8,837 research outputs found
Dynamics measured in a non-Archimedean field
We study dynamical systems using measures taking values in a non-Archimedean
field. The underlying space for such measure is a zero-dimensional topological
space. In this paper we elaborate on the natural translation of several
notions, e.g., probability measures, isomorphic transformations, entropy, from
classical dynamical systems to a non-Archimedean setting.Comment: 12 page
EPR of photochromic Mo3+ in SrTiO3
In single crystals of SrTiO_3, a paramagnetic center, characterized by S =
3/2 and hyperfine interaction with an I = 5/2 nuclear spin has been observed in
the temperature range 4.2K-77K by means of EPR. The impurity center is
attributed to Mo3+. No additional line splitting in the EPR spectrum due to the
105K phase transition has been observed. At 4.2K the following spin Hamiltonian
parameters for this impurity ion were obtained: g = 1.9546\pm0.0010 and A =
(32.0\pm0.05)\times10^-4 cm^-1.Comment: 5 pages, 2 figure
Virtual refinements of the Vafa-Witten formula
We conjecture a formula for the generating function of virtual
-genera of moduli spaces of rank 2 sheaves on arbitrary surfaces with
holomorphic 2-form. Specializing the conjecture to minimal surfaces of general
type and to virtual Euler characteristics, we recover (part of) a formula of C.
Vafa and E. Witten.
These virtual -genera can be written in terms of descendent Donaldson
invariants. Using T. Mochizuki's formula, the latter can be expressed in terms
of Seiberg-Witten invariants and certain explicit integrals over Hilbert
schemes of points. These integrals are governed by seven universal functions,
which are determined by their values on and . Using localization we calculate these functions up to some
order, which allows us to check our conjecture in many cases.
In an appendix by H. Nakajima and the first named author, the virtual Euler
characteristic specialization of our conjecture is extended to include
-classes, thereby interpolating between Vafa-Witten's formula and Witten's
conjecture for Donaldson invariants.Comment: 44 pages. Published version. Appendix C by first named author and H.
Nakajim
Edge reconstruction of the Ihara zeta function
We show that if a graph has average degree , then the
Ihara zeta function of is edge-reconstructible. We prove some general
spectral properties of the edge adjacency operator : it is symmetric for an
indefinite form and has a "large" semi-simple part (but it can fail to be
semi-simple in general). We prove that this implies that if , one can
reconstruct the number of non-backtracking (closed or not) walks through a
given edge, the Perron-Frobenius eigenvector of (modulo a natural
symmetry), as well as the closed walks that pass through a given edge in both
directions at least once.
The appendix by Daniel MacDonald established the analogue for multigraphs of
some basic results in reconstruction theory of simple graphs that are used in
the main text.Comment: 19 pages, 2 pictures, in version 2 some minor changes and now
including an appendix by Daniel McDonal
Donaldson-Thomas invariants of local elliptic surfaces via the topological vertex
We compute the Donaldson-Thomas invariants of a local elliptic surface with
section. We introduce a new computational technique which is a mixture of
motivic and toric methods. This allows us to write the partition function for
the invariants in terms of the topological vertex. Utilizing identities for the
topological vertex proved in arXiv:1603.05271, we derive product formulas for
the partition functions. The connected version of the partition function is
written in terms of Jacobi forms. In the special case where the elliptic
surface is a K3 surface, we get a derivation of the Katz-Klemm-Vafa formula for
primitive curve classes which is independent of the computation of
Kawai-Yoshioka.Comment: 43 pages, 3 figures. Formal methods replaced by much simpler
stratification according to location of embedded points. Published versio
Higher rank sheaves on threefolds and functional equations
We consider the moduli space of stable torsion free sheaves of any rank on a
smooth projective threefold. The singularity set of a torsion free sheaf is the
locus where the sheaf is not locally free. On a threefold it has dimension
. We consider the open subset of moduli space consisting of sheaves
with empty or 0-dimensional singularity set.
For fixed Chern classes and summing over , we show that the
generating function of topological Euler characteristics of these open subsets
equals a power of the MacMahon function times a Laurent polynomial. This
Laurent polynomial is invariant under (upon
replacing ). For some choices of these open
subsets equal the entire moduli space.
The proof involves wall-crossing from Quot schemes of a higher rank reflexive
sheaf to a sublocus of the space of Pandharipande-Thomas pairs. We interpret
this sublocus in terms of the singularities of the reflexive sheaf.Comment: 29 pages. Published versio
A rank 2 Dijkgraaf-Moore-Verlinde-Verlinde formula
We conjecture a formula for the virtual elliptic genera of moduli spaces of
rank 2 sheaves on minimal surfaces of general type. We express our
conjecture in terms of the Igusa cusp form and Borcherds type lifts
of three quasi-Jacobi forms which are all related to the Weierstrass elliptic
function. We also conjecture that the generating function of virtual cobordism
classes of these moduli spaces depends only on and
via two universal functions, one of which is determined by the
cobordism classes of Hilbert schemes of points on . We present
generalizations of these conjectures, e.g. to arbitrary surfaces with
and .
We use a result of J. Shen to express the virtual cobordism class in terms of
descendent Donaldson invariants. In a prequel we used T. Mochizuki's formula,
universality, and toric calculations to compute such Donaldson invariants in
the setting of virtual -genera. Similar techniques allow us to verify
our new conjectures in many cases.Comment: 24 pages. In order to keep the paper self-contained, we recall the
necessary material from the prequel arXiv:1703.07196, which results in some
overlap. Published version. Typo fixe
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