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We show that semi-infinite cohomology of a finite dimensional graded algebra (satisfying some additional requirements) are a particular case of a general categorical construction. The motivating example is provided by small quantum groups…

A generalization of a-maximization is proposed that maximizes a subject to inequalities rather than equalities. The implication of this conjecture is that in the absence of emergent symmetries, there is a maximum R-charge for fields…

We study families of superelliptic curves with fixed automorphism groups. Such families are parametrized with invariants expressed in terms of the coefficients of the curves. Algebraic relations among such invariants determine the lattice…

For the Alt-Caffarelli problem, we study free boundary regularity of energy minimizers. In six dimensions, we show that free boundaries are analytic for generic boundary data. In general, we improve previous generic Hausdorff dimensions of…

We verify a confluence result for the rewriting calculus of the linear category introduced in our previous paper. Together with the termination result proved therein, the generalized coherence theorem for linear category is established.…

We devise a new criterion for linear independence over function fields. Using this tool in the setting of dual t-motives, we find that all algebraic relations among special values of the geometric function field Gamma-function are explained…

Let $A$ be an associative algebra over a field $F$ of characteristic zero and let $L$ be a Lie algebra over $F$. If $L$ acts on $A$ by derivations, then such an action determines an action of its universal enveloping algebra $U(L)$ and in…

Many statistical models are algebraic in that they are defined in terms of polynomial constraints, or in terms of polynomial or rational parametrizations. The parameter spaces of such models are typically semi-algebraic subsets of the…

The Euler characteristic of a finite category is defined and shown to be compatible with Euler characteristics of other types of object, including orbifolds. A formula for the cardinality of the colimit of a diagram of sets is proved,…

In this paper, we introduce the concepts of representation and dual representation for averaging Leibniz algebras. We also develop a cohomology theory for these algebras. Additionally, we explore the infinitesimal and formal deformation…

We describe symmetry structure of a general singular theory (theory with constraints in the Hamiltonian formulation), and, in particular, we relate the structure of gauge transformations with the constraint structure. We show that any…

We carry out a detailed Lie point symmetry group classification of the Li\'enard type equation, $\ddot{x}+f(x)\dot{x}+g(x) = 0$, where $f(x)$ and $g(x)$ are arbitrary smooth functions of $x$. We divide our analysis into two parts. In the…

We generalize the notion of quasielliptic curves, which have infinitesimal symmetries and exist only in characteristic two and three, to a remarkable hierarchy of regular curves having infinitesimal symmetries, defined in all…

We study a class of nonlinear evolution systems of time fractional partial differential equations using Lie symmetry analysis. We obtain not only infinitesimal symmetries but also a complete group classification and a classification of…

Symmetries play an critical role in finding analytic solutions to nonlinear differential equations. A symmetry is a mapping of the solutions of the differential equation into the solutions and have been studied extensively for over a…

This manuscript studies statistical properties of linear classifiers obtained through minimization of an unregularized convex risk over a finite sample. Although the results are explicitly finite-dimensional, inputs may be passed through…

We develop efficient group-theoretical approach to the problem of classification of evolution equations that admit non-local transformation groups (quasi-local symmetries), i.e., groups involving integrals of the dependent variable. We…

We describe all inequalities among generalized diagonals in positive semi-definite matrices. These turn out to be governed by a simple partial order on the symmetric group. This provides an analogue of results of Drake, Gerrish, and…

We start by presenting a generalization of a discrete wave equation that is particularly satisfied by the entries of the matrix coefficients of the refinement equation corresponding to the multiresolution analysis of Alpert. The entries are…

We examine critically the Gambier equation and show that it is the generic linearisable equation containing, as reductions, all the second-order equations which are integrable through linearisation. We then introduce the general discrete…