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Determinental formulae for complete symmetric functions

1992
*
Journal of combinatorial theory. Series A
*

Equations expressing each S,,j. as a determinant involving

doi:10.1016/0097-3165(92)90008-i
fatcat:e4cdsunhsvaslconlfk2guqj2q
*complete**symmetric**functions*are well known [4] . ... We let h, = h,(xl, . . . . x,) denote the*complete**symmetric**function*of degree m in the variables xi, x2, . . . . x, (the sum of all distinct monomials of degree m in these variables). ...##
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Ensemble Kernel Methods, Implicit Regularization and Determinantal Point Processes
[article]

2020
*
arXiv
*
pre-print

Our first empirical results indicate that ensemble of ridgeless regressors can be interesting to use

arXiv:2006.13701v3
fatcat:mq7gejrwdvf67m2mwng5j6zjoa
*for*datasets including redundant information. ... Connections between RLS, DPPs and Christoffel*functions*were explored in [FSS19] . Note that guarantees*for*DPP sampling*for*coresets have been derived in [TBA19] . ... Analogous result*for*kDPP sampling The elementary*symmetric*polynomial e k (K) is proportional to the (n − k)-th coefficients of the characteristic polynomial det(tI − K) = n k=0 (−1) k e k (K)t n−k . ...##
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Correlation Functions for β=1 Ensembles of Matrices of Odd Size

2009
*
Journal of statistical physics
*

Using the method of Tracy and Widom we rederive the correlation

doi:10.1007/s10955-009-9771-8
fatcat:u6jri7tb6ratnm36gygs23szai
*functions**for*\beta=1 Hermitian and real asymmetric ensembles of N x N matrices with N odd. ... Introduction The starting point*for*many results concerning the spectral theory of random matrices is the derivation of a*determinental*or Pfaffian form*for*the correlation*functions*of the eigenvalues ... When β = 2, Fubini's Theorem together with elementary row and column operations on the Vandermonde determinant in the integrand lead to the*determinental**formula*R n (λ) = det [K N (λ j , λ k )] n j,k= ...##
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Schur-finiteness in λ-rings
[article]

2011
*
arXiv
*
pre-print

The authors would like to thank Anders Buch, Alessio Del Padrone and Christophe Soulé

arXiv:1011.1444v2
fatcat:znwcdovku5dc7et4vcnoqnunoy
*for*valuable discussions. ...*Symmetric**functions*We devote this section to a quick study of the ring Λ of*symmetric**functions*, and especially the Schur polynomials s π , referring the reader to [Macd]*for*more information. ... Let A be an idempotent-*complete*exact category which is a QTC*for*⊗.*For*any object A in A, the*symmetric*group Σ n (and hence the group ring Q[Σ n ]) acts on the n-fold tensor product A ⊗n . ...##
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Patterns in eigenvalues: the 70th Josiah Willard Gibbs lecture

2003
*
Bulletin of the American Mathematical Society
*

These same patterns appear in telephone encryption, the zeros of Riemann's zeta

doi:10.1090/s0273-0979-03-00975-3
fatcat:fssgipqlcrcojdsgrcu7cjlrcq
*function*, a variety of physics problems, and in the study of Toeplitz operators. ... Change of basis*formulae*between these two sets of*symmetric**functions*give some information. ... Of course, the coefficients are just the elementary*symmetric**functions*in the eigenvalues, and the traces are the power sum*symmetric**functions*. ...##
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Rational solutions of Painleve systems
[article]

2020
*
arXiv
*
pre-print

The upshot of this analysis is a an explicit

arXiv:2009.11668v1
fatcat:y5ptpi5kfbfa7brnlrdedgwhwi
*determinental*representation*for*rational solutions in terms of classical orthogonal polynomials. ... Although the solutions of Painlevé equations are transcendental in the sense that they cannot be expressed in terms of known elementary*functions*, there do exist rational solutions*for*specialized values ... The classical Jacobi-Trudi identity is a*determinental*representation of the Schur polynomials in terms of*complete**symmetric*polynomials. Proposition 4. ...##
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COMBINATORICS OF THE MODULAR GROUP II THE KONTSEVICH INTEGRALS

1992
*
International Journal of Modern Physics A
*

We study algebraic aspects of Kontsevich integrals as generating

doi:10.1142/s0217751x92002581
fatcat:45mvst3we5af3iaexnyu3kwbhe
*functions**for*intersection theory over moduli space and review the derivation of Virasoro and KdV constraints. 1. ... The main theorem 2.2 Expansion of Z on characters and Schur*functions*2.3 Proof of the first part of the Theorem 3. From Grassmannians to KdV 4. ... Bauer*for*his assistance in algebraic calculations as well as in the elaboration of Lemma 5 and P. Ginsparg*for*a critical reading of the manuscript. ...##
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Wishart and anti-Wishart random matrices

2003
*
Journal of Physics A: Mathematical and General
*

We provide a compact exact representation

doi:10.1088/0305-4470/36/12/343
fatcat:nplkgg6k3bcxdcsuxlvonathoi
*for*the distribution of the matrix elements of the Wishart-type random matrices A^† A,*for*any finite number of rows and columns of A, without any large N approximations ... This representation is of interest*for*a procedure of reconstructing the redundant information hidden in Wishart matrices, with potential applications to numerous models based on biological, social and ... Joint eigenvalue distributions In this section,*for**completeness*, we give the results*for*the joint eigenvalue distributions*for*Wishart and Anti-Wishart random matrices. ...##
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Correlations for the Novak process
[article]

2012
*
arXiv
*
pre-print

We study random lozenge tilings of a certain shape in the plane called the Novak half-hexagon, and compute the correlation

arXiv:1201.4138v1
fatcat:wz75ce2uungc3hwzccfci2vfwq
*functions**for*this process. ... The most difficult step in the present paper is to compute the inverse of the matrix whose (i,j) entry is the binomial coefficient C(A, B_j - i)*for*indeterminate variables A and B_1, ..., B_n. ... It is not at all clear how to algebraically relate (22) with the*formula*in [Joh05b, Theorem 3.1], since the latter is a sum involving products of Hahn polynomials. ...##
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Witt vectors. Part 1
[article]

2008
*
arXiv
*
pre-print

(t) (9.59) From (9.59) one readily obtains a

arXiv:0804.3888v1
fatcat:m2awczhf35bkpen2jpzvrwcgwa
*formula**for*the power sum*symmetric**functions*in terms of the*complete**symmetric**functions*, i.e. a universal*formula**for*the ghost components of an element ... These*determinental**formulas*are exactly the same as those linking power sums and elementary*symmetric**functions*in*symmetric**function*theory ([281] , p. 28), which is as must be because the defining ... Every*symmetric*power series can be uniquely written as a power series in the*complete**symmetric**functions*. ...##
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Integral Geometry on Grassmann Manifolds and Calculus of Invariant Differential Operators

1999
*
Journal of Functional Analysis
*

In particular, we will give eigenvalue

doi:10.1006/jfan.1999.3459
fatcat:pjkezq3blbdinezw5kfpmhq6om
*formulas*and radial part*formulas**for*invariant differential operators. Academic Press ... In this paper, we mainly deal with two problems in integral geometry, the range characterizations and construction of inversion*formulas**for*Radon transforms on higher rank Grassmann manifolds. ... We remark that the above inversion*formula*is the generalization of the Helgason's inversion*formula**for*the Radon transform on the real projective space P n&1 R. (See Helgason [H4].) ...##
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General solutions for tunneling of scalar fields with quartic potentials

1993
*
Physical Review D, Particles and fields
*

We find the solutions numerically and use polynomial fitting

doi:10.1103/physrevd.48.2800
pmid:10016526
fatcat:mwxqikipavbdhchlzb6hhtn4z4
*formulae*to obtain expressions*for*the Euclidean action. ...*For*the theory of a single scalar field φ with a quartic potential V(φ), we find semi-analytic expressions*for*the Euclidean action in both four and three dimensions. ... Watkins*for*useful discussions. This work was supported by NASA Grant No. NAGW-2802 and by funds from the Physics Department at the University of Michigan. ...##
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Weighted permutation problems and Laguerre polynomials

1978
*
Journal of combinatorial theory. Series A
*

The connection between integrals of products of Laguerre polynomials, power series coefficients of certain rational

doi:10.1016/0097-3165(78)90020-1
fatcat:zkwilzpxuvbjdf22qej6z7dvqm
*functions*of several variables, and certain numbers of weighted permutation problems ...*For*a: > --I, let A(n, ,..., nk ; a) = (-l)nl+".+n* s m L",,(x) *.* L",,(x) x"e-u2 -2a, -... -(k -1) u*)a+r ' where CT~ is the jth elementary*symmetric**function*in r1 ,..., rk . ...*Formula*(2.7) can be used to decrease any of the Ai's. ...##
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Correlations for the Novak process

2012
*
Discrete Mathematics & Theoretical Computer Science
*

International audience We study random lozenge tilings of a certain shape in the plane called the Novak half-hexagon, and compute the correlation

doi:10.46298/dmtcs.3070
fatcat:zpldz75w5vatnhsvtuthb7et3i
*functions**for*this process. ... The most difficult step in the present paper is to compute the inverse of the matrix whose (i,j)-entry is the binomial coefficient $C(A, B_j-i)$*for*indeterminate variables $A$ and $B_1, \dots , B_n.$ ... It is not at all clear how to algebraically relate (22) with the*formula*in [Joh05b, Theorem 3.1], since the latter is a sum involving products of Hahn polynomials. ...##
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Vacuum Varieties, Holomorphic Bundles and Complex Structure Stabilization in Heterotic Theories
[article]

2013
*
arXiv
*
pre-print

Thus, we have exchanged the problem of computing bundle holomorphy as a

arXiv:1304.2704v1
fatcat:m6f6optj2ja4zgu7o55h6pip7a
*function*of complex structure*for*that of computing cohomology groups of line bundles as a*function*of those variables. ... We recall that the dimension of a Young tableau may be easily computed from the hook-length*formula*(see [63] ,*for*example). ...
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