Let
 NSFPAR ID:
 10450418
 Date Published:
 Journal Name:
 Memoirs of the American Mathematical Society
 Volume:
 282
 Issue:
 1394
 ISSN:
 00659266
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
More Like this

Abstract E be an elliptic curve over . We conjecture asymptotic estimates for the number of vanishings of$${{\mathbb {Q}}}$$ $Q$ as$$L(E,1,\chi )$$ $L(E,1,\chi )$ varies over all primitive Dirichlet characters of orders 4 and 6, subject to a mild hypothesis on$$\chi $$ $\chi $E . Our conjectures about these families come from conjectures about random unitary matrices as predicted by the philosophy of KatzSarnak. We support our conjectures with numerical evidence. Compared to earlier work by David, Fearnley and Kisilevsky that formulated analogous conjectures for characters of any odd prime order, in the composite order case, we need to justify our use of random matrix theory heuristics by analyzing the equidistribution of the squares of normalized Gauss sums. To do this, we introduce the notion of totally order characters to quantify how quickly the quartic and sextic Gauss sums become equidistributed. Surprisingly, the rate of equidistribution in the full family of quartic (resp., sextic) characters is much slower than in the subfamily of totally quartic (resp., sextic) characters. We provide a conceptual explanation for this phenomenon by observing that the full family of order$$\ell $$ $\ell $ twisted elliptic curve$$\ell $$ $\ell $L functions, with even and composite, is a mixed family with both unitary and orthogonal aspects.$$\ell $$ $\ell $ 
Abstract For every integer k there exists a bound $$B=B(k)$$ B = B ( k ) such that if the characteristic polynomial of $$g\in \textrm{SL}_n(q)$$ g ∈ SL n ( q ) is the product of $$\le k$$ ≤ k pairwise distinct monic irreducible polynomials over $$\mathbb {F}_q$$ F q , then every element x of $$\textrm{SL}_n(q)$$ SL n ( q ) of support at least B is the product of two conjugates of g . We prove this and analogous results for the other classical groups over finite fields; in the orthogonal and symplectic cases, the result is slightly weaker. With finitely many exceptions ( p , q ), in the special case that $$n=p$$ n = p is prime, if g has order $$\frac{q^p1}{q1}$$ q p  1 q  1 , then every nonscalar element $$x \in \textrm{SL}_p(q)$$ x ∈ SL p ( q ) is the product of two conjugates of g . The proofs use the Frobenius formula together with upper bounds for values of unipotent and quadratic unipotent characters in finite classical groups.more » « less

Abstract The limitedmemory BFGS (BroydenFletcherGoldfarbShanno) method is widely used for largescale unconstrained optimization, but its behavior on nonsmooth problems has received little attention. LBFGS (limited memory BFGS) can be used with or without ‘scaling’; the use of scaling is normally recommended. A simple special case, when just one BFGS update is stored and used at every iteration, is sometimes also known as memoryless BFGS. We analyze memoryless BFGS with scaling, using any Armijo–Wolfe line search, on the function $f(x) = ax^{(1)} + \sum _{i=2}^{n} x^{(i)}$, initiated at any point $x_0$ with $x_0^{(1)}\not = 0$. We show that if $a\ge 2\sqrt{n1}$, the absolute value of the normalized search direction generated by this method converges to a constant vector, and if, in addition, $a$ is larger than a quantity that depends on the Armijo parameter, then the iterates converge to a nonoptimal point $\bar x$ with $\bar x^{(1)}=0$, although $f$ is unbounded below. As we showed in previous work, the gradient method with any Armijo–Wolfe line search also fails on the same function if $a\geq \sqrt{n1}$ and $a$ is larger than another quantity depending on the Armijo parameter, but scaled memoryless BFGS fails under a weaker condition relating $a$ to the Armijo parameter than that implying failure of the gradient method. Furthermore, in sharp contrast to the gradient method, if a specific standard Armijo–Wolfe bracketing line search is used, scaled memoryless BFGS fails when $a\ge 2 \sqrt{n1}$regardless of the Armijo parameter. Finally, numerical experiments indicate that the results may extend to scaled LBFGS with any fixed number of updates $m$, and to more general piecewise linear functions.

Generalized permutahedra are polytopes that arise in combinatorics, algebraic geometry, representation theory, topology, and optimization. They possess a rich combinatorial structure. Out of this structure we build a Hopf monoid in the category of species.
Species provide a unifying framework for organizing families of combinatorial objects. Many species carry a Hopf monoid structure and are related to generalized permutahedra by means of morphisms of Hopf monoids. This includes the species of graphs, matroids, posets, set partitions, linear graphs, hypergraphs, simplicial complexes, and building sets, among others. We employ this algebraic structure to define and study polynomial invariants of the various combinatorial structures.
We pay special attention to the antipode of each Hopf monoid. This map is central to the structure of a Hopf monoid, and it interacts well with its characters and polynomial invariants. It also carries information on the values of the invariants on negative integers. For our Hopf monoid of generalized permutahedra, we show that the antipode maps each polytope to the alternating sum of its faces. This fact has numerous combinatorial consequences.
We highlight some main applications:
We obtain uniform proofs of numerous old and new results about the Hopf algebraic and combinatorial structures of these families. In particular, we give optimal formulas for the antipode of graphs, posets, matroids, hypergraphs, and building sets. They are optimal in the sense that they provide explicit descriptions for the integers entering in the expansion of the antipode, after all coefficients have been collected and all cancellations have been taken into account.
We show that reciprocity theorems of Stanley and Billera–Jia–Reiner (BJR) on chromatic polynomials of graphs, order polynomials of posets, and BJRpolynomials of matroids are instances of one such result for generalized permutahedra.
We explain why the formulas for the multiplicative and compositional inverses of power series are governed by the face structure of permutahedra and associahedra, respectively, providing an answer to a question of Loday.
We answer a question of Humpert and Martin on certain invariants of graphs and another of Rota on a certain class of submodular functions.
We hope our work serves as a quick introduction to the theory of Hopf monoids in species, particularly to the reader interested in combinatorial applications. It may be supplemented with Marcelo Aguiar and Swapneel Mahajan’s 2010 and 2013 works, which provide longer accounts with a more algebraic focus.