arXiv research

We study junctions of Wilson lines in refined SU(N) Chern-Simons theory and their local relations. We focus on junctions of Wilson lines in antisymmetric and symmetric powers of the fundamental representation and propose a set of local relations which realize one-parameter deformations of quantum groups $\dot{U}_{q}(\m…

Let ${\mathcal S}_m$ be the set of all $m\times m$ density matrices (Hermitian positively semi-definite matrices of unit trace). Consider a problem of estimation of an unknown density matrix $ρ\in {\mathcal S}_m$ based on outcomes of $n$ measurements of observables $X_1,\dots, X_n\in {\mathbb H}_m$ (${\mathbb H}_m$ bei…

We construct a cohomology theory for oriented links using singular cobordisms and a special type of 2-dimensional Topological Quantum Field Theory (TQFT), categorifying the quantum sl(2) invariant. In particular, we give a description of the universal dot-free sl(2) foam cohomology for links via a TQFT.

New structure for quantum algebra representations.

Ray-based framework classifies high-dimensional structures with reduced data.

This paper describes a method to obtain state model parameters for an infinite series of Links-Gould link invariants LG^{m,n}, based on quantum R matrices associated with the (\dot{0}_m | \dotα_n) representations of the quantum superalgebras U_q[gl(m|n)]. Explicit details of the state models for the cases n=1 and m=1,2…

VQC-MLPNet combines quantum and classical elements for scalable quantum machine learning.

TensorHyper-VQC improves VQC scalability and robustness.

This work constructs a finite-dimensional projective representation for a quantum Teichmüller model.

New quantum algebra connects 3D gravity to complex plane.

We design non-singular cloaks enabling objects to scatter waves like objects with smaller size and very different shapes. We consider the Schrodinger equation which is valid e.g. in the contexts of geometrical and quantum optics. More precisely, we introduce a generalized non-singular transformation for star domains, a…

Optimizes quantum channel mapping between Hilbert spaces.

Using a coordinate free characterization of hyperplanes intersection, we provide explicitly a set of local generators for a smooth affine distribution given by those smooth vector fields $X\in\mathfrak{X}(U)$ defined eventually on an open subset $U\subseteq M$ of a smooth Riemannian manifold $(M,g)$, that verifies the …

The paper examines deformations of simple dotted graphs made of circles.

Using quantum skew-Howe duality, we study the category $\operatorname{Rep}(\mathfrak{gl}(m|n))$ of tensor products of exterior powers of the standard representation of $U_q(\mathfrak{gl}(m|n))$, and prove that it is equivalent to a category of ladder diagrams modulo one extra family of relations. We then construct a ca…

We use the technique of quantum skew Howe duality to investigate the monoidal category of exterior powers of the standard representation of $U_q(\mathfrak{gl}(1|1))$. This produces a complete diagrammatic description of the category in terms of trivalent graphs, with the usual MOY relations plus one additional family o…

The paper refines transformations of lattice diagrams and introduces dotted diagrams.

Formula derived for spherical growth series of specific groups.

Approximating non-linear kernels using feature maps has gained a lot of interest in recent years due to applications in reducing training and testing times of SVM classifiers and other kernel based learning algorithms. We extend this line of work and present low distortion embeddings for dot product kernels into linear…

Estimates kernel eigenvalues for compositional dot-product kernels.

Paper proves link diagrams can be realized for some but not all types of links.

Revisits neural collaborative filtering vs. matrix factorization, showing dot product superiority.

In this paper, we study the topology of real analytic map-germs with isolated critical value $f: (\mathbb{R}^m,0) \to (\mathbb{R}^n,0)$, with $1 <n <m$. We compare the topology of $f$ with the topology of the compositions $π_i^* \circ f$, where $π_i^*: \mathbb{R}^n \to \mathbb{R}^{n-1}$ are the projections $(t_1, \dots…

Game theory applied to splitting surfaces of compact 2-manifolds.

Traditionally, multi-layer neural networks use dot product between the output vector of previous layer and the incoming weight vector as the input to activation function. The result of dot product is unbounded, thus increases the risk of large variance. Large variance of neuron makes the model sensitive to the change o…

Study on the topology of tensorial bodies, showing they are homeomorphic to a product space.

Suppose that $θ_1,θ_2,\dots,θ_n$ are positive numbers and $n\ge 3$. Does there exist a sphere with a spherical metric with $n$ conical singularities of angles $2πθ_1,2πθ_2,\dots,2πθ_n$? A sufficient condition was obtained by Gabriele Mondello and Dmitri Panov (arXiv:1505.01994). We show that it is also necessary when w…

For $g\geq1$ denote by $F_{2g}=\langle x_1, y_1,\dots,x_g,y_g\rangle$ the free group on $2g$ generators and by $B_g=[x_1,y_1]\dots[x_g,y_g]$. For $l,c\geq 1$ and elements $w_1,\dots,w_l\in F_{2g}$ we study orientable quadratic equations of the form $[u_1,v_1]\dots[u_h,v_h]=(B_g^{w_1})^c(B_g^{w_2})^c\dots(B_g^{w_l})^c$ …

Let $E_1,\dots ,E_k$ and $E$ be natural vector bundles defined over the category $\Cal Mf_m^+$ of smooth oriented $m$--dimensional manifolds and orientation preserving local diffeomorphisms, with $m\geq 2$. Let $M$ be an object of $\Cal Mf_m^+$ which is connected. We give a complete classification of all separately con…

IDPGs extend RDPGs with a Poisson process for random latent positions.

Study efficient geodesics in curve complex using dot graphs.

We study a geometry associated with rank 3 distributions in dimension 8, whose symbol algebra is constant and has a simple Lie algebra sp(3,R) as Tanaka prolongation. We restrict our considerations to only those distributions that are defined in terms of a systems of ODEs of the form $\dot{z}_{ij}=\frac{\partial^2 f(\d…

Let $G$ be a countable group that splits as a free product of groups of the form $G=G_1\ast\dots\ast G_k\ast F_N$, where $F_N$ is a finitely generated free group. We identify the closure of the outer space $P\mathcal{O}(G,\{G_1,\dots,G_k\})$ for the axes topology with the space of projective minimal, \emph{very small} …

Improves efficiency of random feature approximations for dot product kernels.

We will look at the entire cycle of the investment process relating to all aspects of, formulating an investment hypothesis, constructing a portfolio based on that, executing the trades to implement it, on-going risk management, periodically measuring the performance of the portfolio, and rebalancing the portfolio eith…

The paper explores equivariant means on topological spaces.

Pulli kolam is a ubiquitous art form in south India. It involves drawing a line looped around a collection of dots (pullis) place on a plane such that three mandatory rules are followed: all line orbits should be closed, all dots are encircled and no two lines can overlap over a finite length. The mathematical foundati…

We show how to use Bar-Natan's `divide and conquer' approach to computations to efficiently compute the universal sl(2) dotted foam cohomology groups, even for big knots and links. We also describe a purely topological version of the sl(2) foam theory, in the sense that no dots are needed on foams.

Polyhedral surfaces can be broken down into parallelograms.

We consider a symmetric multi-players zero-sum game with two strategic variables. There are $n$ players, $n\geq 3$. Each player is denoted by $i$. Two strategic variables are $t_i$ and $s_i$, $i\in \{1, \dots, n\}$. They are related by invertible functions. Using the minimax theorem by \cite{sion} we will show that Nas…

Let $S$ be a closed Riemann surface of genus $g(\geqq 2)$ and set $\dot{S}=S \setminus \{\hat{z}_0 \}$. Then we have the composed map $\varphi\circ r$ of a map $r: T(S) \times U \rightarrow F(S)$ and the Bers isomorphism $\varphi: F(S) \rightarrow T(\dot{S})$, where $F(S)$ is the Bers fiber space of $S$, $T(X)$ is the …

Researchers create crystallizations of lens spaces.

Extends random dot product graph model to handle multiple graphs.

The study counts geodesics on curved surfaces with specific intersections.

In this article we give a criterion for the existence of a metric of curvature $1$ on a $2$-sphere with $n$ conical singularities of prescribed angles $2π\vartheta_1,\dots,2π\vartheta_n$ and non-coaxial holonomy. Such a necessary and sufficient condition is expressed in terms of linear inequalities in $\vartheta_1,\dot…

Convex optimization method infers latent structure in random dot product graphs.

Introduces Lax-Kirchhoff moduli spaces for quivers and Lie groups.

In the present paper, we introduce $\mathbb{Z}_2$-braids and, more generally, $G$-braids for an arbitrary group $G$. They form a natural group-theoretic counterpart of $G$-knots, see \cite{reidmoves}. The underlying idea, used in the construction of these objects --- decoration of crossings with some additional informa…