927 research outputs found
On the Kernel of the Symbol Map for Multiple Polylogarithms
No full textThe symbol map (of Goncharov) takes multiple polylogarithms to a tensor product space where calculations are easier, but where important differential and combinatorial properties of the multiple polylogarithm are retained. Finding linear combinations of multiple polylogarithms in the kernel of the symbol map is an effective way to attempt finding functional equations. We present and utilise methods for finding new linear combinations of multiple polylogarithms (and specifically harmonic polylogarithms) that lie in the kernel of the symbol map.
During this process we introduce a new pictorial construction for calculating the symbol, namely the hook-arrow tree, which can be used to easier encode symbol calculations onto a computer.
We also show how the hook-arrow tree can simplify symbol calculations where the depth of a multiple polylogarithm is lower than its weight and give explicit expressions for the symbol of depth 2 and 3 multiple polylogarithms of any weight. Using this we give the full symbol for I_{2,2,2}(x,y,z). Through similar methods we also give the full symbol of coloured multiple zeta values.
We provide introductory material including the binary tree (of Goncharov) and the polygon dissection (of Gangl, Goncharov and Levin) methods of finding the symbol of a multiple polylogarithm, and give bijections between (adapted forms of) these methods and the hook-arrow tree
Configuration Complexes and Tangential and Infinitesimal versions of Polylogarithmic Complexes
No full textIn this thesis we consider the Grassmannian complex of projective configurations in weight 2 and 3, and Cathelineau's infinitesimal polylogarithmic complexes as well as a tangential complex to the famous Bloch-Suslin complex (in weight 2) and to Goncharov's ``motivic`` complex (in weight 3), respectively, as proposed by Cathelineau [5].
Our main result is a morphism of complexes between the Grassmannian complexes and the associated infinitesimal polylogarithmic complexes as well as the tangential complexes.
In order to establish this connection we introduce an -vector space , which is an intermediate structure between a \varmathbb{Z}-module (scissors congruence group for ) and Cathelineau's -vector space which is an infinitesimal version of it. The structure of is also infinitesimal but it has the advantage of satisfying similar functional equations as the group . We put this in a complex to form a variant of Cathelineau's infinitesimal complex for weight 2. Furthermore, we define for the corresponding infinitesimal complex in weight 3. One of the important ingredients of the proof of our main results is the rewriting of Goncharov's triple-ratios as the product of two projected cross-ratios. Furthermore, we extend Siegel's cross-ratio identity ([21]) for determinants over the truncated polynomial ring . We compute cross-ratios and Goncharov's triple-ratios in and and use them extensively in our computations for the tangential complexes. We also verify a ''projected five-term'' relation in the group which is crucial to prove one of our central statements Theorem 4.3.3
Nulpunten van de Riemann zeta functie
No full textDit verslag geeft als eerste een overzicht van basis resultaten van de Riemann zeta functie. Vervolgens worden twee meer specifieke resultaten laten zien met betrekking tot de nulpunten van deze Riemann zeta functie
Zeta functions connecting multiple zeta values and poly-Bernoulli numbers
No full textWe first review our previous works of Arakawa and the authors on two closely related single-variable zeta functions. Their special values at positive and negative integer arguments are respectively multiple zeta values and poly-Bernoulli numbers. We then introduce, as a generalization of Sasaki's work, level 2-analogue of one of the two zeta functions and prove results analogous to those by Arakawa and the first named author
A New Family of Zeta Type Functions Involving the Hurwitz Zeta Function and the Alternating Hurwitz Zeta Function
No full textIn this paper, we further study the generating function involving a variety of special numbers and ploynomials constructed by the second author. Applying the Mellin transformation to this generating function, we define a new class of zeta type functions, which is related to the interpolation functions of the Apostol–Bernoulli polynomials, the Bernoulli polynomials, and the Euler polynomials. This new class of zeta type functions is related to the Hurwitz zeta function, the alternating Hurwitz zeta function, and the Lerch zeta function. Furthermore, by using these functions, we derive some identities and combinatorial sums involving the Bernoulli numbers and polynomials and the Euler numbers and polynomials
Exponentiation of Motivic Zeta Functions
No full textWe provide a formula for the generating series of the Weil zeta function of symmetric powers \Sym^n X of varieties over finite fields. This realizes the zeta function as an exponentiable measure whose associated Kapranov motivic zeta function takes values in the big Witt ring of . We apply our formula to compute Z(\Sym^n X,t) in a number of explicit cases. Any motivic zeta function of a measure factoring through the Grothendieck ring of Chow motives is itself exponentiable; in fact, this applies to as a motivic measure itself. We prove a condition for which any motivic measure taking values in a -ring has an associated motivic zeta function that is itself an exponentiable measure , and this process is shown to iterate indefinitely. This involves a study of the case of -ring-valued motivic measures. Finally, we provide an understanding of MacDonald's formula in this context
Relativistic double-zeta, triple-zeta, and quadruple-zeta basis sets for the lanthanides La–Lu
No full textRelativistic basis sets of double-zeta, triple-zeta, and quadruple-zeta quality have been optimized for the lanthanide elements La-Lu. The basis sets include SCF exponents for the occupied spinors and for the 6p shell, exponents of correlating functions for the valence shells (4f, 5d and 6s) and the outer core shells (4d, 5s and 5p), and diffuse functions, including functions for dipole polarization of the 4f shell. A finite nuclear size was used in all optimizations. The basis sets are illustrated by calculations on YbF. Prescriptions are given for constructing contracted basis sets. The basis sets are available as an internet archive and from the Dirac program web site, http://dirac. chem. sdu. dk. © 2010 The Author(s)
K(_2) and L-series of elliptic curves over real quadratic fields
No full textThis thesis examines the relationship between the L-series of an elliptic curve evaluated at s = 2 and the image of the regulator map when the curve is defined over a real quadratic field with narrow class number one, thus providing numerical evidence for Beilinson's conjecture. In doing so it provides a practical formula for calculating the L-series for modular elliptic curves over real quadratic fields, and in outline for more general totally real fields, and also provides numerical evidence for the generalization of the Taniyarna-Weil-Shimura conjecture to real quadratic fields
On multiple zeta values and finite multiple zeta values of maximal height
No full text An explicit formula for the height-one multiple zeta values (MZVs) was proved by Kaneko and the second author. We give an alternative proof of this result and its generalization. We also prove its counterpart for the finite multiple zeta values (FMZVs). </jats:p
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Derived Equivalent Varieties and their Zeta Functions
No full textIn order to shed light on Orlov’s conjecture that derived equivalent smooth, projective varieties have isomorphic Chow motives, we examine the zeta functions of derived equivalent varieties over finite fields; in this setting Orlov’s conjecture predicts equality of zeta functions. It is demonstrated that derived equivalent smooth, projective varieties over finite fields that are abelian or satisfy a certain condition on their cohomology. This condition is satisfied, for example, by a surface or Calabi–Yau 3–fold.One of our approaches to comparing the zeta functions of derived equivalent varieties over finite fields comes from using the Lefschetz Fixed Point Theorem to turn the question into one of comparing the -adic ́etale cohomology of varieties. Cohomology groups are not in general preserved under the action of Fourier–Mukai equivalences on cohomology, but cohomological structures we call even and odd Mukai–Hodge structures, which are a realization of the Mukai motive, are preserved. Investigation into when isomorphism of these cohomological structures implies equality of zeta functions gives us our cohomological condition for equality of zeta functions.We also develop a relative version of the map Fourier–Mukai transforms induce on cohomology and define a relative notion of even and odd Mukai–Hodge structures, and show these structures are preserved in a situation arising from the derived equivalence of smooth, projective varieties with semiample (anti-)canonical bundles. Using this result, it is demonstrated that when derived equivalent smooth, projective varieties have semiample (anti-)canonical bundles, the fibers over any fixed geometric point in their shared (anti-)canonical variety must also have isomorphic even and odd Mukai–Hodge structures. Hence, for any such varieties over finite fields, if their geometric fibers satisfy any of the conditions identified for isomorphism of Mukai–Hodge structures to imply equality of zeta functions, then the varieties themselves also have equal zeta functions
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