On Finiteness Conditions in Twisted K-Theory

In a previous paper [14], we have introduced the gauge-equivariant K-theory group $$ K_\mathcal{G}^0 (X)$$ of a bundle πX : X → B endowed with a continuous action of a bundle of compact Lie groups $$ p:\mathcal{G} \to B$$ . These groups are the natural range for the analytic index of a family of gauge-invariant elliptic operators (i.e., a family of elliptic operators invariant with respect to the action of a bundle of compact groups). In this paper, we continue our study of gauge-equivariant K-theory. In particular, we introduce and study products, which helps us establish the Thom isomorphism in gauge-equivariant K-theory. Then we construct push-forward maps and define the topological index of a gauge-invariant family.

This paper explores further the connection between Langlands duality and T-duality for compact simple Lie groups, which appeared in work of Daenzer-Van Erp and Bunke-Nikolaus. We show that Langlands duality gives rise to isomorphisms of twisted K-groups, but that these K-groups are trivial except in the simplest case of SU(2) and SO(3). Along the way we compute explicitly the map on $H^3$ induced by a covering of compact simple Lie groups, which is either 1 or 2 depending in a complicated way on the type of the groups involved. We also give a new method for computing twisted K-theory using the Segal spectral sequence, giving simpler computations of certain twisted K-theory groups of compact Lie groups relevant for D-brane charges in WZW theories and rank-level dualities. Finally we study a duality for orientifolds based on complex Lie groups with an involution.

K-theoretical boundary rings in [formula omitted] coset models

The conditions for sequences $$\{f_{k}\}_{k=1}^{\infty }$$ and $$\{g_{k}\}_{k=1}^{\infty }$$ being Bessel sequences, frames or Riesz bases, can be expressed in terms of the so-called cross-Gram matrix. In this paper, we investigate the cross-Gram operator G, associated to the sequence $$\{\langle f_{k}, g_{j}\rangle \}_{j, k=1}^{\infty }$$ and sufficient and necessary conditions for boundedness, invertibility, compactness and positivity of this operator are determined depending on the associated sequences. We show that invertibility of G is not possible when the associated sequences are frames but not Riesz Bases or at most one of them is Riesz basis. In the special case, we prove that G is a positive operator when $$\{g_{k}\}_{k=1}^{\infty }$$ is the canonical dual of $$\{f_{k}\}_{k=1}^{\infty }$$ .

In this paper we present a model to calculate the stringy product on twisted orbifold K-theory of Adem-Ruan-Zhang for abelian complex orbifolds. In the first part we consider the non-twisted case on an orbifold presented as the quotient of a manifold acted by a compact abelian Lie group. We give an explicit description of the obstruction bundle, we explain the relation with the product defined by Jarvis-Kaufmann-Kimura and, via a Chern character map, with the Chen-Ruan cohomology, we explicitly calculate the stringy product for a weighted projective orbifold. In the second part we consider orbifolds presented as the quotient of a manifold acted by a finite abelian group and twistings coming from the group cohomology. We show a decomposition formula for twisted orbifold K-theory that is suited to calculate the stringy product and we use this formula to calculate two examples when the group is (ℤ/2) 3 .

The problem of memoryless robust stabilization is discussed for a class of nonlinear uncertain systems with time-varying delay. The time-varying delay is assumed to unknown but known bound. The uncertainties are dependent on the delay states and have known bounds. No matching and Lipschitz's boundedness conditions are required. The continuous state feedback controller is proposed based on the Lyapunov direct methods such that the trajectories of the closed-loop nonlinear uncertain time-delay systems are uniformly and ultimately bounded on a compact set of which the diameter can be made arbitrarily large by choosing design parameters. The authors show that several previous results can be considered as a special cases. The effectiveness of the proposed approaches are demonstrated by the simulation results.

We investigate fibrations of the form $Z \to Y \to X$, where two of the three spaces are classifying spaces of compact connected Lie groups. We obtain certain finiteness conditions on the third space which make it also a classifying space. Our results allow to express some of the basic notions in group theory in terms of homotopy theory, i.e., in terms of classifying spaces. As an application we prove that every retract of the classifying space of a compact connected Lie group is again a classifying space.

Concentration inequalities are indispensable tools for studying the generalization capacity of learning models. Hoeffding's and McDiarmid's inequalities are commonly used, giving bounds independent of the data distribution. Although this makes them widely applicable, a drawback is that the bounds can be too loose in some specific cases. Although efforts have been devoted to improving the bounds, we find that the bounds can be further tightened in some distribution-dependent scenarios and conditions for the inequalities can be relaxed. In particular, we propose four types of conditions for probabilistic boundedness and bounded differences, and derive several distribution-dependent extensions of Hoeffding's and McDiarmid's inequalities. These extensions provide bounds for functions not satisfying the conditions of the existing inequalities, and in some special cases, tighter bounds. Furthermore, we obtain generalization bounds for unbounded and hierarchy-bounded loss functions. Finally we discuss the potential applications of our extensions to learning theory.

We consider the ferromagnetic q-state Potts model, with each of the q spin values coupled to an external field. We also introduce a generalized random cluster model, which includes both the Potts model in arbitrary homogeneous external fields and the noninteger q random cluster model as special cases. We establish the FKG property, the finite energy condition, uniqueness of the infinite cluster, and Gibbsianness of limit states for this generalized model. Furthermore, we develop the theory of Gibbs states for the Edwards–Sokal representation of the Potts model in a field, and relate the phase structure in this representation to those in the spin and random cluster representations. Finally, we characterize the possible color(s) of the infinite cluster(s) and show that the correspondence between Edwards–Sokal Gibbs states and their random cluster marginals is bijective, once the color of the infinite cluster is fixed.

We are concerned with dynamical properties of a model of emergence of resistance of cancer cells to chemotherapy. as understood based on recent progress in molecular biology. In some special cases of this model, their asymptotic behavior and the stability problem for the infinite dimensional case were studied. In the case of finite initial condition the stability conditions were derived by asymptotical analysis of the analytical solution to the system of equations. In the case of initial condition with infinite number of elements the stability verification was based on the spectral properties of the infinitesimal generator of the system.

A symplectic Lie group is a Lie group with a left-invariant symplectic form. Its Lie algebra structure is that of a quasi-Frobenius Lie algebra. In this note, we identify the groupoid analogue of a symplectic Lie group. We call the aforementioned structure a $t$-symplectic Lie groupoid; the “$t$" is motivated by the fact that each target fiber of a $t$-symplectic Lie groupoid is a symplectic manifold. For a Lie groupoid $\mathcal{G}\rightrightarrows M$, we show that there is a one-to-one correspondence between quasi-Frobenius Lie algebroid structures on $A\mathcal{G}$ (the associated Lie algebroid) and $t$-symplectic Lie groupoid structures on $\mathcal{G}\rightrightarrows M$. In addition, we also introduce the notion of a symplectic Lie group bundle (SLGB) which is a special case of both a $t$-symplectic Lie groupoid and a Lie group bundle. The basic properties of SLGBs are explored.

Partial differential operators, and in particular elliptic operators, on a Riemannian manifold provide a rich source of problems of non-commutative analysis. In the simplest situation the manifold is a Lie group and one can examine elliptic operators constructed from a vector space basis of the Lie algebra or subelliptic operators constructed from a Lie algebraic basis. In the latter case the choice of algebraic basis affects the characteristic geometric features of the subelliptic operators and their description involves an interesting interplay between analysis and geometry.

Elliptic operators arise naturally in several different mathematical settings, notably in the representation theory of Lie groups, the study of evolution equations, and the examination of Riemannian manifolds. This book develops the basic theory of elliptic operators on Lie groups and thereby extends the conventional theory of parabolic evolution equations to a natural non-commutative context. In order to achieve this goal, the author presents a synthesis of ideas from partial differential equations, harmonic analysis, functional analysis, and the theory of Lie groups. He begins by discussing the abstract theory of general operators with complex coefficients before concentrating on the central case of second-order operators with real coefficients. A full discussion of second-order subellilptic operators is also given. Prerequisites are a familiarity with basic semigroup theory, the elementary theory of Lie groups, and a firm grounding in functional analysis as might be gained from the first year of a graduate course.

Let M be a closed oriented smooth manifold, G a compact Lie group consisting of diffeomorphisms of M , P → Z M,P \to Z a principal G-bundle with a connection and D a G-equivariant elliptic operator. Then a locally constant family of elliptic operators and its determinant line bundle over Z are naturally defined by D. Moreover the holonomy of the determinant line bundle is defined by the connection in P. In this note, we give an explicit formula to calculate the holonomy (Theorem 1.4) and give a proof of the Witten holonomy formula (Theorem 1.7) in the special case above.

The geometric realization of the irreducible square integrable representations for semisimple Lie groups (cf. [3], [6] ) and also for nilpotent Lie groups [5] suggests that, as a general phenomenon, such representations should appear as L -kernels of invariant elliptic operators. One basic problem in this respect is to decide when such a kernel is nonzero. In the compact case the basic tool for this, used in the Borel-Weil-Bott approach, is the Hirzebruch-Riemann-Roch theorem. In the noncompact case one needs an analogue of the index theorem of Atiyah-Singer [2] for noncompact manifolds. When G possesses a discrete cocompact subgroup, the L-index theorem for covering spaces of [1] and [7] provides the required analogue. Our purpose here is to give a general index theorem for homogeneous spaces of arbitrary connected unimodular Lie groups, essentially based on the index theorem for foliations [4]. So let G be a connected unimodular Lie group, and let H be a closed subgroup of G which contains the center Z of G and such that H/Z is compact. Let x be a character of Z, and let E, F be finite-dimensional unitary representations of H whose restrictions to Z are given by xDenote by E, F the corresponding (invariant) induced bundles on the homogeneous space M = G/H, and let D be an invariant elliptic differential operator from E to F. The representation of G in the kernel of D in L(M, E) is square integrable modulo the center of G (see [4]), though not necessarily irreducible. Its formal degree deg(Ker D) (as defined in [4]) is always finite, so that the analytical index of D can be defined as