arXiv subjects

Daniel Herden

Publications and source records attributed to Daniel Herden.

31 records · Page 2Linked to original sources

arXiv2022

Vertex-edge marking score of certain triangular lattices

The vertex-edge marking game is played between two players on a graph, $G=(V,E)$, with one player marking vertices and the other marking edges. The players want to minimize/maximize, respectively, the number of marked edges incident to an unmarked vertex. The vertex-edge coloring number for $G$ is the maximum score achievable with perfect play. Brešar et al., [4], give an upper bound of $5$ for the vertex-edge coloring number for finite planar graphs. It is not known whether the bound is tight. In this paper, in response to questions in [4], we show that the vertex-edge coloring number for the infinite regular triangularization of the plane is 4. We also give two general techniques that allow us to calculate the vertex-edge coloring number in many related triangularizations of the plane.

math.CO↗

arXiv2021

On the absoluteness of $\aleph_1$-freeness

$\aleph_1$-free groups, abelian groups for which every countable subgroup is free, exhibit a number of interesting algebraic and set-theoretic properties. In this paper, we give a complete proof that the property of being $\aleph_1$-free is absolute; that is, if an abelian group $G$ is $\aleph_1$-free in some transitive model $\textbf{M}$ of ZFC, then it is $\aleph_1$-free in any transitive model of ZFC containing $G$. The absoluteness of $\aleph_1$-freeness has the following remarkable consequence: an abelian group $G$ is $\aleph_1$-free in some transitive model of ZFC if and only if it is (countable and) free in some model extension. This set-theoretic characterization will be the starting point for further exploring the relationship between the set-theoretic and algebraic properties of $\aleph_1$-free groups. In particular, this paper will demonstrate how proofs may be dramatically simplified using model extensions for $\aleph_1$-free groups.

math.GR↗

arXiv2021

Partitions With Designated Summands Not Divisible by $2^l$, $2$, and $3^l$ Modulo $2$, $4$, and $3$

Numerous congruences for partitions with designated summands have been proven since first being introduced and studied by Andrews, Lewis, and Lovejoy. This paper explicitly characterizes the number of partitions with designated summands whose parts are not divisible by $2^\ell$, $2$, and $3^\ell$ working modulo $2,\ 4,$ and $3$, respectively, greatly extending previous results on the subject. We provide a few applications of our characterizations throughout in the form of congruences and a computationally fast recurrence. Moreover, we illustrate a previously undocumented connection between the number of partitions with designated summands and the number of partitions with odd multiplicities.

math.CO↗

arXiv2020

Counting the parts divisible by k in all the partitions of n whose parts have multiplicity less than k

Recent results by Andrews and Merca on the number of even parts in all partitions of n into distinct parts, a(n), were derived via generating functions. This paper extends these results to the number of parts divisible by k in all the partitions of n for which the multiplicity of each part is strictly less than k, ak(n). Moreover, a combinatorial proof is provided using an extension of Glaisher's bijection. Finally, we give the generating functions for this new family of integer sequences and use it to verify generalized pentagonal, triangular, and square power recurrence relations.

math.CO↗

arXiv2020

Hilbert series of symplectic quotients by the 2-torus

We compute the Hilbert series of the graded algebra of real regular functions on a linear symplectic quotient by the $2$-torus as well as the first four coefficients of the Laurent expansion of this Hilbert series at $t = 1$. We describe an algorithm to compute the Hilbert series as well as the Laurent coefficients in explicit examples.

math.SG↗

arXiv2019

A theorem of Besicovitch and a generalization of the Birkhoff Ergodic Theorem

A remarkable theorem of Besicovitch is that an integrable function $f$ on $\mathbb{R}^2$ is strongly differentiable if and only if its associated strong maximal function $M_S f$ is finite a.e. We provide an analogue of Besicovitch's result in the context of ergodic theory that provides a generalization of Birkhoff's Ergodic Theorem. In particular, we show that if $f$ is a measurable function on a standard probability space and $T$ is an invertible measure-preserving transformation on that space, then the ergodic averages of $f$ with respect to $T$ converge a.e. if and only if the associated ergodic maximal function $T^*f$ is finite a.e.

math.CA↗

arXiv2019

$\aleph_k$-free cogenerators

We prove in ZFC that an abelian group $C$ is cotorsion if and only if $\operatorname{Ext}(F,C) = 0$ for every $\aleph_k$-free group $F$, and discuss some consequences and related results. This short note includes a condensed overview of the $\barλ$-Black Box for $\aleph_k$-free constructions in ZFC.

math.GR↗

arXiv2018

Hilbert series associated to symplectic quotients by $\operatorname{SU}_2$

We compute the Hilbert series of the graded algebra of real regular functions on the symplectic quotient associated to an $\operatorname{SU}_2$-module and give an explicit expression for the first nonzero coefficient of the Laurent expansion of the Hilbert series at $t = 1$. Our expression for the Hilbert series indicates an algorithm to compute it, and we give the output of this algorithm for representations of dimension at most $10$. Along the way, we compute the Hilbert series of the module of covariants of an arbitrary $\operatorname{Sl}_2$- or $\operatorname{SU}_2$-module as well its first three Laurent coefficients.

math.SG↗

arXiv2018

The Hilbert series of $\operatorname{SL}_2$-invariants

Let $V$ be a finite dimensional representations of the group $\operatorname{SL}_2$ of $2\times 2$ matrices with complex coefficients and determinant one. Let $R=\mathbb{C}[V]^{\operatorname{SL}_2}$ be the algebra of $\operatorname{SL}_2$-invariant polynomials on $V$. We present a calculation of the Hilbert series $\operatorname{Hilb}_R(t)=\sum_{n\ge 0}\dim (R_n)\: t^n$ as well as formulas for the first four coefficients of the Laurent expansion of $\operatorname{Hilb}_R(t)$ at $t=1$.

math.RA↗

arXiv2017

The Laurent coefficients of the Hilbert series of a Gorenstein algebra

By a theorem of R. Stanley, a graded Cohen-Macaulay domain $A$ is Gorenstein if and only if its Hilbert series satisfies the functional equation \[ \operatorname{Hilb}_A(t^{-1})=(-1)^d t^{-a}\operatorname{Hilb}_A(t), \] where $d$ is the Krull dimension and $a$ is the a-invariant of $A$. We reformulate this functional equation in terms of an infinite system of linear constraints on the Laurent coefficients of $\operatorname{Hilb}_A(t)$ at $t=1$. The main idea consists of examining the graded algebra $\mathcal F=\bigoplus_{r\in \mathbb{Z}}\mathcal F_r$ of formal power series in the variable $x$ that fulfill the condition $φ(x/(x-1))=(1-x)^rφ(x)$. As a byproduct, we derive quadratic and cubic relations for the Bernoulli numbers. The cubic relations have a natural interpretation in terms of coefficients of the Euler polynomials. For the special case of degree $r=-(a+d)=0$, these results have been investigated previously by the authors and involved merely even Euler polynomials. A link to the work of H. W. Gould and L. Carlitz on power sums of symmetric number triangles is established.

math.CO↗

arXiv2017

The Hilbert series and $a$-invariant of circle invariants

Let $V$ be a finite-dimensional representation of the complex circle $\mathbb{C}^\times$ determined by a weight vector $\mathbf{a}\in\mathbb{Z}^n$. We study the Hilbert series $\operatorname{Hilb}_{\mathbf{a}}(t)$ of the graded algebra $\mathbb{C}[V]^{\mathbb{C}_{\mathbf{a}}^\times}$ of polynomial $\mathbb{C}^\times$-invariants in terms of the weight vector $\mathbf{a}$ of the $\mathbb{C}^\times$-action. In particular, we give explicit formulas for $\operatorname{Hilb}_{\mathbf{a}}(t)$ as well as the first four coefficients of the Laurent expansion of $\operatorname{Hilb}_{\mathbf{a}}(t)$ at $t=1$. The naive formulas for these coefficients have removable singularities when weights pairwise coincide. Identifying these cancelations, the Laurent coefficients are expressed using partial Schur polynomial that are independently symmetric in two sets of variables. We similarly give an explicit formula for the $a$-invariant of $\mathbb{C}[V]^{\mathbb{C}_{\mathbf{a}}^\times}$ in the case that this algebra is Gorenstein. As an application, we give methods to identify weight vectors with Gorenstein and non-Gorenstein invariant algebras.

math.RA↗

arXiv2017

Local automorphisms of finitary incidence algebras

Let $R$ be a commutative, indecomposable ring with identity and $(P,\le)$ a partially ordered set. Let $FI(P)$ denote the finitary incidence algebra of $(P,\le)$ over $R$. We will show that, in most cases, local automorphisms of $FI(P)$ are actually $R$-algebra automorphisms. In fact, the existence of local automorphisms which fail to be $R$-algebra automorphisms will depend on the chosen model of set theory and will require the existence of measurable cardinals. We will discuss local automorphisms of cartesian products as a special case in preparation of the general result.

math.RA↗

arXiv2014

On compositions with $x^2/(1-x)$

In the past, empirical evidence has been presented that Hilbert series of symplectic quotients of unitary representations obey a certain universal system of infinitely many constraints. Formal series with this property have been called \emph{symplectic}. Here we show that a formal power series is symplectic if and only if it is a formal composite with the formal power series $x^2/(1-x)$. Hence the set of symplectic power series forms a subalgebra of the algebra of formal power series. The subalgebra property is translated into an identity for the coefficients of the even Euler polynomials, which can be interpreted as a cubic identity for the Bernoulli numbers. Furthermore we show that a rational power series is symplectic if and only if it is invariant under the idempotent Möbius transformation $x\mapsto x/(x-1)$. It follows that the Hilbert series of a graded Cohen-Macaulay algebra $A$ is symplectic if and only if $A$ is Gorenstein with its a-invariant and its Krull dimension adding up to zero. It is shown that this is the case for algebras of regular functions on symplectic quotients of unitary representations of tori.

math.SG↗