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We introduce the concept of isolated factorizations of an element of a commutative monoid and study its properties. We give several bounds for the number of isolated factorizations of simplicial affine semigroups and numerical semigroups.…

Let A_0, A_1 be nonnegative matrices in GL(n+1,Z) such that the subsimplexes A_0[Delta], A_1[Delta] split the standard unit n-dimensional simplex Delta in two. We prove that, for every n=1,2,... and up to the natural action of the symmetric…

In this paper we prove a Thomae derivative formula for trigonal curves admitting a non-singular affine model. This formula relates the derivatives of theta functions with rational characteristics on the curve to explicit expressions in the…

The phase diagram of the collapse of a two-dimensional infinite branched polymer interacting with the solvent and with itself through contact interactions is studied from the $q\to 1$ limit of an extension of the $q-$ states Potts model.…

The machinery of zonotopal algebra is linked with two particular polytopes: the Stanley-Pitman polytope and the regular simplex $\mathfrak{Sim}_n(t_1,...,t_n)$ with parameters $t_1,...,t_n\in \mathbb{R}_+^n$, defined by the inequalities…

We study the general structure of Smirnov's axioms on form factors of local operators in integrable models. We find various consistency conditions that the form factor functions have to satisfy. For the special case of the $O(3)$…

We show that for any non-real algebraic number $q$ such that $|q-1|>1$ or $\Re(q)>\frac{3}{2}$ it is \textsc{\#P}-hard to compute a multiplicative (resp. additive) approximation to the absolute value (resp. argument) of the chromatic…

The Poincar\'e polynomial of the complement of an arrangements in a non compact group is a specialization of the $G$-Tutte polynomial associated with the arrangement. In this article we show two unimodular elliptic arrangements (built up…

We propose a new splitting behavior of tree-level string/particle amplitudes for scalars, gluons and gravitons. We identify certain subspaces in the space of Mandelstam variables, where the universal Koba-Nielsen factor splits into two…

We recall the form factors $ f^{(j)}_{N,N}$ corresponding to the $\lambda$-extension $C(N,N; \lambda)$ of the two-point diagonal correlation function of the Ising model on the square lattice and their associated linear differential…

Splitting amplitudes are universal functions governing the collinear behavior of scattering amplitudes for massless particles. We compute the two-loop g -> gg splitting amplitudes in QCD, N=1, and N=4 super-Yang-Mills theories, which…

We study the asymptotic representation for the zeros of the deformed exponential function $\sum\nolimits_{n = 0}^\infty {\frac1{n!}{q^{n(n - 1)/2}{x^n}}} $, $q\in (0,1)$. Indeed, we obtain an asymptotic formula for these zeros: \[x_n=-…

Let $q$ be a prime power, let $\mathbb F_q$ be the finite field with $q$ elements and let $\theta$ be a generator of the cyclic group $\mathbb F_q^*$. For each $a\in \mathbb F_q^*$, let $\log_{\theta} a$ be the unique integer $i\in \{1,…

Let K be an algebraically closed field of characteristic 0. Following Medvedev-Scanlon, a polynomial of degree d > 1 is said to be disintegrated if neither f nor -f is linearly conjugate to x^d or T_d(x) where T_d is the Chebyshev…

We present exact calculations of the Potts model partition function $Z(G,q,v)$ for arbitrary $q$ and temperature-like variable $v$ on $n$-vertex square-lattice strip graphs $G$ for a variety of transverse widths $L_t$ and for arbitrarily…

We present a general method for analytically factorizing the n-fold form factor integrals $f^{(n)}_{N,N}(t)$ for the correlation functions of the Ising model on the diagonal in terms of the hypergeometric functions…

We compute the coefficients of the polynomials $C_n(q)$ defined by the equation \begin{equation*} 1 + \sum_{n\geq 1} \, \frac{C_n(q)}{q^n} \, t^n = \prod_{i\geq 1}\, \frac{(1-t^i)^2}{1-(q+q^{-1})t^i + t^{2i}} \, . \end{equation*} As an…

We prove some distribution results for the $k$-fold divisor function in arithmetic progressions to moduli that exceed the square-root of length $X$ of the sum, with appropriate constrains and averaging on the moduli, saving a power of $X$…

We consider the problem of sampling from the Potts model on random regular graphs. It is conjectured that sampling is possible when the temperature of the model is in the uniqueness regime of the regular tree, but positive algorithmic…

Algebraic topology studies topological spaces with the help of tools from abstract algebra. The main focus of this paper is to show that many concepts from algebraic topology can be conveniently expressed in terms of (normal) factor graphs.…