scholarly journals Quantum $A_r$ $Q$-System Solutions as q-Multinomial Series

10.37236/663 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Philippe Di Francesco
Keyword(s):  
Data Structure ◽  
Initial Data ◽  
Fundamental Solutions ◽  
Generating Series ◽  
Multinomial Coefficients ◽  
Q System ◽  
Explicit Expressions

We derive explicit expressions for the generating series of the fundamental solutions of the $A_r$ quantum $Q$-system of Di Francesco and Kedem [ Non-commutative integrability, paths and quasi-determinants, Adv. in Math. 228(1) (2011) 97–152], expressed in terms of any admissible initial data. These involve products of quantum multinomial coefficients, coded by the initial data structure.

Cauchy and Signaling Problems for the Time-Fractional Diffusion-Wave Equation

2014 ◽  
Vol 136 (5) ◽  
Author(s):  
Yuri Luchko ◽  
Francesco Mainardi
Keyword(s):  
Wave Equation ◽  
Initial Data ◽  
Diffusion Equation ◽  
Fundamental Solutions ◽  
Fractional Diffusion ◽  
The Other ◽  
Diffusion Wave ◽  
One Dimensional ◽  
The One ◽  
Propagation Velocities

In this paper, some known and novel properties of the Cauchy and signaling problems for the one-dimensional time-fractional diffusion-wave equation with the Caputo fractional derivative of order β,1≤β≤2 are investigated. In particular, their response to a localized disturbance of the initial data is studied. It is known that, whereas the diffusion equation describes a process where the disturbance spreads infinitely fast, the propagation velocity of the disturbance is a constant for the wave equation. We show that the time-fractional diffusion-wave equation interpolates between these two different responses in the sense that the propagation velocities of the maximum points, centers of gravity, and medians of the fundamental solutions to both the Cauchy and the signaling problems are all finite. On the other hand, the disturbance spreads infinitely fast and the time-fractional diffusion-wave equation is nonrelativistic like the classical diffusion equation. In this paper, the maximum locations, the centers of gravity, and the medians of the fundamental solution to the Cauchy and signaling problems and their propagation velocities are described analytically and calculated numerically. The obtained results for the Cauchy and the signaling problems are interpreted and compared to each other.

Quantum Q-Systems and Fermionic Sums—The Non-Simply Laced Case

2020 ◽  
Author(s):  
Mingyan Simon Lin
Keyword(s):  
Exact Sequence ◽  
Short Exact Sequence ◽  
Constant Term ◽  
Fusion Product ◽  
Full Generality ◽  
Quantum Cluster ◽  
Constant Term Identity ◽  
Q System ◽  
Explicit Expressions

Abstract In this paper, we seek to prove the equality of the $q$-graded fermionic sums conjectured by Hatayama et al. [ 14] in its full generality, by extending the results of Di Francesco and Kedem [ 9] to the non-simply laced case. To this end, we will derive explicit expressions for the quantum $Q$-system relations, which are quantum cluster mutations that correspond to the classical $Q$-system relations, and write the identity of the $q$-graded fermionic sums as a constant term identity. As an application, we will show that these quantum $Q$-system relations are consistent with the short exact sequence of the Feigin–Loktev fusion product of Kirillov–Reshetikhin modules obtained by Chari and Venkatesh [ 5].

scholarly journals Wavelet coherence as a tool for retrospective analysis of bank activities

2020 ◽  
Vol 2020 (2) ◽  
pp. 32-44
Author(s):  
Oleg Vasiurenko ◽  
◽  
Viacheslav Lyashenko ◽  
◽  
Keyword(s):  
Data Structure ◽  
Initial Data ◽  
Retrospective Analysis ◽  
Statistical Data ◽  
Point Of View ◽  
Wavelet Coherence ◽  
Retrospective Data ◽  
Banking Activity ◽  
Administrative Expenses ◽  
Over Time

The article considers the possibility and expediency of using the apparatus of the theory of wavelets to conduct analysis of banking activities. The authors determine separate stages of the complex application of various tools on the theory of wavelets to analyze the activities of banks based on retrospective data. Among these stages are: decomposition of the initial data by their approximating coefficients and coefficients of detail, and the use of wavelet coherence. Indicated the importance of conducting a retrospective analysis to reveal hidden relationships in the data structure that determine certain aspects of banking. The ad-vantages of using the tools of the theory of wavelets from the point of view of analyzing the activities of banks based on their statistical data are highlighted. Among these advantages, the authors highlight the possibility of studying the relationships be-tween data over time and determining the depth of such relationships. It is noted that this can be done in one research window. Particular attention is focused on the analysis of the reciprocity between the volume of funds in deposit accounts and the volume of loans granted, as one of the key parameters for conducting banking activities. The reciprocity between the volumes of funds in deposit accounts and the volumes of loans granted is revealed in accordance with the volumes of administrative expenses and equity of banks. It is noted that retrospective analysis allows us to identify the consequences of the onset of unwanted events and prevent them in the future. To carry out a corresponding analysis, the content of constructing a description of spatial wavelet coherence is disclosed. Such a description makes it possible to take into account a larger number of parameters than classical approaches for calculating wavelet coherence. This expands the boundaries of the relevant analysis, allows you to explore various mutual influences between individual banks in terms of their individual indicators for banking activities. Such an analysis allows to determine not only the reciprocity between individual indicators of banking activity, but also the depth of influence between individual banks, taking into account such indicators of their activity. Concrete examples are given that prove the feasibility and likelihood of applying the proposed approaches to the analysis of banking activities.

scholarly journals Bessenrodt-Stanley Polynomials and the Octahedron Recurrence

10.37236/4434 ◽  
2015 ◽  
Vol 22 (3) ◽  
Author(s):  
Philippe Di Francesco
Keyword(s):  
Initial Data ◽  
Partition Functions ◽  
Multivariate Polynomials ◽  
Suitable Initial Data ◽  
Octahedron Recurrence ◽  
Explicit Expressions

We show that a family of multivariate polynomials recently introduced by Bessenrodt and Stanley can be expressed as solution of the octahedron recurrence with suitable initial data. This leads to generalizations and explicit expressions as path or dimer partition functions.

scholarly journals Recurrence relations for the sections of the generating series of the solution to the multidimensional difference equation

2021 ◽  
Vol 31 (3) ◽  
pp. 414-423
Author(s):  
A.P. Lyapin ◽  
S.S. Akhtamova
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Difference Equation ◽  
Recurrence Relations ◽  
Multidimensional Analogue ◽  
Recurrent Relations ◽  
Generating Series ◽  
The Cauchy Problem ◽  
Constant Coefficients ◽  
Multidimensional Difference Equation

In this paper, we study the sections of the generating series for solutions to a linear multidimensional difference equation with constant coefficients and find recurrent relations for these sections. As a consequence, a multidimensional analogue of Moivre's theorem on the rationality of sections of the generating series depending on the form of the initial data of the Cauchy problem for a multidimensional difference equation is proved. For problems on the number of paths on an integer lattice, it is shown that the sections of their generating series represent the well-known sequences of polynomials (Fibonacci, Pell, etc.) with a suitable choice of steps.

A Data Structure for Disjoint Sets

Genomic Perl ◽  
2002 ◽  
pp. 313-317
Keyword(s):  
Data Structure ◽  
Disjoint Sets

Latent psychological data structure of secondary driving tasks

2004 ◽  
Author(s):  
Reates Curry
Keyword(s):  
Data Structure ◽  
Psychological Data

Remarks on nonlinear elastic waves with null conditions

2020 ◽  
Vol 26 ◽  
pp. 121
Author(s):  
Dongbing Zha ◽  
Weimin Peng
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Elastic Waves ◽  
Wave Equations ◽  
Alternative Proof ◽  
Classical Solutions ◽  
Small Initial Data ◽  
Hyperelastic Materials ◽  
Variational Structure ◽  
Nonlinear Elastic

For the Cauchy problem of nonlinear elastic wave equations for 3D isotropic, homogeneous and hyperelastic materials with null conditions, global existence of classical solutions with small initial data was proved in R. Agemi (Invent. Math. 142 (2000) 225–250) and T. C. Sideris (Ann. Math. 151 (2000) 849–874) independently. In this paper, we will give some remarks and an alternative proof for it. First, we give the explicit variational structure of nonlinear elastic waves. Thus we can identify whether materials satisfy the null condition by checking the stored energy function directly. Furthermore, by some careful analyses on the nonlinear structure, we show that the Helmholtz projection, which is usually considered to be ill-suited for nonlinear analysis, can be in fact used to show the global existence result. We also improve the amount of Sobolev regularity of initial data, which seems optimal in the framework of classical solutions.

Myocarditis in Paediatric Patients: Age and Gender Are Influencing Incidence, Severity, and Clinical Course: Initial Data Analysis from the German Multi-Centre Registry (“MYKKE”)

2016 ◽  
Vol 64 (S 02) ◽  
Author(s):  
S. Schubert ◽  
B. Opgen-Rhein ◽  
G. Haverkämper ◽  
E. zu Knyphausen ◽  
A. Weise ◽  
...  
Keyword(s):  
Data Analysis ◽  
Initial Data ◽  
Clinical Course ◽  
Age And Gender ◽  
Paediatric Patients ◽  
And Gender
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