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Let K be the product O(n_1) x O(n_2) x ... x O(n_r) of orthogonal groups. Let V the r-fold tensor product of defining representations of each orthogonal factor. We compute a stable formula for the dimension of the K-invariant algebra of…

We investigate the geode and some of its generalizations from the point of view on noncommutative symmetric functions.

It is well known that a direct Lagrangian description of radiation damping is still missing. In this paper a specific approach of this problem was used, which is the standard way to treat the radiation damping problem. A $N=2$…

We describe a correspondence between GL_n-invariant tensors and graphs, and show how this correspondence accomodates various types of symmetries and orientations.

Various properties of a class of braid matrices, presented before, are studied considering $N^2 \times N^2 (N=3,4,...)$ vector representations for two subclasses. For $q=1$ the matrices are nontrivial. Triangularity $(\hat R^2 =I)$…

We develop the theory of linear algebra over a (Z_2)^n-commutative algebra (n in N), which includes the well-known super linear algebra as a special case (n=1). Examples of such graded-commutative algebras are the Clifford algebras, in…

We apply the Clebsch-Gordan and Racah coefficients to calculate the double tensors for two equivalent d electrons. We also obtain the commutation relations for these double tensors and choose certain quantum numbers, which produce a…

We show that the space of rooted tree-based phylogenetic networks is connected under rooted nearest-neighbour interchange (rNNI) moves.

We prove the uniqueness theorem for the solutions to the restricted Weyl commutation relations braiding unitary groups and semi-groups of contractions that are close to unitaries. We also discuss related mathematical problems of continuous…

We give the number of nilpotent orbits in the Lie algebras of orthogonal groups under the adjoint action of the groups over $\tF_{2^n}$. Let $G$ be an adjoint algebraic group of type $B,C$ or $D$ defined over an algebraically closed field…

We study petal diagrams of knots, which provide a method of describing knots in terms of permutations in a symmetric group $S_{2n+1}$. We define two classes of moves on such permutations, called trivial petal additions and crossing…

It is emphasized that equivalent definitions of connections on modules over commutative rings are not so in noncommutative geometry.

This article is divided in two parts. In the first part we review some recent results concerning the expected number of real roots of random system of polynomial equations. In the second part we deal with a different problem, namely, the…

The relation between quantum systems associated to root systems and radial parts of Laplace operators on symmetric spaces is established. From this it follows the complete integrability of some quantum systems.

Algebraic statistics for binary random variables is concerned with highly structured algebraic varieties in the space of 2x2x...x2-tensors. We demonstrate the advantages of representing such varieties in the coordinate system of binary…

This article studies the equation $[A,B]^k = {\rm Id}_n$ for matrices over $\mathbb{C}$, characterizing the pairs $(k,n)$ for which solutions exist via a classical result of Lam and Leung on sums of roots of unity. The problem is next…

We prove simplicity for incomplete rank 2 Kac-Moody groups over algebraic closures of finite fields with trivial commutation relations between root groups corresponding to prenilpotent pairs. We don't use the (yet unknown) simplicity of the…

This paper addresses various questions about pairs of similarity classes of matrices which contain commuting elements. In the case of matrices over finite fields, we show that the problem of determining such pairs reduces to a question…

It is known that the variety parametrizing pairs of commuting nilpotent matrices is irreducible and that this provides a proof of the irreducibility of the punctual Hilbert scheme in the plane. We extend this link to the nilpotent commuting…

Given a smooth projective variety $X$ with a smooth nef divisor $D$ and a positive integer $r$, we construct an $I$-function, an explicit slice of Givental's Lagrangian cone, for Gromov--Witten theory of the root stack $X_{D,r}$. As an…