevier]

Abstract: Whereas results about the number of irreducible modular representations of finite groups are known, those of Bose–Mesner algebras of association schemes are not well understood. In this paper, we consider the number of irreducible modular representations of the adjacency algebra of some and polynomial association schemes. These results should be helpful for investigating the structure of association schemes. [Copyright &y& Elsevier]

Abstract: In this paper we continue the investigation of Cohen–Macaulay projective monomial curves begun in [Les Reid, Leslie G. Roberts, Non-Cohen–Macaulay projective monomial curves, J. Algebra 291 (2005) 171–186]. In the process we introduce maximal curves. Cohen–Macaulay curves are maximal, but not conversely. We show that the number of all curves of degree d that are Cohen–Macaulay grows exponentially, but not as fast as the total number of curves, and also that maximal curves of degree d with sufficiently large embedding dimension relative to d are Cohen–Macaulay. [Copyright &y& Elsevier]

Abstract: In the paper we examine Pexiderized ϕ-homogeneity equation almost everywhere. Assume that G and H are groups with zero, and are a G- and an H-space, respectively. We prove, under some assumption on , that if functions and satisfy Pexiderized ϕ-homogeneity equation almost everywhere in then either almost everywhere in G or almost everywhere in X or there exists a homomorphism such that almost everywhere in G and there exists a function such that and where is a constant. From this result we derive solution of the classical Pexider equation almost everywhere. [Copyright &y& Elsevier]