Related papers: Computational techniques for proving identities in…
Dedicated treatment of symmetries in satisfiability problems (SAT) is indispensable for solving various classes of instances arising in practice. However, the exploitation of symmetries usually takes a black box approach. Typically,…
Galois/monodromy groups attached to parametric systems of polynomial equations provide a method for detecting the existence of symmetries in solution sets. Beyond the question of existence, one would like to compute formulas for these…
We present novel, deterministic, efficient algorithms to compute the symmetries of a planar algebraic curve, implicitly defined, and to check whether or not two given implicit planar algebraic curves are similar, i.e. equal up to a…
Recently the second named author discovered a combinatorial identity in the context of vertex representations of quantum Kac-Moody algebras. We give a direct and elementary proof of this identity. Our method is to show a related identity of…
This article presents a pedagogical probabilistic exploration of the Newton-Girard identities. We show that the coefficients in these classical relations between power sums and elementary symmetric polynomials can be interpreted as the…
We give "hybrid" proofs of the $q$-binomial theorem and other identities. The proofs are "hybrid" in the sense that we use partition arguments to prove a restricted version of the theorem, and then use analytic methods (in the form of the…
In practice symmetries of combinatorial structures are computed by transforming the structure into an annotated graph whose automorphisms correspond exactly to the desired symmetries. An automorphism solver is then employed to compute the…
A probability method is provided to prove three classes of combinatorial identities. The method is extremely simple, only one step after the proper probability setup.
We use geometry to prove a number of new identities among the Littlewood-Richardson coefficients for Schubert polynomials (Schubert classes in a flag manifold). For many of these identities, there is a companion result about the Bruhat…
Proving correctness of distributed or concurrent algorithms is a mind-challenging and complex process. Slight errors in the reasoning are difficult to find, calling for computer-checked proof systems. In order to build computer-checked…
Ontologies usually suffer from the semantic heterogeneity when simultaneously used in information sharing, merging, integrating and querying processes. Therefore, the similarity identification between ontologies being used becomes a…
We obtain identities involving symmetric and doubly symmetric polynomials. These identities provide a way of handling expressions appearing in the Atiyah-Bott-Berline-Vergne formula for Grassmannians. As corollaries, we obtain formulas for…
We give new identities for some symmetric polynomials. As applications of these identities, we obtain some formulas for a higher order analogue of Fibonacci and Lucas numbers.
Let A, B, C, D be given finite sets of pairs of n-by-n complex matrices. We describe an algorithm to determine, with finitely many computations, whether there is a single unitary matrix U such that each pair of matrices in A is unitarily…
Despite the effort put into the detection of academic plagiarism, it continues to be a ubiquitous problem spanning all disciplines. Various tools have been developed to assist human inspectors by automatically identifying suspicious…
A generalization of Newton's identity on symmetric functions is given. Using the generalized Newton identity we give a unified method to show the existence of Hall-Littlewood, Jack and Macdonald polynomials. We also give a simple proof of…
This paper presents both a proof method and a result. The proof method presented is particularly suitable for uniformly proving families of identities satisfied by a family of recursive sequences. To illustrate the method, we study the…
Matching Logic is a framework for specifying programming language semantics and reasoning about programs. Its formulas are called patterns and are built with variables, symbols, connectives and quantifiers. A pattern is a combination of…
Data synthesis has been advocated as an important approach for utilizing data while protecting data privacy. In recent years, a plethora of tabular data synthesis algorithms (i.e., synthesizers) have been proposed. Some synthesizers satisfy…
We present a new identity involving compositions (i.e. ordered partitions of natural numbers). The Formula has its origin in complex dynamical systems and appears when counting, in the polynomial family $\{f_c:z \mapsto z^d + c \}$,…