scholarly journals ALGORITHMS TO IDENTIFY ABUNDANT p-SINGULAR ELEMENTS IN FINITE CLASSICAL GROUPS

2012 ◽  
Vol 86 (1) ◽  
pp. 50-63 ◽  
Author(s):  
ALICE C. NIEMEYER ◽  
TOMASZ POPIEL ◽  
CHERYL E. PRAEGER
Keyword(s):  
Lower Bound ◽  
Absolute Constant ◽  
Classical Group ◽  
A Priori ◽  
Point Of View ◽  
Classical Groups ◽  
Natural Representation ◽  
Constant Multiple ◽  
Computational Point ◽  
Finite Classical Groups

AbstractLet G be a finite d-dimensional classical group and p a prime divisor of ∣G∣ distinct from the characteristic of the natural representation. We consider a subfamily of p-singular elements in G (elements with order divisible by p) that leave invariant a subspace X of the natural G-module of dimension greater than d/2 and either act irreducibly on X or preserve a particular decomposition of X into two equal-dimensional irreducible subspaces. We proved in a recent paper that the proportion in G of these so-called p-abundant elements is at least an absolute constant multiple of the best currently known lower bound for the proportion of all p-singular elements. From a computational point of view, the p-abundant elements generalise another class of p-singular elements which underpin recognition algorithms for finite classical groups, and it is our hope that p-abundant elements might lead to improved versions of these algorithms. As a step towards this, here we present efficient algorithms to test whether a given element is p-abundant, both for a known prime p and for the case where p is not known a priori.

scholarly journals The Regularized Weak Functional Matching Pursuit for linear inverse problems

2019 ◽  
Vol 27 (3) ◽  
pp. 317-340 ◽  
Author(s):  
Max Kontak ◽  
Volker Michel
Keyword(s):  
Inverse Problems ◽  
Matching Pursuit ◽  
A Priori ◽  
Computation Time ◽  
Point Of View ◽  
Computational Point ◽  
Linear Inverse Problems ◽  
Infinite Dimensional ◽  
Frame Condition ◽  
Ill Posed

Abstract In this work, we present the so-called Regularized Weak Functional Matching Pursuit (RWFMP) algorithm, which is a weak greedy algorithm for linear ill-posed inverse problems. In comparison to the Regularized Functional Matching Pursuit (RFMP), on which it is based, the RWFMP possesses an improved theoretical analysis including the guaranteed existence of the iterates, the convergence of the algorithm for inverse problems in infinite-dimensional Hilbert spaces, and a convergence rate, which is also valid for the particular case of the RFMP. Another improvement is the cancellation of the previously required and difficult to verify semi-frame condition. Furthermore, we provide an a-priori parameter choice rule for the RWFMP, which yields a convergent regularization. Finally, we will give a numerical example, which shows that the “weak” approach is also beneficial from the computational point of view. By applying an improved search strategy in the algorithm, which is motivated by the weak approach, we can save up to 90  of computation time in comparison to the RFMP, whereas the accuracy of the solution does not change as much.

scholarly journals Constructing Maximal Subgroups of Classical Groups

2005 ◽  
Vol 8 ◽  
pp. 46-79 ◽  
Author(s):  
Derek F. Holt ◽  
Colva M. Roney-Dougal
Keyword(s):  
Polynomial Time ◽  
Maximal Subgroups ◽  
Classical Groups ◽  
Natural Representation ◽  
Degree Polynomial ◽  
Low Degree ◽  
Finite Classical Groups

AbstractThe maximal subgroups of the finite classical groups are divided by a theorem of Aschbacher into nine classes. In this paper, the authors show how to construct those maximal subgroups of the finite classical groups of linear, symplectic or unitary type that lie in the first eight of these classes. The ninth class consists roughly of absolutely irreducible groups that are almost simple modulo scalars, other than classical groups over the same field in their natural representation. All of these constructions can be carried out in low-degree polynomial time.

The Steinberg Character of Finite Classical Groups

2013 ◽  
Vol 20 (01) ◽  
pp. 163-168
Author(s):  
Xueling Song ◽  
Yanjun Liu
Keyword(s):  
Classical Group ◽  
Classical Groups ◽  
Characteristic P ◽  
Combinatorial Objects ◽  
Finite Classical Group ◽  
Finite Classical Groups ◽  
Steinberg Character

Let G be a finite classical group of characteristic p. In this paper, we give an arithmetic criterion of the primes r ≠ p, for which the Steinberg character lies in the principal r-block of G. The arithmetic criterion is obtained from some combinatorial objects (the so-called partition and symbol).

Uncertainty in Time

2015 ◽  
Vol 25 (01) ◽  
pp. 1540007 ◽  
Author(s):  
José Félix Costa
Keyword(s):  
Lower Bound ◽  
Turing Machine ◽  
Quantum Physics ◽  
Energy Gap ◽  
Point Of View ◽  
Computational Point ◽  
Uncertainty In Measurement ◽  
A Value ◽  
Time And Energy

The relation between scatters of time and energy is well known in Classical and Quantum Physics, but the interpretations of such relation are not well understood. What is the meaning of a time scatter in a measurement? What is the meaning of the correlated energy gap? Why should they combine to give a value greater or equal to a theoretical lower bound? In this paper, we discuss uncertainty in measurement from the computational point of view and introduce the archetype of a generic stochastic oracle to a Turing machine.

COMBINATORIAL POLYNOMIALLY COMPUTABLE CHARACTERISTICS OF SUBSTITUTIONS AND THEIR PROPERTIES

2020 ◽  
Vol 7 (2) ◽  
pp. 34-41
Author(s):  
VLADIMIR NIKONOV ◽  
◽  
ANTON ZOBOV ◽  
Keyword(s):  
Mathematical Expectation ◽  
Point Of View ◽  
Small Range ◽  
Computational Point ◽  
Wide Range ◽  
Bijective Function ◽  
The Difference ◽  
Element Base ◽  
Selection Of

The construction and selection of a suitable bijective function, that is, substitution, is now becoming an important applied task, particularly for building block encryption systems. Many articles have suggested using different approaches to determining the quality of substitution, but most of them are highly computationally complex. The solution of this problem will significantly expand the range of methods for constructing and analyzing scheme in information protection systems. The purpose of research is to find easily measurable characteristics of substitutions, allowing to evaluate their quality, and also measures of the proximity of a particular substitutions to a random one, or its distance from it. For this purpose, several characteristics were proposed in this work: difference and polynomial, and their mathematical expectation was found, as well as variance for the difference characteristic. This allows us to make a conclusion about its quality by comparing the result of calculating the characteristic for a particular substitution with the calculated mathematical expectation. From a computational point of view, the thesises of the article are of exceptional interest due to the simplicity of the algorithm for quantifying the quality of bijective function substitutions. By its nature, the operation of calculating the difference characteristic carries out a simple summation of integer terms in a fixed and small range. Such an operation, both in the modern and in the prospective element base, is embedded in the logic of a wide range of functional elements, especially when implementing computational actions in the optical range, or on other carriers related to the field of nanotechnology.

A Fatal Error of Locke’s?

2017 ◽  
Author(s):  
Galen Strawson
Keyword(s):  
Personal Identity ◽  
John Locke ◽  
A Priori ◽  
Point Of View ◽  
Radical Theory ◽  
Single Person

This chapter argues that the unqualified attribution of the radical theory to John Locke is mistaken if we are to take into account the fact that the theory allows for freaks like [Sₓ]. It first considers [I]-transfer without [P]-transfer—that is, [I]-transfer preserving personal identity—before discussing Locke's response to the idea that personal identity might survive [I]-transfer from an a priori point of view. It suggests that [I]-transfer is possible in such a way that the existence of a single Person [P₁] from t₁ to t₂ can successively (and non-overlappingly) involve the existence of two immaterial substances. It also explains how Locke's claim that [I]-transfer is possible opens up the possibility that it could go wrong, in such a way as to lead to injustice. Finally, it examines Locke's notion of “sensible creature,” which refers to a subject of experience who is a person.

scholarly journals Iterative Variable Selection for High-Dimensional Data: Prediction of Pathological Response in Triple-Negative Breast Cancer

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 222
Author(s):  
Juan C. Laria ◽  
M. Carmen Aguilera-Morillo ◽  
Enrique Álvarez ◽  
Rosa E. Lillo ◽  
Sara López-Taruella ◽  
...  
Keyword(s):  
Breast Cancer ◽  
Variable Selection ◽  
Triple Negative Breast Cancer ◽  
Triple Negative ◽  
A Priori ◽  
Simulated Data ◽  
Point Of View ◽  
High Dimensional ◽  
Whole Genome ◽  
Genome Context

Over the last decade, regularized regression methods have offered alternatives for performing multi-marker analysis and feature selection in a whole genome context. The process of defining a list of genes that will characterize an expression profile remains unclear. It currently relies upon advanced statistics and can use an agnostic point of view or include some a priori knowledge, but overfitting remains a problem. This paper introduces a methodology to deal with the variable selection and model estimation problems in the high-dimensional set-up, which can be particularly useful in the whole genome context. Results are validated using simulated data and a real dataset from a triple-negative breast cancer study.

scholarly journals Power series coefficients for probabilities in finite classical groups

2006 ◽  
Vol 305 (2) ◽  
pp. 1212-1237
Author(s):  
John R. Britnell ◽  
Jason Fulman
Keyword(s):  
Power Series ◽  
Classical Groups ◽  
Finite Classical Groups

ON ROBERTS’ PROOF OF THE TURAEV-WALKER THEOREM

1996 ◽  
Vol 05 (04) ◽  
pp. 427-439 ◽  
Author(s):  
RICCARDO BENEDETTI ◽  
CARLO PETRONIO
Keyword(s):  
Euler Characteristic ◽  
A Priori ◽  
Algebraic Approach ◽  
Fusion Rule ◽  
Point Of View ◽  
Witten Invariant ◽  
Manifolds With Boundary ◽  
3 Dimensional ◽  
Skein Theory ◽  
Natural Way

In this paper we discuss the beautiful idea of Justin Roberts [7] (see also [8]) to re-obtain the Turaev-Viro invariants [11] via skein theory, and re-prove elementarily the Turaev-Walker theorem [9], [10], [13]. We do this by exploiting the presentation of 3-manifolds introduced in [1], [4]. Our presentation supports in a very natural way a formal implementation of Roberts’ idea. More specifically, what we show is how to explicitly extract from an o-graph (the object by which we represent a manifold, see below), one of the framed links in S3 which Roberts uses in the construction of his invariant, and a planar diagrammatic representation of such a link. This implies that the proofs of invariance and equality with the Turaev-Viro invariant can be carried out in a completely “algebraic” way, in terms of a planar diagrammatic calculus which does not require any interpretation of 3-dimensional figures. In particular, when proving the “term-by-term” equality of the expansion of the Roberts invariant with the state sum which gives the Turaev-Viro invariant, we simultaneously apply several times the “fusion rule” (which is formally defined, strictly speaking, only in diagrammatic terms), showing that the “braiding and twisting” which a priori may exist on tetrahedra is globally dispensable. In our point of view the success of this formal “algebraic” approach witnesses a certain efficiency of our presentation of 3-manifolds via o-graphs. In this work we will widely use recoupling theory which was very clearly exposed in [2], and therefore we will avoid recalling notations. Actually, for the purpose of stating and proving our results we will need to slightly extend the class of trivalent ribbon diagrams on which the bracket can be computed. We also address the reader to the references quoted in [2], in particular for the fundamental contributions of Lickorish to this area. In our approach it is more natural to consider invariants of compact 3-manifolds with non-empty boundary. The case of closed 3-manifolds is included by introducing a correction factor corresponding to boundary spheres, as explained in §2. Our main result is actually an extension to manifolds with boundary of the Turaev-Walker theorem: we show that the Turaev-Viro invariant of such a manifold coincides (up to a factor which depends on the Euler characteristic) with the Reshetikhin-Turaev-Witten invariant of the manifold mirrored in its boundary.

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