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In this paper, by studying the maximal good subspaces, we determine the dual Lie coalgebras of the centerless twisted Heisenberg–Virasoro algebra. Based on this, we construct the dual Lie bialgebras structures of the twisted Heisenberg–Virasoro type. As by-products, four new infinite dimensional Lie algebras are obtained. [ABSTRACT FROM AUTHOR]

In [10], Sayed Ahmed recently has shown that there exists an infinite dimensional non-representable quasi-polyadic equality algebra (QPEAω, for short) with a representable cylindric reduct. In this paper we continue related investigations and show that if G⊆ωω is a semigroup containing at least one constant function, then a wide class of representable cylindric algebras occur as the cylindric reduct of some non-representable G-PEAω. More concretely, we prove that if A is an ω-dimensional cylindric set algebra with an infinite base set, then there exists a non-representable G-PEAω whose cylindric reduct is representable and contains an isomorphic copy of A. [ABSTRACT FROM PUBLISHER]

In this paper we investigate the structure of maps on classes of Hilbert space operators leaving the determinant of linear combinations invariant. Our main result is an infinite dimensional version of the famous theorem of Frobenius about determinant preserving linear maps on matrix algebras. In this theorem of ours, we use the notion of (Fredholm) determinant of bounded Hilbert space operators which differ from the identity by an element of the trace class. The other result of the paper describes the structure of those transformations on sets of positive semidefinite matrices which preserve the determinant of linear combinations with fixed coefficients. [ABSTRACT FROM PUBLISHER]

In this paper, a family of infinite dimensional Lie algebras L ˜ is introduced and investigated, called the extended Heisenberg-Virasoro algebra, denoted by L ˜. These Lie algebras are related to the N = 2 superconformal algebra and the Bershadsky-Polyakov algebra. We study restricted modules and associated vertex algebras of the Lie algebra L ˜. More precisely, we construct its associated vertex (operator) algebras V L ˜ (ℓ 123 , 0) , and show that the category of vertex algebra V L ˜ (ℓ 123 , 0) -modules is equivalent to the category of restricted L ˜ -modules of level ℓ 123. Then we give uniform constructions of simple restricted L ˜ -modules. Also, we present several equivalent characterizations of simple restricted modules over L ˜. [ABSTRACT FROM AUTHOR]

In this paper, we use the penalty approach in order to study constrained minimization problems in infinite dimensional spaces. A penalty function is said to have the exact penalty property if there is a penalty coefficient for which a solution of an unconstrained penalized problem is a solution of the corresponding constrained problem. In this paper, we establish the exact penalty property for a large class of inequality-constrained minimization problems. [ABSTRACT FROM AUTHOR]

This paper is devoted to explore the RICCI and MCs (Matter Collineations of the Som-Ray Chaudhary spacetime. The spacetime under consideration is one of the spatially homogeneous and rotating spacetimes. Collineations are the some kinds of the Lie symmetries. To discuss the required collineations we have used the RICCI and energy momentum tensors. As the RICCI tensor is formulated from the metric tensor, it must possess its symmetries. RCs (RICCI Collineations) leads to conservation laws. On the other hand for the distribution of matter in the spacetimes, the symmetries of energy momentum tensor or MCs provides conservation laws on matter field. Throughout this paper, these collineations are discussed by vanishing Lie derivative of RICCI and energy momentum tensors respectively. Complete solution of the RCs and MCs equations, which are formed in the result of vanishing Lie derivative are explored. Studying all these collineations in the said spacetime, it has been shown that RCs of the spacetime form an infinite dimensional vector space where as MCs are Killing vector fields. [ABSTRACT FROM AUTHOR]

This is a call for contributions to a special issue of Journal of Physics A: Mathematical and Theoreticaldedicated to the subject of the conference `IDAQUIS 2007'', 23-27 July 2007 (http://www.ualg.pt/idaquis/welcome.html). Invited speakers and participants at that meeting and other researchers working in the field are invited to submit a research paper to this issue.The Editorial Board has invited Andreas Fring, Petr Kulish, Nenad Manojlovic, Zoltan Nagy, Joana Nunes da Costa and Henning Samtleben to serve as Guest Editors for the special issue. Their criteria for acceptance of contributions are as follows:The subject of the paper should relate to the subject of the workshop (see the website of the conference http://www.ualg.pt/idaquis/welcome.html).Contributions will be refereed and processed according to the usual procedure of the journal.Conference papers may be based on already published work but should either contain significant additional new results and/or insights or give a survey of the present state of the art, a critical assessment of the present understanding of a topic, and a discussion of open problems.Papers submitted by non-participants should be original and contain substantial new results. The guidelines for the preparation of contributions are the following:The DEADLINE for submission of contributions is 31 October 2007. This deadline will allow the special issue to appear in June 2008.There is a nominal page limit of 30 printed pages per contribution for invited speakers and 10 printed pages for all other contributors. For papers exceeding this limit, the Guest Editors reserve the right to request a reduction in length. Further advice on publishing your work in Journal of Physics A: Mathematical and Theoreticalmay be found at www.iop.org/Journals/jphysa.Contributions to the special issue should, if possible, be submitted electronically by web upload at www.iop.org/Journals/jphysaor by e-mail to jphysa@iop.org, quoting `JPhysA Special Issue-IDAQUIS''. Submissions should ideally be in standard LaTeX form. Please see the website for further information on electronic submissions. All contributions should be accompanied by a read-me file or covering letter giving the postal and e-mail addresses for correspondence. The Publishing Office should be notified of any subsequent change of address.This special issue will be published in the paper and online version of the journal. Each participant at the workshop and the corresponding author of each contribution will receive a complimentary copy of the issue. [ABSTRACT FROM AUTHOR]

The present paper is devoted to 2-local derivations on infinitedimensional Lie algebras over a field of characteristic zero. We prove that all 2-local derivations on an infinite-dimensional simple Witt algebra are automatically derivations, and give an example of infinite-dimensional Lie algebra with a 2-local derivation which is not a derivation. [ABSTRACT FROM AUTHOR]

The super-Virasoro algebras, also known as the superconformal algebras, are nontrivial graded extensions of the Virasoro algebra to Lie superalgebra version. In this paper, we classify the compatible left-symmetric superalgebra structures on the N = 2 Ramond and Neveu-Schwarz superconformal algebras under certain conditions, which generalizes the corresponding results for the Witt, Virasoro and super Virasoro algebras. [ABSTRACT FROM AUTHOR]

The aim of this paper is to prove an existence theorem for the Nash equilibria of a noncooperative generalized game with infinite-dimensional strategy spaces. The main peculiarity of this result is the absence of upper semicontinuity assumptions on the constraint multifunctions. Our result is in the same spirit of the paper Cubiotti (J Game Theory 26: 267–273, 1997), where only the case of finite-dimensional strategy spaces was considered. [ABSTRACT FROM AUTHOR]

In the present paper, we prove that every 2-local inner derivation on the Lie ring of skew-symmetric matrices over a commutative ring is an inner derivation. We also apply our technique to various Lie algebras of infinite-dimensional skew-adjoint matrix-valued maps on a set and prove that every 2-local spatial derivation on such algebras is a spatial derivation. A similar technique is applied to the same Lie algebras and proved that every local spatial derivation on such algebras is a spatial derivation. [ABSTRACT FROM AUTHOR]

Copyright of Proceedings of the Estonian Academy of Sciences is the property of Teaduste Akadeemia Kirjastus and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In the present paper, we investigate the dual Lie coalgebras of the centerless W(2, 2) algebra by studying the maximal good subspaces. Based on this, we construct the dual Lie bialgebra structures of the centerless W(2, 2) Lie bialgebra. As by-products, four new infinite dimensional Lie algebras are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we construct two kinds of infinite-dimensional Novikov algebras and give their realizations, respectively. As an application, we describe their sub-adjacent Lie algebras and characterize some of their properties. [ABSTRACT FROM AUTHOR]

This is the third in a series of papers attempting to describe a uniform geometric framework in which many integrable systems can be placed. A soliton hierarchy can be constructed from a splitting of an infinite dimensional group L as positive and negative subgroups $${L_\pm}$$ and a commuting sequence in the Lie algebra $${\mathcal{L}_+}$$ of $${L_+}$$ . Given $${f\in L_-}$$ , there is a formal inverse scattering solution u of the hierarchy. When there is a 2 co-cycle on $${\mathcal{L}}$$ that vanishes on both $${\mathcal{L}_+ {\rm and} \mathcal{L}_-}$$ , Wilson constructed for each $${f\in L_-}$$ a tau function $${\tau_f}$$ for the hierarchy. In this third paper, we prove the following results for the n × n KdV hierarchy: [ABSTRACT FROM AUTHOR]

Little seems to be known about the ergodic dynamics of stochastic systems with fractional noise. This paper is devoted to discern such long time dynamics through the stochastic Lorenz chaotic system (SLCS) with long memory effects. By a truncation technique, the SLCS is proved to generate a continuous stochastic dynamical system Λ. Based on the Krylov-Bogoliubov criterion, the required Lyapunov function is further established to ensure the existence of the invariant measure of Λ. Meanwhile, the uniqueness of the invariant measure of Λ is proved by examining the strong Feller property, together with an irreducibility argument. Therefore, the SLCS has exactly one adapted stationary solution. [ABSTRACT FROM AUTHOR]

This paper studies the existence of pure-strategy Nash equilibria for nonatomic games where players take actions in infinite-dimensional Banach spaces. For any infinite-dimensional Banach space, if the player space is modeled by the Lebesgue unit interval, we construct a nonatomic game which has no pure-strategy Nash equilibrium. But if the player space is modeled by a saturated probability space, there is a pure-strategy Nash equilibrium in every nonatomic game. Finally, if every game with a fixed nonatomic player space and a fixed infinite-dimensional action space has a pure-strategy Nash equilibrium, the underlying player space must be saturated. [ABSTRACT FROM AUTHOR]

We consider the problem of boundary feedback stabilization of a vibrating string that is fixed at one end and with control action at the other end. In contrast to previous studies that have required L²-regularity for the initial position and H-1-regularity for the initial velocity, in this paper we allow for initial positions with L¹-regularity and initial velocities in W-1,1 on the space interval. It is well known that for a certain feedback parameter, for sufficiently regular initial states the classical energy of the closed-loop system with Neumann velocity feedback is controlled to zero after a finite time that is equal to the minimal time where exact controllability holds. In this paper, we present a Dirichlet boundary feedback that yields a well-defined closed-loop system in the (L¹, W-1,1) framework and also has this property. Moreover, for all positive feedback parameters our feedback law leads to exponential decay of a suitably defined L¹-energy. For more regular initial states with (L², H-1) regularity, the proposed feedback law leads to exponential decay of an energy that corresponds to this framework. If the initial states are even more regular with H¹-regularity of the initial position and L²-regularity of the initial velocity, our feedback law also leads to exponential decay of the classical energy. [ABSTRACT FROM AUTHOR]

This paper is concerned with necessary and sufficient second-order conditions for finite-dimensional and infinite-dimensional constrained optimization problems. Using a suitably defined directional curvature functional for the admissible set, we derive no-gap second-order optimality conditions in an abstract functional analytic setting. Our theory not only covers those cases where the classical assumptions of polyhedricity or second-order regularity are satisfied but also allows to study problems in the absence of these requirements. As a tangible example, we consider no-gap second-order conditions for bang-bang optimal control problems. [ABSTRACT FROM AUTHOR]

We revisit the computation of the phase of the Dirac fermion scattering operator in external gauge fields. The computation is through a parallel transport along the path of time evolution operators. The novelty of the present paper compared with the earlier geometric approach by Langmann and Mickelsson (J Math Phys 37(8):3933-3953, ) is that we can avoid the somewhat arbitrary choice in the regularization of the time evolution for intermediate times using a natural choice of the connection form on the space of appropriate unitary operators. [ABSTRACT FROM AUTHOR]

This paper is about solving the output regulation problem for a class of infinite dimensional systems with an unknown exosystem. First, under the unit relative degree assumption, the infinite dimensional system is decomposed into a cascaded one composed of a one-dimensional system and an infinite dimensional system. Then, the problem is solved by combining an adaptive internal model and an adaptive control. The presented results are illustrated via a periodic tracking problem of a heat equation. [ABSTRACT FROM AUTHOR]

This paper investigates coalitional fairness in pure exchange economies with asymmetric information. We study allocations of resources which are immune from envy when comparisons take place between coalitions. The model allows negligible and non-negligible traders, partially informed about the true state of nature at the time of consumption, to exchange any number, possibly infinite, of commodities. Our analysis is based on the Aubin approach to coalitions and cooperation. Namely, on a notion of cooperation allowing traders to take part in one or more coalitions simultaneously, employing only shares of their endowments (participation rates). We introduce and study in detail the notion of coalition fairness with participation rates (or Aubin c-fairness) and show that flexibility in cooperation permits to recover the failure of fairness properties of equilibrium allocations. Our results provide applications to several market outcomes (ex-post core, fine core, ex-post competitive equilibria, rational expectations equilibria) and emphasize the consequences of the convexification effect due to participation rates for models with large traders and infinitely many commodities. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate a constrained optimal control problem governed by a class of semilinear infinite dimensional systems. For a state-quadratic cost functional and a closed convex set of admissible controls, the existence of an optimal control is proven, then this control is characterized for several cases of constraints. An algorithm is developed in order to compute the optimal control. The results are illustrated through simulations of a transport equation and a wave equation. [ABSTRACT FROM AUTHOR]

We study the structure of isotropic ideals of the metric n-Lie algebra (ᴤ, B) over the complex field ℂ for n ≥ 2. Based on the study of isotropic ideals of metric n-Lie algebras in this paper, we obtain the following. (i) Let ᴤ = 픰 ⊕ 픯 be the Levi decomposition and g without semi-simple ideals. If n > 2, ᴤ is perfect and 픯 is isotropic, then 픯 is the unique isomaximal ideal and dim ᴤ = 2m(n + 1), and the metric dimension of ᴤ is m. If 픯 is indecomposable, then 픯 is isomorphic to 픰 as an 픰-module in the regular representation. (ii) If dim ᴤ = n + k with 2 ≼ k ≼ n + 1, then dim ᴤ¹ n + k - 1. If (ᴤ is unsolvable, then (ᴤ is reductive. If the center C (픯) of ᴤ is isotropic, then dim C (픯) = k - 1. (iii) There are only three classes of (n + k)-dimensional metric n-Lie algebras for 2 ≼ k ≼ n + 1 and n > 2, and the concrete multiplications for each class are provided. [ABSTRACT FROM AUTHOR]

This paper is devoted to the biological equilibrium of a stochastic nonlocal PDE population model with state-selective delay. By an improved dissipativity method and a delicate analysis of interaction of stochastic terms and delay terms, we obtain a unique equilibrium, which mixes exponentially. In order to check the validity of the model, we also investigate its stochastic stability. [ABSTRACT FROM AUTHOR]

Multidimensional mechanism design problems have proven difficult to solve by extending techniques from the one-dimensional case. This paper considers mechanism design problems with multidimensional types when the seller's cost function is not separable across buyers. By adapting results obtained by Border [Border, K. 1991. Implementation of reduced form auctions: A geometric approach. Econometrica 59 1175-1187], we transform the seller's problem into a representation that only involves "interim" variables and eliminates the dimensionality dependence on the number of buyers. We show that the associated infinite-dimensional optimization problem posed by the theoretical model can be approximated arbitrarily well by a sequence of finite-dimensional linear programming problems. We provide an efficient--i.e., terminating in polynomial time in the problem size--method to compute the separation oracle associated with the Border constraints and incentive compatibility constraints. This implies that our finite-dimensional approximation is solvable in polynomial time. Finally, we illustrate how the numerical solutions of the finite-dimensional approximations can provide insights into the nature of optimal solutions to the infinite-dimensional problem in particular cases. [ABSTRACT FROM AUTHOR]

In this paper, on a bounded domain $${\Omega\subset {\bf R}^n}$$, we consider a non-local problem of the type Under rather general assumptions on K and f, we prove, in particular, that there exists λ* > 0 such that, for each λ > λ* and each Carathéodory function g with a sub-critical growth, the above problem has at least three weak solutions for every μ ≥ 0 small enough. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to show that ideas of deformation theory can be applied to 'infinite-dimensional geometry'. We develop the deformation theory of Brody curves. A Brody curve is a kind of holomorphic map from the complex plane to the projective space. Since the complex plane is not compact, the parameter space of the deformation can be infinite-dimensional. As an application we prove a lower bound on the mean dimension of the space of Brody curves. [ABSTRACT FROM AUTHOR]

In this paper, we study algebras of Cartan type &Ssline;*, over a field F of characteristic 0, which is a class of infinite dimensional Lie algebras. These algebras are sets of sums of the images of D[SUPh,SUBij] where D[SUPh,SUBij](f) = h{∂[SUBj](f)∂[SUBi] - ∂[SUBi](f)∂[SUBj]} for all f and h in the field F(x[SUB1] ..... x[SUBn],), and occur as subalgebras of the Lie algebra of derivations of the field of rational fuctions. We find the unique minimal ideal of algebras of Cartan type &Ssline;. We also investigate the size of the ideal for some special cases. [ABSTRACT FROM AUTHOR]

In a recent paper [5] S. Solecki proved a finite self-dual Ramsey theorem that extends simultaneously the classical finite Ramsey theorem [4] and the Graham–Rothschild theorem [2] . In this paper we prove the corresponding infinite dimensional version of the self-dual theorem. As a consequence, we extend the classical infinite Ramsey theorem [4] and the Carlson–Simpson theorem [1] . [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate the moving window rule of classification to classify functions under mixing conditions. We consider a random variable X taking values in a metric space (F, ρ) with label Y ∊ {0, 1}. We extend some results on consistency and strong consistency of the moving window rule from the i.i.d. case to the weakly dependent case under mild assumptions. The practical use of the moving window rule will be illustrated through a simulation study. The performance of the moving window rule is investigated. [ABSTRACT FROM AUTHOR]

Let G be a complex simple direct limit group, specifically $$SL(\infty ;\mathbb {C})$$ , $$SO(\infty ;\mathbb {C})$$ or $$Sp(\infty ;\mathbb {C})$$ . Let $$\mathcal {F}$$ be a (generalized) flag in $$\mathbb {C}^\infty $$ . If G is $$SO(\infty ;\mathbb {C})$$ or $$Sp(\infty ;\mathbb {C})$$ we suppose further that $$\mathcal {F}$$ is isotropic. Let $$\mathcal {Z}$$ denote the corresponding flag manifold; thus $$\mathcal {Z}= G/Q$$ where Q is a parabolic subgroup of G. In a recent paper (Penkov and Wolf in Real group orbits on flag ind-varieties of $$SL_{\infty } ({\mathbb {C}})$$ , to appear in Proceedings in Mathematics and Statistics) we studied real forms $$G_0$$ of G and properties of their orbits on $$\mathcal {Z}$$ . Here we concentrate on open $$G_0$$ -orbits $$D \subset \mathcal {Z}$$ . When $$G_0$$ is of hermitian type we work out the complete $$G_0$$ -orbit structure of flag manifolds dual to the bounded symmetric domain for $$G_0$$ . Then we develop the structure of the corresponding cycle spaces $$\mathcal {M}_D$$ . Finally we study the real and quaternionic analogs of these theories. All this extends results from the finite-dimensional cases on the structure of hermitian symmetric spaces and cycle spaces (in chronological order: Wolf in Bull Am Math Soc 75:1121-1237, 1969; Wolf et al. in Ann Math 105:397-448, 1977; Wolf in Ann Math 136:541-555, 1992; Wolf in Compact subvarieties in flag domains, 1994; Wolf and Zierau in Math Ann 316:529-545, 2000; Huckleberry et al. in Journal für die reine und angewandte Mathematik 2001:171-208, 2001; Huckleberry and Wolf in Cycle spaces of real forms of $$SL_n(C)$$ , Springer, New York, pp 111-133, 2002; Wolf and Zierau in J Lie Theory 13:189-191, 2003; Huckleberry and Wolf in Ann Scuola Norm Sup Pisa Cl Sci (5) 9:573-580, 2010). [ABSTRACT FROM AUTHOR]

The present paper is devoted to infinite order decompositions of C*-algebras. It is proved that an infinite order decomposition (IOD) of a C*-algebra forms the complexification of an order unit space, and, if the C*-algebra is monotone complete (not necessarily weakly closed) then its IOD is also monotone complete ordered vector space. Also it is established that an IOD of a C*-algebra is a C*-algebra if and only if this C*-algebra is a von Neumann algebra. As a summary we obtain that the norm of an infinite dimensional matrix is equal to the supremum of norms of all finite dimensional main diagonal submatrices of this matrix and an infinite dimensional matrix is positive if and only if all finite dimensional main diagonal submatrices of this matrix are positive. [ABSTRACT FROM AUTHOR]

Given a dynamical system, we say that a performance function has property P if its time averages along orbits are maximized at a periodic orbit. It is conjectured by several authors that for sufficiently hyperbolic dynamical systems, property P should be typical among sufficiently regular performance functions. In this paper we address this problem using a probabilistic notion of typicality that is suitable to infinite dimension: the concept of prevalence as introduced by Hunt, Sauer, and Yorke. For the one-sided shift on two symbols, we prove that property P is prevalent in spaces of functions with a strong modulus of regularity. Our proof uses Haar wavelets to approximate the ergodic optimization problem by a finite-dimensional one, which can be conveniently restated as a maximum cycle mean problem on a de Bruijin graph. [ABSTRACT FROM AUTHOR]

Motivated by the obstruction to the deformation quantization of Poisson structures in infinite dimensions, we introduce the notion of a quantizable odd Lie bialgebra. The main result of the paper is a construction of the highly non-trivial minimal resolution of the properad governing such Lie bialgebras, and its link with the theory of so-called quantizable Poisson structures. [ABSTRACT FROM AUTHOR]

Classical invariants for representations of one Lie group can often be related to invariants of some other Lie group. Physics suggests that the right objects to consider for these questions are certain refinements of classical invariants known as conformal blocks. Conformal blocks appear in algebraic geometry as spaces of global sections of line bundles on moduli stacks of parabolic bundles on a smooth curve. Rank-level duality connects a conformal block associated to one Lie algebra to a conformal block for a different Lie algebra. In this paper, we prove a rank-level duality for so(2r + 1) conformal blocks on the pointed projective line which was suggested by T. Nakanishi and A. Tsuchiya. [ABSTRACT FROM AUTHOR]

Let H be a Hilbert space with dim H ≥ 2 and Z ∈ B( H) be an arbitrary but fixed operator. In this paper we show that an additive map Φ: B( H) → B( H) satisfies Φ( AB) = Φ( A) B = AΦ( B) for any A, B ∈ B( H) with AB = Z if and only if Φ( AB) = Φ( A) B = AΦ( B), ∀ A, B ∈ B( H), that is, Φ is a centralizer. Similar results are obtained for Hilbert space nest algebras. In addition, we show that Φ( A ) = AΦ( A) = Φ( A) A for any A ∈ B( H) with A = 0 if and only if Φ( A) = AΦ( I) = Φ( I) A, ∀ A ∈ B( H), and generalize main results in Linear Algebra and its Application, 450, 243-249 (2014) to infinite dimensional case. New equivalent characterization of centralizers on B( H) is obtained. [ABSTRACT FROM AUTHOR]

Recently, we developed a method for the study of holonomy properties of non- Riemannian Finsler manifolds and obtained that the holonomy group cannot be a compact Lie group if the Finsler manifold of dimension > 2 has non-zero constant flag curvature. The purpose of this paper is to move further, exploring the holonomy properties of projectively flat Finsler manifolds of non-zero constant flag curvature. We prove in particular that projectively flat Randers and Bryant-Shen manifolds of non-zero constant flag curvature have infinite dimensional holonomy group. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new extragradient method with regularization for finding a common element of the solution set Γ of the split feasibility problem and the set Fix(S) of fixed points of a nonexpansive mapping S in infinitedimensional Hilbert spaces, combining the regularization method and the technique of averaged operator, we prove the sequences generated by the proposed algorithm converge weakly to an element of Fix(S) ∩ Γ under mild conditions. [ABSTRACT FROM AUTHOR]

In this paper, we propose a full characterization of a generalized q-deformed Tamm-Dancoff oscillator algebra and investigate its main mathematical and physical properties. Specifically, we study its various representations and find the condition satisfied by the deformed q-number to define the algebra structure function. Particular Fock spaces involving finite and infinite dimensions are examined. A deformed calculus is performed as well as a coordinate realization for this algebra. A relevant example is exhibited. Associated coherent states are constructed. Finally, some thermodynamics aspects are computed and discussed. [ABSTRACT FROM AUTHOR]

The concept of data depth leads to a center-outward ordering of multivariate data, and it has been effectively used for developing various data analytic tools. While different notions of depth were originally developed for finite dimensional data, there have been some recent attempts to develop depth functions for data in infinite dimensional spaces. In this paper, we consider some notions of depth in infinite dimensional spaces and study their properties under various stochastic models. Our analysis shows that some of the depth functions available in the literature have degenerate behaviour for some commonly used probability distributions in infinite dimensional spaces of sequences and functions. As a consequence, they are not very useful for the analysis of data satisfying such infinite dimensional probability models. However, some modified versions of those depth functions as well as an infinite dimensional extension of the spatial depth do not suffer from such degeneracy and can be conveniently used for analyzing infinite dimensional data. [ABSTRACT FROM AUTHOR]

In this paper, we start from an extension of the notion of holonomy on diffeological bundles, reformulate the notion of regular Lie group or Frolicher Lie groups, state an Ambrose--Singer theorem that enlarges the one stated in [J.-P. Magnot, Structure groups and holonomy in infinite dimensions, Bull. Sci. Math. 128 (2004) 513-529], and conclude with a differential geometric treatment of KP hierarchy. The examples of Lie groups that are studied are principally those obtained by enlarging some graded Frolicher (Lie) algebras such as formal g-series of the quantum algebra of pseudo-differential operators. These deformations can be denned for classical pseudo-differential operators but they are used here on formal pseudo-differential operators in order to get a differential geometric framework to deal with the KP hierarchy that is known to be completely integrable with formal power series. Here, we get an integration of the Zakharov-Shabat connection form by means of smooth sections of a (differential geometric) bundle with structure group, some groups of q-deformed operators. The integration obtained by Mulase [Complete integrability of the Kadomtsev-Petviashvili equation Adv. Math. 54 (1984) 57-66], and the key tools he developed, are totally recovered on the germs of the smooth maps of our construction. The tool coming from (classical) differential geometry used in this construction is the holonomy group, on which we have an Ambrose-Singer-like theorem: the Lie algebra is spanned by the curvature elements. This result is proved for any connection a diffeological principal bundle with structure group a regular Frolicher Lie group. The case of a (classical) Lie group modeled on a complete locally convex topological vector space is also recovered and the work developed in [J.-P. Magnot, Diffeologie du fibre d'Holonomie en dimension infinie, Math. Rep. Canadian Roy. Math. Soc. 28(4) (2006); J.-P. Magnot, Structure groups and holonomy in infinite dimensions, Bull. Sci. Math. 128 (2004) 513-529] is completed. [ABSTRACT FROM AUTHOR]

This paper concerns a study of three families of noncompact type symmetric spaces of infinite dimension. Although they have infinite dimension they have finite rank. More precisely, we show that they have finite telescopic dimension. We also show the existence of Furstenberg maps for some group actions on these spaces. Such maps appear as a first step toward superrigidity results. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present an infinite dimensional associative diagram algebra that satisfies the relations of the generalized Temperley–Lieb algebra having a basis indexed by the fully commutative elements (in the sense of Stembridge) of the Coxeter group of type affine . Moreover, we provide an explicit description of a basis for the diagram algebra. In the sequel to this paper, we show that this diagrammatic representation is faithful. The results of this paper and its sequel will be used to construct a Jones-type trace on the Hecke algebra of type affine , allowing us to non-recursively compute leading coefficients of certain Kazhdan–Lusztig polynomials. [Copyright &y& Elsevier]

In this paper we prove that the profinite completion $\mathcal{\widehat G}$ of the Grigorchuk group $\mathcal{G}$ is not finitely presented as a profinite group. We obtain this result by showing that $H^2(\mathcal{\widehat G}, {\mathbb F}_2)$ is infinite dimensional. Also several results are proven about the finite quotients $\mathcal{G}/St_{\mathcal{G}}(n)$ including minimal presentations and Schur Multipliers. [ABSTRACT FROM AUTHOR]

In this paper, we develop an infinite dimensional Lagrangian duality framework for modeling and analyzing the evolutionary pollution control problem. Specifically, we examine the situation in which different countries aim at determining the optimal investment allocation in environmental projects and the tolerable pollutant emissions, so as to maximize their welfare. We state the equilibrium conditions underlying the model, and provide an equivalent formulation in terms of an evolutionary variational inequality. Moreover, by means of infinite dimensional duality tools, we prove the existence of Lagrange multipliers that play a fundamental role in order to describe countries' decision-making processes. [ABSTRACT FROM AUTHOR]