Modular equations for cubes of the Rogers–Ramanujan and Ramanujan–Göllnitz–Gordon functions and their associated continued fractions

In this paper, we prove modular identities involving cubes of the Rogers–Ramanujan functions. Applications are given to proving relations for the Rogers–Ramanujan continued fraction. Some of our identities are new. We establish analogous results for the Ramanujan–Göllnitz–Gordon functions and the Ramanujan–Göllnitz–Gordon continued fraction. Finally, we offer applications to the theory of partitions.     

New Modular Relations for the Göllnitz-Gordon Functions

We attempt to obtain new modular relations for the Göllnitz-Gordon functions by techniques which have been used by L. J. Rogers, G. N. Watson, and D. Bressoud to prove some of Ramanujan's 40 identities. Also, we give new proofs for some modular relations for the Göllnitz-Gordon functions which have been previously established by using results from L. Rogers and D. Bressoud. Finally, we give applications of those new modular relations to the theory of partitions.     

人口迁移模型的改进及系统动力学仿真预测

求出了Keyfitz模型和Rogers模型的理论解,并以Rogers模型为例,对人口迁移模型进行了改进,使之适用范围更广.在人口迁移模型的基础上,建立了系统动力学模型,并对江西省的人口迁移问题进行了仿真预测,模拟出了江西省农村人口、城镇人口及城市化率的变化趋势.     

Fibration de Hitchin et endoscopie

We propose a geometric interpretation of the theory of elliptic endoscopy, due to Langlands and Kottwitz, in terms of the Hitchin fibration. As applications, we prove a global analog of a purity conjecture, due to Goresky, Kottwitz and MacPherson. For unitary groups, this global purity statement has been used, in a joint work with G. Laumon, to prove the fundamental lemma over a local fields of equal characteristics.      

Rogers Semilattices of Finite Partially Ordered Sets

It is proved that the principal sublattice of a Rogers semilattice of a finite partially ordered set is definable. For this goal to be met, we present a generalization of the Denisov theorem concerning extensions of embeddings of Lachlan semilattices to ideals of Rogers semilattices.     

The Rogers Semilattices of Generalized Computable Enumerations

We study the cardinality and structural properties of the Rogers semilattice of generalized computable enumerations with arbitrary noncomputable oracles and oracles of hyperimmune Turing degree. We show the infinity of the Rogers semilattice of generalized computable enumerations of an arbitrary nontrivial family with a noncomputable oracle. In the case of oracles of hyperimmune degree we prove that the Rogers semilattice of an arbitrary infinite family includes an ideal without minimal elements and establish that the top, if present, is a limit element under the condition that the family contains the inclusion-least set.     

New modular relations involving cubes of the Göllnitz–Gordon functions

Chen and Huang established some elegant modular relations for the Göllnitz–Gordon functions analogous to Ramanujan’s list of forty identities for the Rogers–Ramanujan functions. In this paper, we derive some new modular relations involving cubes of the Göllnitz–Gordon functions. Furthermore, we also provide new proofs of some modular relations for the Göllnitz–Gordon functions due to Gugg.     

Random fixed point theorems for Hardy-Rogers self-random operators with applications to random integral equations

In this paper, we prove some random fixed point theorems for Hardy–Rogers self-random operators in separable Banach spaces and, as some applications, we show the existence of a solution for random nonlinear integral equations in Banach spaces. Some stochastic versions of deterministic fixed point theorems for Hardy–Rogers self mappings and stochastic integral equations are obtained.     

Shadow systems: remarks and extensions

We study some problems from various aspects of convexity, concerning shadow systems. Namely, we extend an old result of Rogers and Shephard, we provide a new simple proof to a result of Reisner, and we study a question related to Geometric Tomography, providing a characterization of central symmetry for convex bodies.     

Commutant lifting theorem and interpolation in discrete nest algebras

Ball in [Ba] showed that the commutant lifting theorem for the nest algebras due to Paulsen and Power gives a unified approach to a wide class of interpolation problems for nest algebras. By restricting our attention to the case when nest algebras associated with the problems are discrete we derive a variant of the commutant lifting theorem which avoids language of representation theory and which is sufficient to treat an analog of the generalized Schur-Nevannlinna-Pick (SNP) problem in the setting of upper triangular operators.     

Real recursive functions and their hierarchy

In the last years, recursive functions over the reals (Theoret. Comput. Sci. 162 (1996) 23) have been considered, first as a model of analog computation, and second to obtain analog characterizations of classical computational complexity classes (Unconventional Models of Computation, UMC 2002, Lecture Notes in Computer Science, Vol. 2509, Springer, Berlin, pp. 1–14). However, one of the operators introduced in the seminal paper by Moore (1996), the minimalization operator, has not been considered: (a) although differential recursion (the analog counterpart of classical recurrence) is, in some extent, directly implementable in the General Purpose Analog Computer of Claude Shannon, analog minimalization is far from physical realizability, and (b) analog minimalization was borrowed from classical recursion theory and does not fit well the analytic realm of analog computation. In this paper, we show that a most natural operator captured from analysis—the operator of taking a limit—can be used properly to enhance the theory of recursion over the reals, providing good solutions to puzzling problems raised by the original model.     

Asymptotic analysis of Emden–Fowler type equation with an application to power flow models

Emden–Fowler type equations are nonlinear differential equations that appear in many fields such as mathematical physics, astrophysics and chemistry. In this paper, we perform an asymptotic analysis of a specific Emden–Fowler type equation that emerges in a queuing theory context as an approximation of voltages under a well-known power flow model. Thus, we place Emden–Fowler type equations in the context of electrical engineering. We derive properties of the continuous solution of this specific Emden–Fowler type equation and study the asymptotic behavior of its discrete analog. We conclude that the discrete analog has the same asymptotic behavior as the classical continuous Emden–Fowler type equation that we consider.     

Dirichlet units and critical points of closed 1-forms

S.P. Novikov developed an analog of the Morse theory for closed 1-forms. In this paper we suggest an analog of the Lusternik-Schnirelman theory for closed 1-forms. For any cohomology class ξ ε H¹(X,R) we define an integer cl(ξ) (the cup-length associated with ξ); we prove that any closed 1-form representing ξ has at least cl(ξ) - 1 critical points. The number cl(ξ) is defined using cup-products in cohomology of some flat line bundles, such that their monodromy is described by complex numbers, which are not Dirichlet units.     

Partitions with fixed largest hook length

Motivated by a recent paper of Straub, we study the distribution of integer partitions according to the length of their largest hook, instead of the usual statistic, namely the size of the partitions. We refine Straub’s analogue of Euler’s Odd-Distinct partition theorem, derive a generalization in the spirit of Alder’s conjecture, as well as a curious analogue of the first Rogers–Ramanujan identity. Moreover, we obtain a partition theorem that is the counterpart of Euler’s pentagonal number theorem in this setting, and connect it with the Rogers–Fine identity. We conclude with some congruence properties.     

Foundations of the theory of bounded cohomology

In this paper we give a new approach to the theory of bounded cohomology. The ideas of relative homological algebra, modified so that they are based on a natural seminorm in the bounded cohomology, play a central role in this approach. Moreover, a new proof is given of the vanishing theorem in the bounded cohomology of simply connected spaces, and also an analog of Leray's theorem on coverings in the theory of bounded cohomology.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 143, pp. 69–109, 1985.     

Shifted versions of the Bailey and Well-Poised Bailey lemmas

The Bailey lemma is a famous tool to prove Rogers–Ramanujan type identities. We use shifted versions of the Bailey lemma to derive m-versions of multisum Rogers–Ramanujan type identities. We also apply this method to the Well-Poised Bailey lemma and obtain a new extension of the Rogers–Ramanujan identities.     

Survival of contact processes on the hierarchical group

We consider contact processes on the hierarchical group, where sites infect other sites at a rate depending on their hierarchical distance, and sites become healthy with a constant recovery rate. If the infection rates decay too fast as a function of the hierarchical distance, then we show that the critical recovery rate is zero. On the other hand, we derive sufficient conditions on the speed of decay of the infection rates for the process to exhibit a nontrivial phase transition between extinction and survival. For our sufficient conditions, we use a coupling argument that compares contact processes on the hierarchical group with freedom two with contact processes on a renormalized lattice. An interesting novelty in this renormalization argument is the use of a result due to Rogers and Pitman on Markov functionals.     

Quasiharmonic classification of infinite networks

As a discrete analog to the quasiharmonic classification of Riemannian manifolds due to Nakai and Sario, we give a characterization of an infinite network by the class of discrete quasiharmonic functions on it. Some potential-theoretic properties of the network will be discussed in this paper.     

Zeta functions of finite graphs and coverings, III

A graph theoretical analog of Brauer-Siegel theory for zeta functions of number fields is developed using the theory of Artin L-functions for Galois coverings of graphs from parts I and II. In the process, we discuss possible versions of the Riemann hypothesis for the Ihara zeta function of an irregular graph.