M. Hoffman | US Naval Academy (original) (raw)

Papers by M. Hoffman

Journal of Algebra

Quasi-shuffle products, introduced by the first author, have been useful in studying multiple zet... more Quasi-shuffle products, introduced by the first author, have been useful in studying multiple zeta values and some of their analogues and generalizations. The second author, together with Kajikawa, Ohno, and Okuda, significantly extended the definition of quasi-shuffle algebras so it could be applied to multiple q-zeta values. This article extends some of the algebraic machinery of the first author's original paper to the more general definition, and demonstrates how various algebraic formulas in the quasi-shuffle algebra can be obtained in a transparent way. Some applications to multiple zeta values, interpolated multiple zeta values, multiple q-zeta values, and multiple polylogarithms are given.

Mediterranean Journal of Mathematics, 2016

A recent paper of O. Furdui and C. Vȃlean proves some results about sums of products of "tails" o... more A recent paper of O. Furdui and C. Vȃlean proves some results about sums of products of "tails" of the series for the Riemann zeta function. We show how such results can be proved with weaker hypotheses using multiple zeta values, and also show how they can be generalized to products of three or more such tails.

Eprint Arxiv Math 0401319, Jan 23, 2004

We present a number of results about (finite) multiple harmonic sums modulo a prime, which provid... more We present a number of results about (finite) multiple harmonic sums modulo a prime, which provide interesting parallels to known results about multiple zeta values (i.e., infinite multiple harmonic series). In particular, we prove a "duality" result for mod p harmonic sums similar to (but distinct from) that for multiple zeta values. We also exploit the Hopf algebra structure of the quasi-symmetric functions to do calculations with multiple harmonic sums mod p, and obtain, for each weight n ≤ 9, a set of generators for the space of weight-n multiple harmonic sums mod p.

The Ramanujan Journal, 2016

We show how infinite series of a certain type involving generalized harmonic numbers can be compu... more We show how infinite series of a certain type involving generalized harmonic numbers can be computed using a knowledge of symmetric functions and multiple zeta values. In particular, we prove and generalize some identities recently conjectured by J. Choi, and give several more families of identities of a similar nature.

A poset can be regarded as a category in which there is at most one morphism between objects, and... more A poset can be regarded as a category in which there is at most one morphism between objects, and such that at most one of Hom(c, c ′) and Hom(c ′ , c) is nonempty for c = c ′. If we keep in place the latter axiom but allow for more than one morphism between objects, we have a sort of generalized poset in which there are multiplicities attached to the covering relations, and possibly nontrivial automorphism groups. We call such a category an "updown category." In this paper we give a precise definition of such categories and develop a theory for them. We also give a detailed account of ten examples, including updown categories of integer partitions, integer compositions, planar rooted trees, and rooted trees.

For a connected topological manifold M we define the noncoincidence index of M, a topological inv... more For a connected topological manifold M we define the noncoincidence index of M, a topological invariant reflecting the abundance of fixed-point-free self-maps of A/. We give some theorems on noncoincidence index and compute the noncoincidence index of the homogeneous manifold U(n)/H, where H is conjugate to U(l) k X U(n-k).

Let M be a compact oriented connected topological manifold. We show that if the Euler characteris... more Let M be a compact oriented connected topological manifold. We show that if the Euler characteristic χ{M) Φ 0 and M admits no degree zero self-maps without fixed points, then there is a finite number r such that any set of r or more fixed-point-free self-maps of M has a coincidence (i.e. for two of the maps / and g there exists x e M so that f(x)-g(x)). We call r the noncoincidence index of M. More generally, for any manifold M with χ(M) Φ 0 there is a finite number r (called the restricted noncoincidence index of M) so that any set of r or more fixed-point-free nonzero degree self-maps of M has a coincidence. We investigate how these indices change as one passes from a space to its orbit space under a free action. We compute the restricted noncoincidence index for certain products and for the homogeneous spaces SU n /K, K a closed connected subgroup of maximal rank; in some cases these computations also give the noncoincidence index of the space.

Suppose P is a partially ordered set that is locally finite, has a least element, and admits a ra... more Suppose P is a partially ordered set that is locally finite, has a least element, and admits a rank function. We call P a weighted-relation poset if all the covering relations of P are assigned a positive integer weight. We develop a theory of covering maps for weighted-relation posets, and in particular show that any weighted-relation poset P has a universal cover P → P , unique up to isomorphism, so that 1. P → P factors through any other covering map P → P ; 2. every principal order ideal of P is a chain; and 3. the weight assigned to each covering relation of P is 1. If P is a poset of "natural" combinatorial objects, the elements of its universal cover P often have a simple description as well. For example, if P is the poset of partitions ordered by inclusion of their Young diagrams, then the universal cover P is the poset of standard Young tableaux; if P is the poset of rooted trees ordered by inclusion, then P consists of permutations. We discuss several other examples, including the posets of necklaces, bracket arrangements, and compositions.

We define a homomorphism ζ from the algebra of quasi-symmetric functions to the reals which invol... more We define a homomorphism ζ from the algebra of quasi-symmetric functions to the reals which involves the Euler constant and multiple zeta values. Besides advancing the study of multiple zeta values, the homomorphism ζ appears in connection with two Hirzebruch genera of almost complex manifolds: the Γ-genus (related to mirror symmetry) and theΓ-genus (related to an S 1-equivariant Euler class). We decompose ζ into its even and odd factors in the sense of Aguiar, Bergeron, and Sottille, and demonstrate the usefulness of this decomposition in computing ζ on the subalgebra of symmetric functions (which suffices for computations of the Γ-andΓ-genera).

For k ≤ n, let E(2n, k) be the sum of all multiple zeta values with even arguments whose weight i... more For k ≤ n, let E(2n, k) be the sum of all multiple zeta values with even arguments whose weight is 2n and whose depth is k. Of course E(2n, 1) is the value ζ(2n) of the Riemann zeta function at 2n, and it is well known that E(2n, 2) = 3 4 ζ(2n). Recently Z. Shen and T. Cai gave formulas for E(2n, 3) and E(2n, 4) in terms ζ(2n) and ζ(2)ζ(2n − 2). We give two formulas for E(2n, k), both valid for arbitrary k ≤ n, one of which generalizes the Shen-Cai results; by comparing the two we obtain a Bernoulli-number identity. We also give an explicit generating function for the numbers E(2n, k).

The Connes-Kreimer Hopf algebra of rooted trees, its dual, and the Foissy Hopf algebra of planar ... more The Connes-Kreimer Hopf algebra of rooted trees, its dual, and the Foissy Hopf algebra of planar rooted trees are related to each other and to the well-known Hopf algebras of symmetric and quasi-symmetric functions via a pair of commutative diagrams. We show how this point of view can simplify computations in the Connes-Kreimer Hopf algebra and its dual, particularly for combinatorial Dyson-Schwinger equations.

In a recent paper, A. Libgober showed that the multiplicative sequence {Q i (c 1 ,. .. , c i)} of... more In a recent paper, A. Libgober showed that the multiplicative sequence {Q i (c 1 ,. .. , c i)} of Chern classes corresponding to the power series Q(z) = Γ(1 + z) −1 appears in a relation between the Chern classes of certain Calabi-Yau manifolds and the periods of their mirrors. We show that the polynomials Q i can be expressed in terms of multiple zeta values.

Multiple harmonic sums appear in the perturbative computation of various quantities of interest i... more Multiple harmonic sums appear in the perturbative computation of various quantities of interest in quantum field theory. In this article we introduce a class of Hopf algebras that describe the structure of such sums, and develop some of their properties that can be exploited in calculations.

Let P n and Q n be the polynomials obtained by repeated differentiation of the tangent and secant... more Let P n and Q n be the polynomials obtained by repeated differentiation of the tangent and secant functions respectively. From the exponential generating functions of these polynomials we develop relations among their values, which are then applied to various numerical sequences which occur as values of the P n and Q n. For example, P n (0) and Q n (0) are respectively the nth tangent and secant numbers, while P n (0) + Q n (0) is the nth André number. The André numbers, along with the numbers Q n (1) and P n (1) − Q n (1), are the Springer numbers of root systems of types A n , B n , and D n respectively, or alternatively (following V. I. Arnol'd) count the number of "snakes" of these types. We prove this for the latter two cases using combinatorial arguments. We relate the values of P n and Q n at √ 3 to certain "generalized Euler and class numbers" of D. Shanks, which have a combinatorial interpretation in terms of 3-signed permutations as defined by R. Ehrenborg and M. A. Readdy. Finally, we express the values of Euler polynomials at any rational argument in terms of P n and Q n , and from this deduce formulas for Springer and Shanks numbers in terms of Euler polynomials.

We begin with a compact figure that can be folded into smaller replicas of itself, such as the in... more We begin with a compact figure that can be folded into smaller replicas of itself, such as the interval or equilateral triangle. Such figures are in one-to-one correspondence with affine Weyl groups. For each such figure in n-dimensional Euclidean space, we construct a sequence of polynomials P/c: Rn-► Rn so that the mapping P^ is conjugate to stretching the figure by a factor A; and folding it back onto itself. If re = 1 and the figure is the interval, this construction yields the Chebyshev polynomials (up to conjugation). The polynomials Pk are orthogonal with respect to a suitable measure and can be extended in a natural way to a complete set of orthogonal polynomials.

We begin by considering the graded vector space with a basis consisting of rooted trees, with gra... more We begin by considering the graded vector space with a basis consisting of rooted trees, with grading given by the count of non-root vertices. We define two linear operators on this vector space, the growth and pruning operators, which respectively raise and lower grading; their commutator is the operator that multiplies a rooted tree by its number of vertices, and each operator naturally associates a multiplicity to each pair of rooted trees. By using symmetry groups of trees we define an inner product with respect to which the growth and pruning operators are adjoint, and obtain several results about the associated multiplicities. Now the symmetric algebra on the vector space of rooted trees (after a degree shift) can be endowed with a coproduct to make a Hopf algebra; this was defined by Kreimer in connection with renormalization. We extend the growth and pruning operators, as well as the inner product mentioned above, to Kreimer's Hopf algebra. On the other hand, the vector space of rooted trees itself can be given a noncommutative multiplication: with an appropriate coproduct, this leads to the Hopf algebra of Grossman and Larson. We show that the inner product on rooted trees leads to an isomorphism of the Grossman-Larson Hopf algebra with the graded dual of Kreimer's Hopf algebra, correcting an earlier result of Panaite.

We consider several identities involving the multiple harmonic series v^ 1 " ii 1 ' 1 #t' 2 n ik ... more We consider several identities involving the multiple harmonic series v^ 1 " ii 1 ' 1 #t' 2 n ik ' n ι >n 2 >->n k >l n \ n 2 n k which converge when the exponents /, are at least 1 and i\ > 1. There is a simple relation of these series with products of Riemann zeta functions (the case k = 1) when all the i } exceed 1. There are also two plausible identities concerning these series for integer exponents, which we call the sum and duality conjectures. Both generalize identities first proved by Euler. We give a partial proof of the duality conjecture, which coincides with the sum conjecture in one family of cases. We also prove all cases of the sum and duality conjectures when the sum of the exponents is at most 6.

Journal of Algebraic Combinatorics, 2000

Given a locally finite graded set A and a commutative, associative operation on A that adds degre... more Given a locally finite graded set A and a commutative, associative operation on A that adds degrees, we construct a commutative multiplication * on the set of noncommutative polynomials in A which we call a quasi-shuffle product; it can be viewed as a generalization of the shuffle product III. We extend this commutative algebra structure to a Hopf algebra (U, *, Δ); in the case where A is the set of positive integers and the operation on A is addition, this gives the Hopf algebra of quasi-symmetric functions. If rational coefficients are allowed, the quasi-shuffle product is in fact no more general than the shuffle product; we give an isomorphism exp of the shuffle Hopf algebra (U, III, Δ) onto (U, *, Δ) the set L of Lyndon words on A and their images { exp(w) ∣ w ∈ L} freely generate the algebra (U, *). We also consider the graded dual of (U, *, Δ). We define a deformation *q of * that coincides with * when q = 1 and is isomorphic to the concatenation product when q is not a root o...