10,870 research outputs found
Classification of multiplicity free quasi-Hamiltonian manifolds
A quasi-Hamiltonian manifold is called multiplicity free if all of its
symplectic reductions are 0-dimensional. In this paper, we classify compact,
multiplicity free, twisted quasi-Hamiltonian manifolds for simply connected,
compact Lie groups. Thereby, we recover old and find new examples of these
structures.Comment: v1: 35 pages, this is a complete revision of arxiv:1612.03843. Since
some omitted parts have already been cited, I opted for a new submission
under a new title. v2: 39 pages, revised according to the advice of a very
helpful refere
Towards A Practical High-Assurance Systems Programming Language
Writing correct and performant low-level systems code is a notoriously demanding job, even for experienced developers. To make the matter worse, formally reasoning about their correctness properties introduces yet another level of complexity to the task. It requires considerable expertise in both systems programming and formal verification. The development can be extremely costly due to the sheer complexity of the systems and the nuances in them, if not assisted with appropriate tools that provide abstraction and automation.
Cogent is designed to alleviate the burden on developers when writing and verifying systems code. It is a high-level functional language with a certifying compiler, which automatically proves the correctness of the compiled code and also provides a purely functional abstraction of the low-level program to the developer. Equational reasoning techniques can then be used to prove functional correctness properties of the program on top of this abstract semantics, which is notably less laborious than directly verifying the C code.
To make Cogent a more approachable and effective tool for developing real-world systems, we further strengthen the framework by extending the core language and its ecosystem. Specifically, we enrich the language to allow users to control the memory representation of algebraic data types, while retaining the automatic proof with a data layout refinement calculus. We repurpose existing tools in a novel way and develop an intuitive foreign function interface, which provides users a seamless experience when using Cogent in conjunction with native C. We augment the Cogent ecosystem with a property-based testing framework, which helps developers better understand the impact formal verification has on their programs and enables a progressive approach to producing high-assurance systems. Finally we explore refinement type systems, which we plan to incorporate into Cogent for more expressiveness and better integration of systems programmers with the verification process
Beam scanning by liquid-crystal biasing in a modified SIW structure
A fixed-frequency beam-scanning 1D antenna based on Liquid Crystals (LCs) is designed for application in 2D scanning with lateral alignment. The 2D array environment imposes full decoupling of adjacent 1D antennas, which often conflicts with the LC requirement of DC biasing: the proposed design accommodates both. The LC medium is placed inside a Substrate Integrated Waveguide (SIW) modified to work as a Groove Gap Waveguide, with radiating slots etched on the upper broad wall, that radiates as a Leaky-Wave Antenna (LWA). This allows effective application of the DC bias voltage needed for tuning the LCs. At the same time, the RF field remains laterally confined, enabling the possibility to lay several antennas in parallel and achieve 2D beam scanning. The design is validated by simulation employing the actual properties of a commercial LC medium
Harder's conjecture II
Let be a primitive form of weight for , and let
be a prime ideal of the Hecke field of . We denote by
the Siegel modular group of degree . Suppose that and that divides the algebraic part of
. Put . Then under certain mild
conditions, we prove that there exists a Hecke eigenform in the space of
modular forms of weight for such that is
congruent to modulo . Here, is
the Klingen-Eisenstein lift of the Saito-Kurokawa lift of to the
space of modular forms of weight for , and is
a certain lift of to the space of cusp forms of weight for
. As an application, we prove Harder's conjecture on the congruence
between the Hecke eigenvalues of and some quantities related to the Hecke
eigenvalues of
On the representation theory of Schur algebras in type
We study the representation theory of the type B Schur algebra
with unequal parameters introduced in work of Lai, Nakano
and Xiang. For generic values of , this algebra is semi-simple and Morita
equivalent to the Hecke algebra, but for special values, its category of
modules is more complicated. We study this representation theory by comparison
with the cyclotomic -Schur algebra of Dipper, James and Mathas, and use this
to construct a cellular algebra structure on .
This allows us to index the simple -modules as a subset of
the set of bipartitions of . For large, this will be all bipartitions of
if and only if is quasi-hereditary, in which case,
is Morita equivalent to the cyclotomic -Schur algebra. We
prove a modified version of a conjecture of Lai, Nakano and Xiang giving the
values of where this holds: if is large and odd, for
all satisfying ; if is large and even, for all satisfying We also prove two strengthenings of this
result: an indexing of the simple modules when is not a root of unity, and
a characterization of the quasi-hereditary blocks of .Comment: preliminary version, comments welcom
Geometric G-functions and Atypicality
We describe a general method for giving -adic interpretations of
-functions arising from degenerating periods of smooth projective algebraic
varieties. Using this, we are able to implement a strategy due to Andr\'e for
bounding heights of moduli points where period functions acquire unusual
algebraic relations. This leads to new results on Galois lower bounds for
special moduli, and new cases of the Zilber-Pink conjecture. In particular, we
establish the first Galois-orbit lower bounds on CM moduli in non-Shimura
settings.
As a more technical contribution, we introduce a refinement of the
Pila-Zannier strategy capable of handling Zilber-Pink-type atypical
intersection problems in arbitrary dimension and for arbitrary smooth
projective families.Comment: v2: Updated and streamlined introduction, giving a clearer
explanation of proof strategy and ideas. Significant updates to section 7.2,
including a discussion on Hodge-theoretic Siegel sets, orbits of Siegel sets
on Hodge-theoretic idempotents, and a height theorem for such orbits. The
last of these replaces a mistaken Siegel-set argument in the previous
version. Other smaller changes/fixe
Some Properties of Internal Locale Morphisms Externalised
We study morphisms of internal locales of Grothendieck toposes externally:
treating internal locales and their morphisms as sheaves and natural
transformations. We characterise those morphisms of internal locales that
induce surjective geometric morphisms and geometric embeddings, demonstrating
that both can be computed `pointwise'. We also show that the co-frame
operations on the co-frame of internal sublocales can also be computed
`pointwise' too.Comment: 46 pages. Updated version for submission. New sections on examples
and applications, in addition to general improvement
Recommended from our members
Rare-Event Estimation and Calibration for Large-Scale Stochastic Simulation Models
Stochastic simulation has been widely applied in many domains. More recently, however, the rapid surge of sophisticated problems such as safety evaluation of intelligent systems has posed various challenges to conventional statistical methods. Motivated by these challenges, in this thesis, we develop novel methodologies with theoretical guarantees and numerical applications to tackle them from different perspectives.
In particular, our works can be categorized into two areas: (1) rare-event estimation (Chapters 2 to 5) where we develop approaches to estimating the probabilities of rare events via simulation; (2) model calibration (Chapters 6 and 7) where we aim at calibrating the simulation model so that it is close to reality.
In Chapter 2, we study rare-event simulation for a class of problems where the target hitting sets of interest are defined via modern machine learning tools such as neural networks and random forests. We investigate an importance sampling scheme that integrates the dominating point machinery in large deviations and sequential mixed integer programming to locate the underlying dominating points. We provide efficiency guarantees and numerical demonstration of our approach.
In Chapter 3, we propose a new efficiency criterion for importance sampling, which we call probabilistic efficiency. Conventionally, an estimator is regarded as efficient if its relative error is sufficiently controlled. It is widely known that when a rare-event set contains multiple "important regions" encoded by the dominating points, importance sampling needs to account for all of them via mixing to achieve efficiency. We argue that the traditional analysis recipe could suffer from intrinsic looseness by using relative error as an efficiency criterion. Thus, we propose the new efficiency notion to tighten this gap. In particular, we show that under the standard Gartner-Ellis large deviations regime, an importance sampling that uses only the most significant dominating points is sufficient to attain this efficiency notion.
In Chapter 4, we consider the estimation of rare-event probabilities using sample proportions output by crude Monte Carlo. Due to the recent surge of sophisticated rare-event problems, efficiency-guaranteed variance reduction may face implementation challenges, which motivate one to look at naive estimators. In this chapter we construct confidence intervals for the target probability using this naive estimator from various techniques, and then analyze their validity as well as tightness respectively quantified by the coverage probability and relative half-width.
In Chapter 5, we propose the use of extreme value analysis, in particular the peak-over-threshold method which is popularly employed for extremal estimation of real datasets, in the simulation setting. More specifically, we view crude Monte Carlo samples as data to fit on a generalized Pareto distribution. We test this idea on several numerical examples. The results show that in the absence of efficient variance reduction schemes, it appears to offer potential benefits to enhance crude Monte Carlo estimates.
In Chapter 6, we investigate a framework to develop calibration schemes in parametric settings, which satisfies rigorous frequentist statistical guarantees via a basic notion that we call eligibility set designed to bypass non-identifiability via a set-based estimation. We investigate a feature extraction-then-aggregation approach to construct these sets that target at multivariate outputs. We demonstrate our methodology on several numerical examples, including an application to calibration of a limit order book market simulator.
In Chapter 7, we study a methodology to tackle the NASA Langley Uncertainty Quantification Challenge, a model calibration problem under both aleatory and epistemic uncertainties. Our methodology is based on an integration of distributionally robust optimization and importance sampling. The main computation machinery in this integrated methodology amounts to solving sampled linear programs. We present theoretical statistical guarantees of our approach via connections to nonparametric hypothesis testing, and numerical performances including parameter calibration and downstream decision and risk evaluation tasks
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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