scholarly journals Hidden Symmetries of Weighted Lozenge Tilings

10.37236/9498 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Igor Pak ◽  
Fedor Petrov
Keyword(s):  
Partition Function ◽  
Schur Functions ◽  
Rational Functions ◽  
Hidden Symmetries ◽  
Lozenge Tilings ◽  
Large Families ◽  
Algebraic Proofs ◽  
Weighted Partition

We study the weighted partition function for lozenge tilings, with weights given by multivariate rational functions originally defined by Morales, Pak and Panova (2019) in the context of the factorial Schur functions. We prove that this partition function is symmetric for large families of regions. We employ both combinatorial and algebraic proofs.

scholarly journals Solution of a multiple Nevanlinna–Pick problem for Schur functions via orthogonal rational functions

Analysis ◽  
2008 ◽  
Vol 28 (1) ◽  
Author(s):  
Adhemar Bultheel ◽  
Andreas Lasarow
Keyword(s):  
Unique Solution ◽  
Interpolation Problem ◽  
Schur Functions ◽  
Rational Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Orthogonal Rational Functions ◽  
All Solutions ◽  
Pick Problem ◽  
Complex Valued

An interpolation problem of Nevanlinna–Pick type for complex-valued Schur functions in the open unit disk is considered. We prescribe the values of the function and its derivatives up to a certain order at finitely many points. Primarily, we study the case that there exist many Schur functions fulfilling the required conditions. For this situation, an application of the theory of orthogonal rational functions is used to characterize the set of all solutions of the problem in question. Moreover, we treat briefly the case of exactly one solution and present an explicit description of the unique solution in that case.

Inverse Problems for Partition Functions

2001 ◽  
Vol 53 (4) ◽  
pp. 866-896
Author(s):  
Yifan Yang
Keyword(s):  
Asymptotic Behavior ◽  
Partition Function ◽  
Inverse Problems ◽  
Large Class ◽  
Generating Function ◽  
Riemann Hypothesis ◽  
Arithmetic Function ◽  
Partition Functions ◽  
Class Of Functions ◽  
Weighted Partition

AbstractLet pw(n) be the weighted partition function defined by the generating function , where w(m) is a non-negative arithmetic function. Let be the summatory functions for pw(n) and w(n), respectively. Generalizing results of G. A. Freiman and E. E. Kohlbecker, we show that, for a large class of functions Φ(u) and λ(u), an estimate for Pw(u) of the formlog Pw(u) = Φ(u){1 + Ou(1/λ(u))} (u→∞) implies an estimate forNw(u) of the formNw(u) = Φ*(u){1+O(1/ log ƛ(u))} (u→∞) with a suitable function Φ*(u) defined in terms of Φ(u). We apply this result and related results to obtain characterizations of the Riemann Hypothesis and the Generalized Riemann Hypothesis in terms of the asymptotic behavior of certain weighted partition functions.

scholarly journals SCHUR–NEVANLINNA SEQUENCES OF RATIONAL FUNCTIONS

2007 ◽  
Vol 50 (3) ◽  
pp. 571-596 ◽  
Author(s):  
Adhemar Bultheel ◽  
Andreas Lasarow
Keyword(s):  
Unit Circle ◽  
Interpolation Problem ◽  
Schur Functions ◽  
Rational Functions ◽  
The Other ◽  
Complex Numbers ◽  
Orthogonal Rational Functions ◽  
All Solutions ◽  
The One

AbstractWe study certain sequences of rational functions with poles outside the unit circle. Such kinds of sequences are recursively constructed based on sequences of complex numbers with norm less than one. In fact, such sequences are closely related to the Schur–Nevanlinna algorithm for Schur functions on the one hand, and to orthogonal rational functions on the unit circle on the other. We shall see that rational functions belonging to a Schur–Nevanlinna sequence can be used to parametrize the set of all solutions of an interpolation problem of Nevanlinna–Pick type for Schur functions.

Generalized factorial Schur functions, biorthogonal rational functions and the RIIchain

2011 ◽  
Vol 44 (39) ◽  
pp. 395202 ◽  
Author(s):  
Ryosuke Miyaura ◽  
Atsushi Mukaihira
Keyword(s):  
Schur Functions ◽  
Rational Functions

scholarly journals Universal character, phase model and topological strings on $$\pmb {\mathbb {C}^3}$$

2019 ◽  
Vol 79 (11) ◽  
Author(s):  
Na Wang ◽  
Chuanzhong Li
Keyword(s):  
Partition Function ◽  
Topological Strings ◽  
Monodromy Matrix ◽  
Topological String ◽  
Schur Functions ◽  
Phase Model ◽  
Vertex Operators ◽  
Strongly Correlated

AbstractIn this paper, we consider two different subjects: the algebra of universal characters $$S_{[\lambda ,\mu ]}(\mathbf{x},\mathbf{y})$$S[λ,μ](x,y) (a generalization of Schur functions) and the phase model of strongly correlated bosons. We find that the two-site generalized phase model can be realized in the algebra of universal characters, and the entries in the monodromy matrix of the phase model can be represented by the vertex operators $$\Gamma _i^\pm (z) (i=1,2)$$Γi±(z)(i=1,2) which generate universal characters. Meanwhile, we find that these vertex operators can also be used to obtain the A-model topological string partition function on $$\mathbb {C}^3$$C3.

scholarly journals On a certain weighted partition function

1950 ◽  
Vol 1 (2) ◽  
pp. 192-192 ◽  
Author(s):  
Nelson A. Brigham
Keyword(s):  
Partition Function ◽  
Weighted Partition

scholarly journals Asymptotics for the number of standard tableaux of skew shape and for weighted lozenge tilings

2021 ◽  
pp. 1-24
Author(s):  
Alejandro H. Morales ◽  
Igor Pak ◽  
Martin Tassy
Keyword(s):  
Variational Principle ◽  
Partition Function ◽  
Stable Limit ◽  
Young Tableaux ◽  
Limit Shape ◽  
Lozenge Tilings ◽  
Standard Young Tableaux

Abstract We prove and generalise a conjecture in [MPP4] about the asymptotics of $\frac{1}{\sqrt{n!}} f^{\lambda/\mu}$ , where $f^{\lambda/\mu}$ is the number of standard Young tableaux of skew shape $\lambda/\mu$ which have stable limit shape under the $1/\sqrt{n}$ scaling. The proof is based on the variational principle on the partition function of certain weighted lozenge tilings.

Factorial-type Schur functions, orthogonal rational functions, and discrete dressing chains

2016 ◽  
Vol 57 (5) ◽  
pp. 052701
Author(s):  
Ryosuke Miyaura ◽  
Atsushi Mukaihira
Keyword(s):  
Schur Functions ◽  
Rational Functions ◽  
Orthogonal Rational Functions

The Hebridean Blackhouse on the Isle of Barra

2000 ◽  
Vol 22 (1) ◽  
pp. 1-16 ◽  
Author(s):  
KEITH BRANIGAN ◽  
COLIN MERRONY
Keyword(s):  
Nineteenth Century ◽  
Internal Space ◽  
Use Of Space ◽  
Large Families ◽  
Human Use

The Hebridean blackhouse is a well-known part of the eighteenth and nineteenth century landscape of the Western Isles, described by numerous early travellers and preserved for posterity at Arnol in Lewis. Survey and excavation of blackhouses on the Isle of Barra, however, suggests that here at least, the majority of blackhouses did not conform to the 'norm' of a long building with accommodation shared by animals and humans. Despite the large families of the Catholic population of Barra, the houses are shorter and provide less internal space than blackhouses further north in the island chain. Animals were more often housed in separate byres. Similarly, the human use of space in the Barra blackhouses shows some variations from the pattern described by nineteenth century sources. As to the origins of the blackhouse, unexcavated sites on Barra suggest two possible future routes of enquiry.

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