We introduce a class of parameter-dependent q-series arising from the moduli of certain pencils of elliptic curves and rank 19 lattice polarized K3 surfaces. These series serve to ‘link’ the various Hauptmoduln which arise as modular mirror maps (as characterized in [C. F. Doran, in The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), CRM Proc. Lect. Notes 24, 257–281 (2000; Zbl 0972.14033); Commun. Math. Phys. 212, 625–647 (2000; Zbl 1047.11043)]), and can in principle be computed from the functional invariants of the pencils. Each corresponds to an algebraic solution to an isomonodromic deformation equation which parametrizes the Hurwitz space for those invariants. The discussion is made more concrete in the context of the Painlevé VI equation.
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Article Full-text available June 2013 · Journal of Logic and Computation
A famous result, conjectured by Gödel in 1932 and proved by McKinsey and Tarski in 1948, says that ϕ ϕ is a theorem of intuitionistic propositional logic IPC iff its Gödel-translation ϕ ′ ϕ′ is a theorem of modal logic S4. In this article, we extend an intuitionistic version of modal logic S1 + SP, introduced in
our previous paper , to a classical modal logic L and prove the following: a
... [Show full abstract] propositional formula ϕ ϕ is a theorem of IPC iff □ ϕ □ϕ is a theorem of L (actually, we show: Φ ⊢ I P C ϕ Φ⊢IPCϕ iff □ Φ ⊢ L □ ϕ □Φ⊢L□ϕ, for propositional Φ , ϕ Φ,ϕ). Thus, the map ϕ ↦ □ ϕ ϕ↦□ϕ is an embedding of IPC into L, i.e. L contains a copy of IPC. Moreover, L is a conservative extension of classical propositional
logic CPC. In this sense, L is an amalgam of CPC and IPC. We show that L is sound and complete w.r.t. a class of special Heyting
algebras with a (non-normal) modal operator. View full-text October 2009 · Communications in Algebra
In this article, I will prove that assuming Schanuel's conjecture, an exponential polynomial with algebraic coefficients can have only finitely many algebraic roots. Furthermore, this proof demonstrates that there are no unexpected algebraic roots of any such exponential polynomial. This implies a special case of Shapiro's conjecture: if p(x) and q(x) are two exponential polynomials with
... [Show full abstract] algebraic coefficients, each involving only one iteration of the exponential map, and they have common factors only of the form exp(g) for some exponential polynomial g, then p and q have only finitely many common zeros. Read more December 2012 · International Mathematics Research Notices
We investigate, in this article, a generalization of the Riemann–Roch theorem for graphs of Baker and Norine, with a view
toward identifying new objects for which a two-variable zeta-function can be defined. To a lattice Λ of rank n−1 in and perpendicular to a positive integer vector R, we define the notions of g-number and of canonical vector, in analogy with the notions of genus and canonical
... [Show full abstract] class in the theory of algebraic curves.
When Λ is the full sublattice of perpendicular to R, its g-number turns out to be the classical Frobenius number of the coefficients of R. We investigate the existence of canonical vectors—in particular, in the context of arithmetical graphs—where we obtain an
existence theorem using methods from arithmetic geometry. We show that a two-variable zeta-function can be defined when a
canonical vector exists. Read more Article Full-text available December 2011 · Information (Switzerland)
Information is usually related to knowledge. However, the recent development of information theory demonstrated that information is a much broader concept, being actually present in and virtually related to everything. As a result, many unknown types and kinds of information have been discovered. Nevertheless, information that acts on knowledge, bringing new and updating existing knowledge, is of
... [Show full abstract] primary importance to people. It is called epistemic information, which is studied in this paper based on the general theory of information and further developing its mathematical stratum. As a synthetic approach, which reveals the essence of information, organizing and encompassing all main directions in information theory, the general theory of information provides efficient means for such a study. Different types of information dynamics representation use tools of mathematical disciplines such as the theory of categories, functional analysis, mathematical logic and algebra. Here we employ algebraic structures for exploration of information and knowledge dynamics. In Introduction (Section 1), we discuss previous studies of epistemic information. Section 2 gives a compressed description of the parametric phenomenological definition of information in the general theory of information. In Section 3, anthropic information, which is received, exchanged, processed and used by people is singled out and studied based on the Componential Triune Brain model. One of the basic forms of anthropic information called epistemic information, which is related to knowledge, is analyzed in Section 4. Mathematical models of epistemic information are studied in Section 5. In Conclusion, some open problems related to epistemic information are given. View full-text February 2017 · Journal of Number Theory
In this paper we introduce the notion of global equivariant hom-Lie algebra. This is a Lie algebra-like structure associated with twisted derivations. We prove several results on the structure of modules of twisted derivations and how they form global equivariant hom-Lie algebras. Particular emphasis is put on examples and results in arithmetic geometry.
Read more November 2006 · Nuclear Physics B
We define a superspace over a ring R as a functor on a subcategory of the category of supercommutative R-algebras. As an application the notion of a p-adic superspace is introduced and used to give a transparent construction of the Frobenius map on p-adic cohomology of a smooth projective variety over Zp (the ring of p-adic integers).
Read more November 2009 · Annals of Pure and Applied Logic
Glivenko-type theorems for substructural logics (over FL) are comprehensively studied in the paper [N. Galatos, H. Ono, Glivenko theorems for substructural logics over FL, Journal of Symbolic Logic 71 (2006) 1353-1384]. Arguments used there are fully algebraic, and based on the fact that all substructural logics are algebraizable (see [N. Galatos, H. Ono, Algebraization, parametrized local
... [Show full abstract] deduction theorem and interpolation for substructural logics over FL, Studia Logica 83 (2006) 279-308] and also [N. Galatos, P. Jipsen, T. Kowalski, H. Ono, Residuated Lattices: An Algebraic Glimpse at Substructural Logics, in: Studies in Logic and the Foundations of Mathematics, vol. 151, Elsevier, 2007] for the details). Read more Article Full-text available July 2016 · Compositio Mathematica
Our aim is to clarify the relationship between Kudla's and Bruinier's Green functions attached to special cycles on Shimura varieties of orthogonal and unitary type. These functions play a key role in the arithmetic geometry of the special cycles in the context of Kudla's program. In particular, we show that the generating series obtained by taking the differences of the two families of Green
... [Show full abstract] functions is a modular form with trivial holomorphic projection. Along the way, we construct a section of the Maass lowering operator for moderate growth modular forms valued in the Weil representation using a regularized theta lift, which may be of independent interest. We also consider arithmetic-geometric applications to integral models of U(n, 1) Shimura varieties. Each family of Green functions gives rise to a formal arithmetic theta function, valued in an arithmetic Chow group, that is conjectured to be modular; our main result is the modularity of the difference of the two arithmetic theta functions. Finally, we relate the arithmetic heights of the special cycles to special derivatives of Eisenstein series, as predicted by Kudla's conjecture, and describe a refinement of a theorem of Bruinier-Howard-Yang on arithmetic intersections against CM points. View full-text August 2008
There is a huge amount of work on different kinds of theta functions, the theta correspondence, cohomology classes coming from special Schwartz classes via theta distribution, and much more. The aim of this text is to try to find joint construction principles while often leaving aside relevant but cumbersome details. The next steps after this prehistoric Part I will be directed to a description
... [Show full abstract] of the Howe operators introduced by Kudla and Millson and their special Schwartz forms and classes. This has as attractor the fact that the modular and automorphic forms arising naturally in context with these classes find very nice geometric interpretations of their Fourier coefficients and thus lead to an intriguing intertwining of elements of representation theory with algebraic and arithmetic geometry. The presentation here is in the spirit of my book on representations of linear groups. Though it may be seen as just another chapter, it has it has its own raison d’être and Read more December 2009
This thesis explores a variety of topics in two-dimensional arithmetic geometry, including the further development of I. Fesenko's adelic analysis and its relations with ramification theory, model-theoretic integration on valued fields, and Grothendieck duality on arithmetic surfaces.
I. Fesenko's theories of integration and harmonic analysis for higher dimensional local fields are extended to
... [Show full abstract] an arbitrary valuation field F whose residue field is a local field; applications to local zeta integrals are considered.
The integral is extended to F^n, where a linear change of variables formula is proved, yielding a translation-invariant integral on GL_n(F).
Non-linear changes of variables and Fubini's theorem are then examined. An interesting example is presented in which imperfectness of a positive characteristic local field causes Fubini's theorem to unexpectedly fail.
It is explained how the motivic integration theory of E. Hrushovski and D. Kazhdan can be modified to provide a model-theoretic approach to integration on two-dimensional local fields. The possible unification of this work with A. Abbes and T. Saito's ramification theory is explored.
Relationships between Fubini's theorem, ramification theory, and Riemann-Hurwitz formulae are established in the setting of curves and surfaces over an algebraically closed field.
A theory of residues for arithmetic surfaces is developed, and the reciprocity law around a point is established. The residue maps are used to explicitly construct the dualising sheaf of the surface. Read more Last Updated: 04 Jul 2022
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