Some Parity Results for 16-Cores

Kolitsch and Sellers showed recently that a₈(n), the number of 8-core partitions of n, is even when n belongs to certain arithmetic progressions. We prove a similar result for 16-cores. In doing so, we prove the surprising result that the a₁₆(n), given by

Elementary Proofs of Infinitely Many Congruences for 8-Cores

Using a very elementary argument, we prove the congruences where a₈(n) is the number of 8-core partitions of n. We also exhibit two infinite families of congruences modulo 2 for 8-cores.     

Lengths on semigroups and groups

We give a characterization of those semigroups with topology arising from a collection of left sub-invariant pseudometrics or quasimetrics. We also characterize those with topology arising from sub-invariant pseudometrics or quasimetrics.     

Upper and Lower Bounds for the Lengths of Steiner Trees in 3-Space

We present a method of determining upper and lower bounds for the length of a Steiner minimal tree in 3-space whose topology is a given full Steiner topology, or a degenerate form of that full Steiner topology. The bounds are tight, in the sense that they are exactly satisfied for some configurations. This represents the first nontrivial lower bound to appear in the literature. The bounds are developed by first studying properties of Simpson lines in both two and three dimensional space, and then introducing a class of easily constructed trees, called midpoint trees, which provide the upper and lower bounds. These bounds can be constructed in quadratic time. Finally, we discuss strategies for improving the lower bound.Supported by a grant from the Australia Research Council.     

Generalized Sets of Lengths

LetRbe a Dedekind domain and (R) the set of irreducible elements ofR. In this paper, we study the sets _R(n) = {m | α₁,…,α_n, β₁,…,β_m (R) such that α₁,…,α_n = β₁,…,β_m}, wherenis a positive integer. We show, in constrast to indications in some earlier work, that the sets _R(n) are not completely determined by the Davenport constant of the class group ofR. We offer some specific constructions for Dedekind domains with small class groups, and show how these sets are generalizations of the sets studied earlier by Geroldinger [[9], [10]].     

Hook formula and related identities

A new proof of the hook formula for the dimension of representations of the symmetric group is given with the help of identities which are of independent interest. A probabilistic interpretation of the proof and new formulas relating the parameters of the Young diagrams are given.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Akad. Nauk SSSR, Vol. 172, pp. 3–20, 1989.     

Polygons in Buildings and their Refined Side Lengths

As in a symmetric space of noncompact type, one can associate to an oriented geodesic segment in a Euclidean building a vector valued length in the Euclidean Weyl chamber Δ _euc . In addition to the metric length it contains information on the direction of the segment. In this paper we study restrictions on the Δ _euc -valued side lengths of polygons in Euclidean buildings. The main result is that for thick Euclidean buildings X the set Pn(X){\mathcal{P}n(X)} of possible Δ _euc -valued side lengths of oriented n-gons depends only on the associated spherical Coxeter complex. We show moreover that it coincides with the space of Δ _euc -valued weights of semistable weighted configurations on the Tits boundary ∂ _Tits X. The side lengths of polygons in symmetric spaces of noncompact type are studied in the related paper [KLM1]. Applications of the geometric results in both papers to algebraic group theory are given in [KLM2].     

Lengths of edges in carrier graphs

We show that if X is a minimal length carrier graph in a hyperbolic 3-manifold, M, then if X contains a sufficiently short edge, it must contain a short circuit, as well. The meaning of ??short?? depends only on the rank of ?? ₁(M). We also expand the class of manifolds which are known to have minimal length carrier graphs.     

两跳蚤移行路径长度比较

讨论有关文献在谈到曲线弧长公式时所提出的一个关于两跳蚤移行路径长度的问题.通过理论推导并借助图形分析显示,只是在一些离散的特殊时刻,两跳蚤移行的摆线长度才相等.     

Lengths of cycles in halin graphs

A Halin graph is a plane graph H = T U C, where T is a plane tree with no vertex of degree two and at least one vertex of degree three or more, and C is a cycle connecting the endvertices of T in the cyclic order determined by the embedding of T. We prove that such a graph on n vertices contains cycles of all lengths l, 3 ≤ l n, except, possibly, for one even value m of l. We prove also that if the tree T contains no vertex of degree three then G is pancyclic.     

Chain Lengths in the Tamari Lattice

We find the size of the largest union of two or three chains in the Tamari lattice.AMS Subject Classification: 06A07.     

Inequalities for Edge Lengths and Volumes of Two Simplexes

In this paper, a class of geometric inequalities for the edge lengths and volumes of two n-dimensional simplexes are established. As a corollary, a generalization to several dimensions of the Neuberg–Pedoe inequality is given.     

2-Cell Embeddings with Prescribed Face Lengths and Genus

Let n be a positive integer, let d ₁, . . . , d _n be a sequence of positive integers, and let ${{q = \frac{1}{2}\sum^{n}_{i=1} d_{i}\cdot}}$ . It is shown that there exists a connected graph G on n vertices, whose degree sequence is d ₁, . . . , d _n and such that G admits a 2-cell embedding in every closed surface whose Euler characteristic is at least n ? q?+?1, if and only if q is an integer and q ?? n ? 1. Moreover, the graph G can be required to be loopless if and only if d _i ?? q for i = 1, . . . , n. This, in particular, answers a question of Skopenkov.     

Layer Lengths, Torsion Theories and the Finitistic Dimension

Let $\mathcal{C}$ be a length-category. Generalizing the Loewy length, we propose the layer length associated with a torsion theory, which is a new measure for objects of $\mathcal{C}$ . As an application, we use the layer lengths and the Igusa–Todorov function to get a theorem (see Theorem 6.4) having as corollaries the main results of Huard et al. (Bull Lond Math Soc 41:367–376, 2009) and Wang (Commun Algebra 22(7):419–449, 1994).     

Lengths of geodesics on non-orientable hyperbolic surfaces

We give an identity involving sums of functions of lengths of simple closed geodesics, known as a McShane identity, on any non-orientable hyperbolic surface with boundary which generalises Mirzakhani’s identities on orientable hyperbolic surfaces with boundary.      

Lengths of involutions in finite Coxeter groups

Let W be a finite Coxeter group and X a subset of W. The length polynomial $L_{W,X}(t)$ is defined by $L_{W,X}(t)={\sum }_{x\in X}{t}^{?(x)}$, where ? is the length function on W. If $X=\{x\in W:x^2=1\}$ then we call $L_{W,X}(t)$ the involution length polynomial of W. In this article we derive expressions for the length polynomial where X is any conjugacy class of involutions, and the involution length polynomial, in any finite Coxeter group W. In particular, these results correct errors in [11] for the involution length polynomials of Coxeter groups of type $B_n$ and $D_n$. Moreover, we give a counterexample to a unimodality conjecture stated in [11].     

Steepest Descent Method with Random Step Lengths

The paper studies the steepest descent method applied to the minimization of a twice continuously differentiable function. Under certain conditions, the random choice of the step length parameter, independent of the actual iteration, generates a process that is almost surely R-convergent for quadratic functions. The convergence properties of this random procedure are characterized based on the mean value function related to the distribution of the step length parameter. The distribution of the random step length, which guarantees the maximum asymptotic convergence rate independent of the detailed properties of the Hessian matrix of the minimized function, is found, and its uniqueness is proved. The asymptotic convergence rate of this optimally created random procedure is equal to the convergence rate of the Chebyshev polynomials method. Under practical conditions, the efficiency of the suggested random steepest descent method is degraded by numeric noise, particularly for ill-conditioned problems; furthermore, the asymptotic convergence rate is not achieved due to the finiteness of the realized calculations. The suggested random procedure is also applied to the minimization of a general non-quadratic function. An algorithm needed to estimate relevant bounds for the Hessian matrix spectrum is created. In certain cases, the random procedure may surpass the conjugate gradient method. Interesting results are achieved when minimizing functions having a large number of local minima. Preliminary results of numerical experiments show that some modifications of the presented basic method may significantly improve its properties.