scholarly journals On the index of Dirac operators on arithmetic quotients

1997 ◽  
Vol 56 (3) ◽  
pp. 489-497 ◽  
Author(s):  
Anton Deitmar
Keyword(s):  
Trace Formula ◽  
Special Interest ◽  
Error Term ◽  
Dirac Operators ◽  
Index Formula ◽  
Real Rank ◽  
Present Evidence ◽  
Index Theorems ◽  
Compact Case ◽  
Rational Rank

The aim of this note is to show how the trace formula of Arthur-Selberg can be used to derive index theorems for noncompact arithmetic manifolds. Of special interest is the question, under which circumstances there is an index formula without error term, that is, of the same shape as in the compact case. We shall thus present evidence for the hypothesis that the error term for the Euler operator vanishes in the case that the rational rank is smaller than the real rank.

scholarly journals Index Theory and Non-Commutative Geometry II. Dirac Operators and Index Bundles

2007 ◽  
Vol 1 (2) ◽  
pp. 305-356 ◽  
Author(s):  
Moulay-Tahar Benameur ◽  
James L. Heitsch
Keyword(s):  
Chern Character ◽  
Index Theory ◽  
Dirac Operators ◽  
Type Operator ◽  
Index Formula ◽  
Dirac Type ◽  
Dirac Type Operator ◽  
Index Bundle

AbstractWhen the index bundle of a longitudinal Dirac type operator is transversely smooth, we define its Chern character in Haefliger cohomology and relate it to the Chern character of the K—theory index. This result gives a concrete connection between the topology of the foliation and the longitudinal index formula. Moreover, the usual spectral assumption on the Novikov-Shubin invariants of the operator is improved.

scholarly journals Index theorems on manifolds with straight ends

2012 ◽  
Vol 148 (6) ◽  
pp. 1897-1968 ◽  
Author(s):  
Werner Ballmann ◽  
Jochen Brüning ◽  
Gilles Carron
Keyword(s):  
Sectional Curvature ◽  
Finite Volume ◽  
Riemannian Manifolds ◽  
Dirac Operators ◽  
Important Class ◽  
Index Theorems ◽  
Complete Riemannian Manifolds ◽  
Index Formulas ◽  
Negative Sectional Curvature

AbstractWe study Fredholm properties and index formulas for Dirac operators over complete Riemannian manifolds with straight ends. An important class of examples of such manifolds are complete Riemannian manifolds with pinched negative sectional curvature and finite volume.

scholarly journals The Selberg trace formula for Dirac operators

2006 ◽  
Vol 47 (11) ◽  
pp. 112104 ◽  
Author(s):  
Jens Bolte ◽  
Hans-Michael Stiepan
Keyword(s):  
Trace Formula ◽  
Dirac Operators ◽  
Selberg Trace Formula

scholarly journals An Index Formula for Perturbed Dirac Operators on Lie Manifolds

2013 ◽  
Vol 24 (4) ◽  
pp. 1808-1843 ◽  
Author(s):  
Catarina Carvalho ◽  
Victor Nistor
Keyword(s):  
Dirac Operators ◽  
Index Formula ◽  
Lie Manifolds

A note on K -theoretical index theorems for orbifolds

1992 ◽  
Vol 437 (1900) ◽  
pp. 429-431 ◽  
Keyword(s):  
Index Theory ◽  
Index Theorem ◽  
Index Formula ◽  
Index Theorems ◽  
Cohomological Index

In this note we study index theory for orbifolds. We deduce from the K -theoretical index theorem of Farsi ( Q . J1 Math. 43, 183-200 (1992)) the orbifold index theorem of T. Kawasaki using cyclic theory (A. Connes and G. Yu). Our main result is an extension of Theorem 14 in the Farsi paper, namely we derive the cohomological index formula for orbifolds of T. Kawasaki from our K -theoretical index theorem by using the methods introduced by A. Connes and G. Yu. In §1 we review some notation and describe the K -theoretical index theorems for orbifolds and in §2 we prove our main result.

scholarly journals On Some Classical andWeighted Estimates for SL4

2020 ◽  
Vol 15 ◽  
Keyword(s):  
Error Term ◽  
Symmetric Spaces ◽  
Compact Riemann Surface ◽  
Real Rank ◽  
Classical Approach ◽  
Locally Symmetric Spaces ◽  
Compact Symmetric Space ◽  
Rank One ◽  
Prime Geodesic Theorem ◽  
Correct Estimate

The purpose of this paper is two-sided. First, we obtain the correct estimate of the error term in the classical prime geodesic theorem for compact symmetric space SL4. As it turns out, the corrected error term depends on the degree of a certain polynomial appearing in the functional equation of the attached zeta function. This is in line with the known result in the case of compact Riemann surface, or more generally, with the corresponding result in the case of compact locally symmetric spaces of real rank one. Second, we derive a weighted form of the theorem. In particular, we prove that the aforementioned error term can be significantly improved when the classical approach is replaced by its higher level analogue.

SLOW–FAST DYNAMICS GENERATED BY NONINVERTIBLE PLANE MAPS

2005 ◽  
Vol 15 (11) ◽  
pp. 3509-3534 ◽  
Author(s):  
CHRISTIAN MIRA ◽  
ANDREY SHILNIKOV
Keyword(s):  
Special Interest ◽  
Phase Plane ◽  
Chaotic Attractors ◽  
Real Rank ◽  
Rank One ◽  
Fast Motion ◽  
Close Curve ◽  
Chaotic Transients ◽  
Noninvertible Maps ◽  
Stable Invariant

The present paper focuses on the two time scale dynamics generated by 2D polynomial noninvertible maps T of (Z0 - Z2) and (Z1 - Z3 - Z1) types. This symbolism, specific to noninvertible maps, means that the phase plane is partitioned into zones Zk, where each point possesses the k real rank-one preimages. Of special interest here is the structure of slow and fast motion sets of such maps. The formation mechanism of a stable invariant close curve through the interaction of fast and slow dynamics, as well as its transformation into a canard are studied. A few among the plethora of chaotic attractors and chaotic transients produced by such maps are described as well.

Crystalloid Inclusions in Hepatocyte Mitochondria and Their Similarity to Cytoplasmic Crystalloids

1974 ◽  
Vol 32 ◽  
pp. 198-199
Author(s):  
Odell T. Minick ◽  
Hidejiro Yokoo
Keyword(s):  
Liver Disease ◽  
Alcoholic Liver Disease ◽  
Special Interest ◽  
Structural Variations ◽  
Mitochondrial Alterations ◽  
The Matrix ◽  
Crystalloid Inclusions ◽  
Liver Biopsies

Mitochondrial alterations were studied in 25 liver biopsies from patients with alcoholic liver disease. Of special interest were the morphologic resemblance of certain fine structural variations in mitochondria and crystalloid inclusions. Four types of alterations within mitochondria were found that seemed to relate to cytoplasmic crystalloids.Type 1 alteration consisted of localized groups of cristae, usually oriented in the long direction of the organelle (Fig. 1A). In this plane they appeared serrated at the periphery with blind endings in the matrix. Other sections revealed a system of equally-spaced diagonal lines lengthwise in the mitochondrion with cristae protruding from both ends (Fig. 1B). Profiles of this inclusion were not unlike tangential cuts of a crystalloid structure frequently seen in enlarged mitochondria described below.

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