Alexander Berkovich

University of Florida | UF · Department of Mathematics

Using the Euler-Maclaurin formula, the authors compute finite-size corrections to the ground- and excited-state energies and momenta. This enables them to obtain all possible operator scaling dimensions at the critical point (T=0) and surface exponents for a variety of boundary conditions. They extend the predictions of conformal invariance to incl...

A method based on the Euler-Maclaurin formula is proposed to obtain all finite-size corrections to the energies of the ground and excited states for the one-dimensional Bose gas with delta function interactions. Scaling dimensions for all gapless states and some states having a macroscopic momentum are obtained.

Several new transformations for q-binomial coefficients are found, which have the special feature that the kernel is a polynomial with nonnegative coefficients. By studying the group-like properties of these positivity preserving transformations, as well as their connection with the Bailey lemma, many new summation and transformation formulas for b...

In this paper we give a computer proof of a new polynomial identity, which extends a recent result of Alladi and the first author. In addition, we provide computer proofs for new finite analogs of Jacobi and Euler formulas. All computer proofs are done with the aid of the new computer algebra package qMultiSum developed by the second author. qMulti...

Given integers i,j,k,L,M, we establish a new double bounded q-series identity from which the three parameter (i,j,k) key identity of Alladi-Andrews-Gordon for Goellnitz's (big) theorem follows if L, M tend to infinity. When L = M, the identity yields a strong refinement of Goellnitz's theorem with a bound on the parts given by L. This is the first...

In 2009, Berkovich and Garvan introduced a new partition statistic called the GBG-rank modulo t which is a generalization of the well-known BG-rank. In this paper, we use the Littlewood decomposition of partitions to study partitions with bounded largest part and the fixed integral value of GBG-rank modulo primes. As a consequence, we obtain new el...

Motivated by recent research of Krattenthaler and Wang, we propose five new "Borwein-type" conjectures modulo 3 and two new "Borwein-type" conejctures modulo 5.

In this paper, we conjecture an extension to Bressoud's 1996 generalization of Borwein's famous 1990 conjecture. We then state two infinite hierarchies of non-negative $q$-series identities which are interesting and elegant examples of our proposed conjecture and Bressoud's generalized conjecture respectively. Finally, using certain positivity-pres...

In 2009, Berkovich and Garvan introduced a new partition statistic called the GBG-rank modulo $t$ which is a generalization of the well-known BG-rank. In this paper, we use the Littlewood decomposition of partitions to study partitions with bounded largest part and fixed integral value of GBG-rank modulo primes. As a consequence, we obtain new eleg...

I use polynomial analogue of the Jacobi triple product identity together with the Eisenstein formula for the Legendre symbol modulo 3 . to prove six identities involving the $q$-binomial coefficients. These identities are then extended to the new infinite hierarchies of q-series identities by means of the special case of Bailey's lemma. Some of the...

We revisit Bressoud's generalized Borwein conjecture. Making use of certain positivity-preserving transformations for q-binomial coefficients, we establish the truth of infinitely many new cases of the Bressoud conjecture. In addition, we prove new doubly-bounded refinement of the Foda-Quano identities. Finally, we discuss new companions to the Bre...

We refine Schmidt's problem and a partition identity related to 2-color partitions which we will refer to as Uncu-Andrews-Paule theorem. We will approach the problem using Boulet-Stanley weights and a formula on Rogers-Szeg\H{o} polynomials by Berkovich-Warnaar, and present various Schmidt's problem alike theorems and their refinements. Our new Sch...

I revisit Bressoud's generalised Borwein conjecture. Making use of certain positivity-preserving transformations for q-binomial coefficients, I establish the truth of infinitely many new cases of the Bressoud conjecture. In addition, I prove new doubly-bounded refinement of the Foda-Quano identities. Finally, I discuss new companions to the Bressou...

We prove a new polynomial refinement of the Capparelli's identities. Using a special case of Bailey's lemma we prove many infinite families of sum-product identities that root from our finite analogues of Capparelli's identities. We also discuss the q↦1/q duality transformation of the base identities and some related partition theoretic relations.

We prove a new polynomial refinement of the Capparelli's identities. Using a special case of Bailey's lemma we prove many infinite families of sum-product identities that root from our finite analogues of Capparelli's identities. We also discuss the $q\mapsto 1/q$ duality transformation of the base identities and some related partition theoretic re...

We use the q-binomial theorem to prove three new polynomial identities involving q-trinomial coefficients. We then use summation formulas for the q-trinomial coefficients to convert our identities into another set of three polynomial identities, which imply Capparelli’s partition theorems when the degree of the polynomial tends to infinity. This wa...

We revisit Bressoud’s generalized Borwein conjecture. Making use of the new positivity-preserving transformations for q-binomial coefficients we establish the truth of infinitely many cases of the Bressoud conjecture. In addition, we prove new bounded versions of Lebesgue’s identity and of Euler’s Pentagonal Number Theorem. Finally, we discuss new...

We will prove an identity involving refined q-trinomial coefficients. We then extend this identity to two infinite families of doubly bounded polynomial identities using transformation properties of the refined q-trinomials in an iterative fashion in the spirit of Bailey chains. One of these two hierarchies contains an identity which is equivalent...

I revisit Bressoud's generalized Borwein conjecture. Making use of new positivity-preserving transformations for q-binomial coefficients I establish the truth of infinitely many cases of the Bressoud conjecture. In addition, I prove new bounded version of Lebesgue's identity and of Euler's Pentagonal Number Theorem. Finally, I discuss new companion...

We used the MACH2 supercomputer to study coefficients in the q-series expansion of (1 − q)(1 − q 2). .. (1 − q n), for all n ≤ 75000. As a result, we were able to conjecture some periodic properties associated with the before unknown location of the maximum coefficient of these polynomials with odd n. Remarkably, the observed period is 62,624.

We used the MACH2 supercomputer to study coefficients in the $q$-series expansion of $(1-q)(1-q^2)\dots(1-q^n)$, for all $n\leq 75000$. As a result, we were able to conjecture some periodic properties associated with the before unknown location of the maximum coefficient of these polynomials with odd $n$. Remarkably the observed period is 62,624.

We use the q-binomial theorem to prove three new polynomial identities involving q-trinomial coefficients. We then use summation formulas for the q-trinomial coefficients to convert our identities into another set of three polynomial identities, which imply Capparelli’s partition theorems when the degree of the polynomial tends to infinity. This wa...

We prove various inequalities between the number of partitions with the bound on the largest part and some restrictions on occurrences of parts. We explore many interesting consequences of these partition inequalities. In particular, we show that for L≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \u...

In this paper, we give new evaluations for Ramanujan's circular summation function. We also provide simpler proofs for known evaluations and give some generalizations. We discover modular relations among circular summation function partition function and give uniform proof of Ramanujan's partition congruences for the moduli 5, 7 and 11. We also pro...

We propose and recursively prove polynomial identities which imply Capparelli's partition theorems. We also find perfect companions to the results of Andrews, and Alladi, Andrews and Gordon involving q-trinomial coefficients. We follow Kurşungöz's ideas to provide direct combinatorial interpretations of some of our expressions. We make use of the t...

We will prove an identity involving refined $q$-trinomial coefficients. We then extend this identity to two infinite families of doubly bounded polynomial identities using transformation properties of the refined $q$-trinomials in an iterative fashion in the spirit of Bailey chains. One of these two hierarcies contains an identity which is equivale...

We use $q$-binomial theorem to prove three new polynomial identities involving $q$-trinomial coefficients. We then use summation formulas for the $q$-trinomial coefficients to convert our identities into another set of three polynomial identities, which imply Capparelli's partition theorems when the degree of the polynomial tends to infinity. This...

We propose and recursively prove polynomial identities which imply Capparelli's partition theorems. We also find perfect companions to the results of Andrews, and Alladi, Andrews and Gordon involving $q$-trinomial coefficients. We follow Kur\c{s}ung\"oz's ideas to provide direct combinatorial interpretations of some of our expressions. We use of th...

We prove various inequalities between the number of partitions with the bound on the largest part and some restrictions on occurrences of parts. We explore many interesting consequences of these partition inequalities. In particular, we show that for $L\geq 1$, the number of partitions with $l-s \leq L$ and $s=1$ is greater than the number of parti...

We will show that (1 − q)(1 − q²) … (1 − qm) is a polynomial in q with coefficients from {−1, 0, 1} iff m = 1, 2, 3, or 5 and explore some interesting consequences of this result. We find explicit formulas for the q-series coefficients of (1−q²)(1−q³)(1−q⁴)(1−q⁵) … and (1−q³)(1−q⁴)(1−q⁵)(1− q⁶) …. In doing so, we extend certain observations made by...

We will show that $(1-q)(1-q^2)\dots (1-q^m)$ is a polynomial in $q$ with coefficients from $\{-1,0,1\}$ iff $m=1,\ 2,\ 3,$ or $5$ and explore some interesting consequences of this result. We find explicit formulas for the $q$-series coefficients of $(1-q^2)(1-q^3)(1-q^4)(1-q^5)\dots$ and $(1-q^3)(1-q^4)(1-q^5)(1-q^6)\dots$. In doing so, we extend...

We utilize false theta function results of Nathan Fine to discover four new partition identities involving weights. These relations connect Göllnitz–Gordon type partitions and partitions with distinct odd parts, partitions into distinct parts and ordinary partitions, and partitions with distinct odd parts where the smallest positive integer that is...

In this article, we define functions analogous to Ramanujan's function $f(m)$ defined in his famous paper "Modular equations and approximations to $\pi$". We then show how these new functions lead to simpler formulas for series for $1/\pi$ associated with the classical, cubic and quartic base.

In this article, we define functions analogous to Ramanujan's function $f(n)$ defined in his famous paper "Modular equations and approximations to $\pi$". We then use these new functions to study Ramanujan's series for $1/\pi$ associated with the classical, cubic and quartic bases.

We use the $q$-binomial theorem, the $q$-Gauss sum, and the ${}_2\phi_1 \rightarrow {}_2\phi_2$ transformation of Jackson to discover and prove many new weighted partition identities. These identities involve unrestricted partitions, overpartitions, and partitions with distinct even parts. Smallest part of the partitions plays an important role in...

Abstract.We utilize false theta function results of Nathan Fine to discover three new partitionidentities involving weights. These relations connect G ̈ollnitz–Gordon type partitions and partitionswith distinct odd parts, partitions into distinct parts and ordinary partitions, and partitions withdistinct odd parts where the smallest positive intege...

We utilize false theta function results of Nathan Fine to discover three new partition identities involving weights. These relations connect G\"ollnitz--Gordon type partitions and partitions with distinct odd parts, partitions into distinct parts and ordinary partitions, and partitions with distinct odd parts where the smallest positive integer tha...

This article is an extensive study of partitions with fixed number of odd and even-indexed odd parts. We use these partitions to generalize recent results of C. Savage and A. Sills. Moreover, we derive explicit formulas for generating functions for partitions with bounds on the largest part, the number of parts and with a fixed value of BG-rank or...

We discuss a new companion for Capparelli's identities. Capparelli's identities for m=1,2 state that the number of partitions of n into distinct parts not congruent to m,-m modulo 6 is equal to the number of partitions of n into distinct parts not equal to m, where the difference between parts is greater than or equal to 4 unless consecutive parts...

We discuss a very interesting new companion to Capparelli's identities.

I use the 1907 Hurwitz formula along with the Jacobi triple product identity to under-stand representation properties of two JP (Jones-Pall) forms of Kaplansky: 9x 2 + 16y 2 + 36z 2 + 16yz + 4xz + 8xy and 9x 2 + 17y 2 + 32z 2 − 8yz + 8xz + 6xy. I also discuss three nontrivial analogues of the Gauss EΥPHKA theorem. My technique can be applied to all...

I use the 1907 Hurwitz formula along with the Jacobi triple product identity to understand representation properties of two JP (Jones-Pall) forms of Kaplansky: 9x^2+ 16y^2 +36z^2 + 16yz+ 4xz + 8xy and 9x^2+ 17y^2 +32z^2 -8yz+ 8xz + 6xy. I also discuss three nontrivial analogues of the Gauss EYPHKA theorem. My technique can be applied to all known s...

In this paper we generalize the idea of "essentially unique" representations by ternary quadratic forms. We employ the Siegel-Weil formula along with the complete classification of imaginary quadratic fields of class number less than or equal to 8, to deduce the set of integers which are represented in essentially one way by a given form which is a...

I discuss a variety of results involving s(n), the number of representations of n as a sum of three squares. One of my objectives is to reveal numerous interesting connections between the properties of this function and certain modular equations of degree 3 and 5. In particular, I show that $$\displaystyle{ s(25n) = \left (6 -\left (-n\vert 5\right...

We use an injection method to prove a new class of partition inequalities involving certain $q$-products with two to four finitization parameters. Our new theorems are a substantial generalization of work by Andrews and of previous work by Berkovich and Grizzell. We also briefly discuss how our products might relate to lecture hall partitions.

In this paper we provide proofs of two new theorems that provide a broad class of partition inequalities and that illustrate a nave version of Andrews' anti-telescoping technique quite well. These new theorems also put to rest any notion that including parts of size 1 is somehow necessary in order to have a valid irreducible partition inequality. I...

In this paper we provide proofs of two new theorems that provide a broad class of partition inequalities and that illustrate a na\"ive version of Andrews' anti-telescoping technique quite well. These new theorems also put to rest any notion that including parts of size 1 is somehow necessary in order to have a valid irreducible partition inequality...

We state and prove an identity which represents the most general eta-products of weight 1 by binary quadratic forms. We discuss the utility of binary quadratic forms in finding a multiplicative completion for certain eta-quotients. We then derive explicit formulas for the Fourier coefficients of certain eta-quotients of weight 1 and level 47, 71, 1...

In a handwritten manuscript published with his lost notebook, Ramanujan stated without proofs forty identities for the Rogers-Ramanujan functions. We observe that the functions that appear in Ramanujan's identities can be obtained from a Hecke action on a certain family of eta products. We establish further Hecke-type relations for these functions...

Let s(n) be the number of representations of n as the sum of three squares. We prove a remarkable new identity for s(np^2 ) − ps(n) with p being an odd prime. This identity makes nontrivial use of ternary quadratic forms with discriminants p^2, 16p^2. These forms are related by Watson’s transformations. To prove this identity we employ the Siegel–W...

In this paper we revisit a 1987 question of Rabbi Ehrenpreis. Among many, things, we provide an elementary injective proof that P-1(L, y, n) >= P-2(L, y, n) for any L, n > 0 and any odd y > 1. Here, P-1 (L, y, n) denotes the number of partitions of n into parts congruent to 1, y + 2, or 2y (mod 2y + 2) with the largest part not exceeding (2y + 2)L...

We revisit old conjectures of Fermat and Euler regarding the representation of integers by binary quadratic form x 2+5y 2. Making use of Ramanujan’s 1 ψ 1 summation formula, we establish a new Lambert series identity for ån,m=-¥¥qn2+5m2\sum_{n,m=-\infty }^{\infty}q^{n^{2}+5m^{2}} . Conjectures of Fermat and Euler are shown to follow easily from th...

In this paper we derive an explicit formula for the number of representations of an integer by the sextenary form x(2) + y(2) + z(2) + 7s(2) + 7t(2) + 7u(2). We establish the following intriguing inequalities 2 omega(n + 2) >= a(7) (n) >= omega(n + 2) for n not equal 0, 2, 6, 16. Here a(7)(n) is the number of partitions of n that are 7-cores and om...

Let r_j(\pi,s) denote the number of cells, colored j, in the s-residue diagram of partition \pi. The GBG-rank of \pi mod s is defined as r_0+r_1*w_s+r_2*w_s^2+...+r_(s-1)*w_s^(s-1), where w_s=exp(2*\Pi*I/s). We will prove that for (s,t)=1, v(s,t) <= binomial(s+t,s)/(s+t), where v(s,t) denotes a number of distinct values that GBG-rank mod s of t-cor...

We prove that the Fourier coefficients of a certain general eta product considered by K. Saito are nonnegative. The proof is elementary and depends on a multidimensional theta function identity. The z=1 case is an identity for the generating function for p-cores due to Klyachko [A.A. Klyachko, Modular forms and representations of symmetric groups,...

In this paper we derive an explicit formula for the number of representations of an integer by the sextenary form x^2+y^2+z^2+ 7s^2+7t^2+ 7u^2. We establish the following intriguing inequalities 2b(n)>=a_7(n)>=b(n) for n not equal to 0,2,6,16. Here a_7(n) is the number of partitions of n that are 7-cores and b(n) is the number of representations of...

Let π denote a partition into parts . In a 2006 paper we defined BG-rank(π) as This statistic was employed to generalize and refine the famous Ramanujan modulo 5 partition congruence. Let pj(n) denote the number of partitions of n with BG-rank=j. Here, we provide a combinatorial proof that by showing that the residue of the 5-core crank mod 5 divid...

I revisit an automated proof of Andrews' pentagonal number theorem found by Riese. I uncover a simple polynomial identity hidden behind his proof. I explain how to use this identity to prove Andrews' result along with a variety of new formulas of similar type. I reveal an interesting relation between the tri-pentagonal theorem and items (19), (20),...

We prove that the Fourier coefficients of a certain general eta product considered by K. Saito are nonnegative. The proof is elementary and depends on a multidimensional theta function identity. The z=1 case is an identity for the generating function for p-cores due to Klyachko [17] and Garvan, Kim and Stanton [10]. A number of other infinite produ...

We revisit old conjectures of Fermat and Euler regarding representation of integers by binary quadratic form x^2+5y^2. Making use of Ramanujan's_1\psi_1 summation formula we establish a new Lambert series identity for \sum_{n,m=-\infty}^{\infty} q^{n^2+5m^2}. Conjectures of Fermat and Euler are shown to follow easily from this new formula. But we d...

We prove that the Fourier coefficients of a certain general eta product considered by K. Saito are nonnegative. The proof is elementary and depends on a multidimensional theta function identity. The z = 1 case is an identity for the generating function for p-cores due to Klyachko [12] and Garvan, Kim and Stanton [7].

A q-series with nonnegative power series coefficients is called positive. The partition statistics BG-rank is defined as an alternating sum of parities of parts of a partition. It is known that the generating function for the number of partitions of n that are 7-cores with given BG-rank can be written as certain sum of multi-theta functions. We giv...

Let \pi be a partition. In [2] we defined BG-rank(\pi) as an alternating sum of parities of parts. This statistic was employed to generalize and refine the famous Ramanujan modulo 5 partition congruence. Let p_j(n)(a_{t,j}(n)) denote a number of partitions (t-cores) of n with BG-rank=j. Here, we provide an elegant combinatorial proof that 5|p_j(5n+...

In a recent study of sign-balanced, labelled posets, Stanley introduced a new integral partition statistic srank(π) = script O sign(π) - script O sign(π′), where script O sign(π) denotes the number of odd parts of the partition π and π′ is the conjugate of π. In a forthcoming paper, Andrews proved the following refinement of Ramanujan's partition c...

Let p(n) denote the number of unrestricted partitions of n. For i=0, 2, let pi(n) denote the number of partitions π of n such that . Here O(π) denotes the number of odd parts of the partition π and π′ is the conjugate of π. Stanley [Amer. Math. Monthly 109 (2002) 760; Adv. Appl. Math., to appear] derived an infinite product representation for the g...

Let p(n) denote the number of unrestricted partitions of n. For i=0, 2, let p[i](n) denote the number of partitions pi of n such that O(pi) - O(pi') = i mod 4. Here O(pi) denotes the number of odd parts of the partition pi and pi' is the conjugate of pi. R. Stanley [13], [14] derived an infinite product representation for the generating function of...

In a recent study of sign-balanced, labelled posets Stanley, introduced a new integral partition statistic srank(pi) = O(pi) - O(pi'), where O(pi) denotes the number of odd parts of the partition pi and pi' is the conjugate of pi. Andrews proved the following refinement of Ramanujan's partition congruence mod 5: p[0](5n +4) = p[2](5n + 4) = 0 (mod...

In a recent study of sign-balanced, labelled posets Stanley [13], introduced a new integral partition statistic srank(pi) = O(pi) - O(pi'), where O(pi) denotes the number of odd parts of the partition pi and pi' the conjugate of pi. In [1] Andrews proved the following refinement of Ramanujan's partition congruence mod 5: p[0](5n +4) = p[2](5n + 4)...

We utilize Dyson' concept of the adjoint of a partition to derive an infinite family of new polynomial analogs of Euler's Pentagonal Number Theorem. We streamline Dyson's bijection relating partitions with crank ⩽k and those with k in the Rank-Set of partitions. Also, we extend Dyson's adjoint of a partition to MacMahon's “modular” partitions with...

We utilize Dyson's concept of the adjoint of a partition to derive an infinite family of new polynomial analogues of Euler's Pentagonal Number Theorem. We streamline Dyson's bijection relating partitions with crank <= k and those with k in the Rank-Set of partitions. Also, we extend Dyson's adjoint of a partition to MacMahon's ``modular'' partition...

The object of this paper is to propose and prove a new generalization of the Andrews-Gordon Identities, extending a recent result of Garrett, Ismail and Stanton. We also give a combinatorial discussion of the finite form of their result which appeared in the work of Andrews, Knopfmacher, and Paule.

A few years ago Foda, Quano, Kirillov and Warnaar proposed and proved various nite analogs of the celebrated Andrews-Gordon identities. In this paper we use these polynomial identities along with the combinatorial techniques introduced in our recent paper to derive Garrett, Ismail, Stanton type formulas for two variants of the Andrews-Gordon identi...

Given integers i,j,k,L,M, we establish a new double bounded q-series identity from which the three parameter (i,j,k) key identity of Alladi-Andrews-Gordon for Goellnitz's (big) theorem follows if L, M tend to infinity. When L = M, the identity yields a strong refinement of Goellnitz's theorem with a bound on the parts given by L. This is the first...

Schur's partition theorem states that the number of partitions of n into distinct parts congruent 1, 2 (mod 3) equals the number of partitions of n into parts which differ by >= 3, where the inequality is strict if a part is a multiple of 3. We establish a double bounded refined version of this theorem by imposing one bound on the parts congruent 0...

The author shows that the finite-temperature equal-time field correlators of the exactly integrable (1+1)-dimensional gas of impenetrable fermions in a magnetic field can be expressed as the first Fredholm minor of a completely integrable linear operator. This result enables the author to derive differential equations for quantum correlators and to...

We demonstrate the equality between the universal chiral partition function, which was first found in the context of conformal field theory and Rogers-Ramanujan identities, and the exclusion statistics introduced by Haldane in the study of the fractional quantum Hall effect. The phenomena of multiple representations of the same conformal field theo...

We present and prove Rogers–Schur–Ramanujan (Bose/Fermi) type identities for the Virasoro characters of the minimal model M(p,p′). The proof uses the continued fraction decomposition of p′/p introduced by Takahashi and Suzuki for the study of the Bethe's Ansatz equations of the XXZ model and gives a general method to construct polynomial generaliza...

In this talk we present the discoveries made in the theory of Rogers-Ramanujan identities in the last five years which have been made because of the interchange of ideas between mathematics and physics. We find that not only does every minimal representation M(p,p ' ) of the Virasoro algebra lead to a Rogers-Ramanujan identity but that different co...

We derive the fermionic polynomial generalizations of the characters of the integrable perturbations $\phi_{2,1}$ and $\phi_{1,5}$ of the general minimal $M(p,p')$ conformal field theory by use of the recently discovered trinomial analogue of Bailey's lemma. For $\phi_{2,1}$ perturbations results are given for all models with $2p>p'$ and for $\phi_...

We present generalized Rogers-Ramanujan identities which relate the Fermi and Bose forms of all the characters of the superconformal model SM(2, 4ν). In particular, we show that to each bosonic form of the character there is an infinite family of distinct fermionic q-series representations.

We use the recently established higher-level Bailey lemma and Bose-Fermi polynomial identities for the minimal models $M(p,p')$ to demonstrate the existence of a Bailey flow from $M(p,p')$ to the coset models $(A^{(1)}_1)_N\times (A^{(1)}_1)_{N'}/(A^{(1)}_1)_{N+N'}$ where $N$ is a positive integer and $N'$ is fractional, and to obtain Bose-Fermi id...

We propose and prove a trinomial version of the celebrated Bailey's lemma. As an application we obtain new fermionic representations for characters of some unitary as well as nonunitary models of N= 2 superconformal field theory (SCFT). We also establish interesting relations between N= 1 and N= 2 models of SCFT with central charges \(\) and \(\)....

We use the recently established higher-level Bailey lemma and Bose-Fermi polynomial identities for the minimal models M ( p , p ' ) to demonstrate the existence of a Bailey flow from M ( p , p ' ) to the coset models (A^{(1)}_1)_N x (A^{(1)}_1)_{N'}/((A^{(1)}_1)_{N+N'} , where N is a positive integer and N' is fractional, and to obtain Bose-Fermi i...

We demonstrate that the Bailey pair formulation of Rogers-Ramanujan identities unifies the calculations of the characters of $N=1$ and $N=2$ supersymmetric conformal field theories with the counterpart theory with no supersymmetry. We illustrate this construction for the $M(3,4)$ (Ising) model where the Bailey pairs have been given by Slater. We th...

The Hilbert space of an RSOS-model, introduced by Andrews, Baxter, and Forrester, can be viewed as a space of sequences (paths) {a_0,a_1,...,a_L}, with a_j-integers restricted by 1<=a_j<=\nu, |a_j-a_{j+1}|=1, a_0=s, a_L=r. In this paper we introduce different basis which, as shown here, has the same dimension as that of an RSOS-model. Following McC...

In this paper we consider a class of the 2D integrable models. These models are higher spin XXZ chains with an extra condition of the commensurability between spin and anisotropy. The mathematics underlying this commensurability is provided by the quantum groups with deformation parameter being an Nth root of unity. Our discussion covers a range of...

We study the thermodynamic properties of a family of integrable 1D spin chain hamiltonians associated with quantum groups at roots of unity. These hamiltonians depend for each primitive root of unit on a parameter $s$ which plays the role of a continuous spin. The model exhibits ferrimagnetism even though the interaction involved is between nearest...

We define a new class of integrable vertex models associated to quantum groups at roots of unit

We define a new class of integrable vertex models associated to quantum groups at roots of unit

The quantum nonlinear Schrödinger model (one-dimensional Bose gas) is analyzed by means of conformal field theory. We show that conformal algebra, which describes the long-distance asymptotic behavior of the model, in conjunction with a specific choice of the cut-off procedure, can be used to solve the problem of exact correlators and determine una...

The authors analyse the nonlinear Schrodinger model (the one-dimensional Bose gas) by means of conformal field theory in conjunction with a newly developed cutoff procedure. This makes it possible to obtain exact expressions for current correlators in the model. A number of new results are presented, including the expression for the two-point curre...

A one-dimensional system of bosons, interacting via a delta-function potential, is analyzed by means of a conformal field theory approach. A systematic method for deriving the operator content of the critical Hamiltonian of the model is proposed. In the case of impenetrable bosons it is shown that the Hamiltonian consists of only two commuting part...

The two-point, equal-time correlation function for a one-dimensional, finite-density gas of impenetrable fermions is derived and expressed in terms of a solution of the Painlevé equation of the fifth kind. The asymptotic large-distance behavior is found to be quite different from that found in the bosonic case.