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The paper deals with constructing an asymptotic one-phase soliton-like solution to the Benjamin--Bona--Mahony equation with variable coefficients and a strong singularity making use of the non-linear WKB technique. The influence of the small-parameter value on the structure and the qualitative properties of the asymptotic solution, as well as the accuracy with which the solution satisfies the considerable equation, have been analyzed. It was demonstrated that due to the strong singularity, it is possible to write explicitly not only the main term of the asymptotics but at least its first-order term.

In the study of nonlinear oscillations, the question on the dependence of the frequency or the period on the energy often arises. In this paper, we find conditions under which the frequency depends on the energy monotonically. In addition, for oscillations near separatrix trajectories, an asymptotics of the period with respect to the energy is constructed.

In this paper, we consider systems of linear differential equations with periodic coefficients depending on a small parameter. We propose new approaches to the problem of constructing a monodromy matrix that lead to new effective formulas for calculating multipliers of the system studies. We present a number of applications in problems of the perturbation theory of linear operators, in the analysis of stability of linear differential equations with periodic coefficients, in the problem of constructing the stability domains of linear dynamical systems, etc.

In this paper, we consider adiabatic deformations of Riemann surfaces that preserve the integrability of the corresponding dynamic system, which leads to the appearance of modulated quasiperiodic motions, similar to the effect of parametric resonance. We show that in this way it is possible to control the amplitude and frequency of nonlinear modes. We consider several examples of the dynamics of top-type systems.

In this paper, we prove that for a linearly ordered local ring R with 1/2 the group of quotients of the semigroup of invertible nonnegative matrices Gn(R) for n ≥ 3 coincides with the group GLn(R).

This paper belongs to our series of works on algebraic geometry over arbitrary algebraic structures. In this one, there will be investigated seven equivalences (namely: geometric, universal geometric, quasi-equational, universal, elementary, and combinations thereof) in specific classes of algebraic structures (equationally Noetherian, qω-compact, uω-compact, equational domains, equational co-domains, etc.). The main questions are the following: (1) Which equivalences coincide inside a given class K, which do not? (2) With respect to which equivalences a given class K is invariant, with respect to which it is not?

This paper introduces the concept of the Artin semigroup of a large (extra large) type. We prove their injective embedding in the corresponding Artin groups of large (extra large) type and the solvability of the word conjugacy problem in this class of semigroups.

This paper defines Artin groups with m-gon structure and proves that for m > 3 in this class the problems of equality and conjugacy of words are solvable.

Among the finitely generated groups of diffeomorphisms of the line and the circle, groups that act freely on the orbit of almost every point of the line (circle) are allocated. The paper is devoted to the study of the structure of the set of finitely generated groups of orientation-preserving diffeomorphisms of the line and the circle of C(1) smoothness with a given number of generators and the property noted above. It is shown that such a set contains a massive subset (contains a countable intersection of open everywhere dense subsets). Such a result for finitely generated groups of orientation-preserving diffeomorphisms of the circle, in the case of C(2) smoothness, was obtained by the author earlier.

This is the first paper of a conceived series under the common title “The boundary control method in inverse problems.” The aim of the series is to expound systematically an approach to inverse problems based upon its relationship with control theory. The 1d-variant of the method is shown with the example of the classical problem of recovering the density of an inhomogeneous string, and both dynamical and spectral statements of the problem are considered. The paper is written in such a way as to serve as an introduction to the multidimensional BC-method: the basic tools and constructions are amenable to further generalization to multidimensional problems. Bibliography: 31 titles.

This paper is a sequel to the author's papers [1–8] devoted to lower multiplicative bounds on L 1 norms and their applications. Now we give estimates for L 1 norms of exponential sums and prove the result announced in [8].

Let A be a class of Abelian groups, A ∈ A, and End(A) be the additive endomorphism group of the group A. The group A is said to be defined by its endomorphism group in the class {ie208-01} if for every group B ∈ B such that End(B) ≅ End(A) the isomorphism B ≅ A holds. The paper considers the problem of definability of a periodic Abelian group A such that End-End(A) ≅ End(A). The classes of periodical Abelian groups, of divisible Abelian groups, of reduced Abelian groups, of nonreduced Abelian groups, and of all Abelian groups are investigated in this paper.

This paper is devoted to relations between the Kurosh problem and the Shirshov height theorem. The central point and main technical tool is the identity of algebraicity. The main result of this paper is the following. Let A be a finitely generated PI-algebra and Y be a finite subset of A. For any Noetherian associative and commutative ring {ie125-01}, let any factor of R ⊗ A such that all projections of elements from Y are algebraic over π(R) be a Noetherian R-module. Then A has bounded essential height over Y. If, furthermore, Y generates A as an algebra, then A has bounded height over Y in the Shirshov sense.

In this paper, we consider reduced semirings with supplementary conditions of annihilators, namely these are Rickart and weakly Rickart semirings. The main aim of the paper is to study functional representations of semirings. We build two sheaves of semirings and prove that a reduced Rickart semiring is presented by sections of these sheaves.

The paper is devoted to two new directions in developing the classical geometric subjects related to studying the problem of unique determination of closed convex surfaces by their intrinsic metrics. The first of these directions is the study of unique determination of domains (i.e., open connected sets) in Euclidean spaces by relative metrics of the boundaries of these domains. It appeared about 25–30 years ago and was developed owing to the efforts of Russian scientists. The first part of the paper (Secs. 3–7) contains an overview of the results referring to this direction.

In this paper, a hyperbolicity criterion for periodic solutions of nonlinear functional-differential equations is constructed in terms of zeros of the characteristic function. In the earlier papers in this area, necessary and sufficient conditions were different from each other. Moreover, it was assumed that if the period of the investigated solution is irrational, then that solution admits a rational approximation. In this paper, we obtain necessary and sufficient conditions of the hyperbolicity. It is proved (and the proof is constructive) that a rational approximation exists for any irrational period. All the results are obtained for the case of several rational delays.

The paper is concerned with a new way of deriving computable estimates for the difference between the exact solutions of elliptic variational inequalities and arbitrary functions in the corresponding energy space that satisfy the main (Dirichlet) boundary conditions. Unlike the method derived earlier, the estimates are obtained by certain transformations of variational inequalities without using duality arguments. For linear elliptic and parabolic problems, this method was suggested by the author in previous papers. The present paper deals with two different types of variational inequalities (also called variational inequalities of the first and second kind). The techniques discussed can be applied to other nonlinear problems related to variational inequalities. Bibliography: 20 titles.

We study rings over which all right modules are I0-modules. All rings are assumed to be associative and with nonzero identity element. For a module M , a submodule N of M is said to be superfluous if N +P = M for every proper submodule P of the module M . A module M is called an I0-module if every cyclic submodule of M either is superfluous in M or contains a nonzero direct summand of the module M . A ring A is called a right (left) I0-ring if AA (respectively, AA) is a right (respectively, left) I0-module. I0-modules and I0-rings were studied in [8; 11, Chap. 3; 1–3; 7; 6] and other works. In the present paper, we study rings over which all right modules are I0-modules. The main result of the present paper is Theorem 1. Theorem 1. For a ring A, the following conditions are equivalent. (1) Every right A-module is an I0-module. (2) For every right A-module M , we have that J(M) is a semisimple module and if J(M) = 0, then every nonzero submodule of the module M contains a nonzero direct summand of the module M . (3) For every right A-module M , either M has a nonzero injective direct summand or M is a semisimple module and is contained in the Jacobson radical of the injective hull of M . (4) Every cyclic right A-module either has a nonzero injective direct summand or is a semisimple module. The residue ring Z/4Z is an example of a nonsemisimple ring that satisfies the conditions of Theorem 1. In Example 11 of the present paper, we give an example of a ring A such that all right A-modules are I0-modules and A contains an infinite set of orthogonal idempotents (therefore, A is not Noetherian). It can also be proved that A is a left semi-Artinian ring and left A-modules are not necessarily I0-modules. I0-modules are close to regular modules and semiregular modules. A module M is said to be regular if every cyclic submodule of M is a direct summand of the module M . A module M is said to be semiregular if for every cyclic submodule N ofM , there exists a direct decomposition M = M1⊕M2 such thatM1 ⊆ N and N ∩ M2 is a superfluous submodule in M2. Semiregular modules were studied in [9; 10, Chap. B; 11, Chap. 4; 12, 14] and other works. It is easy to verify that every semiregular module is an I0-module, every regular module is semiregular, and every semiprimitive, semiregular module is regular. The cyclic group of order 4 is a semiregular nonregular module over the rings Z and Z/4Z. Lemma 4(4) contains an example of a semiprimitive I0-module that is not a semiregular module. The proof of Theorem 1 is decomposed into a series of assertions; some of the assertions are of independent interest. We present the necessary notation and definitions. The intersection of all maximal submodules of the module M is denoted by J(M); it is called the Jacobson radical of the module M . It is well known that J(M) coincides with the sum of all superfluous submodules of the module M (see, e.g., [13, 21.5]). A module M is said to be semiprimitive if J(M) = 0. A module M is said to be semi-Artinian if every nonzero submodule of the module M contains a simple submodule. A ring A is said to be a right V -ring if every simple right A-module is injective (this is equivalent to the property that every right A-module is semiprimitive [4, 7.32A]). A module M is said to be uniserial if any two submodules of M are comparable with respect to inclusion. A direct sum of uniserial modules is called a serial module. A module M is said to be semisimple if every submodule of M is a direct summand of Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 8, pp. 233–241, 2006. 3928 1072–3374/08/1522–3928 c © 2008 Springer Science+Business Media, Inc. the module M . A submodule N of the module M is said to be essential if, for every submodule X of the module M , the relation X ∩N = 0 implies the relation X = 0. A module M is said to be injective if for every module X and each submodule Y of the module X, any homomorphism Y → M can be extended to a homomorphism X → M . If M is an injective module and N is an essential submodule of the module M , then the module M is called the injective hull of the module N . Every module has an injective hull, which is unique up to isomorphism. Lemma 2. Let M be a nonzero right module over a ring A. (1) The module M is an I0-module if and only if every submodule of the module M either is contained in J(M) or contains a nonzero direct summand of the module M . (2) M is a semiprimitive I0-module if and only if every nonzero submodule of the module M contains a nonzero direct summand of the module M . (3) If A is a right V -ring, then M is an I0-module if and only if every nonzero submodule of the module M contains a nonzero direct summand of the module M . (4) If M is an essential extension of a semisimple module and every simple submodule of the moduleM is injective, then M is a semiprimitive I0-module. (5) If A is a right semi-Artinian right V -ring, then M is a semiprimitive I0-module. Proof. (1) The sufficiency follows from the property that J(M) contains all superfluous submodules of the module M . We prove the necessity. Let N be a submodule of the module M that is not contained in J(M). There exists a cyclic submodule X of the module N that is not contained in J(M). Since J(M) is the sum of all superfluous submodules of the module M , the module X is not a superfluous submodule of the module M . By condition (1), some nonzero direct summand Y of the module M is contained in X. Then Y ⊆ N . (2) The assertion follows from (1). (3) The assertion follows from (2) and the property that every right module over any right V -ring is semiprimitive. (4) Since M is an essential extension of a semisimple module, every nonzero submodule N of the module M contains some simple submodule S. By assumption, the module S is injective. Therefore, S is a nonzero direct summand of the module M . (5) Since A is a right semi-Artinian ring, M is an essential extension of a semisimple module. Since A is a right V -ring, every simple submodule of the module M is injective. By (4), M is a semiprimitive I0-module. Lemma 3. For a ring A, the following conditions are equivalent. (1) A is a semiprimitive right I0-ring. (2) Every nonzero right ideal of the ring A contains a nonzero idempotent. (3) Every nonzero principal right ideal of the ring A contains a nonzero idempotent. Lemma 3 follows from Lemma 2(2). Lemma 4. Let A be a ring, B be a unitary subring of the ring A, {Ai}i=1 be a countable set of copies of the ring A, D be the direct product of the rings Ai, and R be the subring in D generated by the ideal ∞ ⊕ i=1 Ai and by the subring B′ ≡ {(b, b, b, . . .) | b ∈ B}. (1) The identity elements ei of the rings Ai are central idempotents of the ring D and ei are contained in the ring R, R = {(a1, . . . , an, b, b, b, . . .) | ai ∈ A, b ∈ B}, where the positive integer n depends on the element (a1, . . . , an, b, b, b, . . .), and R has the factor ring R/ ( ∞ ⊕

This paper is devoted to the introduction of a D∞-differential analog of the notion of an E∞-(co)algebra and to the construction of generalized Steenrod operations in terms of multiplicative spectral sequences. In this paper, we investigate basic homotopy properties of D∞-differential E∞-(co)algebras and construct a spectral sequence of a D∞-differential E∞-(co)algebra.

This paper is devoted to singular points of the so-called lifted vector fields, which arise in studying systems of implicit differential equations by using the method of lifting the equation to a surface, a generalization of the construction used by Poincare for a single implicit equation. The author studies the phase portraits and renormal forms of such fields in a neighborhood of their singular points. In conclusion, this paper considers the lifted vectors fields generated by Euler-Lagrange and Euler-Poisson equations and fast-slow systems.

In this paper, we prove the existence of time-periodic weak solutions for the wave equation with homogeneous boundary conditions. This paper deals with the cases where a nonlinear term has a superlinear and sublinear growth.

Let X be an irreducible projective variety over an algebraically closed field of characteristic zero. For ≥ 3, if every (r−2)-plane $$\overline {x_1 , \ldots ,x_{r - 1} } $$ , where the x i are generic points, also meets X in a point x r different from x 1,..., x r−1, then X is contained in a linear subspace L such that codim L X ≥ r − 2. In this paper, our purpose is to present another derivation of this result for r = 3 and then to introduce a generalization to nonequidimensional varieties. For the sake of clarity, we shall reformulate our problem as follows. Let Z be an equidimensional variety (maybe singular and/or reducible) of dimension n, other than a linear space, embedded into ℙr, where r ≥ n + 1. The variety of trisecant lines of Z, say V 1,3(Z), has dimension strictly less than 2n, unless Z is included in an (n + 1)-dimensional linear space and has degree at least 3, in which case dim V 1,3(Z) = 2n. This also implies that if dim V 1,3(Z) = 2n, then Z can be embedded in ℙ n + 1. Then we inquire the more general case, where Z is not required to be equidimensional. In that case, let Z be a possibly singular variety of dimension n, which may be neither irreducible nor equidimensional, embedded into ℙr, where r ≥ n + 1, and let Y be a proper subvariety of dimension k ≥ 1. Consider now S being a component of maximal dimension of the closure of $$\{ l \in \mathbb{G}(1,r)|\exists p \in Y, q_1 , q_2 \in Z\backslash Y, q_1 , q_2 ,p \in l\} $$ . We show that S has dimension strictly less than n + k, unless the union of lines in S has dimension n + 1, in which case dim S = n + k. In the latter case, if the dimension of the space is strictly greater than n + 1, then the union of lines in S cannot cover the whole space. This is the main result of our paper. We also introduce some examples showing that our bound is strict.

This paper compares sequences of independent, mean zero random variables in a rearrangement-invariant space X on [0, 1] with sequences of disjoint copies of individual terms in the corresponding rearrangement-invariant space Z X 2 on [0, ∞). The principal results of the paper show that these sequences are equivalent in X and Z X 2 , respectively, if and only if X possesses the (so-called) Kruglov property. We also apply our technique to complement well-known results concerning the isomorphism between rearrangement-invariant spaces on [0, 1] and [0, ∞). Bibliography: 20 titles.

Let G = G(Φ, K) be a Chevalley group over a field K of characteristic ≠ 2. In the present paper, we classify the subgroups of G generated by triples of long root subgroups, two of which are opposite, up to conjugacy. For finite fields, this result is contained in papers by B. Cooperstein on the geometry of root subgroups, whereas for SL (n, K) it is proved in a paper by L. Di Martino and the first-named author. All interesting items of our list appear in deep geometric results on abstract root subgroups and quadratic actions by F. Timmesfeld and A. Steinbach, and also by E. Bashkirov. However, for applications to the groups of type E l, we need a detailed justification of this list, which we could not extract from the published papers. That is why in the present paper, we produce a direct elementary proof based on the reduction to D 4, where the question is settled by straightforward matrix calculations. Bibliography: 73 titles.

The Euler problem on the stability of a thin elastic rod under compressing forces is widely known. In a number of papers this problem is generalized to finding the shape of such a rod under the simultaneous action of a compressing force and a torsional moment. The shape is determined by solving a complex nonlinear boundary value problem by numerical methods. In this paper, an approach providing a full analytical solution of the problem for a broad range of external conditions is discussed.

Retrial queueing models are often used for the performance and reliability modeling of computer systems and communication networks. The reason is that the return of customers plays a special role in many of these systems as well as in other practical applications, and it has a nonneglectable negative effect on the performance measures. For some applications of retrial queues, see, for example, [1–4], and for some fundamental results on finite-source retrial queueing systems, refer to [5–9]. Usually, the components of computer systems are subject to random breakdowns, which has a substantial influence on the performance measures, so it is of practical importance to investigate nonreliable retrial queueing systems, too. Nonreliable, infinite-source retrial queues were studied in [10–12] and finite-source retrial queues with a single nonreliable server were studied in [13]. The purpose of this paper is to generalize the model of [9, 13] and to give the main stationary performance measures of the nonreliable multiserver model described in the next section. Furthermore, our aim is to illustrate graphically the effect of the nonreliability of the servers on the steady-state systems’s measures. Because of the fact that the state space of the Markov chain described is very large, it is difficult to calculate the system measures in the traditional way of solving the system of steady-state equations. To simplify this procedure, we used the software tool MOSEL (Modeling, Specification, and Evaluation Language) (see [14]) to formulate the model and to obtain the performance measures. With the help of MOSEL, we can use various performance tools (such as SPNP — Stochastic Petri Net Package) to get these characteristics. The results of the tool can graphically be displayed using IGL (Intermediate Graphical Language), which is a part of MOSEL. The organization of the paper is as follows. Section 2 contains an accurate description of the investigated retrial queueing model and the derivation of the main steady-state performance measures. Section 3 is devoted to the validation of the results of the tool and some graphically displayed numerical results. The paper ends with Comments and Conclusions.

In the present paper, we characterize ⋀n(GL(n, R)) over any commutative ring R as the connected component of the stabilizer of the Plucker ideal. This folk theorem is classically known for algebraically closed fields and should also be well known in general. However, we are not aware of any obvious reference, so we produce a detailed proof, which follows a general scheme developed by W.C.Waterhouse. The present paper is a technical preliminary to a subsequent paper, where we construct the decomposition of transvections in polyvector representations of GL n. Bibliography: 50 titles.

In this paper, the controlled surgery sequence of Ranicki, Pedersen, and Quinn is applied to the solution of surgery problems in dimension four when the fundamental group is not known to be good. Our examples concern free non-Abelian fundamental groups, surface fundamental groups, and special knot groups. Using results from our earlier paper (joint with Spaggiari), we state a general result from which our examples follow.

In the Hilbert space setting (i.e., in the framework of C∗-algebra theory), analogous objects are closely related with a crossed product (see, e.g., [4]), and many works are devoted to the description of their structures. In particular, Landstad [3] found necessary and sufficient conditions (in terms of duality theory) under which a C∗-algebra is isomorphic to the crossed product (of a certain algebra and a locally compact automorphism group). In the case of a discrete group, in [1, Chap. 2], there are necessary and sufficient conditions under which a C∗-algebra is isomorphic to a crossed product in terms of the action of the group (the so-called topologically free action (see Sec. 2.8 of the present paper) and also in terms of fulfillment of a certain inequality (Property (∗), (2.1) of the present paper) that guarantees the existence of an expectation of the algebra B(A, Tg) on the algebra A (see (2.2) and (2.3)). The goal of the present paper is the study of interrelations between the properties (topologically free action, Property (∗), and dual action of the group) mentioned above in the general Banach space setting. Along with this, we study property (∗∗) (Sec. 2.2 ) of reconstructing an element of the algebra B(A, Tg) by its “Fourier coefficients,” which naturally arises here.

The present paper is one in a series of papers devoted to the classification of some classes of tame algebras up to stable category equivalence. In this paper, we study symmetric algebras (their stable categories have a structure of triangulated categories) and the simplest class of tame algebras-the class of special biserial algebras (SB-algebras). In the paper, we give a relevant version of the “diagrammatic method” and study the structure of the triangulated category “in a neighborhood” of the periodic part (with respect to Ω) of the stable category. Thus we prove the invariance of the collection of lengths of G-cycles under equivalence of stable categories (see Theorem 2.12). Then we use the invariance stated above, together with some properties of the Cartan matrix of a symmetric SB-algebra, to prove that the number of A-cycles (but not their lengths!) is also an invariant of stable equivalence. Bibliography: 8 titles.

This paper provides a method for the computation of Yoneda algebras for algebras of dihedral type. The Yoneda algebras for one infinite family of algebras of dihedral type (the family \(D(3\mathcal{Q})\)) in K. Erdmann’s notation) are computed. The minimal projective resolutions of simple modules were calculated by an original computer program implemented by one of the authors in the C++ language. The algorithm of the program is based on a diagrammatic method presented in this paper and inspired by that of D. Benson and J. Carlson.

The paper considers the approach game problem of one control object moving in the space with the other whose motion is executed in the horizontal plane. In this case, the horizontal plane plays the role of state constraints for the pursuer, who, therefore, can move only in the upper half-space. The dynamics of the player models the motion of different-type objects in a medium with friction. The goal of the pursuer is the approach of geometric coordinates and the velocities of the players (soft landing) at a certain finite instant of time. The paper distinguishes initial states of the pursuer and also establishes sufficient conditions on the parameters of the conflict-control process under which the “soft landing” problem is solvable at a finite time. Moreover, the authors use a method that allows them to reduce the game problem to an equivalent control problem. Based on a detailed study of the reachable set of the latter problem, the authors construct pursuer controls allowing one to solve the initial problem in explicit (analytical) form. Moreover, in the first stage, based on the N. N. Krasovskii extremal aiming principle, the authors align the velocity of the players, and in the concluding stage, they directly perform the “soft landing.” At each stage, the time needed for solution of the problem can be found a priori. In conclusion, the authors discuss the results of modeling the “soft landing” process.

Many problems of conflict-control theory can be reduced to approach-avoidance games with a certain terminal set. One of the main approaches to solution of such problems is the approach suggested by N. N. Krasovskii, which is based on positional constructions. The basis of these constructions consists of the extremal aiming principle at stable bridges. In this connection, the problem of constructing the maximal stable bridge, the set of all positions from which the problem of approaching the terminal set (which is the main task of one of the problem) is solvable, is important. The paper considers the approach-avoidance game on a finite interval of time in which the first player must ensure the attainment by the state vector of the control system of the terminal set on this interval, and the second player must ensure the avoidance of the terminal set. The main subject of the study is the maximal stable bridge, which is the set of positional consumption in this game. The problem of exactly constructing the set of positional consumption is solvable only in simple cases, and it is more realistic to consider and solve the problem of approximate construction of this set. The paper proposes approaches to approximate construction of the set of positional consumption based on sampling the time interval of the game and the technique of backward constructions, which has been developed at the scientific school of N. N. Krasovskii since the 1980s.

In this paper, we consider linear (in dynamics) conflict control problems (linear antagonistic differential games) with a fixed instant of termination and a continuous terminal cost function. We formulate and prove assertions on sufficient conditions under which such a method guarantees the obtaining of a result close to optimal by the minimizing player and has the stability property. In the concluding part of the paper, we give a brief description of publications devoted to computer modeling by using the proposed control method.

The paper is devoted to the theory of singly generated multivalued groups. We construct new classes of such groups and find criteria that allow to check whether such a group is a coset group. For this purpose, we introduce the construction of a σ-extension of a singly generated multivalued group with Hermitian generator. This construction is based on the relation of the theory of such groups with the theory of symmetric graphs. The main result of the paper is as follows: the suggested construction of σ-extensions of singly generated bicoset multivalued groups with Hermitian generators is equivariant with respect to morphisms of graph-theoretic nature. Bibliography: 11 titles.

∞n=0 ∞ pnF ∗n (x), where F ∗n (x) denotes the n-fold convolution of F (x) and where F ∗0 (x) denotes the unit mass at 0. The d.f. W (x )i s called subordinate to F (x) with subordinator {pn}. As in the univariate case in the paper, we shall assume that N satisfies condition (A): N has a generating function P (z )= E(z N ) that is analytic at z =1 . In the present paper, we discuss the relation between the asymptotic behavior of 1 − F (x) and that of 1 − F ∗n (x) and 1 − W (x). It turns out that, as in the univariate case, there are many cases in which 1 − F ∗n (x) asymptotically behaves as n(1 − F (x)) and 1 − W (x) behaves as E(N )(1 − F (x)). To specify the precise kind of asymptotic behavior, we present a form of multivariate subexponentiality. The paper is organized as follows. In Sec. 2, we briefly recall some basic properties and definitions concerning univariate subexponential d.f. In Sec. 3, we introduce and study multivariate subexponential d.f.’s. In Sec. 4, we discuss the relation with regular variation and, in Sec. 5, we provide some extensions. In our main results, we obtain first-order estimates for 1 − F ∗n (x) and 1 − W (x). In a forthcoming paper, we discuss second-order estimates. Without further comment, in the paper, we shall assume that all random vectors X, Y, Z, etc. are positive and have infinite support, i.e., the d.f. satisfies F (0+) = 0 and F (x) < 1, ∀x ∈ R d . We also use the notation F (x )=1 − F (x), and for vectors x and a ,w e setx ◦ = min(xi )a nda ∗ x =( a1x1 ,a 2x2 ,...,a dxd). 2. Univariate Subexponential Distributions In the one-dimensional case, many papers have been devoted to the tail behavior of subordinated d.f.’s. In doing so, the class of subexponential d.f.’s (notation: S) plays an important role. Extending the class S, Chover et al. [6, 7], introduced the class S(γ), where γ ≥ 0. To define these classes, let F (x) denote a d.f. in R such that F (0+) = 0 and F (x) < 1, ∀x ∈ R. Also, let f (s )= E(e −sX ) denote the generating function of X or F (x). The d.f. F (x) belongs to the subexponential class S (notation: F ∈ S) if it satisfies lim

In every variety of algebras Θ, we can consider its logic and its algebraic geometry. In previous papers, geometry in equational logic, i.e., equational geometry, has been studied. Here we describe an extension of this theory to first-order logic (FOL). The algebraic sets in this geometry are determined by arbitrary sets of FOL formulas. The principal motivation of such a generalization lies in the area of applications to knowledge science. In this paper, the FOL formulas are considered in the context of algebraic logic. For this purpose, we define special Halmos categories. These categories in algebraic geometry related to FOL play the same role as the category of free algebras Θ0 play in equational algebraic geometry. This paper consists of three parts. Section 1 is of introductory character. The first part (Secs. 2–4) contains background on algebraic logic in the given variety of algebras Θ. The second part is devoted to algebraic geometry related to FOL (Secs. 5–7). In the last part (Secs. 8–9), we consider applications of the previous material to knowledge science.

In this paper, we consider the generalized Hilbert symbol in a higher local field of charactersitic 0 with first residue field of characteristic 0 as well and with perfect last residue field of positive characteristic p, which originates from higher local p-class field theory developed by I. Fesenko. Using the descent to a subfield of mixed characteristic, from the generalized Artin-Hasse and Iwasawa formulas proved in a previous paper we deduce respective Artin-Hasse and Iwasawa explicit reciprocity laws in the case under consideration. Bibliography: 6 titles.

This paper is the first in a series of three, the aim of which is to lay the foundations of algebraic geometry over the free metabelian Lie algebra $F$. In the current paper we introduce the notion of a metabelian Lie $U$-algebra and establish connections between metabelian Lie $U$-algebras and special matrix Lie algebras. We define the $\Delta $-localisation of a metabelian Lie $U$-algebra $A$ and the direct module extension of the Fitting's radical of $A$ and show that these algebras lie in the universal closure of $A$., Comment: 34 pages

In one of his early papers, D. Grigoriev analyzed the decidability and computational complexity of different physical theories. This analysis was motivated by the hope that it would help physicists. In this paper, we survey several similar ideas that may be of help to physicists. We hope that further research may lead to useful physical applications. Bibliography: 41 titles.

A new method, which enables us to compute rather efficiently the Galois group of a polynomial over ℚ or over ℤ, is presented. Reductions of this polynomial with respect to different prime modules are studied, and the information obtained is used for the calculation of the Galois group of the initial polynomial. This method uses an original modification of the Chebotarev density theorem, and it is in essence a probabilistic method. The irreducibility of the polynomial under consideration is not assumed. The appendix to this paper contains tables, which enable us to find the Galois group of polynomials of degree less than or equal to 10 as a subgroup of the symmetric group. Here the final part of the paper is published. The first part is contained in a previous issue (see Vol. 134, No. 6 (2006)). Bibliography: 10 titles.

In a paper by Cook and Reckhow (1979), it is shown that any two classical Frege systems polynomially simulate each other. The same proof does not work for intuitionistic Frege systems, since they can have nonderivable admissible rules. (The rule A/B is derivable if the formula A → B is derivable. The rule A/B is admissible if for all substitutions σ, if σ(A) is derivable, then σ(B) is derivable.) In this paper, we polynomially simulate a single admissible rule. Therefore any two intuitionistic Frege systems polynomially simulate each other. Bibliography: 20 titles.

The paper may be viewed as an addendum to a paper of Thomason and Throbaugh, where the K-theory of algebraic varieties is equipped with relative K-groups. It is proved that this enriched K-theory satisfies the Panin—Smirnov axioms for ring cohomology theories of algebraic varieties. In particular, it is proved that the Leibniz formula, describing the interaction between multiplication and differential, holds in this case. The language of symmetric spectra and of monoidal model categories is used. Bibliography: 7 titles.

In the paper, formal groups over the rings of integers of σ-fields are studied. These fields were constructed by the first author in a previous paper. They are a generalization of the inertia field of a classical local field to an arbitrary complete discrete valuation field of characteristic zero. An analog of Honda’s theory for such formal groups is constructed. The arithmetic of the group of points in an extension of a σ-field that contains sufficiently many torsion points is studied. Using the classification of formal groups and the arithmetic results obtained, an explicit formula for the Hilbert pairing for formal groups over σ-fields is proved. Bibliography: 16 titles.

The paper deals with an equilibrium characterization of so-called totally contractual allocations. As a consequence of the characterization obtained, a rather strong coalitional stability of equilibrium allocations is established. In view of the complicated logical structure of contractual blocking, we pay strong attention to pure descriptive aspects of the concepts under consideration. Quite simple sufficient conditions guaranteeing the coincidence of the totally contractual core and the set of Walrasian equilibrium allocations are established, and the structure of the domination relations induced by several rules of breaking contracts is studied. The game-theoretic approach elaborated in the paper rests on the reduction of the original blocking to some simpler domination relations in cooperative games associated with the contractual blocking in question. Bibliography: 3 titles.

This paper gives a comprehensive treatment of EVPI-based sequential importance sampling algorithms for dynamic (multistage) stochastic programming problems. Both theory and computational algorithms are discussed. Under general assumptions it is shown that both an expected value of perfect information (EVPI) process and the corresponding marginal EVPI process (the supremum norm of the conditional expectation of its generalized derivative) are nonanticipative nonnegative supermartingales. These processes are used as importance criteria in the class of sampling algorithms treated in the paper. When their values are negligible at a node of the current sample problem scenario tree, scenarios descending from the node are replaced by a single scenario at the next iteration. On the other hand, high values lead to increasing the number of scenarios descending from the node. Both the small sample and asymptotic properties of the sample problem estimates arising from the algorithms are established, and the former are evaluated numerically in the context of a financial planning problem. Finally, current and future research is described. Bibliography: 49 titles.

The paper presents a survey of new results in general equilibrium theory with linear vector lattice commodity space (Kantorovich space). The importance of order structures and the Riesz-Kantorovich formula is clarified. The main novelty of the paper is new characterizations of elements of the fuzzy core in an exchange economy. Then we apply these characterizations to prove a new theorem on the existence of quasi-equilibrium for a linear vector lattice economy. This theorem, based on the E-properness of preferences by Podczeck-Florenzano-Marakulin, develops the Florenzano-Marakulin approach and generalizes previous Tourky's results. Bibliography: 29 titles.

In this paper, we shall show how to interpret (3) by means of the convolution integral of the Mellin transform, and how to use the tools of the Mellin-Barnes integrals to treat the subordination for the class of self-similar stochastic processes, which are governed by Levy strictly stable probability distributions. The paper is divided as follows. In Sec. 2, we provide the reader with the essential notions and notations concerning the Mellin transforms. In particular, we recall the fundamental property of the Mellin convolution to represent the p.d.f. of the product of two independent random variables and consequently to interpret the subordination formula involving a self-similar parent process. In Sec. 3, we first recall the Mellin-Barnes integral representation of the whole class of Levy (symmetric and not symmetric) strictly stable densities, as derived in previous papers (3, 5). Then, through the Mellin convolution, we derive for these densities a subordination formula involving, as directing p.d.f.'s, those of unilateral stable densities. In so doing, we clarify and generalize known results of Feller, proven via Fourier and Laplace transforms. Finally, in Sec. 4, we draw the main conclusions and outline the direction for future work.

In the present paper, the enumeration of a certain class of directed lattice paths is based on the analysis of dynamical correlation functions of the exactly solvable XX0 model. This model is the zero anisotropy limit of one of the basic models of the theory of integrable systems, the XXZ Heisenberg magnet. It is demonstrated that the considered correlation functions under different boundary conditions are the exponential generating functions of various types of paths, in particular, Dyck and Motzkin paths.

We present two-loop renormalization of $\phi^3$-model effective action by using the background field method and cutoff momentum regularization. In this paper, we also study a derivation of the quantum equation of motion and its application to the renormalization., Comment: LaTeX, 15 pages, 3 figures; The work has been published three years ago. In this version we have made some corrections and added calculations for the counterterm