Related papers: On Stanley's reciprocity theorem for rational cone…

It will be shown that Pascal's Theorem is equivalent to the associativity of a natural binary operation on conic sections. A novel proof for Pascal's Theorem will then be given by showing that this binary operation is associative…

Let G be a simple complex algebraic group and let K be a reductive subgroup of G such that the coordinate ring of G/K is a multiplicity free G-module. We consider the G-algebra structure of C[G/K], and study the decomposition into…

We show that, if $\Gamma$ is a point group of $\mathbb{R}^{k+1}$ of order two for some $k\geq 2$ and $\mathcal S$ is a $k$-pseudomanifold which has a free automorphism of order two, then either $\mathcal S$ has a $\Gamma$-symmetric…

Let U be a simply connected domain on the Riemann sphere whose complement K contains more than one point. We establish a characterization of local connectivity of K at a point in terms of the prime ends whose impressions contain this point.…

Consider a commutative monoid $(M,+,0)$ and a biadditive binary operation $\mu \colon M \times M \to M$. We will show that under some additional general assumptions, the operation $\mu$ is automatically both associative and commutative. The…

Wang et al. [1] demonstrated different power transmission coefficients for forward and backward propagation in simulation and experiment. From such a demonstration, the central claim of their paper is that "the spatial inversion symmetry…

An identity is proved connecting two finite sums of inverse tangents. This identity is discretized version of Jacobi's imaginary transformation for the modular angle from the theory of elliptic functions. Some other related identities are…

We prove that the integer part of the reciprocal of the tail of $\zeta(s)$ at a rational number $s=\frac{1}{p}$ for any integer with $p \geq 5$ or $s=\frac{2}{p}$ for any odd integer with $p \geq 5$ can be described essentially as the…

We introduce the notions of $\mathbb{K}$-framings, based $\mathbb{K}$-framings and relative $\mathbb{K}$-framings of a compact connected oriented surface $\Sigma$ for any commutative ring $\mathbb{K}$ with unit, and a map which maps a based…

Let $\mathbb{K}$ be a field, and let $S=\mathbb{K}[X_1, ..., X_n]$ be the polynomial ring. Let $I$ be a monomial ideal of $S$ with up to 5 generators. In this paper, we present a computational experiment which allows us to prove that…

Given a partition $\lambda$ of $n$, the {\it Schur functor} $\mathbb{S}_\lambda$ associates to any complex vector space $V$, a subspace $\mathbb{S}_\lambda(V)$ of $V^{\otimes n}$. Hermite's reciprocity law, in terms of the Schur functor,…

We show that the infinite symmetric product of a connected graded-commutative algebra over the rationals is naturally isomorphic to the free graded-commutative algebra on the positive degree subspace of the original algebra. In particular,…

We prove the following result. Let k be an infinite perfect field of positive characteristic and assume that strong resolution of singularities holds over k. Let R be a localization of a commutative d-dimensional k-algebra of finite type…

The existence theorem for replica-symmetry breaking (RSB) in the transverse field Sherrington-Kirkpatrick (SK) model is extended to the model with a general random exchange interactions. The relation between the expectation value of the…

Rohatgi and the author recently proved a shuffling theorem for lozenge tilings of `doubly-dented hexagons' (arXiv:1905.08311). The theorem can be considered as a hybrid between two classical theorems in the enumeration of tilings:…

We prove a genus zero Givental-style mirror theorem for all complete intersections in proper toric Deligne-Mumford stacks, which provides an explicit slice called big $I-$function on Givental's Lagrangian cone for such targets. In…

We prove the inequality sum_{k=1}^infty (-1)^{k+1} r^k cos(k*phi) (k+2)^{-1} < sum_{k=1}^infty(-1)^{k+1} r^k (k+2)^{-1} for 0 < r <= 1 and 0 < phi < pi. For the case r = 1 we give two proofs. The first one is by means of a general numerical…

The characters of the (total) Springer representations are identified with the Green functions by Kazhdan [Israel J. Math. {\bf 28} (1977)], and the latter are identified with Hall-Littlewood's $Q$-functions by Green [Trans. Amer. Math.…

Let $\Delta$ be a simplicial complex of a matroid $M$. In this paper, we explicitly compute the regularity of all the symbolic powers of a Stanley-Reisner ideal $I_\Delta$ in terms of combinatorial data of the matroid $M$. In order to do…

We use the action of Atkin-Lehner operators to generate a family of reciprocity formulas for newform Dedekind sums. This family of reciprocity formulas provides symmetries which we use to investigate the kernel of these Dedekind sums.