Functional equations for higher logarithms

Following earlier work by Abel and others, Kummer gave in 1840 functional equations for the polylogarithm function Li _m (z) up to m = 5, but no example for larger m was known until recently. We give the first genuine 2-variable functional equation for the 7-logarithm. We investigate and relate identities for the 3-logarithm given by Goncharov and Wojtkowiak and deduce a certain family of functional equations for the 4-logarithm.     

The zeros of a quadratic form at square-free points

Let F(x₁,…,xn) be a nonsingular indefinite quadratic form, n=3 or 4. For n=4, suppose the determinant of F is a square. Results are obtained on the number of solutions of

F(x₁,…,xn)=0     

Algebraic independence of arithmetic gamma values and Carlitz zeta values

We consider the values at proper fractions of the arithmetic gamma function and the values at positive integers of the zeta function for Fq[θ] and provide complete algebraic independence results for them.     

A restricted sum formula among multiple zeta values

For positive integers α₁,α₂,…,αr with αr?2, the multiple zeta value or r-fold Euler sum is defined as

     

Universal factorization property of certain polycyclic groups

Let H be a torsion-free strongly polycyclic (torsion-free virtually polycyclic, resp.) group. Let G be any group with maximal condition. We show that there exists a torsion-free strongly polycyclic (torsion-free virtually polycyclic, resp.) group and an epimorphism such that for any homomorphism ?:G→H, it factors through , i.e., there exists a homomorphism such that . We show that this factorization property cannot be extended to any finitely generated group G. As an application of factorization, we give necessary and sufficient conditions for N(f,g)=R(f,g) to hold for maps f,g:X→Y between closed orientable n-manifolds where π₁(X) has the maximal condition, Y is an infra-solvmanifold, N(f,g) and R(f,g) denote the Nielsen and Reidemeister coincidence numbers, respectively.     

Constructing one-parameter families of elliptic curves with moderate rank

We give several new constructions for moderate rank elliptic curves over Q(T). In particular we construct infinitely many rational elliptic surfaces (not in Weierstrass form) of rank 6 over Q using polynomials of degree two in T. While our method generates linearly independent points, we are able to show the rank is exactly 6 without having to verify the points are independent. The method generalizes; however, the higher rank surfaces are not rational, and we need to check that the constructed points are linearly independent.     

Discrete Morse theory for totally non-negative flag varieties

In a seminal 1994 paper Lusztig (1994) [26], Lusztig extended the theory of total positivity by introducing the totally non-negative part (G/P)_?0 of an arbitrary (generalized, partial) flag variety G/P. He referred to this space as a “remarkable polyhedral subspace”, and conjectured a decomposition into cells, which was subsequently proven by the first author Rietsch (1998) [33]. In Williams (2007) [40] the second author made the concrete conjecture that this cell decomposed space is the next best thing to a polyhedron, by conjecturing it to be a regular CW complex that is homeomorphic to a closed ball. In this article we use discrete Morse theory to prove this conjecture up to homotopy-equivalence. Explicitly, we prove that the boundaries of the cells are homotopic to spheres, and the closures of cells are contractible. The latter part generalizes a result of Lusztig's (1998) [28], that (G/P)_?0 - the closure of the top-dimensional cell - is contractible. Concerning our result on the boundaries of cells, even the special case that the boundary of the top-dimensional cell (G/P)_>0 is homotopic to a sphere, is new for all G/P other than projective space.     

Identities for the Ramanujan zeta function

We prove formulas for special values of the Ramanujan tau zeta function. Our formulas show that L(Δ,k)$L(\mathrm{\Delta },k)$ is a period in the sense of Kontsevich and Zagier when k?12$k?12$. As an illustration, we reduce L(Δ,k)$L(\mathrm{\Delta },k)$ to explicit integrals of hypergeometric and algebraic functions when k∈{12,13,14,15}$k\in \{12,13,14,15\}$.     

Multiple harmonic series related to multiple Euler numbers

In this paper, we define the multiple Euler numbers and consider some multiple harmonic series of Mordell-Tornheim's type, which is a partial sum of the Mordell-Tornheim zeta series defined by Matsumoto. Indeed, we prove a certain reducibility of these series as well as the multiple zeta values.     

The 26th power of Dedekind's η-function

We prove a simple and explicit formula, which expresses the 26th power of Dedekind's η-function as a double series. The proof relies on properties of Ramanujan's Eisenstein series P, Q and R, and parameters from the theory of elliptic functions.The formula reveals a number of properties of the product , for example its lacunarity, the action of the Hecke operator, and sufficient conditions for a coefficient to be zero.     

Ramanujan-type supercongruences

We present several supercongruences that may be viewed as p-adic analogues of Ramanujan-type series for 1/π and 1/π², and prove three of these examples.     

A (very short) introduction to buildings

This is an informal elementary introduction to buildings—what they are and where they come from.     

New notions in derived categories of local rings

In the derived category of a local commutative noetherian ring, we define irreducible chain complexes, atomic chain complexes, minimal atomic chain complexes and chain complexes having no mod m detectable homology. Also, we define nuclear chain complexes and core of chain complexes. After defining these notions, we establish the connection between them.     

When does the Haver property imply selective screenability?

We point out that in metric spaces Haver's property is not equivalent to the property introduced by Addis and Gresham. We prove that they are equal when the space has the Hurewicz property. We prove several results about the preservation of Haver's property in products. We show that if a separable metric space has the Haver property, and the nth power has the Hurewicz property, then the nth power has the Addis-Gresham property. R. Pol showed earlier that this is not the case when the Hurewicz property is replaced by the weaker Menger property. We introduce new classes of weakly infinite dimensional spaces.     

Selective screenability in topological groups

We examine the selective screenability property in topological groups. In the metrizable case we also give characterizations of Sc(Onbd,O) and Smirnov-Sc(Onbd,O) in terms of the Haver property and finitary Haver property respectively relative to left-invariant metrics. We prove theorems stating conditions under which Sc(Onbd,O) is preserved by products. Among metrizable groups we characterize the countable dimensional ones by a natural game.     

On Fox?s m-dimensional category and theorems of Bochner type

We show that catm(X)=cat(jm), where catm(X) is Fox?s m-dimensional category, jm:X→X[m] is the mth Postnikov section of X and cat(X) is the Lusternik-Schnirelmann category of X. This characterization is used to give new “Bochner-type” bounds on the rank of the Gottlieb group and the first Betti number for manifolds of non-negative Ricci curvature. Finally, we apply these methods to obtain Bochner-type theorems for manifolds of almost non-negative sectional curvature.     

A constructive proof of Ky Fan's generalization of Tucker's lemma

We present a proof of Ky Fan's combinatorial lemma on labellings of triangulated spheres that differs from earlier proofs in that it is constructive. We slightly generalize the hypotheses of Fan's lemma to allow for triangulations of Sⁿ that contain a flag of hemispheres. As a consequence, we can obtain a constructive proof of Tucker's lemma that holds for a more general class of triangulations than the usual version.     

Evaluation of double L-values

In this paper, a theorem of Zagier concerning double zeta evaluations is generalized to the double L-values. In addition, fast computation of the double L-values is demonstrated, extending the method of Crandall. The PARI commands are available electronically.     

L-S category of quaternionic Stiefel manifolds

By calculating certain generalized cohomology theory, lower bounds for the L-S category of quaternionic Stiefel manifolds are given.