Some modular relations for the Göllnitz–Gordon functions by an even–odd method

In this paper, we derive some new modular relations which only involve Göllnitz–Gordon functions by using an even–odd method. We also give new proofs of some modular relations of the same nature established earlier by Huang and Chen.     

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.     

A motivated proof of the Göllnitz–Gordon–Andrews identities

We present what we call a “motivated proof” of the Göllnitz–Gordon–Andrews identities. A similar motivated proof of the Rogers–Ramanujan identities was previously given by G. E. Andrews and R. J. Baxter, and was subsequently generalized to Gordon’s identities by J. Lepowsky and M. Zhu. We anticipate that the present proof of the Göllnitz–Gordon–Andrews identities will illuminate certain twisted vertex-algebraic constructions.     

Some modular relations for the Göllnitz-Gordon functions and Ramanujan’s modular equations

Huang used the methods of Rogers, Watson and Bressoud to derive some new modular relations involving the Göllnitz-Gordon functions. In this paper, using Ramanujan’s modular equations, we present a uniform method to prove these modular relations established by Huang.     

Extending Theorems of Göllnitz,A New Family of Partition Identities

A new family of partition identities is given which include as special cases two theorems of Göllnitz. We show, also, a relation between our result and a theorem given by Sylvester.     

Structure of certain level 2 standard modules for $$A_5^{(2)}$$ and the Göllnitz–Gordon identities

We employ the technique of Lepowsky–Wilson Z-algebras to analyze the structure of certain level 2 standard modules for the affine Lie algebra \(A_5^{(2)}\) that are contained in the tensor product of two inequivalent level 1 standard modules for \(A_5^{(2)}\). As a corollary, we obtain a vertex-operator-theoretic interpretation of the Göllnitz–Gordon identities.     

Some properties of certain multivalent analytic functions involving the Cho–Kwon–Srivastava operator

By making use of the principle of subordination between analytic functions and the Cho–Kwon–Srivastava operator, we introduce a certain subclass of multivalent analytic functions. Such results as subordination and superordination properties, convolution properties, inclusion relationships, distortion theorems, inequality properties and sufficient conditions for multivalent starlikeness are proved. The results presented here would provide extensions of those given in earlier works. Several other new results are also obtained.     

Two modular equations for squares of the Rogers–Ramanujan functions with applications

Two modular identities of Gordon, McIntosh, and Robins are shown to be connected to the Rogers–Ramanujan continued fraction R(q), and in particular to Ramanujan’s parameter k:=R(q)R ²(q ²). Using this connection, we give new modular relations for R(q), and offer new and uniform proofs of several results of Ramanujan. In particular, we give a new proof of a famous and fundamental modular identity satisfied by the Rogers–Ramanujan continued fraction. We furthermore show that many analogous results hold for Ramanujan’s parameters μ:=R(q)R(q ⁴) and ν:=R ²(q ^1/2)R(q)/R(q ²). New proofs are offered for modular relations connecting R(q) to R(−q), R(q ²), and R(q ⁴), and new relations connecting R(q) at these arguments are offered. Eleven identities for the Rogers–Ramanujan functions are proved, including four new identities.      

Uniqueness in the characteristic Cauchy problem of the Klein–Gordon equation and tame restrictions of generalized functions

We show that every tempered distribution, which is a solution of the (homogenous) Klein–Gordon equation, admits a “tame” restriction to the characteristic (hyper)surface {x ⁰ + x ⁿ = 0} in (1 + n)-dimensional Minkowski space and is uniquely determined by this restriction. The restriction belongs to the space which we have introduced in (Ullrich in J. Math. Phys. 45, 2004). Moreover, we show that every element of appears as the “tame” restriction of a solution of the (homogeneous) Klein–Gordon equation.     

A complete generalization of Göllnitz’s “big” theorem

In the spirit of Göllnitz’s “big” partition theorem of 1967, we present a new mod-6 partition identity. Alladi et al. provided a four-parameter refinement of Göllnitz’s big theorem in 1995 via a key identity of generating functions and the method of weighted words. By means of this technique, two similar mod-6 identities of this type were discovered—one by Alladi in 1999 and one by Alladi and Andrews in 2015. We finish the picture by presenting and proving the fourth and final possible mod-6 identity in this spirit. Furthermore, we provide a complete generalization of mod-n identities of this type. Finally, we apply a similar argument to generalize an identity of Alladi et al. from 2003.

     

Characterizing the elementary recursive functions by a fragment of Gödel's T

Let T be G?del's system of primitive recursive functionals of finite type in a combinatory logic formulation. Let be the subsystem of T in which the iterator and recursor constants are permitted only when immediately applied to type 0 arguments. By a Howard-Schütte-style argument the -derivation lengths are classified in terms of an iterated exponential function. As a consequence a constructive strong normalization proof for is obtained. Another consequence is that every -representable number-theoretic function is elementary recursive. Furthermore, it is shown that, conversely, every elementary recursive function is representable in . The expressive weakness of compared to the full system T can be explained as follows: In contrast to , computation steps in never increase the nesting-depth of and at recursion positions. Received: 3 July 1996/ Revised version: 24 September 1999     

On Hölder maps of cubes

A map of metric spaces f: X → Y satisfying the inequality $$ \left| {f(x) - f(y)} \right| \leqslant C\left| {x - y} \right|^\alpha $$ for some C and α and all x, y ∈ X is called a Hölder map with exponent α. V. I. Arnold posed the following problem: Does there exist a Höldermap from the square onto the cube with exponent 2/3? The firstmain theorem of this paper gives a general method for constructing Höldermaps of compact metric spaces. This construction yields, in particular, a dimension-raising map f: I ⁿ → I ^m with Hölder exponent arbitrarily close to m/n for m > n > 1 and a map I ¹ → I ^m with Hölder exponent 1/m. The second main theorem states the nonexistence of a regular fractal map f: I ⁿ → I ^m with Hölder exponent n/m from the n-cube onto the m-cube for m < 2n.     

The geometry of the Gauss–Picard modular group

We give a construction of a fundamental domain for PU(2,1,mathbbZ [i]){{rm PU}(2,1,mathbb{Z} [i])}, that is the group of holomorphic isometries of complex hyperbolic space with coefficients in the Gaussian ring of integers mathbbZ [i]{mathbb{Z} [i]}. We obtain from that construction a presentation of that lattice and relate it, in particular, to lattices constructed by Mostow.     

Modular relations for the Rogers–Ramanujan functions with applications to partitions

The Ramanujan Journal - In this paper, we obtain several new modular relations for the Rogers–Ramanujan functions. Furthermore, we give partition theoretic interpretations for some of our...     

Exact solutions of the generalized Sinh–Gordon equation

In this paper, we successfully derive a new exact traveling wave solutions of the generalized Sinh–Gordon equation by new application of the homogeneous balance method. This method could be used in further works to establish more entirely new solutions for other kinds of nonlinear evolution equations arising in physics.     

Hydrodynamic limits of the nonlinear Klein–Gordon equation

We perform the mathematical derivation of the compressible and incompressible Euler equations from the modulated nonlinear Klein–Gordon equation. Before the formation of singularities in the limit system, the nonrelativistic-semiclassical limit is shown to be the compressible Euler equations. If we further rescale the time variable, then in the semiclassical limit (the light speed kept fixed), the incompressible Euler equations are recovered. The proof involves the modulated energy introduced by Brenier (2000) [1].