Related papers: Exponential bases on multi-rectangles in $\mathbb{…
In this note, we presented a new decomposition of elements of finite fields of even order and illustrated that it is an effective tool in evaluation of some specific exponential sums over finite fields, the explicit value of some…
We extend the concept of exponential B-spline to complex orders. This extension contains as special cases the class of exponential splines and also the class of polynomial B-splines of complex order. We derive a time domain representation…
We provide a construction of sets of (d/2+1) mutually unbiased bases (MUBs) in dimensions d=4,8 using maximal commuting classes of Pauli operators. We show that these incomplete sets cannot be extended further using the operators of the…
We extend Bravyi and Smolin's construction for obtaining unextendible maximally entangled bases (UMEBs) from equiangular lines. We show that equiangular real projections of rank more than 1 also exhibit examples of UMEBs. These projections…
We study the collection of group structures that can be realized as a group of rational points on an elliptic curve over a finite field (such groups are well known to be of rank at most two). We also study various subsets of this collection…
We provide a complete and explicit characterization of the exposed extreme rays of the cone of sums of nonnegative circuit (SONC) polynomials. The criterion we derive is purely combinatorial and depends only on the existence of certain…
We prove asymptotic formulas for the number of rational points of bounded height on certain equivariant compactifications of the affine plane.
The theory of modular forms and spherical harmonic analysis are applied to establish new best bounds towards the counting and equidistribution of rational points on spheres and other higher dimensional ellipsoids, in what may be viewed as a…
The perturbative expansion of two-point functions of lowest dimension supersymmetric operators in $\mathcal{N}=4$ SYM and ABJM theory exhibits uniform transcendental weight. Inspired by this, we construct an explicit basis of uniformly…
In this article, we revisit some block matrix construction methods and use them to derive various general expansion formulas for calculating the ranks of matrix expressions. As applications, we derive a variety of interesting rank…
The asymptotic behaviour is studied of exponentially bounded sequences of codimensions of identities of algebras with unity. A series of algebras is constructed for which the base of the exponential increases by exactly one when an outer…
Using matrix identities, we construct explicit pseudo-exponential-type solutions of linear Dirac, Loewner and Schr\"odinger equations depending on two variables and of nonlinear wave equations depending on three variables.
We obtain sufficient conditions for an exponential type entire function not to have zeros in the open lower half-plane. An exact inequality containing the real and imaginary parts of such functions and their derivatives restricted to the…
We give an estimate of exponential sums over singular binary quintic forms in a characteristic-free form, based on the Waring decomposition of binary forms. This extends the method on our preceding result on the space of binary quartics to…
I provide a systematic construction of points, defined over finite radical extensions, on any Legendre curve over any field of characteristic not equal two. This includes as special case Douglas Ulmer's construction of rational points over…
Here I present several theorems about trapezoids tilings. The first one is related to trapezoids with rational base relation, the other ones are related to those with base relation from quadratic number field.
Suppose that for some unit vectors $b_1,\ldots b_n$ in $\mathbb C^d$ we have that for any $j\neq k$ $b_j$ is either orthogonal to $b_k$ or $|\langle b_j,b_k\rangle|^2 = 1/d$ (i.e. $b_j$ and $b_k$ are unbiased). We prove that if $n=d(d+1)$,…
A determination of the fixed components, base points and irregularity is made for arbitrary numerically effective divisors on any smooth projective rational surface having an effective anticanonical divisor. All of the results are proven…
Mutually Unbiased Bases (MUBs) are closely connected with quantum physics, and the structure has a rich mathematical background. We provide equivalent criteria for extending a set of MUBs for $C^n$ by studying real points of a certain…
Let $\mathbb F_{q^2}$ be the finite field with $q^2$ elements. We provide a simple and effective method, using reciprocal polynomials, for the construction of algebraic curves over $\mathbb F_{q^2}$ with many rational points. The curves…