1,788 research outputs found
Symmetries of Riemann surfaces and magnetic monopoles
This thesis studies, broadly, the role of symmetry in elucidating structure. In particular, I investigate the role that automorphisms of algebraic curves play in three specific contexts; determining the orbits of theta characteristics, influencing the geometry of the highly-symmetric Bring’s curve, and in constructing magnetic monopole solutions. On theta characteristics, I show how to turn questions on the existence of invariant characteristics into questions of group cohomology, compute comprehensive tables of orbit decompositions for curves of genus 9 or less, and prove results on the existence of infinite families of curves with invariant characteristics. On Bring’s curve, I identify key points with geometric significance on the curve, completely determine the structure of the quotients by subgroups of automorphisms, finding new elliptic curves in the process, and identify the unique invariant theta characteristic on the curve. With respect to monopoles, I elucidate the role that the Hitchin conditions play in determining monopole spectral curves, the relation between these conditions and the automorphism group of the curve, and I develop the theory of computing Nahm data of symmetric monopoles. As such I classify all 3-monopoles whose Nahm data may be solved for in terms of elliptic functions
2023-2024 Catalog
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Northeastern Illinois University, Academic Catalog 2023-2024
https://neiudc.neiu.edu/catalogs/1064/thumbnail.jp
Effective bounds for the measure of rotations
A fundamental question in dynamical systems is to identify regions of phase/parameter space satisfying a given property (stability, linearization, etc). Given a family of analytic circle diffeomorphisms depending on a parameter, we obtain effective (almost optimal) lower bounds of the Lebesgue measure of the set of parameters that are conjugated to a rigid rotation. We estimate this measure using an a posteriori KAM scheme that relies on quantitative conditions that are checkable using computer-assistance. We carefully describe how the hypotheses in our theorems are reduced to a finite number of computations, and apply our methodology to the case of the Arnold family. Hence we show that obtaining non-asymptotic lower bounds for the applicability of KAM theorems is a feasible task provided one has an a posteriori theorem to characterize the problem. Finally, as a direct corollary, we produce explicit asymptotic estimates in the so called local reduction setting (à la Arnold) which are valid for a global set of rotations
Geometric optimization problems in quantum computation and discrete mathematics: Stabilizer states and lattices
This thesis consists of two parts:
Part I deals with properties of stabilizer states and their convex
hull, the stabilizer polytope. Stabilizer states, Pauli measurements
and Clifford unitaries are the three building blocks of the stabilizer
formalism whose computational power is limited by the Gottesman-
Knill theorem. This model is usually enriched by a magic state to get
a universal model for quantum computation, referred to as quantum
computation with magic states (QCM). The first part of this thesis
will investigate the role of stabilizer states within QCM from three
different angles.
The first considered quantity is the stabilizer extent, which provides
a tool to measure the non-stabilizerness or magic of a quantum state.
It assigns a quantity to each state roughly measuring how many stabilizer
states are required to approximate the state. It has been shown
that the extent is multiplicative under taking tensor products when
the considered state is a product state whose components are composed
of maximally three qubits. In Chapter 2, we will prove that
this property does not hold in general, more precisely, that the stabilizer
extent is strictly submultiplicative. We obtain this result as
a consequence of rather general properties of stabilizer states. Informally
our result implies that one should not expect a dictionary to be
multiplicative under taking tensor products whenever the dictionary
size grows subexponentially in the dimension.
In Chapter 3, we consider QCM from a resource theoretic perspective.
The resource theory of magic is based on two types of quantum
channels, completely stabilizer preserving maps and stabilizer operations.
Both classes have the property that they cannot generate additional
magic resources. We will show that these two classes of quantum
channels do not coincide, specifically, that stabilizer operations are a
strict subset of the set of completely stabilizer preserving channels.
This might have the consequence that certain tasks which are usually
realized by stabilizer operations could in principle be performed better
by completely stabilizer preserving maps.
In Chapter 4, the last one of Part I, we consider QCM via the polar
dual stabilizer polytope (also called the Lambda-polytope). This polytope
is a superset of the quantum state space and every quantum state
can be written as a convex combination of its vertices. A way to
classically simulate quantum computing with magic states is based on
simulating Pauli measurements and Clifford unitaries on the vertices
of the Lambda-polytope. The complexity of classical simulation with respect
to the polytope is determined by classically simulating the updates
of vertices under Clifford unitaries and Pauli measurements. However,
a complete description of this polytope as a convex hull of its vertices is
only known in low dimensions (for up to two qubits or one qudit when
odd dimensional systems are considered). We make progress on this
question by characterizing a certain class of operators that live on the
boundary of the Lambda-polytope when the underlying dimension is an odd
prime. This class encompasses for instance Wigner operators, which
have been shown to be vertices of Lambda. We conjecture that this class
contains even more vertices of Lambda. Eventually, we will shortly sketch
why applying Clifford unitaries and Pauli measurements to this class
of operators can be efficiently classically simulated.
Part II of this thesis deals with lattices. Lattices are discrete subgroups
of the Euclidean space. They occur in various different areas of
mathematics, physics and computer science. We will investigate two
types of optimization problems related to lattices.
In Chapter 6 we are concerned with optimization within the space of
lattices. That is, we want to compare the Gaussian potential energy
of different lattices. To make the energy of lattices comparable we
focus on lattices with point density one. In particular, we focus on
even unimodular lattices and show that, up to dimension 24, they are
all critical for the Gaussian potential energy. Furthermore, we find
that all n-dimensional even unimodular lattices with n 24 are local
minima or saddle points. In contrast in dimension 32, there are even
unimodular lattices which are local maxima and others which are not
even critical.
In Chapter 7 we consider flat tori R^n/L, where L is an n-dimensional
lattice. A flat torus comes with a metric and our goal is to approximate
this metric with a Hilbert space metric. To achieve this, we
derive an infinite-dimensional semidefinite optimization program that
computes the least distortion embedding of the metric space R^n/L into
a Hilbert space. This program allows us to make several interesting
statements about the nature of least distortion embeddings of flat tori.
In particular, we give a simple proof for a lower bound which gives
a constant factor improvement over the previously best lower bound
on the minimal distortion of an embedding of an n-dimensional flat
torus. Furthermore, we show that there is always an optimal embedding
into a finite-dimensional Hilbert space. Finally, we construct
optimal least distortion embeddings for the standard torus R^n/Z^n and
all 2-dimensional flat tori
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Efficient NIZKs for Algebraic Sets
Significantly extending the framework of (Couteau and Hartmann, Crypto 2020), we propose a general methodology to construct NIZKs for showing that an encrypted vector belongs to an algebraic set, i.e., is in the zero locus of an ideal of a polynomial ring. In the case where is principal, i.e., generated by a single polynomial , we first construct a matrix that is a ``quasideterminantal representation\u27\u27 of and then a NIZK argument to show that . This leads to compact NIZKs for general computational structures, such as polynomial-size algebraic branching programs. We extend the framework to the case where \IDEAL is non-principal, obtaining efficient NIZKs for R1CS, arithmetic constraint satisfaction systems, and thus for . As an independent result, we explicitly describe the corresponding language of ciphertexts as an algebraic language, with smaller parameters than in previous constructions that were based on the disjunction of algebraic languages. This results in an efficient GL-SPHF for algebraic branching programs
(Un)Solvable Loop Analysis
Automatically generating invariants, key to computer-aided analysis of
probabilistic and deterministic programs and compiler optimisation, is a
challenging open problem. Whilst the problem is in general undecidable, the
goal is settled for restricted classes of loops. For the class of solvable
loops, introduced by Kapur and Rodr\'iguez-Carbonell in 2004, one can
automatically compute invariants from closed-form solutions of recurrence
equations that model the loop behaviour. In this paper we establish a technique
for invariant synthesis for loops that are not solvable, termed unsolvable
loops. Our approach automatically partitions the program variables and
identifies the so-called defective variables that characterise unsolvability.
Herein we consider the following two applications. First, we present a novel
technique that automatically synthesises polynomials from defective monomials,
that admit closed-form solutions and thus lead to polynomial loop invariants.
Second, given an unsolvable loop, we synthesise solvable loops with the
following property: the invariant polynomials of the solvable loops are all
invariants of the given unsolvable loop. Our implementation and experiments
demonstrate both the feasibility and applicability of our approach to both
deterministic and probabilistic programs.Comment: Extended version of the conference paper `Solving Invariant
Generation for Unsolvable Loops' published at SAS 2022 (see also the preprint
arXiv:2206.06943). We extended both the text and results. 36 page
Transporting Higher-Order Quadrature Rules: Quasi-Monte Carlo Points and Sparse Grids for Mixture Distributions
Integration against, and hence sampling from, high-dimensional probability
distributions is of essential importance in many application areas and has been
an active research area for decades. One approach that has drawn increasing
attention in recent years has been the generation of samples from a target
distribution using transport maps: if
is the pushforward
of an easily-sampled probability distribution under
the transport map , then the application of to
-distributed samples yields
-distributed samples. This paper proposes the
application of transport maps not just to random samples, but also to
quasi-Monte Carlo points, higher-order nets, and sparse grids in order for the
transformed samples to inherit the original convergence rates that are often
better than , being the number of samples/quadrature nodes. Our
main result is the derivation of an explicit transport map for the case that
is a mixture of simple distributions, e.g.\ a
Gaussian mixture, in which case application of the transport map requires
the solution of an \emph{explicit} ODE with \emph{closed-form} right-hand side.
Mixture distributions are of particular applicability and interest since many
methods proceed by first approximating by a mixture
and then sampling from that mixture (often using importance reweighting).
Hence, this paper allows for the sampling step to provide a better convergence
rate than for all such methods.Comment: 24 page
2023-2024 Boise State University Undergraduate Catalog
This catalog is primarily for and directed at students. However, it serves many audiences, such as high school counselors, academic advisors, and the public. In this catalog you will find an overview of Boise State University and information on admission, registration, grades, tuition and fees, financial aid, housing, student services, and other important policies and procedures. However, most of this catalog is devoted to describing the various programs and courses offered at Boise State
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