We prove polynomial identities for the N = 1 superconformal model SM (2, 4ν) which generalize and... more We prove polynomial identities for the N = 1 superconformal model SM (2, 4ν) which generalize and extend the known Fermi/Bose character identities. Our proof uses the qtrinomial coefficients of Andrews and Baxter on the bosonic side and a recently introduced very general method of producing recursion relations for q-series on the fermionic side. We use these polynomials to demonstrate a dual relation under q → q −1 between SM (2, 4ν) and M . We also introduce a generalization of the Witten index which is expressible in terms of the Rogers false theta functions.
We prove polynomial identities for the N= 1 superconformal model SM(2, 4v) which generalize and e... more We prove polynomial identities for the N= 1 superconformal model SM(2, 4v) which generalize and extend the known Fermi/Bose character identities. Our proof uses the q-trinomial coefficients of Andrews and Baxter on the bosonic side and a recently introduced very general method of producing recursion relations for q-series on the fermionic side. We use these polynomials to demonstrate a dual relation under q___,q-t between SM(2,4v) and M(2v-1, 4v). We also introduce a generalization of the Witten index which is expressible in terms of the Rogers false theta functions.
ABSTRACT Certain field theories have especially large regions of analyticity which allow computat... more ABSTRACT Certain field theories have especially large regions of analyticity which allow computations which generically cannot be done. This phenomenon is especially pronounced in the chiral Potts model. We study here the perturbation theory for the general non-integrable chiral Potts model depending on two chiral angles and a strength parameter and show how the analyticity of the ground state energy and correlation functions dramatically increases when the angles and the strength parameter satisfy the integrability condition.
We present analytic expressions for the single particle excitation energies of the eight quasi-pa... more We present analytic expressions for the single particle excitation energies of the eight quasi-particles in the lattice Es Ising model and demonstrate that all excitations have an extended Brillouin zone which, depending on the excitation, ranges from 0 < P < 4n to 0 < P < 12~. These are compared with exact diagonalizations for systems through size 10 and with the E, fermionic representations of the characters of the critical system in order to study the counting statistics. 0 1997 Published by Elsevier Science B.V. '
Journal of Statistical Planning and Inference, 2008
Two matrices with elements taken from the set {−1, 1} are Hadamard equivalent if one can be conve... more Two matrices with elements taken from the set {−1, 1} are Hadamard equivalent if one can be converted into the other by a sequence of permutations of rows and columns, and negations of rows and columns. In this paper we summarize what is known about the number of equivalence classes of matrices having maximal determinant. We establish that there are 7 equivalence classes for matrices of order 21 and that there are at least 9,884 equivalence classes for matrices of order 26. The latter result is obtained primarily using a switching technique for producing new designs from old.
We prove polynomial identities for the N= 1 superconformal model SM(2, 4v) which generalize and e... more We prove polynomial identities for the N= 1 superconformal model SM(2, 4v) which generalize and extend the known Fermi/Bose character identities. Our proof uses the q-trinomial coefficients of Andrews and Baxter on the bosonic side and a recently introduced very general method of producing recursion relations for q-series on the fermionic side. We use these polynomials to demonstrate a dual relation under q___,q-t between SM(2,4v) and M(2v-1, 4v). We also introduce a generalization of the Witten index which is expressible in terms of the Rogers false theta functions.
The Hadamard maximal determinant problem asks for the largest n × n determinant with entries ±1. ... more The Hadamard maximal determinant problem asks for the largest n × n determinant with entries ±1. When n ≡ 1(mod 4), the maximal excess construction of Farmakis and Kounias [The excess of Hadamard matrices and optimal designs, Discrete Math. 67 (1987) 165-176] produces many large (though seldom maximal) determinants. For certain small n, still larger determinants have been known; e.g., see [W.P. Orrick, B. Solomon, R. Dowdeswell, W.D. Smith, New lower bounds for the maximal determinant problem, arXiv preprint math.CO/0304410]. Here, we define "3-normalized" n × n Hadamard matrices, and construct large (n + 1) × (n + 1) determinants from them. Our constructions give most of the previous "small n" records, and set new records when n = 37, 49, 65, 73, 77, 85, 89, 93, 97, and 101, most of which exceed the reach of the maximal excess technique. We suspect that our n = 37 determinant, 72 × 9 17 × 2 36 is best possible.
Abstract: We report new world records for the maximal determinant of an n-by-n matrix with entrie... more Abstract: We report new world records for the maximal determinant of an n-by-n matrix with entries+/-1. Using various techniques, we beat existing records for n= 22, 23, 27, 29, 31, 33, 34, 35, 39, 45, 47, 53, 63, 69, 73, 77, 79, 93, and 95, and we present the record-breaking ...
We have made substantial advances in elucidating the properties of the susceptibility of the squa... more We have made substantial advances in elucidating the properties of the susceptibility of the square lattice Ising model. We discuss its analyticity properties, certain closed form expressions for subsets of the coefficients, and give an algorithm of complexity O(N 6 ) to determine its first N coefficients. As a result, we have generated and analyzed series with more than 300 terms in both the high-and low-temperature regime. We quantify the effect of irrelevant variables to the scaling-amplitude functions. In particular, we find and quantify the breakdown of simple scaling, in the absence of irrelevant scaling fields, arising first at order |T − T c | 9/4 , though high-low temperature symmetry is still preserved. At terms of order |T − T c | 17/4 and beyond, this symmetry is no longer present. The short-distance terms are shown to have the form (T − T c ) p (log |T − T c |) q with p ≥ q 2 . Conjectured exact expressions for some correlation functions and series coefficients in terms of elliptic theta functions also foreshadow future developments.
We derive self-reciprocity properties for a number of polyomino generating functions, including s... more We derive self-reciprocity properties for a number of polyomino generating functions, including several families of column-convex polygons, three-choice polygons and staircase polygons with a staircase hole. In so doing, we establish a connection between the reciprocity results known to combinatorialists and the inversion relations used by physicists to solve models in statistical mechanics. For several classes of convex polygons, the inversion (reciprocity) relation, augmented by certain symmetry and analyticity properties, completely determines the anisotropic perimeter generating function.
We construct integrable lattice realizations of conformal twisted boundary conditions for s (2) u... more We construct integrable lattice realizations of conformal twisted boundary conditions for s (2) unitary minimal models on a torus. These conformal field theories are realized as the continuum scaling limit of critical A-D-E lattice models with positive spectral parameter. The integrable seam boundary conditions are labelled by (r, s, ζ ) ∈ (A g−2 , A g−1 , Γ ) where Γ is the group of automorphisms of the graph G and g is the Coxeter number of G = A, D, E. Taking symmetries into account, these are identified with conformal twisted boundary conditions of Petkova and Zuber labelled by (a, b, γ ) ∈ (A g−2 ⊗ G, A g−2 ⊗ G, Z 2 ) and associated with nodes of the minimal analog of the Ocneanu quantum graph. Our results are illustrated using the Ising (A 2 , A 3 ) and 3-state Potts (A 4 , D 4 ) models. 2001 Elsevier Science B.V. All rights reserved. 0370-2693/01/$ -see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 -2 6 9 3 ( 0 1 ) 0 0 9 8 2 -0
We prove polynomial identities for the N = 1 superconformal model SM (2, 4ν) which generalize and... more We prove polynomial identities for the N = 1 superconformal model SM (2, 4ν) which generalize and extend the known Fermi/Bose character identities. Our proof uses the qtrinomial coefficients of Andrews and Baxter on the bosonic side and a recently introduced very general method of producing recursion relations for q-series on the fermionic side. We use these polynomials to demonstrate a dual relation under q → q −1 between SM (2, 4ν) and M . We also introduce a generalization of the Witten index which is expressible in terms of the Rogers false theta functions.
We prove polynomial identities for the N= 1 superconformal model SM(2, 4v) which generalize and e... more We prove polynomial identities for the N= 1 superconformal model SM(2, 4v) which generalize and extend the known Fermi/Bose character identities. Our proof uses the q-trinomial coefficients of Andrews and Baxter on the bosonic side and a recently introduced very general method of producing recursion relations for q-series on the fermionic side. We use these polynomials to demonstrate a dual relation under q___,q-t between SM(2,4v) and M(2v-1, 4v). We also introduce a generalization of the Witten index which is expressible in terms of the Rogers false theta functions.
ABSTRACT Certain field theories have especially large regions of analyticity which allow computat... more ABSTRACT Certain field theories have especially large regions of analyticity which allow computations which generically cannot be done. This phenomenon is especially pronounced in the chiral Potts model. We study here the perturbation theory for the general non-integrable chiral Potts model depending on two chiral angles and a strength parameter and show how the analyticity of the ground state energy and correlation functions dramatically increases when the angles and the strength parameter satisfy the integrability condition.
We present analytic expressions for the single particle excitation energies of the eight quasi-pa... more We present analytic expressions for the single particle excitation energies of the eight quasi-particles in the lattice Es Ising model and demonstrate that all excitations have an extended Brillouin zone which, depending on the excitation, ranges from 0 < P < 4n to 0 < P < 12~. These are compared with exact diagonalizations for systems through size 10 and with the E, fermionic representations of the characters of the critical system in order to study the counting statistics. 0 1997 Published by Elsevier Science B.V. '
Journal of Statistical Planning and Inference, 2008
Two matrices with elements taken from the set {−1, 1} are Hadamard equivalent if one can be conve... more Two matrices with elements taken from the set {−1, 1} are Hadamard equivalent if one can be converted into the other by a sequence of permutations of rows and columns, and negations of rows and columns. In this paper we summarize what is known about the number of equivalence classes of matrices having maximal determinant. We establish that there are 7 equivalence classes for matrices of order 21 and that there are at least 9,884 equivalence classes for matrices of order 26. The latter result is obtained primarily using a switching technique for producing new designs from old.
We prove polynomial identities for the N= 1 superconformal model SM(2, 4v) which generalize and e... more We prove polynomial identities for the N= 1 superconformal model SM(2, 4v) which generalize and extend the known Fermi/Bose character identities. Our proof uses the q-trinomial coefficients of Andrews and Baxter on the bosonic side and a recently introduced very general method of producing recursion relations for q-series on the fermionic side. We use these polynomials to demonstrate a dual relation under q___,q-t between SM(2,4v) and M(2v-1, 4v). We also introduce a generalization of the Witten index which is expressible in terms of the Rogers false theta functions.
The Hadamard maximal determinant problem asks for the largest n × n determinant with entries ±1. ... more The Hadamard maximal determinant problem asks for the largest n × n determinant with entries ±1. When n ≡ 1(mod 4), the maximal excess construction of Farmakis and Kounias [The excess of Hadamard matrices and optimal designs, Discrete Math. 67 (1987) 165-176] produces many large (though seldom maximal) determinants. For certain small n, still larger determinants have been known; e.g., see [W.P. Orrick, B. Solomon, R. Dowdeswell, W.D. Smith, New lower bounds for the maximal determinant problem, arXiv preprint math.CO/0304410]. Here, we define "3-normalized" n × n Hadamard matrices, and construct large (n + 1) × (n + 1) determinants from them. Our constructions give most of the previous "small n" records, and set new records when n = 37, 49, 65, 73, 77, 85, 89, 93, 97, and 101, most of which exceed the reach of the maximal excess technique. We suspect that our n = 37 determinant, 72 × 9 17 × 2 36 is best possible.
Abstract: We report new world records for the maximal determinant of an n-by-n matrix with entrie... more Abstract: We report new world records for the maximal determinant of an n-by-n matrix with entries+/-1. Using various techniques, we beat existing records for n= 22, 23, 27, 29, 31, 33, 34, 35, 39, 45, 47, 53, 63, 69, 73, 77, 79, 93, and 95, and we present the record-breaking ...
We have made substantial advances in elucidating the properties of the susceptibility of the squa... more We have made substantial advances in elucidating the properties of the susceptibility of the square lattice Ising model. We discuss its analyticity properties, certain closed form expressions for subsets of the coefficients, and give an algorithm of complexity O(N 6 ) to determine its first N coefficients. As a result, we have generated and analyzed series with more than 300 terms in both the high-and low-temperature regime. We quantify the effect of irrelevant variables to the scaling-amplitude functions. In particular, we find and quantify the breakdown of simple scaling, in the absence of irrelevant scaling fields, arising first at order |T − T c | 9/4 , though high-low temperature symmetry is still preserved. At terms of order |T − T c | 17/4 and beyond, this symmetry is no longer present. The short-distance terms are shown to have the form (T − T c ) p (log |T − T c |) q with p ≥ q 2 . Conjectured exact expressions for some correlation functions and series coefficients in terms of elliptic theta functions also foreshadow future developments.
We derive self-reciprocity properties for a number of polyomino generating functions, including s... more We derive self-reciprocity properties for a number of polyomino generating functions, including several families of column-convex polygons, three-choice polygons and staircase polygons with a staircase hole. In so doing, we establish a connection between the reciprocity results known to combinatorialists and the inversion relations used by physicists to solve models in statistical mechanics. For several classes of convex polygons, the inversion (reciprocity) relation, augmented by certain symmetry and analyticity properties, completely determines the anisotropic perimeter generating function.
We construct integrable lattice realizations of conformal twisted boundary conditions for s (2) u... more We construct integrable lattice realizations of conformal twisted boundary conditions for s (2) unitary minimal models on a torus. These conformal field theories are realized as the continuum scaling limit of critical A-D-E lattice models with positive spectral parameter. The integrable seam boundary conditions are labelled by (r, s, ζ ) ∈ (A g−2 , A g−1 , Γ ) where Γ is the group of automorphisms of the graph G and g is the Coxeter number of G = A, D, E. Taking symmetries into account, these are identified with conformal twisted boundary conditions of Petkova and Zuber labelled by (a, b, γ ) ∈ (A g−2 ⊗ G, A g−2 ⊗ G, Z 2 ) and associated with nodes of the minimal analog of the Ocneanu quantum graph. Our results are illustrated using the Ising (A 2 , A 3 ) and 3-state Potts (A 4 , D 4 ) models. 2001 Elsevier Science B.V. All rights reserved. 0370-2693/01/$ -see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 -2 6 9 3 ( 0 1 ) 0 0 9 8 2 -0
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