Related papers: Quantifying singularities with differential operat…

The signature kernel is a positive definite kernel for sequential data. It inherits theoretical guarantees from stochastic analysis, has efficient algorithms for computation, and shows strong empirical performance. In this short survey…

We find sufficient conditions which imply equality of the finitistic test ideal and test ideal in rings of prime characteristic. Utilizing recent progress from the prime characteristic minimal model program we equate the notions of…

We give a new sufficient condition which allows to test primality of Fermat's numbers. This characterization uses uniquely values at most equal to tested Fermat number. The robustness of this result is due to a strict use of elementary…

The notion of $F$-signature is defined by C. Huneke and G. Leuschke and this numerical invariant characterizes some singularities. This notion is extended to finitely generated modules and called dual $F$-signature. In this paper, we…

We explore the singularity classes $F$-nilpotent, weakly $F$-nilpotent, and generalized weakly $F$-nilpotent under faithfully flat local ring maps. As an application, we show that the loci of primes in a Noetherian ring of prime…

Inspired by methods in prime characteristic in commutative algebra, we introduce and study combinatorial invariants of seminormal monoids. We relate such numbers with the singularities and homological invariants of the semigroup ring…

Invariants withstand transformations and, therefore, represent the essence of objects or phenomena. In mathematics, transformations often constitute a group action. Since the 19th century, studying the structure of various types of…

The entanglement characteristics of two qubits are encoded in the invariants of the adjoint action of SU(2) x SU(2) group on the space of density matrices defined as the space of positive semi-definite Hermitian matrices. The corresponding…

The local zero structure of a smooth map may qualitatively change, when the map is subjected to small perturbations. The changes may include births and/or deaths of zeros. The qualitative properties are defined as the invariances of an…

A singularity in characteristic zero is said to be of dense F-pure type if its modulo p reduction is locally F-split for infinitely many p. We prove that if $x \in X$ is an isolated log canonical singularity with $\mu(x \in X) \le 2$ (see…

We characterize the set of properties of Boolean-valued functions on a finite domain $\mathcal{X}$ that are testable with a constant number of samples. Specifically, we show that a property $\mathcal{P}$ is testable with a constant number…

The causal character of singularities is often studied in relation to the existence of naked singularities and the subsequent possible violation of the cosmic censorship conjecture. Generally one constructs a model in the framework of…

This paper studies properties of certain hypersurfaces in prime characteristic: we give a sufficient and necessary conditions for some classes of such hypersurfaces to have Finite $F$-representation Type (FFRT) and we compute the…

Given a finite group G and a field F, a G-set X gives rise to an F[G]-permutation module F[X]. This defines a map from the Burnside ring of G to its representation ring over F. It is an old problem in representation theory, with…

For a real matrix $M$, we denote by $sp(M)$ the spectrum of $M$ and by $\left \vert M\right \vert $ its absolute value, that is the matrix obtained from $M$ by replacing each entry of $M$ by its absolute value. Let $A$ be a nonnegative real…

The qualitative properties of local random invariant manifolds for stochastic partial differential equations with quadratic nonlinearities and multiplicative noise is studied by a cut off technique. By a detail estimates on the Perron fixed…

The map x -> x^x modulo p is related to a variation of the digital signature scheme in a similar way to the discrete exponentiation map, but it has received much less study. We explore the number of fixed points of this map by a statistical…

Let R=k[x_1,...,x_d] be the polynomial ring in d independent variables, where k is a field of characteristic p>0. Let D be the ring of k-linear differential operators of R and let f be a polynomial in R. In this work we prove that the…

Given a non-zero polynomial $f$ in a polynomial ring $R$ with coefficients in a finite field of prime characteristic $p$, we present an algorithm to compute a differential operator $\delta$ which raises $1/f$ to its $p$th power. For some…

In characteristic zero, Zinovy Reichstein and the author generalized the usual relationship between irreducible Zariski closed subsets of the affine space, their defining ideals, coordinate rings, and function fields, to a non-commutative…