The Efficient Evaluation of the Hypergeometric Function of a Matrix Argument

We present new algorithms that efficiently approximate the hypergeometric function of a matrix argument through its expansion as a series of Jack functions. Our algorithms exploit the combinatorial properties of the Jack function, and have complexity that is only linear in the size of the matrix.Comment: 14 pages, 3 figure

Computing with rational symmetric functions and applications to invariant theory and PI-algebras

Let the formal power series f in d variables with coefficients in an arbitrary field be a symmetric function decomposed as a series of Schur functions, and let f be a rational function whose denominator is a product of binomials of the form (1 - monomial). We use a classical combinatorial method of Elliott of 1903 further developed in the Partition Analysis of MacMahon in 1916 to compute the generating function of the multiplicities (i.e., the coefficients) of the Schur functions in the expression of f. It is a rational function with denominator of a similar form as f. We apply the method to several problems on symmetric algebras, as well as problems in classical invariant theory, algebras with polynomial identities, and noncommutative invariant theory.Comment: 37 page

Eigenvalue distributions of beta-Wishart matrices

We derive explicit expressions for the distributions of the extreme eigenvalues of the Beta-Wishart random matrices in terms of the hypergeometric function of a matrix argument. These results generalize the classical results for the real (β = 1), complex (β = 2), and quaternion (β = 4) Wishart matrices to any β > 0

Smart-entrepreneurship education in training of the hotel business specialists

The article is devoted to the problems of introducing the SMART-education technology in the training and development of personnel of hotel complexes and business activities in the field of hotel business. The methodological and organizational bases for the application of SMART-education in staff training were identified; the leading qualitative features and development trends of this type of entrepreneurial educational activity were outlined. The principles of SMART-education of staff in the field of hotel business and its applied features in the service sector were developed. A model of SMART- education of hotel complex staff was developed based on solving case problems and practical mastering of professional content

Accurate and Efficient Expression Evaluation and Linear Algebra

We survey and unify recent results on the existence of accurate algorithms for evaluating multivariate polynomials, and more generally for accurate numerical linear algebra with structured matrices. By "accurate" we mean that the computed answer has relative error less than 1, i.e., has some correct leading digits. We also address efficiency, by which we mean algorithms that run in polynomial time in the size of the input. Our results will depend strongly on the model of arithmetic: Most of our results will use the so-called Traditional Model (TM). We give a set of necessary and sufficient conditions to decide whether a high accuracy algorithm exists in the TM, and describe progress toward a decision procedure that will take any problem and provide either a high accuracy algorithm or a proof that none exists. When no accurate algorithm exists in the TM, it is natural to extend the set of available accurate operations by a library of additional operations, such as $x+y+z$, dot products, or indeed any enumerable set which could then be used to build further accurate algorithms. We show how our accurate algorithms and decision procedure for finding them extend to this case. Finally, we address other models of arithmetic, and the relationship between (im)possibility in the TM and (in)efficient algorithms operating on numbers represented as bit strings.Comment: 49 pages, 6 figures, 1 tabl

Eigenvalue distributions for some correlated complex sample covariance matrices

The distributions of the smallest and largest eigenvalues for the matrix product $Z^\dagger Z$, where $Z$ is an $n \times m$ complex Gaussian matrix with correlations both along rows and down columns, are expressed as $m \times m$ determinants. In the case of correlation along rows, these expressions are computationally more efficient than those involving sums over partitions and Schur polynomials reported recently for the same distributions.Comment: 11 page

Parentheticality, assertion strength, and discourse

Sentences with so-called SLIFTING PARENTHETICALS (e.g. The dean, Jill said, flirted with the secretary; Ross 1973) grammaticalize an intriguing interaction between speech act function and conventional meaning, one that is not found in regular embedding constructions (e.g. Jill said that the dean flirted with the secretary). In such sentences, the main clause is independently asserted and at the same time is interpreted in the scope of the parenthetical, which typically serves an evidential function. The discourse effect of this pragmasemantic setup is that slifting parentheticals modulate the strength with which the main part of the sentence is asserted (Urmson 1952, Asher 2000, Rooryck 2001, Davis et al. 2007, Simons 2007, Maier and Bary 2015). Building on Davis et al. (2007), this paper proposes a probabilistic discourse model that captures the role of parentheticality as a language tool for qualifying speaker’s commitments. The model also derives two empirical properties that set apart slifting parentheticals from regular embedding constructions, i.e. (i) the fact that slifting parentheticals invariably express upward entailing operators and (ii) the fact that they usually do not occur in subordinate clauses