arXiv research

The paper finds discrete real specializations of braid group representations using Salem numbers.

Discrete approximation solves Björling's minimal surface problem.

We study the limit set of discrete subgroups arising from Anosov representations. Specially we study the limit set of discrete groups arising from strictly convex real projective structures and Anosov representations from a finitely generated word hyperbolic group into a semisimple Lie group.

This classification is found by analyzing the action of a normal subgroup of $B_3$ as hyperbolic isometries. This paper gives an example of an unfaithful specialization of the Burau representation on $B_4$ that is faithful when restricted to $B_3$, as well as examples of unfaithful specializations of $B_3$.

We propose a unified definition for discrete analogues of constant mean curvature surfaces in spaces of constant curvature as a special case of discrete special isothermic nets. Bäcklund transformations and Lawson's correspondence are discussed. It is shown that the definition generalizes previous definitions and a con…

Isothermic nets created from special maps for smooth surfaces.

Permutability of surface transforms yields discrete analogs.

Study of discrete Koenigs nets and their properties.

Discretizes special surfaces using Koenigs nets.

Discrete linear Weingarten surfaces in space forms are characterized as special discrete $Ω$-nets, a discrete analogue of Demoulin's $Ω$-surfaces. It is shown that the Lie-geometric deformation of $Ω$-nets descends to a Lawson transformation for discrete linear Weingarten surfaces, which coincides with the well-known L…

The paper finds formulas for special surface shapes in 3D space.

The paper examines special Q-nets that terminate after a finite number of Laplace steps.

In the search for appropriate discretizations of surface theory it is crucial to preserve such fundamental properties of surfaces as their invariance with respect to transformation groups. We discuss discretizations based on Möbius invariant building blocks such as circles and spheres. Concrete problems considered in t…

We propose a discrete surface theory in $\mathbb R^3$ that unites the most prevalent versions of discrete special parametrizations. This theory encapsulates a large class of discrete surfaces given by a Lax representation and, in particular, the one-parameter associated families of constant curvature surfaces. The theo…

New Y-systems for Miquel dynamics are Möbius invariant.

Survey on Coxeter groups for Lie group examples.

The optimization of expensive to evaluate, black-box, mixed-variable functions, i.e. functions that have continuous and discrete inputs, is a difficult and yet pervasive problem in science and engineering. In Bayesian optimization (BO), special cases of this problem that consider fully continuous or fully discrete doma…

In this review we establish various connections between complex networks and symmetry. While special types of symmetries (e.g., automorphisms) are studied in detail within discrete mathematics for particular classes of deterministic graphs, the analysis of more general symmetries in real complex networks is far less de…

Discretizes Hodge-Dirac operators on a torus.

This paper develops a discrete theory of real Riemann surfaces using quad-graphs and linear discretization.

Paper proves rigidity of discrete conformal structures on polyhedral surfaces.

Let H be a discrete cocompact subgroup of SL_2(C). We conjecture that the quotient manifold X=SL_2(C)/H contains infinitely many non-isogeneous elliptic curves and prove that this is indeed the case if Schanuel's conjecture holds. We also prove it in the special case where the intersection of H and SL_2(R) is cocompact…

We study a special type of almost complex structures, called pure and full and introduced by T.J. Li and W. Zhang, in relation to symplectic structures and Hard Lefschetz condition. We provide sufficient conditions to the existence of the above type of almost complex structures on compact quotients of Lie groups by dis…

We study Christoffel and Darboux transforms of discrete isothermic nets in 4-dimensional Euclidean space: definitions and basic properties are derived. Analogies with the smooth case are discussed and a definition for discrete Ribaucour congruences is given. Surfaces of constant mean curvature are special among all iso…

We present an axiomatic approach to finite- and infinite-dimensional differential calculus over arbitrary infinite fields (and, more generally, suitable rings). The corresponding basic theory of manifolds and Lie groups is developed. Special attention is paid to the case of mappings between topological vector spaces ov…

New framework models time-uncertain point processes for better event prediction.

Structured latent attribute models (SLAMs) are a special family of discrete latent variable models widely used in social and biological sciences. This paper considers the problem of learning significant attribute patterns from a SLAM with potentially high-dimensional configurations of the latent attributes. We address …

We describe a Groebner basis of relations among conditional probabilities in a discrete probability space, with any set of conditioned-upon events. They may be specialized to the partially-observed random variable case, the purely conditional case, and other special cases. We also investigate the connection to generali…

In this paper we introduce a discrete integrable system generalizing the discrete (real) cross-ratio system in $S^4$ to complex values of a generalized cross-ratio by considering $S^4$ as a real section of the complex Plücker quadric, realized as the space of two-spheres in $S^4.$ We develop the geometry of the Plücker…

New discrete curves defined in space forms with geometric properties.

Complex hyperbolic triangle groups are discrete when certain conditions are met.

Finding appropriate notions of discrete holomorphic maps and, more generally, conformal immersions of discrete Riemann surfaces into 3-space is an important problem of discrete differential geometry and computer visualization. We propose an approach to discrete conformality that is based on the concept of holomorphic l…

Hybrid RL method optimizes trading by balancing continuous and discrete actions.

Real analytic solutions found for special Lagrangian equation.

Paper develops Bayesian inference for discrete-choice mnp models with Gaussian priors.

We consider a general theory of curvatures of discrete surfaces equipped with edgewise parallel Gauss images, and where mean and Gaussian curvatures of faces are derived from the faces' areas and mixed areas. Remarkably these notions are capable of unifying notable previously defined classes of surfaces, such as discre…

Discrete conformal maps on surfaces with vertex decorations are studied.

Study on the limits of projective special real manifolds and their symmetries.

Homogeneous magnetic trajectories in a special linear group proven.

In 1990, Hitchin's proved a component of the space of representations of a surface group in SL(n,R) is homeomorphic to a ball. For n=2,3 this component has been identified with the holonomies of geometric structures (hyperbolic for n=2, or real projective for n=3). In the preprint "Anosov flows, Surface groups and Curv…

Hybrid quantum-classical RL model solves standard benchmark tasks and proves quantum advantage.

Classifies special quartic curves up to equivalence.

The study examines discrete subgroups of Lie groups and their residual finiteness.

Following the previous authors works (joint with I.A.Dynnikov) we develop a theory of the discrete analogs of the differential-geometrical (DG) connections in the triangulated manifolds. We study a nonstandard discretization based on the interpretation of DG Connection as linear first order (''triangle'') difference eq…

Authors prove an asymptotic expansion for spectral zeta functions on discrete tori.

Discrete analogues of ellipsoids with preserved circular cross sections.

We introduce a setup of model uncertainty in discrete time. In this setup we derive dual expressions for the super--replication prices of game options with upper semicontinuous payoffs. We show that the super--replication price is equal to the supremum over a special (non dominated) set of martingale measures, of the c…

This paper constructs real algebraic maps that are topologically special generic maps.