Related papers: Real enumerative geometry and effective algebraic …
An infinite dimensional algebra, which is useful for deriving exact solutions of the generalized pairing problem, is introduced. A formalism for diagonalizing the corresponding Hamiltonian is also proposed. The theory is illustrated with…
The algebras for all possible Lorentzian and Euclidean kinematics with $\frak{so}(3)$ isotropy except static ones are re-classified. The geometries for algebras are presented by contraction approach. The relations among the geometries are…
The real number system is geometrically extended to include three new anticommuting square roots of plus one, each such root representing the direction of a unit vector along the orthonormal coordinate axes of Euclidean 3-space. The…
Given positive real numbers, we prove two inequalities involving their potential energy and their power sums. We also prove an inequality involving the energy and the discriminant and apply it to deduce a result on totally positive…
We study the existence of formal power series solutions to q-algebraic equations. When a solution exists, we give a sufficient condition on the equation for this solution to have a positive radius of convergence. We emphasize on the case…
On logarithmic paper some real algebraic curves look like smoothed broken lines. Moreover, the broken lines can be obtained as limits of those curves. The corresponding deformation can be viewed as a quantization, in which the broken line…
We enumerate complex curves on toric surfaces of any given degree and genus, having a single cusp and nodes as their singularities, and matching appropriately many point constraints. The solution is obtained via tropical enumerative…
We investigate the existence and uniqueness of solutions for second-order semi-linear partial differential equations defined on a Riemannian manifold $M$. By combining differential geometry and analysis techniques, we establish the…
A point visibility graph is a graph induced by a set of points in the plane, where every vertex corresponds to a point, and two vertices are adjacent whenever the two corresponding points are visible from each other, that is, the open…
We associate with any simplicial complex $\K$ and any integer $m$ a system of linear equations and inequalities. If $\K$ has a simplicial embedding in $\R^m$ then the system has an integer solution. This result extends the work of I. Novik…
A multidimensional extremal problem in the idempotent algebra setting is considered which consists in minimizing a nonlinear functional defined on a finite-dimensional semimodule over an idempotent semifield. The problem integrates two…
We derive "numerical" criteria for the existence of embeddings of representations of finite dimensional algebras.
The real numbers are important in both mathematics and computation theory. Computationally, real numbers can be represented in several ways; most commonly using inexact floating-point data-types, but also using exact arbitrary-precision…
We prove that each real semisimple Lie algebra G has a Q-form, such that every real representation of G can be realized over the rational numbers Q. This was previously proved by M.S.Raghunathan (and rediscovered by P.Eberlein) in the…
In this work, we study the computability of topological graphs, which are obtained by gluing arcs and rays together at their endpoints. We prove that every semicomputable graph in a computable metric space can be approximated, with…
The Mukhin-Tarasov-Varchenko Theorem (previously the Shapiro Conjecture) asserts that a Schubert problem has all solutions distinct and real if the Schubert varieties involved osculate a rational normal curve at real points. This sparked…
This paper contains an attempt to formulate rigorously and to check predictions in enumerative geometry of curves following from Mirror Symmetry. The main tool is a new notion of stable map. We give an outline of a contsruction of…
Determining whether two graphs are structurally identical is a fundamental problem with applications spanning mathematics, computer science, chemistry, and network science. Despite decades of study, graph isomorphism remains a challenging…
Some notions of algebraic geometry can be defined for arbitrary varieties of algebras. This leads to universal algebraic geometry. The main idea of the presented theory is to consider interactions between algebra, logic and geometry in…
The theory of classical realizability is a framework in which we can develop the proof-program correspondence. Using this framework, we show how to transform into programs the proofs in classical analysis with dependent choice and the…