Combinatorial proofs of some determinantal identities

In this paper, we give combinatorial proofs of some determinantal identities. In fact, we give a combinatorial proof of a theorem of R. P. Stanley regarding the enumeration of paths in acyclic digraphs along with some interesting applications. We also give an almost visual proof of a recent result of Oliver Knill, namely ‘The generalized Cauchy–Binet Theorem.’     

On some determinantal identities formation laws

We show that Muir’s law of extensible minors, Cayley’s law of complementaries and Jacobi’s identity for minors of the adjugate [Determinantal identities Linear Algebra and its Applications 52/53 (1983) pp. 769–791] are equivalent. We also show our generalization of Mühlbach/Muir’s extension principle [A generalization of Mühlbach’s extension principle for determinantal identities. Linear and Multilinear Algebra 61 (10) (2013) pp. 1363–1376] is equivalent to its previous form derived by Mühlbach. As a corollary, we show that Mühlbach–Gasca–(Lopez-Carmona)–Ramirez identity [A generalization of Sylvester’s identity on determinants and some applications. Linear Algebra and its Applications 66 (1985) pp. 221–234/On extending determinantal identities. Linear Algebra and its Applications 132 (1990) pp. 145–162] is equivalent to its generalization found by Beckermann and Mühlbach [A general determinantal identity of Sylvester type and some applications. Linear Algebra and its Applications 197,198 (1994) pp. 93–112].     

A problem on semigroups satisfying permutation identities

Let σ be a nontrivial permutation of ordern. A semigroupS is said to be σ-permutable ifx ₁ x ₂ ...x _n =x _σ(1) x _σ(2) ...x _σ(n) , for every (x ₁ ,x ₂,...,x _n )∈S ⁿ . A semigroupS is called(r,t)-commutative, wherer,t are in ℕ^*, ifx ₁ ...x _r x _r+1 ...x _r+t =x _r+1 ...x _r+t x ₁ ...x _r , for every (x ₁ ,x ₂,...,x _r+t ∈S ^r+t . According to a result of M. Putcha and A. Yaqub ([11]), if σ is a fixed-point-free permutation andS is a σ-permutable semigroup then there existsk ∈ ℕ^* such thatS is (1,k)-commutative. In [8], S. Lajos raises up the problem to determine the leastk=k(n) ∈ ℕ^* such that, for every fixed-point-free permutation σ of ordern, every σ-permutable semigroup is also (1,k)-commutative. In this paper this problem is solved for anyn less than or equal to eight and also whenn is any odd integer. For doing this we establish that if a semigroup satisfies a permutation identity of ordern then inevitably it also satisfies some permutation identities of ordern+1.     

Abundant semigroups whose idempotents satisfy permutation identities

The aim of this paper is to study abundant semigroups whose idempotents satisfy permutation identities. After some properties are obtained, the quasi-spined product structure of such semigroups is established while some special cases are investigated. In particular, the structure ofPI-abundant semigroups is obtained.     

New analogues of Ramanujan's partition identities

We establish several new analogues of Ramanujan's exact partition identities using the theory of modular functions.     

Some formulas related to the Rogers-Ramanujan identities

Summary The paper contains a number of identities related to theRogers-Ramanujan identities. In particular, the formulas (13) and (14) are generalizations of those identities.     

More hypergeometric identities related to Ramanujan-type series

We find new hypergeometric identities which, in a certain aspect, are stronger than others of the same style found by the author in a previous paper. The identities in Sect. 3 are related to some Ramanujan-type series for 1/π. We derive them by using WZ-pairs associated to some interesting formulas by Wenchang Chu. The identities we prove in Sect. 4 are of the same style but related to Ramanujan-like series for 1/π ².     

Two identities related to dirichlet character of polynomials

Let q be a positive integer, χ denote any Dirichlet character mod q. For any integer m with (m, q) = 1, we define a sum C(χ, k,m; q) analogous to high-dimensional Kloosterman sums as follows: , where a · ā ≡ 1 mod q. The main purpose of this paper is to use elementary methods and properties of Gauss sums to study the computational problem of the absolute value |C(χ, k,m; q)|, and give two interesting identities for it.     

Further determinant identities related to classical root systems

By introducing polynomials in matrix entries, six determinants are evaluated which may be considered extensions of Vandermonde-like determinants related to the classical root systems.     

Four identities related to third-order mock theta functions

The Ramanujan Journal - Ramanujan presented four identities for third-order mock theta functions in his Lost Notebook. In 2005, with the aid of complex analysis, Yesilyurt first proved these four...     

Polynomial identities of related rings

The set of polynomial identities of a ringA is considered, as well as some types of minimal identities. The change which occurs in these identities upon passage to related rings is then studied. This paper was written while both authors were doing their Ph. D. these at the Hebrew University of Jerusalem under the supervision of S. A. Amitsur, to whom they wish to express their warm thanks.     

Obstructions to determinantal representability

There has recently been ample interest in the question of which sets can be represented by linear matrix inequalities (LMIs). A necessary condition is that the set is rigidly convex, and it has been conjectured that rigid convexity is also sufficient. To this end Helton and Vinnikov conjectured that any real zero polynomial admits a determinantal representation with symmetric matrices. We disprove this conjecture. By relating the question of finding LMI representations to the problem of determining whether a polymatroid is representable over the complex numbers, we find a real zero polynomial such that no power of it admits a determinantal representation. The proof uses recent results of Wagner and Wei on matroids with the half-plane property, and the polymatroids associated to hyperbolic polynomials introduced by Gurvits.     

Hook formula and related identities

A new proof of the hook formula for the dimension of representations of the symmetric group is given with the help of identities which are of independent interest. A probabilistic interpretation of the proof and new formulas relating the parameters of the Young diagrams are given.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Akad. Nauk SSSR, Vol. 172, pp. 3–20, 1989.     

A family of identities related to zero-sum and team games

We list and prove a family of binomial identities by calculating in two ways the probabilities of approximate saddlepoints occurring in random m×n matrices. The identities are easily seen to be equivalent to the evaluation of a family of Gauss ₂F₁ polynomials according to a formula of Vandermonde. We also consider some implications concerning the number of approximate pure strategy Nash equilibria we can expect in large matrix zero-sum and team games.