Related papers: Equivariant Hilbert series for hierarchical models
We study the Hilbert series of a family of ideals J_\phi generated by powers of linear forms in k[x_1,...,x_n]. Using the results of Emsalem-Iarrobino, we formulate this as a question about fatpoints in P^{n-1}. In the three variable case…
We study an operation in matroid theory that allows one to transition a given matroid into another with more bases via relaxing a \emph{stressed subset}. This framework provides a new combinatorial characterization of the class of split…
In this study, we introduce a method for learning group (known or unknown) equivariant functions by learning the associated quadratic form $x^T A x$ corresponding to the group from the data. Certain groups, known as orthogonal groups,…
We develop the theory of categories of measurable fields of Hilbert spaces and bounded fields of bounded operators. We examine classes of functors and natural transformations with good measure theoretic properties, providing in the end a…
We encode the binomials belonging to the toric ideal $I_A$ associated with an integral $d \times n$ matrix $A$ using a short sum of rational functions as introduced by Barvinok \cite{bar,newbar}. Under the assumption that $d,n$ are fixed,…
We develop a Hilbert space framework for a number of general multi-scale problems from dynamics. The aim is to identify a spectral theory for a class of systems based on iterations of a non-invertible endomorphism. We are motivated by the…
We obtain multiplicity results for a class of first-order superquadratic Hamiltonian systems and a class of indefinite superquadratic elliptic systems which lead to the study of strongly indefinite functionals. There is no assumption to the…
Continuous first-order logic is used to apply model-theoretic analysis to analytic structures (e.g. Hilbert spaces, Banach spaces, probability spaces, etc.). Classical computable model theory is used to examine the algorithmic structure of…
Let BG be the classifying space of an algebraic group over the complex field C. We compute a new stable rational invariant defined by the difference of two coniveau filtrations (by Benoist and Ottem) of a (projective) approximation for BG.
We introduce an efficient way, called Newton algorithm, to study arbitrary ideals in C[[x,y]], using a finite succession of Newton polygons. We codify most of the data of the algorithm in a useful combinatorial object, the Newton tree. For…
Given any finite quiver, we consider a complete flag of vector spaces over each vertex. Consider the unipotent invariant subalgebra of the coordinate ring of the filtered quiver representation subspace. We prove that the dimension of the…
We compute the Hilbert series of the graded algebra of regular functions on a symplectic quotient of a unitary circle representation. Additionally, we elaborate explicit formulas for the lowest coefficients of the Laurent expansion of such…
This work is concerned with the convex analysis of functions defined on (not necessarily finite-dimensional) Hilbert spaces whose values depend solely on a certain ``spectrum'' of the arguments, a class we term ``spectral functions.'' We…
We find the complete equivalence group of a class of (1+1)-dimensional second-order evolution equations, which is infinite-dimensional. The equivariant moving frame methodology is invoked to construct, in the regular case of the…
In this paper, we give a way to construct graded filtrations of graded modules. We then apply it to the Sally module, which describes a correction term of the Hilbert function. As a result, we obtain the inequality of the Hilbert…
We study and classify topologically invariant $\sigma$-ideals with a Borel base on the Hilbert cube and evaluate their cardinal characteristics. One of the results of this paper solves (positively) a known problem whether the minimal…
We consider a Fermat curve $F_n:x^n+y^n+z^n=1$ over an algebraically closed field $k$ of characteristic $p\geq0$ and study the action of the automorphism group $G=\left(\mathbb{Z}/n\mathbb{Z}\times\mathbb{Z}/n\mathbb{Z}\right)\rtimes S_3$…
We develop a new approach to the representation theory of quantum algebras supporting a torus action via methods from the theory of finite-state automata and algebraic combinatorics. We show that for a fixed number $m$, the torus-invariant…
In modular invariant models of flavor, observables must be modular invariant. The observables discussed so far in the literature are functions of the modulus $\tau$ and its conjugate, $\bar\tau$. We point out that certain combinations of…
This article presents an overview of the theory of integrable systems with symmetries, focusing on toric systems, semitoric systems, and their classifications via decorated polygons. We discuss certain one-parameter families of integrable…