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Abstract: In this paper we discuss a certain class of sets of generators for a full Hilbert -module which characterize Hilbert -modules over the -algebra of compact operators. An example of such sets of generators are orthonormal bases. [Copyright &y& Elsevier]

Abstract: In this paper, we compute the congruences of Catalan and Motzkin numbers modulo 4 and 8. In particular, we prove the conjecture proposed by Deutsch and Sagan that no Motzkin number is a multiple of 8. [Copyright &y& Elsevier]

Abstract: The principal aim of this paper is to shed some light on the algebraic content of the Tratado da arte de arismetica by Bento Fernandes, which was published in Porto in 1555 and is the earliest treatise of a Portuguese author that has come down to us in which algebra is studied. Since it therefore constitutes an important testimony of the state of development of algebra in Portugal in the middle of the 16th century, it deserves special attention. At a time when Pacioli''s Summa, the first printed text that includes algebraic methods, was already so diffuse, it is surprising that it turns out not to have been the source of the algebraic material of Bento Fernandes. The comparative study I have carried out between the Tratado da arte de arismetica and a number of abacus books from the 14th and the 15th centuries shows that Bento Fernandes''s algebra had its origin in abacus manuscripts antedating the Summa. [Copyright &y& Elsevier]

Abstract: A standard matrix representation A of a matroid M represents M relative to a fixed basis B. Deleting rows and columns of A correspond to contracting elements of B and deleting elements of . If M is 3-connected, it is often desirable to perform such an element removal from M while maintaining 3-connectivity. This paper proves that this is always possible provided M has no 4-element fans. We also show that, subject to a mild essential restriction, this element removal can be done so as to retain a copy of a specified 3-connected minor of M. [Copyright &y& Elsevier]

Abstract: In this paper we consider partial actions of groups on algebras and partial skew group rings. After some general results we prove two versions of Maschke''s theorem and then we study von Neumann regularity, the prime radical and the Jacobson radical of partial skew group rings. In this way we extend many results which are known for skew group rings. [Copyright &y& Elsevier]

Abstract: In a variety of dynamical systems, formations of motion patterns occur. Observing colonies of animals, for instance, for the scientist it is not only of interest which kinds of formations these animals show, but also how they altogether move around. In order to analyse motion patterns for the purpose of making predictions, to describe the behaviour of systems, or to index databases of moving objects, methods are required for dealing with them. This becomes increasingly important since a number of technologies have been devised which allow objects precisely to get traced. However, the indeterminacy of spatial information in real world environments also requires techniques to approximate reasoning, for example, in order to compensate for small and unimportant distinctions which are due to noisy measurements. As a consequence, precise as well as coarse motion patterns have to be dealt with. A set of 16 atomic motion patterns is proposed. On the one hand, a relation algebra is defined on them. On the other hand, these 16 relations form the basis of a visual language using which motion patterns can easily be dealt with in a diagrammatic way. The relations are coarse but crisp and they allow imprecise knowledge about motion patterns to be dealt with, while their diagrammatic realisation also allow precise patterns to get handled. While almost all approaches consider motion patterns along arbitrary time intervals, this paper in particular focuses on short-term motion patterns as we permanently observe them in our everyday life. The bottom line of the current work, however, is yet more general. While it has been widely argued that it makes sense to use both sentential and diagrammatic representations in order to represent different things in the same system adequately (and hence differently), we argue that it makes even sense to represent the same things differently in order to grasp different aspects of one and the same object of interest from different viewpoints. We demonstrate this by providing both a sentential and a diagrammatic representation for the purpose of grasping different aspects of motion patterns. It shows that both representations complement each other. [Copyright &y& Elsevier]

Abstract: The paper has its origin in an attempt to answer the following question: Given an arbitrary finite dimensional associative K-algebra A, does there exist a quasi-hereditary algebra B such that the subcategories of all A-modules and all B-modules, filtered by the corresponding standard modules are equivalent. Such an algebra will be called a quasi-hereditary approximation of A. The question is answered in the appropriate language of standardly stratified algebras: For any K-algebra A, there is a uniquely defined basic algebra such that is Δ-filtered and the subcategories and of all Δ-filtered modules are equivalent; similarly there is a uniquely defined basic algebra such that is -filtered and the subcategories and of all -filtered modules are equivalent. These subcategories play a fundamental role in the theory of stratified algebras. Since, in general, it is difficult to localize these subcategories in the category of all A-modules, the construction of and often helps to describe them explicitly. By applying consecutively the operators Σ and Ω for an algebra, we get a sequence of standardly stratified algebras which, after a finite number of steps, stabilizes in a properly stratified algebra. Thus, all standardly stratified algebras are partitioned into (generally infinite) trees, indexed by properly stratified algebras (as their roots). [Copyright &y& Elsevier]

Abstract: We conclude our classification of powerful 2-Engel groups of class three that are minimal in the sense that every proper powerful section is nilpotent of class at most two. In the predecessor to this paper we obtained three families of minimal groups. Here we get a fourth family of minimal examples that is described in terms of irreducible polynomials over the field of three elements. We also get one isolated minimal example of rank 5 and exponent 27. The last one has a related algebraic structure that we call a “symplectic alternating algebra.” To each symplectic alternating algebra over the field of three elements there corresponds a unique 2-Engel group of exponent 27. [Copyright &y& Elsevier]

Abstract: The notion of a topological Jordan decomposition of a compact element of a reductive p-adic group has proven useful in many contexts. In this paper, we generalise it to groups defined over fairly general discretely valued fields and prove the usual existence and uniqueness properties, as well as an analogue of a fixed-point result of Prasad and Yu. [Copyright &y& Elsevier]

Abstract: In this paper, we offer a general Prime Ideal Principle for proving that certain ideals in a commutative ring are prime. This leads to a direct and uniform treatment of a number of standard results on prime ideals in commutative algebra, due to Krull, Cohen, Kaplansky, Herstein, Isaacs, McAdam, D.D. Anderson, and others. More significantly, the simple nature of this Prime Ideal Principle enables us to generate a large number of hitherto unknown results of the “maximal implies prime” variety. The key notions used in our uniform approach to such prime ideal problems are those of Oka families and Ako families of ideals in a commutative ring, defined in (2.1) and (2.2). Much of this work has also natural interpretations in terms of categories of cyclic modules. [Copyright &y& Elsevier]

Abstract: This paper deals with the smoothness of the transition map between two sections transverse to the fast flow of a singularly perturbed vector field (one fast, multiple slow directions). Orbits connecting both sections are canard orbits, i.e. they first move rapidly towards the attracting part of a critical surface, then travel a distance near this critical surface, even beyond the point where the orbit enters the repelling part of the critical surface, and finally repel away from the surface. We prove that the transition map is smooth. In a transcritical situation however, where orbits from an attracting part of one critical manifold follow the repelling part of another critical manifold, the smoothness of the transition map may be limited, due to resonance phenomena that are revealed by blowing up the turning point! We present a polynomial example in . [Copyright &y& Elsevier]

Abstract: In this paper, we study the free boundary problem for 1D compressible Navier–Stokes equations with density-dependent viscosity. We focus on the case where the viscosity coefficient vanishes on vacuum. We prove the global existence and uniqueness for discontinuous solutions to the Navier–Stokes equations when the initial density is a bounded variation function, and give a decay result for the density as . [Copyright &y& Elsevier]

Abstract: In this paper, we prove an important inequality and investigate some new calculating problems of random fixed point index, and generalize famous theorem by means of the theory of random fixed point index. [Copyright &y& Elsevier]

Abstract: By a theorem due to the first author, the bounded derived category of a finite dimensional algebra over a field embeds fully faithfully into the stable category over its repetitive algebra. This embedding is an equivalence if the algebra is of finite global dimension. The purpose of this paper is to investigate the relationship between the derived category and the stable category over the repetitive algebra from various points of view for algebras of infinite global dimension. The most satisfactory results are obtained for Gorenstein algebras, especially for selfinjective algebras. [Copyright &y& Elsevier]

Abstract: In this paper, we consider dynamical behavior of non-autonomous plate-type evolutionary equations with critical nonlinearity. We prove the existence of a uniform attractor in the space . [Copyright &y& Elsevier]

Abstract: In this paper we study the global asymptotic stability for a class of delay logistic equations. A sufficient and necessary condition is derived. As a result, we obtain a full affirmative answer to the Gopalsamy and Liu'' conjecture. [Copyright &y& Elsevier]

Abstract: This paper investigates the regularity of solutions of convolution equations in the frame of classes of ultradifferentiable functions and ultradistributions. We improve previous work by Bonet, Chou, Fernández, Galbis, Meise and others. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a new generalized discontinuous Galerkin (GDG) method for Schrödinger equations with nonsmooth solutions. The numerical method is based on a reformulation of Schrödinger equations, using split distributional variables and their related integration by parts formulae to account for solution jumps across material interfaces. The proposed GDG method can handle time dependent and nonlinear jump conditions . Numerical results for 1D and 2D time dependent Schrödinger equations validate the high order accuracy and the flexibility of the method for various types of interface conditions. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to derive two transformation formulae which imply relations between basic and bilateral basic hypergeometric series. Some special cases of them are also discussed. [Copyright &y& Elsevier]

Abstract: A language is called a relatively regular language if its syntactic monoid has finite ideals. In this paper, we show that there are close relationships between the relatively regular languages and some other classes of languages such as (generalized) disjunctive languages, fd-domains and 2-codes. In particular, we prove that every relatively regular 2-code is thin. Thus, the well known result of Bestel and Perrin in 1985 becomes an easy corollary of our result. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a relation between the Hall–Janko graph and the Witt system . We will reconstruct the Hall–Janko graph from the group and the hexacode. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a generalization for the class of laura algebras, called almost laura. We show that this new class of algebras retains most of the essential features of laura algebras, especially concerning the important role played by the non-semiregular components in their Auslander–Reiten quivers. Also, we study more intensively the left supported almost laura algebras, showing that these are characterized by the presence of a generalized standard, convex and faithful component. Finally, we prove that almost laura algebras behave well with respect to full subcategories, split-by-nilpotent extensions and skew group algebras. [Copyright &y& Elsevier]

Abstract: In [Ch. Brouder, A. Frabetti, Renormalization of QED with planar binary trees, Eur. Phys. J. C 19 (2001) 715–741; Ch. Brouder, A. Frabetti, QED Hopf algebras on planar binary trees, J. Algebra 267 (2003) 298–322] we introduced three Hopf algebras on planar binary trees related to the renormalization of quantum electrodynamics. One of them, the algebra , is commutative, and is therefore the ring of coordinate functions of a proalgebraic group . The other two algebras, and , are free non-commutative. Therefore their abelian quotients are the coordinate rings of two proalgebraic groups and . In this paper we describe explicitly these groups. Using two monoidal structures and a set-operad structure on planar binary trees, we show that these groups can be realized on formal series expanded over trees, and that the group laws are generalizations of the multiplication and the composition of usual series in one variable. Therefore we obtain some new groups of invertible tree-expanded series and of tree-expanded formal diffeomorphisms respectively. The Hopf algebra describing the renormalization of the electric charge corresponds to the subgroup of tree-expanded formal diffeomorphisms formed of the translations, which fix the zero, by some particular tree-expanded series which remind the proper correlation functions in quantum field theory. In turn, the group of tree-expanded formal diffeomorphisms and some of its subgroups give rise to new Hopf algebras on trees. All the constructions are done in a general operad-theoretic setting, and then applied to the specific duplicial operad on trees. [Copyright &y& Elsevier]

Abstract: Let Φ be a root system of type , and let G be a group generated by non-trivial subgroups , , satisfying some generalized Steinberg relations, which are also satisfied by root subgroups corresponding to a Moufang hexagon. These relations can be considered as a generalization of the element-wise commutator relations in Chevalley groups. The Steinberg presentation specifies the groups satisfying the Chevalley commutator relations. In the present paper some sort of generalized Steinberg presentation for groups with root system of type is provided. As a main result we classify the possible structures for G. [Copyright &y& Elsevier]

Abstract: Let be a Henselian valued field satisfying the following conditions, for a given prime number p: (i) central division K-algebras of (finite) p-primary dimensions have Schur indices equal to their exponents; (ii) the value group properly includes its subgroup . The paper shows that if is the residue field of and is an intermediate field of the maximal p-extension , then the natural homomorphism of Brauer groups maps surjectively the p-component on . It proves that is divisible, if or is a nonreal field, and that is of order 2 when is formally real. We also obtain that embeds as a -subalgebra in a central division -algebra if and only if the degree divides the index of . [Copyright &y& Elsevier]

Abstract: This paper compares the expressiveness of different fragments of ambient calculi via leader election problems. We consider Mobile Ambients (MA), Safe Ambients (SA) and the Push and Pull Ambient Calculus (PAC). Cardelli and Gordon encoded the asynchronous π-calculus into MA. Zimmer has shown that the synchronous π-calculus without choice can be encoded in pure (no communication) SA. We show that pure MA without restriction has symmetric electoral systems, that is, it is possible to solve the problem of electing a leader in a symmetric network. By the work of Palamidessi, this implies that this fragment of MA is not encodable (under certain conditions) in the π-calculus with separate choice. Moreover, we use the same technique to show that fragments of SA and PAC are not encodable (under certain conditions) in the π-calculus with separate choice. We also show that particular fragments of ambient calculi do not admit a solution to leader election problems, in the same way as the π-calculus with separate choice. This yields a fine-grained hierarchy within ambient calculi. [Copyright &y& Elsevier]

Abstract: Let G be a finite abelian group of order n and let be non-empty. Generalizing a well-known constant, we define the Davenport constant of G with weight A, denoted by , to be the least natural number k such that for any sequence with , there exists a non-empty subsequence and such that . Similarly, for any such set A, is defined to be the least such that for all sequences with , there exist indices , and with . In the present paper, we establish a relation between the constants and under certain conditions. Our definitions are compatible with the previous generalizations for the particular group and the relation we establish had been conjectured in that particular case. [Copyright &y& Elsevier]

Abstract: Separating hash families are useful combinatorial structures which are studied in a general form in this paper. Necessary and sufficient conditions for the existence of certain types of generalized hash functions are considered. [Copyright &y& Elsevier]

Abstract: Let be a polynomial ring and let be a graded ideal. In [T. Römer, Betti numbers and shifts in minimal graded free resolutions, arXiv: AC/070119], Römer asked whether under the Cohen–Macaulay assumption the ith Betti number can be bounded above by a function of the maximal shifts in the minimal graded free R-resolution of as well as bounded below by a function of the minimal shifts. The goal of this paper is to establish such bounds for graded Cohen–Macaulay algebras when I is a standard determinantal ideal of arbitrary codimension. We also discuss other examples as well as when these bounds are sharp. [Copyright &y& Elsevier]

Abstract: We show that, under some mild conditions, a bialgebra in an abelian and coabelian braided monoidal category has a weak projection onto a formally smooth (as a coalgebra) sub-bialgebra with antipode; see Theorem 1.14. In the second part of the paper we prove that bialgebras with weak projections are cross product bialgebras; see Theorem 2.12. In the particular case when the bialgebra A is cocommutative and a certain cocycle associated to the weak projection is trivial we prove that A is a double cross product, or biproduct in Madjid''s terminology. The last result is based on a universal property of double cross products which, by Theorem 2.15, works in braided monoidal categories. We also investigate the situation when the right action of the associated matched pair is trivial. [Copyright &y& Elsevier]

Abstract: We discuss a notion of large complex structure for elliptic K3 surfaces with section inspired by the eight-dimensional F-theory/heterotic duality in string theory. This concept is naturally associated with the Type II Mumford partial compactification of the moduli space of periods for these structures. The paper provides an explicit Hodge-theoretic condition for the complex structure of an elliptic K3 surface with section to be large. We also establish certain geometric consequences of this large complex structure condition in terms of the Kodaira types of the singular fibers of the elliptic fibration. [Copyright &y& Elsevier]

Abstract: For any vertex algebra and any subalgebra , there is a new subalgebra of known as the commutant of in . This construction was introduced by Frenkel–Zhu, and is a generalization of an earlier construction due to Kac–Peterson and Goddard–Kent–Olive known as the coset construction. In this paper, we interpret the commutant as a vertex algebra notion of invariant theory. We present an approach to describing commutant algebras in an appropriate category of vertex algebras by reducing the problem to a question in commutative algebra. We give an interesting example of a Howe pair (i.e., a pair of mutual commutants) in the vertex algebra setting. [Copyright &y& Elsevier]

Abstract: In this paper we investigate the class of the connected graded algebras which are finitely generated in degree 1, which are finitely presented with relations of degrees greater or equal to 2 and which are of finite global dimension D and Gorenstein. For D greater or equal to 4 we add the condition that these algebras are homogeneous and Koszul. It is shown that each such algebra is completely characterized by a multilinear form satisfying a twisted cyclicity condition and some other nondegeneracy conditions depending on the global dimension D. This multilinear form plays the role of a volume form and canonically identifies in the quadratic case to a nontrivial Hochschild cycle of maximal degree. Several examples including the Yang–Mills algebra and the extended 4-dimensional Sklyanin algebra are analyzed in this context. Actions of quantum groups are also investigated. [Copyright &y& Elsevier]

Abstract: In this paper, we verify Dade''s invariant conjecture for Steinberg''s triality groups in the defining characteristic, i.e., in characteristic 2. Together with the results in [J. An, Dade''s conjecture for Steinberg triality groups in non-defining characteristics, Math. Z. 241 (2002) 445–469] and [J. An, F. Himstedt, S. Huang, Uno''s invariant conjecture for Steinberg''s triality groups in defining characteristic, in preparation], this completes the proof of Dade''s conjecture for Steinberg''s triality groups. Furthermore, we show that the Isaacs–Malle–Navarro version of the McKay conjecture holds for in the defining characteristic, i.e., is good for the prime 2 in the sense of Isaacs, Malle and Navarro. [Copyright &y& Elsevier]

Abstract: The present paper is concerned with asymptotic behaviours of the solutions to the micropolar fluid motion equations in . Upper and lower bounds are derived for the decay rates of higher order derivatives of solutions to the micropolar fluid flows. The findings are mainly based on the basic estimates of the linearized micropolar fluid motion equations and generalized Gronwall type argument. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a food-limited population model with impulsive effect. Several explicit sufficient conditions are established for oscillation and nonoscillation of solutions of the equations. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to give necessary and sufficient conditions for rings for which every right self-small module is finitely generated. It is proved that: semi-simple rings, commutative perfect rings and right nonsingular Σ-extending rings have this property; a right nonsingular semi-prime ring has the property if and only if it is semi-simple; a commutative noetherian ring has the property if and only if it is artinian. [Copyright &y& Elsevier]

Abstract: A pro-p group G is a PF-group if it has central series of closed subgroups with trivial intersection satisfying and . In this paper, we prove that a finitely generated pro-p group G is a p-saturable group, in the sense of Lazard, if and only if it is a torsion free PF-group. Using this characterization, we study certain families of subgroups of p-saturable groups. For example, we prove that any normal subgroup of a p-saturable group contained in the Frattini is again p-saturable. [Copyright &y& Elsevier]

Abstract: In this paper we investigate the Lie structure of the Lie superalgebra K of skew elements of a semiprime associative superalgebra A with superinvolution. We show that if U is a Lie ideal of K, then either there exists an ideal J of A such that the Lie ideal is nonzero and contained in U, or A is a subdirect sum of , , where the image of U in is central, and is a subdirect product of orders in simple superalgebras, each at most 16-dimensional over its center. [Copyright &y& Elsevier]

Abstract: Let e be a real number and be a set of n distinct positive integers. The set S is said to be gcd-closed (respectively lcm-closed) if (respectively ) for all . The matrix having eth power of the least common multiple of and as its -entry is called the eth power least common multiple (LCM) matrix, denoted by (or abbreviated by ). In this paper, we show that for any real number and , the power LCM matrix defined on any gcd-closed (respectively lcm-closed) set is nonsingular. This confirms partially two conjectures raised by Hong in [S. Hong, Nonsingularity of matrices associated with classes of arithmetical functions, J. Algebra 281 (2004) 1–14]. Similar results are established for reciprocal real number power GCD matrices. [Copyright &y& Elsevier]

Abstract: Two relevant theorems of B.H. Neumann characterize groups with finite conjugacy classes of subgroups and groups in which every subgroup has finite index in its normal closure as central-by-finite groups and finite-by-abelian groups, respectively. These results have later been extended to the case of groups with similar restrictions on abelian subgroups. Moreover, Romalis and Sesekin have studied groups in which all non-abelian subgroups are normal, and in this paper we consider groups with normality conditions of Neumann''s type for non-abelian subgroups. [Copyright &y& Elsevier]

Abstract: For a commutative ring R, the zero-divisor graph of R is the graph whose vertices are the nonzero zero-divisors of R such that the vertices x and y are adjacent if and only if . In this paper, we classify the zero-divisor graphs of Boolean rings, as well as those of Boolean rings that are rationally complete. We also provide a complete list of those rings whose zero-divisor graphs have the property that every vertex is either an end or adjacent to an end. [Copyright &y& Elsevier]

Abstract: Let be the Hilbert scheme parametrizing the closed subschemes of with Hilbert polynomial over a field K of characteristic zero. By bounding below the cohomological Hilbert functions of the points of we define locally closed subspaces of the Hilbert scheme. The aim of this paper is to show that some of these subspaces are connected. For this we exploit the edge ideals constructed by D. Mall in [D. Mall, Connectedness of Hilbert function strata and other connectedness results, J. Pure Appl. Algebra 150 (2000) 175–205]. It turns out that these ideals are sequentially Cohen–Macaulay and that their initial ideals with respect to the reverse lexicographic term order are generic initial ideals. [Copyright &y& Elsevier]

Abstract: In this paper, the notion of generalized umbrella rings is introduced. We show that every generalized umbrella ring is a stably coherent ring. As an application, we show that every finitely generated projective module over is free if R is a generalized umbrella ring with weak global dimension two. [Copyright &y& Elsevier]

Abstract: In this paper we describe graded antiautomorphisms of finite order on matrix algebras endowed with a group gradings by a finite abelian group over an arbitrary algebraically closed field of characteristic different from 2. [Copyright &y& Elsevier]

Abstract: In this paper, we observe that a certain local property on highest weight crystal graphs forces a more global property. In type A, this statement says that if a node has a single parent and single grandparent, then there is a unique walk from the highest weight node to it. This crystal observation was motivated by certain representation-theoretic behavior of the affine Hecke algebra of type A. In other classical types, there is a similar statement. This walk is obtained from the associated level 1 perfect crystal, . (It is unique unless the Dynkin diagram contains that of as a subdiagram.) [Copyright &y& Elsevier]

Abstract: In this paper, we study the weak convergence of “Leray-type” solutions of a magneto-hydro-dynamic systems. The proofs are based on weak compactness arguments and linear algebra methods. [Copyright &y& Elsevier]

Abstract: In this paper we give an alternative review over the subject of strata and simple characters as more generally already considered in the references [P. Broussous, M. Grabitz, Pure elements and intertwining classes of simple strata in local central simple algebras, Comm. Algebra 28 (11) (2000) 5405–5442] and [V. Sécherre, Représentations lisses de , I: charactères simples, Bull. Soc. Math. France 132 (3) (2004) 327–396]. Finally, we prove the intertwining implies conjugacy property of simple characters as known in the split case [C.J. Bushnell, P.C. Kutzko, The Admissible Dual of via Compact Open Subgroups, Ann. of Math. Stud., vol. 129, Princeton Univ. Press, 1993]. [Copyright &y& Elsevier]

Abstract: In this paper we prove that two quantities relating to the length of permutations defined on trees are independent of the structures of trees. We also find that these results are closely related to the results obtained by Graham and Pollak [R.L. Graham, H.O. Pollak, On the addressing problem for loop switching, Bell System Tech. J. 50 (1971) 2495–2519] and by Bapat, Kirkland, and Neumann [R. Bapat, S.J. Kirkland, M. Neumann, On distance matrices and Laplacians, Linear Algebra Appl. 401 (2005) 193–209]. [Copyright &y& Elsevier]

Abstract: Let L be a finite-dimensional simple Lie algebra over an algebraically closed field F of characteristic . We prove in this paper that if for every torus T of maximal dimension in the p-envelope of adL in DerL the centralizer of T in adL acts triangulably on L, then L is either classical or of Cartan type. As a consequence we obtain that any finite-dimensional simple Lie algebra over an algebraically closed field of characteristic is either classical or of Cartan type. This settles the last remaining case of the generalized Kostrikin–Shafarevich conjecture (the case where ). [Copyright &y& Elsevier]