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We address the problem of computing a dynamic visualization of a geometric graph $G$ as a sequence of frames. Each frame shows only a portion of the graph but their union covers $G$ entirely. The two main requirements of our dynamic…
Assuming that the differential field $(K,\delta)$ is differentially large, in the sense of Le\'on S\'anchez and Tressl, and "bounded" as a field, we prove that for any linear differential algebraic group $G$ over $K$, the differential…
We introduce the notion of G-hypergeometric function, where G is a complex Lie group. In the case when G is a complex torus, this notion amounts to the notion of Gelfand's A-hypergeometric function. We show that the integral $\int…
Solvability of the rational quantum integrable systems related to exceptional root spaces $G_2, F_4$ is re-examined and for $E_{6,7,8}$ is established in the framework of a unified approach. It is shown the Hamiltonians take algebraic form…
We study algebraic integrability of complex planar polynomial vector fields $X=A (x,y)(\partial/\partial x) + B(x,y) (\partial/\partial y) $ through extensions to Hirzebruch surfaces. Using these extensions, each vector field $X$ determines…
We present a new family of zero-field Ising models over $N$ binary variables/spins obtained by consecutive "gluing" of planar and $O(1)$-sized components and subsets of at most three vertices into a tree. The polynomial-time algorithm of…
We consider the Rosenfeld-Groebner algorithm for computing a regular decomposition of a radical differential ideal generated by a set of ordinary differential polynomials in n indeterminates. For a set of ordinary differential polynomials…
We point out the relevance of the Differential Galois Theory of linear differential equations for the exact semiclassical computations in path integrals in quantum mechanics. The main tool will be a necessary condition for complete…
We apply a recently suggested new strategy to solve differential equations for Feynman integrals. We develop this method further by analyzing asymptotic expansions of the integrals. We argue that this allows the systematic application of…
Motivated by mechanical problems where external forces are non-smooth, we consider the differential inclusion problem \[ \begin{cases} -\Delta u(x)\in \partial F(u(x))+\lambda \partial G(u(x))\ \mbox{in}\ \Omega \newline u\geq 0\ \mbox{in}\…
A classical result of F.Klein states that, given a finite primitive group $G\subseteq SL_2(\mathbb{C})$, there exists a hypergeometric equation such that any second order LODE whose differential Galois group is isomorphic to $G$ is…
We compute the Galois group of the splitting field $F$ of any irreducible and separable polynomial $f(x)=x^6+ax^3+b$ with $a,b\in K$, a field with characteristic different from two. The proofs require to distinguish between two cases:…
In this survey we present the parameterized Galois theory of difference equations, as introduced by Hardouin-Singer. The purpose of this theory is to give a systematic approach to differential transcendence, also called hypertranscendence.…
We consider a class of sums over products of Z-sums whose arguments differ by a symbolic integer. Such sums appear, for instance, in the expansion of Gauss hypergeometric functions around integer indices that depend on a symbolic parameter.…
We present simpler algorithms for two closely related morphing problems, both based on the barycentric interpolation paradigm introduced by Floater and Gotsman, which is in turn based on Floater's asymmetric extension of Tutte's classical…
We prove estimates in H\"{o}lder spaces for some Cauchy-type integral operators representing holomorphic functions in Cartesian and symmetric products of planar domains. As a consequence, we obtain information on the boundary regularity in…
We present a symbolic algorithm for treating perturbative expansions of Hamiltonians with general two-body interactions. The method, formally equivalent to determinant Monte Carlo methods, merges well-known analytics with the recently…
Let C be an algebraically closed field and X a projective curve over C. Consider an ordinary linear differential equation, or a linear differ- ence equation, with coefficients in the field of rational functions of X, and assume that its…
This paper continues the studies of symbolic integration by focusing on the stability problems on D-finite functions. We introduce the notion of stability index in order to investigate the order growth of the differential operators…
We study differential-difference equation of the form $$ \frac{d}{dx}t(n+1,x)=f(t(n,x),t(n+1,x),\frac{d}{dx}t(n,x)) $$ with unknown $t(n,x)$ depending on continuous and discrete variables $x$ and $n$. Equation of such kind is called Darboux…