Related papers: Exponential tropical varieties and complex Monge-A…
In this paper we consider a fractional analogue of the Monge-Amp\`ere operator. Our operator is a concave envelope of fractional linear operators of the form $ \inf_{A\in \mathcal{A}}L_Au, $ where the set of operators corresponds to all…
We introduce variable exponent versions of Morreyfied Triebel-Lizorkin spaces. To that end, we prove an important convolution inequality which is a replacement for the Hardy-Littlewood maximal inequality in the fully variable setting. Using…
We extend the tropical intersection theory to tropicalizations of germs of analytic sets. In particular, we construct a (not entirely obvious) local version of the ring of tropical fans with a nondegenerate intersection pairing. As an…
We show how tropical varieties of ideals I over a field K with non-trivial valuation can be traced back to tropical varieties of ideals in R[[t]][x] over some dense subring R in its ring of integers. Moreover, for homogeneous ideals, we…
We study the asymptotic behavior of a multiple series of Mordell-Tornheim type and its integral analogue at x=0. Our approach is to show a relation between the multiple series and its integral analogue by using Abel's summation formula, and…
Consider the space $M_n^{nor}$ of square normal matrices $X=(x_{ij})$ over $\mathbb{R}\cup\{-\infty\}$, i.e., $-\infty\le x_{ij}\le0$ and $x_{ii}=0$. Endow $M_n^{nor}$ with the tropical sum $\oplus$ and multiplication $\odot$. Fix a real…
This paper supplements [17], showing that categorically the layered theory is the same as the theory of ordered monoids (e.g. the max-plus algebra) used in tropical mathematics. A layered theory is developed in the context of categories,…
We prove a convergence result for a mixed finite element method for the Monge-Ampere equation to its weak solution in the sense of Aleksandrov. The unknowns in the formulation are the scalar variable and the Hessian matrix.
Toric quiver varieties (moduli spaces of quiver representations) are studied. Given a quiver and a weight there is an associated quasiprojective toric variety together with a canonical embedding into projective space. It is shown that for a…
In this paper we continue the program to develop the algebraic foundations of tropical (algebraic) geometry. We give strong characterizations of prime congruences containing a given congruence on a toric semiring. We give four applications…
Auto-encoding Variational Bayes (AEVB) is a powerful and general algorithm for fitting latent variable models (a promising direction for unsupervised learning), and is well-known for training the Variational Auto-Encoder (VAE). In this…
We examine a multidimensional optimisation problem in the tropical mathematics setting. The problem involves the minimisation of a nonlinear function defined on a finite-dimensional semimodule over an idempotent semifield subject to linear…
In this note we study the relationship between ideals and congruences of the tropical polynomial and Laurent polynomial semirings. We show that the variety of a non-zero prime ideal of the tropical (Laurent) polynomial semiring consists of…
A tropical curve in $\mathbb R^{3}$ contributes to Gromov-Witten invariants in all genus. Nevertheless, we present a simple formula for how a given tropical curve contributes to Gromov-Witten invariants when we encode these invariants in a…
In this paper, we study parameter deformations of matrix valued orthogonal polynomials (MVOPs). These deformations are built on the use of certain matrix valued operators which are symmetric with respect to the matrix valued inner product…
Taking symmetric powers of varieties can be seen as a functor from the category of varieties to the category of varieties with an action by the symmetric group. We study a corresponding map between the Grothendieck groups of these…
We study random exponential sums of the form $\sum_{k=1}^nX_k\times\ex p\{i(\lambda_k^{(1)}t_1+...+\lambda_k^{(s)}t_s)\}$, where $\{X_n\}$ is a sequence of random variables and $\{\lambda_n^{(i)}:1\leq i\leq s\}$ are sequences of real…
In this paper we show that each polynomial exponential functor on complex finite-dimensional inner product spaces is defined up to equivalence of monoidal functors by an involutive solution to the Yang-Baxter equation (an involutive…
A combinatorial construction is used to analyze the properties of polyhedral products and generalized moment-angle complexes with respect to certain operations on CW pairs including exponentiation. This allows for the construction of…
In this paper, we establish an innovative framework in logarithmic Hodge theory for toroidal varieties, introducing weighted toroidal structures and developing a systematic obstruction theory for Hodge classes. Building upon recent advances…