936 research outputs found
Elementary abelian subgroups: from algebraic groups to finite groups
We describe a new approach for classifying conjugacy classes of elementary abelian subgroups in simple algebraic groups over an algebraically closed field, and understanding the normaliser and centraliser structure of these. For toral subgroups, we give an effective classification algorithm. For non-toral elementary abelian subgroups, we focus on algebraic groups of exceptional type with a view to future applications, and in this case we provide tables explicitly describing the subgroups and their local structure. We then describe how to transfer results to the corresponding finite groups of Lie type using the Lang-Steinberg Theorem; this will be used in forthcoming work to complete the classification of elementary abelian p-subgroups for torsion primes p in finite groups of exceptional Lie type. Such classification results are important for determining the maximal p-local subgroups and p-radical subgroups, both of which play a crucial role in modular representation theory
Quantum Algorithms for Interpolation and Sampling
Gibbs sampling from continuous real-valued functions is a challenging problem of interest in machine learning. Here we leverage quantum Fourier transforms to build a quantum algorithm for this task when the function is periodic. We use the quantum algorithms for solving linear ordinary differential equations to solve the Fokker--Planck equation and prepare a quantum state encoding the Gibbs distribution. We show that the efficiency of interpolation and differentiation of these functions on a quantum computer depends on the rate of decay of the Fourier coefficients of the Fourier transform of the function. We view this property as a concentration of measure in the Fourier domain, and also provide functional analytic conditions for it. Our algorithm makes zeroeth order queries to a quantum oracle of the function. Despite suffering from an exponentially long mixing time, this algorithm allows for exponentially improved precision in sampling, and polynomial quantum speedups in mean estimation in the
general case, and particularly under geometric conditions we identify for the critical points of the energy function
Generalized Implicit Factorization Problem
The Implicit Factorization Problem (IFP) was first introduced by May and Ritzenhofen at PKC\u2709, which concerns the factorization of two RSA moduli and , where and share a certain consecutive number of least significant bits. Since its introduction, many different variants of IFP have been considered, such as the cases where and share most significant bits or middle bits at the same positions. In this paper, we consider a more generalized case of IFP, in which the shared consecutive bits can be located at positions in each prime, not necessarily required to be located at the same positions as before. We propose a lattice-based algorithm to solve this problem under specific conditions, and also provide some experimental results to verify our analysis
Graph GOSPA metric: a metric to measure the discrepancy between graphs of different sizes
This paper proposes a metric to measure the dissimilarity between graphs that
may have a different number of nodes. The proposed metric extends the
generalised optimal subpattern assignment (GOSPA) metric, which is a metric for
sets, to graphs. The proposed graph GOSPA metric includes costs associated with
node attribute errors for properly assigned nodes, missed and false nodes and
edge mismatches between graphs. The computation of this metric is based on
finding the optimal assignments between nodes in the two graphs, with the
possibility of leaving some of the nodes unassigned. We also propose a lower
bound for the metric, which is also a metric for graphs and is computable in
polynomial time using linear programming. The metric is first derived for
undirected unweighted graphs and it is then extended to directed and weighted
graphs. The properties of the metric are demonstrated via simulated and
empirical datasets
Generalized Implicit Factorization Problem
The Implicit Factorization Problem was first introduced by May and
Ritzenhofen at PKC'09. This problem aims to factorize two RSA moduli
and when their prime factors share a certain number
of least significant bits (LSBs). They proposed a lattice-based algorithm to
tackle this problem and extended it to cover RSA moduli. Since then,
several variations of the Implicit Factorization Problem have been studied,
including the cases where and share some most significant bits
(MSBs), middle bits, or both MSBs and LSBs at the same position.
In this paper, we explore a more general case of the Implicit Factorization
Problem, where the shared bits are located at different and unknown positions
for different primes. We propose a lattice-based algorithm and analyze its
efficiency under certain conditions. We also present experimental results to
support our analysis
2023- The Twenty-seventh Annual Symposium of Student Scholars
The full program book from the Twenty-seventh Annual Symposium of Student Scholars, held on April 18-21, 2023. Includes abstracts from the presentations and posters.https://digitalcommons.kennesaw.edu/sssprograms/1027/thumbnail.jp
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Intersection Theory on Weighted Blowups of F-theory Vacua
Generalizing the results of 1211.6077 and 1703.00905, we prove a formula for
the pushforward of an arbitrary analytic function of the exceptional divisor
class of a weighted blowup of an algebraic variety centered at a smooth
complete intersection with normal crossing. We check this formula extensively
by computing the generating function of intersection numbers of a weighted
blowup of the generic SU(5) Tate model over arbitrary smooth base, and
comparing the answer to known results. Motivated by applications to
four-dimensional F-theory flux compactifications, we use our formula to compute
the intersection pairing on the vertical part of the middle cohomology of
elliptic Calabi-Yau 4-folds resolving the generic F and Sp(6) Tate models
with non-minimal singularities. These resolutions lead to non-flat fibrations
in which certain fibers contain 3-fold (divisor) components, whose physical
interpretation in M/F-theory remains to be fully explored.Comment: 37 pages plus an appendix. v2: Minor clarifications to Sections 3 and
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