Related papers: The Minimum Number of Rotations About Two Axes for…

It is well known that a rigid motion of the Euclidean plane can be written as the composition of at most three reflections. It is perhaps not so widely known that a similar result holds for Euclidean space in any number of dimensions. The…

The approximate discrete Radon transform (ADRT) is a hierarchical multiscale approximation of the Radon transform. In this paper, we factor the ADRT into a product of linear transforms that resemble convolutions and derive an explicit…

We give an almost-complete description of orthogonal matrices $M$ of order $n$ that "rotate a non-negligible fraction of the Boolean hypercube $C_n=\{-1,1\}^n$ onto itself," in the sense that $$P_{x\in C_n}(Mx\in C_n) \ge n^{-C},\mbox{ for…

For a tuple $T$ of Hilbert space operators, the 'commuting dilation constant' is the smallest number $c$ such that the operators of $T$ are a simultaneous compression of commuting normal operators of norm at most $c$. We present numerical…

Rota's basis conjecture, open since 1989, states that if B_1, B_2, ..., B_n are n bases of a vector space of rank n, then there is an nxn grid of vectors such that the vectors in the ith row are precisely the elements of B_i and such that…

For the second fundamental representation of the general linear group over a commutative ring $R$ we construct straightforward and uniform polynomial expressions of elementary generators as products of elementary conjugates of an arbitrary…

A recurrence scheme is presented to decompose an $n$-qubit unitary gate to the product of no more than $N(N-1)/2$ single qubit gates with small number of controls, where $N = 2^n$. Detailed description of the recurrence steps and formulas…

What is the smallest number of random transpositions (meaning that we swap given pairs of elements with given probabilities) that we can make on an $n$-point set to ensure that each element is uniformly distributed -- in the sense that the…

Let R be a unital commutative ring and let $M$ be an $R$-module that is generated by $k$ elements but not less. Let $E_n(R)$ be the subgroup of $GL_n(R)$ generated by the elementary matrices. In this paper we study the action of $E_n(R)$ by…

We consider generators of algebraic curvature tensors R which can be constructed by a Young symmetrization of product tensors U*w or w*U, where U and w are covariant tensors of order 3 and 1. We assume that U belongs to a class of the…

We propose a simple physical implementation of the quantum Householder reflection (QHR) M(v)=I-2|v><v| in a quantum system of N degenerate states (forming a qunit) coupled simultaneously to an ancillary (excited) state by N resonant or…

We reduce the problem of integer quantum Hall transition to a random rotation of an N-dimensional vector by an su(N) algebra, where only N specially selected generators of the algebra are nonzero. The group-theoretical structure revealed in…

Let $A$ be an element of the copositive cone ${\cal C}_n$. A zero $u$ of $A$ is a nonzero nonnegative vector such that $u^TAu = 0$. The support of $u$ is the index set $\mbox{supp}u \subset \{1,\dots,n\}$ corresponding to the positive…

We provide a new way to represent numerical semigroups by showing that the position of every Ap\'ery set of a numerical semigroup $S$ in the enumeration of the elements of $S$ is unique, and that $S$ can be re-constructed from this…

We construct, for each irrational number $\alpha$, a minimal $C^1$-diffeomorphism of the circle with rotation number $\alpha$ which admits a measur

Irrespective of whether n is prime, prime power with exponent >1, or composite, the group U_n of units of Z_n can sometimes be obtained as the direct product of cyclic groups generated by x, x+k and x+2k, for x, k in Z_n. Indeed, for many…

We have developed a numerical code to study the deformation ($\varepsilon = (I_{zz}-I_{xx})/I_{zz}$, where $I_{ii}$ are the moments of inertia) of neutron stars in rapidly rotation in a fully general relativistic calculation. We have found…

Uniform random rotations are a useful primitive in applications such as fast Johnson-Lindenstrauss embeddings, kernel approximation, communication-efficient learning, and recent AI compression pipelines, but they are computationally…

Any unitary transformation of quantum computational networks is explicitly decomposed, in an exact and unified form, into a sequence of a limited number of one-qubit quantum gates and the two-qubit diagonal gates that have diagonal unitary…

Let $G$ be a group. The orbits of the natural action of $Aut(G)$ on $G$ are called the automorphism orbits of $G$, and their number is denoted by $\omega(G)$. Let $\mathbb{F}$ be an infinite field, and let $UT_n(\mathbb{F})$ denote the…