777 research outputs found
Notes on TQFT Wire Models and Coherence Equations for SU(3) Triangular Cells
After a summary of the TQFT wire model formalism we bridge the gap from
Kuperberg equations for SU(3) spiders to Ocneanu coherence equations for
systems of triangular cells on fusion graphs that describe modules associated
with the fusion category of SU(3) at level k. We show how to solve these
equations in a number of examples.Comment: 44 figure
Conformal Invariance in the Long-Range Ising Model
We consider the question of conformal invariance of the long-range Ising
model at the critical point. The continuum description is given in terms of a
nonlocal field theory, and the absence of a stress tensor invalidates all of
the standard arguments for the enhancement of scale invariance to conformal
invariance. We however show that several correlation functions, computed to
second order in the epsilon expansion, are nontrivially consistent with
conformal invariance. We proceed to give a proof of conformal invariance to all
orders in the epsilon expansion, based on the description of the long-range
Ising model as a defect theory in an auxiliary higher-dimensional space. A
detailed review of conformal invariance in the d-dimensional short-range Ising
model is also included and may be of independent interest.Comment: 52pp; V2: refs added; V3: ref added, published versio
The power-series algorithm:A numerical approach to Markov processes
Abstract: The development of computer and communication networks and flexible manufacturing systems has led to new and interesting multidimensional queueing models. The Power-Series Algorithm is a numerical method to analyze and optimize the performance of such models. In this thesis, the applicability of the algorithm is extended. This is illustrated by introducing and analyzing a wide class of queueing networks with very general dependencies between the different queues. The theoretical basis of the algorithm is strengthened by proving analyticity of the steady-state distribution in light traffic and finding remedies for previous imperfections of the method. Applying similar ideas to the transient distribution renders new analyticity results. Various aspects of Markov processes, analytic functions and extrapolation methods are reviewed, necessary for a thorough understanding and efficient implementation of the Power-Series Algorithm.
Differentially Private Numerical Vector Analyses in the Local and Shuffle Model
Numerical vector aggregation plays a crucial role in privacy-sensitive
applications, such as distributed gradient estimation in federated learning and
statistical analysis of key-value data. In the context of local differential
privacy, this study provides a tight minimax error bound of
, where represents the dimension of the
numerical vector and denotes the number of non-zero entries. By converting
the conditional/unconditional numerical mean estimation problem into a
frequency estimation problem, we develop an optimal and efficient mechanism
called Collision. In contrast, existing methods exhibit sub-optimal error rates
of or . Specifically,
for unconditional mean estimation, we leverage the negative correlation between
two frequencies in each dimension and propose the CoCo mechanism, which further
reduces estimation errors for mean values compared to Collision. Moreover, to
surpass the error barrier in local privacy, we examine privacy amplification in
the shuffle model for the proposed mechanisms and derive precisely tight
amplification bounds. Our experiments validate and compare our mechanisms with
existing approaches, demonstrating significant error reductions for frequency
estimation and mean estimation on numerical vectors.Comment: Full version of "Hiding Numerical Vectors in Local Private and
Shuffled Messages" (IJCAI 2021
An Algorithm for the Computation of Eigenvalues, Spectral Zeta Functions and Zeta-Determinants on Hyperbolic Surfaces
We present a rigorous scheme that makes it possible to compute eigenvalues of
the Laplace operator on hyperbolic surfaces within a given precision. The
method is based on an adaptation of the method of particular solutions to the
case of locally symmetric spaces and on explicit estimates for the
approximation of eigenfunctions on hyperbolic surfaces by certain basis
functions. It can be applied to check whether or not there is an eigenvalue in
an \epsilon-neighborhood of a given number \lambda>0. This makes it possible to
find all the eigenvalues in a specified interval, up to a given precision with
rigorous error estimates. The method converges exponentially fast with the
number of basis functions used. Combining the knowledge of the eigenvalues with
the Selberg trace formula we are able to compute values and derivatives of the
spectral zeta function again with error bounds. As an example we calculate the
spectral determinant and the Casimir energy of the Bolza surface and other
surfaces.Comment: 48 pages, 8 figures, LaTeX, some more typos corrected, more Figures
added, some explanations are more detailed now, Fenchel-Nielsen coordinates
and numbers for the surface with symmetry group of order 10 corrected,
datafiles are now available as ancillary file
Gentzen’s “cut rule” and quantum measurement in terms of Hilbert arithmetic. Metaphor and understanding modeled formally
Hilbert arithmetic in a wide sense, including Hilbert arithmetic in a narrow sense consisting by two dual and anti-isometric Peano arithmetics, on the one hand, and the qubit Hilbert space (originating for the standard separable complex Hilbert space of quantum mechanics), on the other hand, allows for an arithmetic version of Gentzen’s cut elimination and quantum measurement to be described uniformy as two processes occurring accordingly in those two branches. A philosophical reflection also justifying that unity by quantum neo-Pythagoreanism links it to the opposition of propositional logic, to which Gentzen’s cut rule refers immediately, on the one hand, and the linguistic and mathematical theory of metaphor therefore sharing the same structure borrowed from Hilbert arithmetic in a wide sense. An example by hermeneutical circle modeled as a dual pair of a syllogism (accomplishable also by a Turing machine) and a relevant metaphor (being a formal and logical mistake and thus fundamentally inaccessible to any Turing machine) visualizes human understanding corresponding also to Gentzen’s cut elimination and the Gödel dichotomy about the relation of arithmetic to set theory: either incompleteness or contradiction. The metaphor as the complementing “half” of any understanding of hermeneutical circle is what allows for that Gödel-like incompleteness to be overcome in human thought
- …