A Generalized Goursat Lemma

Bauer, Kristine; Sen, Debasis; Zvengrowski, Peter

doi:10.48550/arxiv.1109.0024

Cited by 5 publications

(8 citation statements)

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“…However, the reverse argument is not true, i.e., some subgroup of Z × n cannot be written as a product of subgroups of Z p k i i . To go beyond subgroup products, we may resort to the fundamental theorem of finite Abelian group (see e.g., Section 5.2 of [23]), which help decompose Z × n to direct groups of Z b (see Theorem 1.4.1 of [42]) and then the subgroup enumeration problem becomes that of counting the subgroups of a finite abelian group, where the case of product of 2 groups Z b 1 × Z b 2 is fully solved and otherwise open [43][44][45] (for analytic solutions of some simple cases, see [46]). Finally, the total number of subgroups of Z n has order O( log n log log n ) [47].…”

Section: Appendixmentioning

confidence: 99%

Expand-and-Randomize: An Algebraic Approach to Secure Computation

Zhao,

Sun

2020

Preprint

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We consider the secure computation problem in a minimal model, where Alice and Bob each holds an input and wish to securely compute a function of their inputs at Carol without revealing any additional information about the inputs. For this minimal secure computation problem, we propose a novel coding scheme built from two steps. First, the function to be computed is expanded such that it can be recovered while additional information might be leaked. Second, a randomization step is applied to the expanded function such that the leaked information is protected. We implement this expand-and-randomize coding scheme with two algebraic structures -the finite field and the modulo ring of integers, where the expansion step is realized with the addition operation and the randomization step is realized with the multiplication operation over the respective algebraic structures.

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Section: Appendixmentioning

confidence: 99%

Expand-and-Randomize: An Algebraic Approach to Secure Computation

Zhao,

Sun

2020

Preprint

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“…(Indeed, it is traditional in some papers to describe joint surjectivity results as applications of Goursat's lemma.) Lemma 3.3 (Goursat [7,21]). Let G 1 and G 2 be groups and let H ≤ G 1 × G 2 be a subgroup that surjects onto each factor G i .…”

Section: Joint Surjectivitymentioning

confidence: 99%

Computational complexity and 3–manifolds and zombies

Kuperberg

Samperton

2018

Geom. Topol.

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We show the problem of counting homomorphisms from the fundamental group of a homology 3-sphere M to a finite, non-abelian simple group G is #P-complete, in the case that G is fixed and M is the computational input. Similarly, deciding if there is a non-trivial homomorphism is NP-complete. In both reductions, we can guarantee that every non-trivial homomorphism is a surjection. As a corollary, for any fixed integer m ≥ 5, it is NP-complete to decide whether M admits a connected m-sheeted covering.Our construction is inspired by universality results in topological quantum computation. Given a classical reversible circuit C, we construct M so that evaluations of C with certain initialization and finalization conditions correspond to homomorphisms π 1 (M) → G. An intermediate state of C likewise corresponds to a homomorphism π 1 (Σ g ) → G, where Σ g is a pointed Heegaard surface of M of genus g. We analyze the action on these homomorphisms by the pointed mapping class group MCG * (Σ g ) and its Torelli subgroup Tor * (Σ g ). By results of Dunfield-Thurston, the action of MCG * (Σ g ) is as large as possible when g is sufficiently large; we can pass to the Torelli group using the congruence subgroup property of Sp(2g, Z). Our results can be interpreted as a sharp classical universality property of an associated combinatorial (2 + 1)-dimensional TQFT.

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“…In order to proceed further with computations of U (G) , where G = Γ × O(2) and Γ is a finite group, we need to develop a description of subgroups its subgroups and conjugacy classes. For completeness of this paper, following [8], we will discuss a description of subgroups of G × G ′ . The classical result, Goursat's Lemma allows to characterize all subgroups H of the direct product G × G ′ of groups G, G ′ in terms of the isomorphisms between their quotients.…”

Section: 3mentioning

confidence: 99%

Multiple periodic solutions for Γ-symmetric Newtonian systems

Da̧bkowski

Krawcewicz

et al. 2017

Journal of Differential Equations

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The existence of periodic solutions in Γ-symmetric Newtonian systemsẍ = −∇f (x) can be effectively studied by means of the Γ × O(2)-equivariant gradient degree with values in the Euler ring U (Γ × O(2)). In this paper we show that in the case of Γ being a finite group, the Euler ring U (Γ × O(2)) and the related basic degrees are effectively computable using Euler ring homomorphisms, the Burnside ring A(Γ × O(2)), and the reduced Γ × O(2)-degree with no free parameters. We present several examples of Newtonian systems with various symmetries, for which we show existence of multiple periodic solutions. We also provide exact value of the equivariant topological invariant for those problems.1 for example a difference of the equivariant gradient degrees on large and small ball). Such cases are for example when the system (1) is asymptotically linear or satisfy a Nagumo-type growth condition. Then clearly the existence of non-trivial solutions (i.e. outside the small ball) can be concluded by the fact that ω = 0. However, one can be also interested to predict the existence of multiple 2π-periodic solutions with different types of symmetry. In such a case, the coefficients of ω corresponding to the so-called maximal orbit type can provide the crucial information in order to formulate such results. But the maximality of such obit types implies that it is a generator of the Burnside ring A(G), therefore it can actually be detected by the Γ × O(2)-equivariant degree with no free parameter, which can be much easier computed than the equivariant gradient degree. Similar arguments apply to the system (2), which we can consider as a bifurcation problem with a parameter λ. More precisely, in this case we are looking for critical values λ o of the parameter λ > 0, to which we can associate the Γ × O(2)-equivariant gradient bifurcation invariants ω(λ o ) ∈ U (Γ×O(2)) classifying the bifurcation of 2π-periodic solutions from the zero solution. The existence and multiplicity of such bifurcating branches of 2π-periodic solutions can be described from the information contained in the invariants ω(λ o ). Consequently, all the essential information needed to establish the existence and multiplicity results for the systems (1) and (2) can be extracted from the Γ × O(2)-equivariant degree (with no free parameter) of J which takes values in the Burnside ring A(G). It is clear that the Γ × O(2)-equivariant degree without free parameter can be easily computed (without getting entangled in complicated technical details), has similar properties and provides enough information for analyzing these problems.Nevertheless, let us emphasize that only the equivariant invariants ω ∈ U (G) (without truncation of its coefficients) provide a complete equivariant topological classification for the related solution sets to (1) or (2).To illustrate the usage and the computations of the associated with the systems (1) and (2) equivariant invariants, in section 7 we present several examples of symmetric Newtonian systems, for which the exact values of the a...

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A Generalized Goursat Lemma

Cited by 5 publications

References 0 publications

Expand-and-Randomize: An Algebraic Approach to Secure Computation

Expand-and-Randomize: An Algebraic Approach to Secure Computation

Computational complexity and 3–manifolds and zombies

Multiple periodic solutions for Γ-symmetric Newtonian systems

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