Related papers: On nonnegative invariant quartics in type A

We consider finite-dimensional real vector spaces $X$ ordered by a closed cone $X_+$ with non-empty interior and study eventual nonnegativity of matrix semigroups $(e^{tA})_{t \ge 0}$ with respect to this cone. Our first contribution is the…

Let $0< n,\alpha,\beta\in\mathbb{N}$ be such that $\gcd{(\alpha,\beta)}=1$. We carry out the evaluation of the convolution sums $\underset{\substack{ {(k,l)\in\mathbb{N}^{2}} \\ {\alpha\,k+\beta\,l=n} } }{\sum}\sigma(k)\sigma_{3}(l)$ and…

We verify a conjecture of Emil Artin, for the case of a Cubic and Quadratic form over any $p$-adic field, provided the cardinality of the residue class field exceeds 293. That is any Cubic and Quadratic form with at least 14 variables has a…

We compute the Dijkgraaf-Witten invariants of surfaces in terms of projective representations of groups. As an application we prove that the complex Dijkgraaf-Witten invariants of surfaces of positive genus are positive integers.

Quadric complexes are square complexes satisfying a certain combinatorial nonpositive curvature condition. These complexes generalize 2-dimensional CAT(0) cube complexes and are a square analog of systolic complexes. We introduce and study…

We establish a lower bound for the frequency with which an irreducible monic cubic polynomial with negative discriminant can be expressed as a sum of two squares ($\square_{2}$). This provides a quantitative answer to a question posed by…

In our previous papers we introduced categorical invariants, which are, roughly speaking, sets of triangulated subcategories in a given triangulated category and their quotients. Here is extended the list of examples, where these sets are…

We define and study virtual representation spaces having both positive and negative dimensions at the vertices of a quiver without oriented cycles. We consider the natural semi-invariants on these spaces which we call virtual…

Given a surface F, we are interested in Z/2 valued invariants of immersions of F into R^3, which are constant on each connected component of the complement of the quadruple point discriminant in Imm(F,R^3). Such invariants will be called…

This paper is concerned with the problem of finding $n$ distinct squares such that, on excluding any one of them, the sum of the remaining $n-1$ squares is a square. While parametric solutions are known when $n=3$ and $n=4$, when $n > 4$,…

For every positive integer k, it is shown that there exists a positive definite diagonal quaternary integral quadratic form that represents all positive integers except for precisely those which lie in k arithmetic progressions. For k=1,…

Local unitary invariance and the notion of negativity fonts are used as the principle tools to construct four qubit invariants of degree 8, 12, and 24. A degree 8 polynomial invariant that is non-zero on pure four qubit states with…

An important theorem by Timofte states that nonnegativity of real $n$-variate symmetric polynomials of degree $d$ can be decided at test sets given by all points with at most $\lfloor\frac{d}{2}\rfloor$ distinct components. However, if the…

We have shown quant-ph/0206104 (Lett. Nuovo Cimento, 1972, 4, 344) that for free particles and antiparticles with mass m>0 and arbitrary spin s>0, in the framework of the Poincare group P(1,3), there exist three types of nonequivalent…

We prove the precise inversion of adjunction formula for finite linear group quotients of complete intersection varieties defined by semi-invariant equations. As an application, we prove the semi-continuity of minimal log discrepancies for…

Using the theory of signatures of hermitian forms over algebras with involution, developed by us in earlier work, we introduce a notion of positivity for symmetric elements and prove a noncommutative analogue of Artin's solution to…

We classify holomorphic as well as algebraic torus equivariant principal $G$-bundles over a nonsingular toric variety $X$, where $G$ is a complex linear algebraic group. It is shown that any such bundle over an affine, nonsingular toric…

We characterize mutation-finite cluster algebras of rank at least 3 using positive semi-definite quadratic forms. In particular, we associate with every unpunctured bordered surface a positive semi-definite quadratic space $V$, and with…

For a monic polynomial $f$ over a commutative, unitary ring $A$ the splitting algebra $A_f$ is the universal $A$-algebra such that $f$ splits in $A_f$. The symmetric group acts on the splitting algebra by permuting the roots of $f$. It is…

We study the eta invariants of compact flat spin manifolds of dimension n with holonomy group cyclic of odd prime order p. We find explicit expressions for the twisted and relative eta invariants and show that the reduced eta invariant is…