Tangential extremal principles for finite and infinite systems of sets, I: basic theory

In this paper we develop new extremal principles in variational analysis that deal with finite and infinite systems of convex and nonconvex sets. The results obtained, unified under the name of tangential extremal principles, combine primal and dual approaches to the study of variational systems being in fact first extremal principles applied to infinite systems of sets. The first part of the paper concerns the basic theory of tangential extremal principles while the second part presents applications to problems of semi-infinite programming and multiobjective optimization.     

Extremal infinite graph theory

We survey various aspects of infinite extremal graph theory and prove several new results. The lead role play the parameters connectivity and degree. This includes the end degree. Many open problems are suggested.     

An infinite element theory

A technique for solution of plane problems in mathematical physics or mechanics is presented. The method was developed initially for elastic plate problems with transverse load concentrations but is here extended to other two-dimensional problems. Draw-down problems with a number of wells in a porous medium, heat-flow problems in a plate with a distributed source, and with discrete sinks, or membrane displacements under concentrated forces, are considered in particular.The technique is to treat the two-dimensional medium as infinite: each source or sink is treated separately and equilibrated by an auxiliary source function. Superposition allows the effects of the separate sources and sinks to be added. The residual effect of the auxiliary source, after superposition, is negligible so that the summed effects of the individual sources and sinks gives the required solution. The method is approximate but the error may be made arbitrarily small. It is simple and well suited to the class of problems above, and to either manual or computer based analyses. Examples of its use are given.     

Operator renewal theory for continuous time dynamical systems with finite and infinite measure

We develop operator renewal theory for flows and apply this to obtain results on mixing and rates of mixing for a large class of finite and infinite measure semiflows. Examples of systems covered by our results include suspensions over parabolic rational maps of the complex plane, and nonuniformly expanding semiflows with indifferent periodic orbits. In the finite measure case, the emphasis is on obtaining sharp rates of decorrelations, extending results of Gouëzel and Sarig from the discrete time setting to continuous time. In the infinite measure case, the primary question is to prove results on mixing itself, extending our results in the discrete time setting. In some cases, we obtain also higher order asymptotics and rates of mixing.     

Multiplicative infinite loop space theory

Let G be a finitely presented group given by its pre-abelian presentation <X1,…,Xm; Xe11ζ1,…,Xemmζ,ζm+1,…>, where ei≥0 for i = 1,…, m and ζj?G′ for j≥1. Let N be the subgroup of G generated by the normal subgroups [xeii, G] for i = 1,…, m. Then Dn+2(G)≡γn+2(G) (modNG′) for all n≥0, where G” is the second commutator subgroup of G,γn+2(G) is the (n+2)th term of the lower central series of G and Dn+2(G) = G∩(1+△n+2(G)) is the (n+2)th dimension subgroup of G.     

Clifford theory for infinite dimensional modules

Clifford theory of possibly infinite dimensional modules is studied.     

Dissipative sequences in infinite ergodic theory

Let T be an invertible, ergodic, measure-preserving transformation on a nonatomic, infinite, σ-finite measure space . A set is weakly wandering under the sequence of integers W={n_i}_i=1^∞ if the sets T^n_iA are disjoint. If, for a given sequence W, there exists a weakly wandering set of positive measure, then we call W a weakly wandering sequence. A sequence {k_i}_i=1^∞ is dissipative if, for every function , we have , almost everywhere. Every weakly wandering sequence is dissipative. In 1978, Hajian and Ito constructed a dissipative sequence which is not weakly wandering. An example of a dissipative sequence which is not a finite union of weakly wandering sequences is presented. Also, some further properties of dissipative sequences are studied.     

The infinite time horizon dynamic pricing problem with multi-unit demand

This paper studies a dynamic pricing problem for a monopolist selling multiple identical items to potential buyers arriving over time, where the time horizon is infinite, the goods are imperishable and the buyers’ arrival follows a renewal process. Each potential buyer has some private information about his purchasing will, and this private information is unknown to the seller and therefore characterized as a random variable in this paper. Thus, the buyers may have multi-unit demand. Meanwhile, the seller needs to determine the optimal posted price such that his expected discounted revenue is maximized. This problem is formulated as a stochastic dynamic programming in this paper and then how to obtain the solution is explored. A numerical study shows that the optimal posted price performs better than that of optimal fixed price, and this advantage becomes obvious as the interest rate and/or the number of initial items increases.     

Approximate solutions to infinite dimensional LQ problems over infinite time horizon

This paper is addressed to develop an approximate method to solve a class of infinite dimensional LQ optimal regulator problems over infinite time horizon. Our algorithm is based on a construction of approximate solutions which solve some finite dimensional LQ optimal regulator problems over finite time horizon, and it is shown that these approximate solutions converge strongly to the desired solution in the double limit sense.     

Ordinal machines and admissible recursion theory

We generalize standard Turing machines, which work in time ω on a tape of length ω, to α-machines with time α and tape length α, for α some limit ordinal. We show that this provides a simple machine model adequate for classical admissible recursion theory as developed by G. Sacks and his school. For α an admissible ordinal, the basic notions of α-recursive or α-recursively enumerable are equivalent to being computable or computably enumerable by an α-machine, respectively. We emphasize the algorithmic approach to admissible recursion theory by indicating how the proof of the Sacks–Simpson theorem, i.e., the solution of Post’s problem in α-recursion theory, could be based on α-machines, without involving constructibility theory.     

The new interpretation of support vector machines on statistical learning theory

This paper is concerned with the theoretical foundation of support vector machines (SVMs). The purpose is to develop further an exact relationship between SVMs and the statistical learning theory (SLT). As a representative, the standard C-support vector classification (C-SVC) is considered here. More precisely, we show that the decision function obtained by C-SVC is just one of the decision functions obtained by solving the optimization problem derived directly from the structural risk minimization principl...     

Post’s Problem for ordinal register machines: An explicit approach

We provide a positive solution for Post’s Problem for ordinal register machines, and also prove that these machines and ordinal Turing machines compute precisely the same partial functions on ordinals. To do so, we construct ordinal register machine programs which compute the necessary functions. In addition, we show that any set of ordinals solving Post’s Problem must be unbounded in the writable ordinals.     

Enlargeability and index theory: infinite covers

In Hanke and Schick (J Differ Geom 74(2):293–320, 2006) we showed non-vanishing of the universal index elements in the K-theory of the maximal C*-algebras of the fundamental groups of enlargeable spin manifolds. The underlying notion of enlargeability was the one from Gromov and Lawson (Ann Math 111(2):209–230, 1980), involving contracting maps defined on finite covers of the given manifolds. In the paper at hand, we weaken this assumption to the one in Gromov and Lawson (Publ IHES 58:83–196, 1983) where infinite covers are allowed. The new idea is the construction of a geometrically given C*-algebra with trace which encodes the information given by these infinite covers. Along the way we obtain an easy proof of a relative index theorem relevant in this context. We thank S. Stolz and A. Thom for useful conversations regarding the research in this paper. Both authors are members of the DFG emphasis programme “Globale Differentialgeometrie” whose support is gratefully acknowledged.     

Distributional limit theorems in infinite ergodic theory

We present a unified approach to the Darling-Kac theorem and the arcsine laws for occupation times and waiting times for ergodic transformations preserving an infinite measure. Our method is based on control of the transfer operator up to the first entrance to a suitable reference set rather than on the full asymptotics of the operator. We illustrate our abstract results by showing that they easily apply to a significant class of infinite measure preserving interval maps. We also show that some of the tools introduced here are useful in the setup of pointwise dual ergodic transformations.     

An infinite dimensional Morse theory with applications

In this paper we construct an infinite dimensional (extraordinary) cohomology theory and a Morse theory corresponding to it. These theories have some special properties which make them useful in the study of critical points of strongly indefinite functionals (by strongly indefinite we mean a functional unbounded from below and from above on any subspace of finite codimension). Several applications are given to Hamiltonian systems, the one-dimensional wave equation (of vibrating string type) and systems of elliptic partial differential equations.