936 research outputs found

    Elementary abelian subgroups: from algebraic groups to finite groups

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    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

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    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

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    The Implicit Factorization Problem (IFP) was first introduced by May and Ritzenhofen at PKC\u2709, which concerns the factorization of two RSA moduli N1=p1q1N_1=p_1q_1 and N2=p2q2N_2=p_2q_2, where p1p_1 and p2p_2 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 p1p_1 and p2p_2 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 anyany 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

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    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

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    The Implicit Factorization Problem was first introduced by May and Ritzenhofen at PKC'09. This problem aims to factorize two RSA moduli N1=p1q1N_1=p_1q_1 and N2=p2q2N_2=p_2q_2 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 k>2k>2 RSA moduli. Since then, several variations of the Implicit Factorization Problem have been studied, including the cases where p1p_1 and p2p_2 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

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    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

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    Intersection Theory on Weighted Blowups of F-theory Vacua

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    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 F4_4 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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