Enumeration of maps

2018 ◽  
Author(s):  
Grigori Olshanski
Keyword(s):  
Matrix Model ◽  
Matrix Theory ◽  
Polynomial Potential ◽  
Quadratic Potential ◽  
The Matrix ◽  
Formal Expansion ◽  
Enumeration Problem ◽  
The One ◽  
Map Enumeration ◽  
The Relationship

This article discusses the relationship between random matrices and maps, i.e. graphs drawn on surfaces, with particular emphasis on the one-matrix model and how it can be used to solve a map enumeration problem. It first provides an overview of maps and related objects, recalling the basic definitions related to graphs and defining maps as graphs embedded into surfaces before considering a coding of maps by pairs of permutations. It then examines the connection between matrix integrals and maps, focusing on the Hermitian one-matrix model with a polynomial potential and how the formal expansion of its free energy around a Gaussian point (quadratic potential) can be represented by diagrams identifiable with maps. The article also illustrates how the solution of the map enumeration problem can be deduced by means of random matrix theory (RMT). Finally, it explains how the matrix model result can be translated into a bijective proof.

scholarly journals Divergent ⇒ complex amplitudes in two dimensional string theory

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Ashoke Sen
Keyword(s):  
Scattering Amplitudes ◽  
String Theory ◽  
Matrix Model ◽  
Two Dimensional ◽  
Full Matrix ◽  
Instanton Contribution ◽  
The Matrix ◽  
Theory Result ◽  
The One ◽  
Remarkable Agreement

Abstract In a recent paper, Balthazar, Rodriguez and Yin found remarkable agreement between the one instanton contribution to the scattering amplitudes of two dimensional string theory and those in the matrix model to the first subleading order. The comparison was carried out numerically by analytically continuing the external energies to imaginary values, since for real energies the string theory result diverges. We use insights from string field theory to give finite expressions for the string theory amplitudes for real energies. We also show analytically that the imaginary parts of the string theory amplitudes computed this way reproduce the full matrix model results for general scattering amplitudes involving multiple closed strings.

A Portrait of the Artist

2014 ◽  
pp. 1-79
Author(s):  
David Gillis
Keyword(s):  
Central Feature ◽  
Jewish Law ◽  
Knowledge Of God ◽  
Great Thing ◽  
Literary Art ◽  
Philosophy And Religion ◽  
The Matrix ◽  
The One ◽  
The Individual ◽  
The Relationship

This introductory chapter provides a background of Maimonides and his code of Jewish law, the Mishneh torah. Maimonides applied the highest literary art to the highest of tasks: to bequeath, as philosopher-statesman, a law that would regulate the life of the individual and of society and move people closer to the knowledge of God. The result of that art is a book to be read and experienced, not just consulted. The central feature of Mishneh torah as a work of art is the casting of the commandments of the law in the form of the cosmos. The microcosmic form suggests, in the first place, that studying Mishneh torah, like the study of the universe, can be a way to the knowledge and love of God. On the plane of ideas, this form embodies the relationship between the ‘small thing’ and the ‘great thing’, between halakhah, on the one hand, and physics and metaphysics on the other. It depicts philosophy as the matrix of halakhah, reflecting the view of the relationship between philosophy and religion in the Islamic philosophers.

TOPOLOGICAL σ MODELS AND LARGE N MATRIX INTEGRAL

1995 ◽  
Vol 10 (29) ◽  
pp. 4203-4224 ◽  
Author(s):  
TOHRU EGUCHI ◽  
KENTARO HORI ◽  
SUNG-KIL YANG
Keyword(s):  
Matrix Model ◽  
Riemann Surfaces ◽  
Holomorphic Maps ◽  
Integrable Structure ◽  
Toda Hierarchy ◽  
Matrix Integral ◽  
Large N ◽  
The Matrix ◽  
Lax Operator ◽  
The One

In this paper we describe in some detail the representation of the topological CP1 model in terms of a matrix integral which we have introduced in a previous article. We first discuss the integrable structure of the CP1 model and show that it is governed by an extension of the one-dimensional Toda hierarchy. We then introduce a matrix model which reproduces the sum over holomorphic maps from arbitrary Riemann surfaces onto CP1. We compute intersection numbers on the moduli space of curves using a geometrical method and show that the results agree with those predicted by the matrix model. We also develop a Landau-Ginzburg (LG) description of the CP1 model using a superpotential eX + et0,Q e-X given by the Lax operator of the Toda hierarchy (X is the LG field and t0,Q is the coupling constant of the Kähler class). The form of the superpotential indicates the close connection between CP1 and N=2 supersymmetric sine-Gordon theory which was noted sometime ago by several authors. We also discuss possible generalizations of our construction to other manifolds and present an LG formulation of the topological CP2 model.

scholarly journals COVARIANT MATRIX MODEL OF SUPERPARTICLE IN THE PURE SPINOR FORMALISM

2003 ◽  
Vol 18 (15) ◽  
pp. 1023-1035 ◽  
Author(s):  
ICHIRO ODA
Keyword(s):  
Matrix Model ◽  
Brst Symmetry ◽  
Explicit Calculation ◽  
Pure Spinor ◽  
First Order ◽  
Free Theory ◽  
The Matrix ◽  
The One ◽  
Spinor Formalism ◽  
Pure Spinor Formalism

On the basis of the Berkovits pure spinor formalism of covariant quantization of supermembrane, we attempt to construct a M(atrix) theory which is covariant under SO(1, 10) Lorentz group. We first construct a bosonic M(atrix) theory by starting with the first-order formalism of bosonic membrane, which precisely gives us a bosonic sector of M(atrix) theory by BFSS. Next we generalize this method to the construction of M(atrix) theory of supermembranes. However, it seems to be difficult to obtain a covariant and supersymmetric M(atrix) theory from the Berkovits pure spinor formalism of supermembrane because of the matrix character of the BRST symmetry. Instead, in this paper, we construct a supersymmetric and covariant matrix model of 11D superparticle, which corresponds to a particle limit of covariant M(atrix) theory. By an explicit calculation, we show that the one-loop effective potential is trivial, thereby implying that this matrix model is a free theory at least at the one-loop level.

EXCITATIONS AND INTERACTIONS IN d=1 STRING THEORY

1991 ◽  
Vol 06 (11) ◽  
pp. 1961-1984 ◽  
Author(s):  
ANIRVAN M. SENGUPTA ◽  
SPENTA R. WADIA
Keyword(s):  
Matrix Model ◽  
Periodic Wave ◽  
Density Fluctuations ◽  
Dirac Fermion ◽  
Unitary Matrix ◽  
Additional Parameter ◽  
Continuum Approach ◽  
The Matrix ◽  
The One ◽  
The Continuum

We discuss the singlet sector of the d=1 matrix model in terms of a Dirac fermion formalism. The leading order two- and three-point functions of the density fluctuations are obtained by this method. This allows us to construct the effective action to that order and hence provide the equation of motion. This equation is compared with the one obtained from the continuum approach. We also compare continuum results for correlation functions with the matrix model ones and discuss the nature of gravitational dressing for this regularization. Finally, we address the question of boundary conditions within the framework of the d=1 unitary matrix model, considered as a regularized version of the Hermitian model, and study the implications of a generalized action with an additional parameter (analogous to the θ parameter) which give rise to quasi-periodic wave functions.

The Application Research of the Matrix in Multi-Parameter Measurement FBGs

2012 ◽  
Vol 461 ◽  
pp. 702-706
Author(s):  
Xiao Xia Wang ◽  
Chun Ying Wu ◽  
Win Lin Wang
Keyword(s):  
Condition Number ◽  
Matrix Theory ◽  
Parameter Measurement ◽  
Application Research ◽  
Fbg Sensor ◽  
The Matrix ◽  
The Relationship ◽  
Different Thickness

The sensitivity of the FBG sensor based on multi-parameter measurement was established and determined by the matrix theory. The condition number of matrix was proposed to deduced the relationship among the measurement multi-parameters of the coated FBGs. The ill-conditioned matrix parameters can be removed, and the relationship between the FBGs sensitivities and many attribute parameters of the coated-FBG was found. As indicated by the experiment, when measure the temperature and the pressure at the same time, the sensitivities of FBG is higher by coated with different thickness of copper,and the second radius is less than 0.4mm,and the FBGs sensitivities can be improved to 5~10 times.

scholarly journals Fundamental Strings and D-Strings in the IIB Matrix Model

1997 ◽  
Vol 12 (18) ◽  
pp. 1301-1315 ◽  
Author(s):  
B. Sathiapalan
Keyword(s):  
Matrix Model ◽  
Scaling Limit ◽  
Duality Group ◽  
Fundamental String ◽  
Large N ◽  
The Matrix ◽  
Model Derivation ◽  
The One ◽  
The Right ◽  
String Worldsheet

The matrix model for IIB superstring proposed by Ishibashi, Kawai, Kitazawa and Tsuchiya is investigated. Consideration of planar and non-planar diagrams suggests that large-N perturbative expansion is consistent with the double scaling limit proposed by the above authors. We write down a Wilson loop that can be interpreted as a fundamental string vertex operator. The one-point tadpole in the presence of a D-string has the right form and this can be viewed as a matrix model derivation of the boundary conditions that define a D-string. We also argue that if worldsheet coordinates σ and τ are introduced to the fundamental string, then the conjugate variable d/dσ and d/dτ can be interpreted as the D-string worldsheet coordinates. In this way the SL (2Z) duality group of the IIB superstring becomes identified with the symplectic group acting on (p,q).

scholarly journals Exploring the effect of different modes of ICT use on school performance including social background

2018 ◽  
Vol 30 (4) ◽  
pp. 181-190 ◽  
Author(s):  
Anikó Vincze
Keyword(s):  
School Performance ◽  
Social Background ◽  
The Internet ◽  
Educational Inequalities ◽  
Ict Use ◽  
The Matrix ◽  
Cultural Status ◽  
Pisa Data ◽  
The One ◽  
The Relationship

Nowadays the use of ICT, the Internet is indispensible in everyday life. We are supposed to go online for administration, working, entertainment and for learning as well. The different modes of use, whether for entertainment, recreation, or for learning or work, influence our position in the matrix of digital inequalities. Digital inequalities at the same time have an effect on educational inequalities. Therefore our paper focuses on the effect of ICT use for different purposes on school performance to reveal the correlation between digital inequalities and educational inequalities. Both types of inequalities are strongly influenced by social background. We intend to explain the relationship between these factors by showing the effect of ICT use on school performance when taking into consideration the socio-economic and cultural status. First we introduce the main theories and results from previous researches on the tie between on the one hand social background and academic achievement, on the other hand between social background and ICT use. Then we present the main outcomes of our analysis conducted on the Hungarian subsample of the latest PISA data from 2015. Finally conclusions are summarized and further research possibilities are suggested.

scholarly journals CORRELATORS OF THE KAZAKOV-MIGDAL MODEL

1993 ◽  
Vol 08 (25) ◽  
pp. 2387-2401 ◽  
Author(s):  
M. I. DOBROLIUBOV ◽  
YU. MAKEENKO ◽  
G. W. SEMENOFF
Keyword(s):  
Matrix Model ◽  
Explicit Solution ◽  
Scalar Fields ◽  
Algebraic Equations ◽  
Quadratic Potential ◽  
Large N ◽  
Gaussian Case ◽  
The One ◽  
Non Gaussian

We derive loop equations for the one-link correlators of gauge and scalar fields in the Kazakov-Migdal model. These equations determine the solution of the model in the large-N limit and are similar to analogous equations for the Hermitian two-matrix model. We give an explicit solution of the equations for the case of a Gaussian, quadratic potential. We also show how similar calculations in a non-Gaussian case reduce to purely algebraic equations.

The density matrix in many-electron quantum mechanics I. Generalized product functions. Factorization and physical interpretation of the density matrices

1959 ◽  
Vol 253 (1273) ◽  
pp. 242-259 ◽  
Keyword(s):  
Density Matrix ◽  
Matrix Theory ◽  
Particle System ◽  
Intermolecular Forces ◽  
Density Matrices ◽  
Hartree Fock ◽  
Interaction Terms ◽  
The Matrix ◽  
Group Functions ◽  
The One

Many-electron wave functions are usually constructed from antisymmetrized products of one-electron orbitals (determinants) and energy calculations are based on the matrix element expressions due to Slater (1931). In this paper, the orbitals in such a product are replaced by ‘group functions’, each describing any number of electrons, and the necessary generalization of Slater’s results is carried out. It is first necessary to develop the density matrix theory of N -particle systems and to show that for systems described by ‘generalized product functions’ the density matrices of the whole system may be expressed in terms of those of the component electron groups. The matrix elements of the Hamiltonian between generalized product functions are then given by expressions which resemble those of Slater, the ‘coulomb’ and ‘exchange’ integrals being replaced by integrals containing the one-electron density matrices of the various groups. By setting up an ‘effective’ Hamiltonian for each electron group in the presence of the others, the discussion of a many-particle system in which groups or ‘shells’ can be distinguished (e. g. atomic K, L, M , ..., shells) can rigorously be reduced to a discussion of smaller subsystems. A single generalized product (cf. the single determinant of Hartree—Fock theory) provides a convenient first approximation; and the effect of admitting ‘excited’ products (cf. configuration interaction) can be estimated by a perturbation method. The energy expression may then be discussed in terms of the electon density and ‘pair’ functions. The energy is a sum of group energies supplemented by interaction terms which represent (i) electrostatic repulsions between charge clouds, (ii) the polarization of each group in the field of the others, and (iii) ‘dispersion’ effects of the type defined by London. All these terms can be calculated, for group functions of any kind, in terms of the density matrices of the separate groups. Applications to the theory of intermolecular forces and to π -electron systems are also discussed.

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