scholarly journals Products of several commutators in a Lie nilpotent associative algebra

2017 ◽  
Vol 27 (08) ◽  
pp. 1027-1040 ◽  
Author(s):  
Galina Deryabina ◽  
Alexei Krasilnikov
Keyword(s):  
General Formula ◽  
Associative Algebra ◽  
Present Note ◽  
Commutator Formula ◽  
Positive Integers

Let [Formula: see text] be a field of characteristic [Formula: see text] and let [Formula: see text] be a unital associative [Formula: see text]-algebra. Define a left-normed commutator [Formula: see text] [Formula: see text] recursively by [Formula: see text], [Formula: see text] [Formula: see text]. For [Formula: see text], let [Formula: see text] be the two-sided ideal in [Formula: see text] generated by all commutators [Formula: see text] ([Formula: see text]. Define [Formula: see text]. Let [Formula: see text] be integers such that [Formula: see text], [Formula: see text]. Let [Formula: see text] be positive integers such that [Formula: see text] of them are odd and [Formula: see text] of them are even. Let [Formula: see text]. The aim of the present note is to show that, for any positive integers [Formula: see text], in general, [Formula: see text]. It is known that if [Formula: see text] (that is, if at least one of [Formula: see text] is even), then [Formula: see text] for each [Formula: see text] so our result cannot be improved if [Formula: see text]. Let [Formula: see text]. Recently, Dangovski has proved that if [Formula: see text] are any positive integers then, in general, [Formula: see text]. Since [Formula: see text], Dangovski’s result is stronger than ours if [Formula: see text] and is weaker than ours if [Formula: see text]; if [Formula: see text], then [Formula: see text] so both results coincide. It is known that if [Formula: see text] (that is, if all [Formula: see text] are odd) then, for each [Formula: see text], [Formula: see text] so in this case Dangovski’s result cannot be improved.

scholarly journals Congruence successions in compositions

2014 ◽  
Vol Vol. 16 no. 1 (Combinatorics) ◽  
Author(s):  
Toufik Mansour ◽  
Mark Shattuck ◽  
Mark Wilson
Keyword(s):  
General Formula ◽  
Generating Function ◽  
Asymptotic Estimate ◽  
International Audience ◽  
Limiting Case ◽  
Positive Integers

Combinatorics International audience A composition is a sequence of positive integers, called parts, having a fixed sum. By an m-congruence succession, we will mean a pair of adjacent parts x and y within a composition such that x=y(modm). Here, we consider the problem of counting the compositions of size n according to the number of m-congruence successions, extending recent results concerning successions on subsets and permutations. A general formula is obtained, which reduces in the limiting case to the known generating function formula for the number of Carlitz compositions. Special attention is paid to the case m=2, where further enumerative results may be obtained by means of combinatorial arguments. Finally, an asymptotic estimate is provided for the number of compositions of size n having no m-congruence successions.

A note on the Entscheidungsproblem

1936 ◽  
Vol 1 (1) ◽  
pp. 40-41 ◽  
Author(s):  
Alonzo Church
Keyword(s):  
Present Note ◽  
Arithmetic Function ◽  
Arithmetic Functions ◽  
Symbolic Logic ◽  
Recursive Definition ◽  
System L ◽  
Preceding Paragraph ◽  
Individual Variables ◽  
Definition Of ◽  
Positive Integers

In a recent paper the author has proposed a definition of the commonly used term “effectively calculable” and has shown on the basis of this definition that the general case of the Entscheidungsproblem is unsolvable in any system of symbolic logic which is adequate to a certain portion of arithmetic and is ω-consistent. The purpose of the present note is to outline an extension of this result to the engere Funktionenkalkul of Hilbert and Ackermann.In the author's cited paper it is pointed out that there can be associated recursively with every well-formed formula a recursive enumeration of the formulas into which it is convertible. This means the existence of a recursively defined function a of two positive integers such that, if y is the Gödel representation of a well-formed formula Y then a(x, y) is the Gödel representation of the xth formula in the enumeration of the formulas into which Y is convertible.Consider the system L of symbolic logic which arises from the engere Funktionenkalkül by adding to it: as additional undefined symbols, a symbol 1 for the number 1 (regarded as an individual), a symbol = for the propositional function = (equality of individuals), a symbol s for the arithmetic function x+1, a symbol a for the arithmetic function a described in the preceding paragraph, and symbols b1, b2, …, bk for the auxiliary arithmetic functions which are employed in the recursive definition of a; and as additional axioms, the recursion equations for the functions a, b1, b2, …, bk (expressed with free individual variables, the class of individuals being taken as identical with the class of positive integers), and two axioms of equality, x = x, and x = y →[F(x)→F(y)].

A Combinatorial Interpretation of Ramanujan's Continued Fraction

1968 ◽  
Vol 11 (3) ◽  
pp. 405-408 ◽  
Author(s):  
G. Szekeres
Keyword(s):  
Continued Fraction ◽  
Present Note ◽  
Combinatorial Interpretation ◽  
Image Position ◽  
Positive Integers ◽  
Ramanujan's Continued Fraction

The purpose of the present note is to give a combinatorial interpretation of the coefficients of expansion of the Ramanujan continued fraction ([1], p. 295)The result is expressed by formula (12) below.The enumeration of distinct score vectors of a tournament leads to the following problem: (Erdős and Moser, see Moon [2], p. 68). Given n ≥ 1, k ≥ 0, determine the number of distinct sequences of positive integers

An Extension of the Hájek-Rényi inequality for the case without moment conditions

1968 ◽  
Vol 5 (2) ◽  
pp. 481-483 ◽  
Author(s):  
C. C. Heyde
Keyword(s):  
Present Note ◽  
Random Variables ◽  
Independent Random Variables ◽  
Extended Version ◽  
Moment Conditions ◽  
General Bound ◽  
Image Position ◽  
Positive Integers ◽  
Standard Techniques

In the paper [1], Hájek and Rényi established an inequality which they formulated in the following way: X1,X2,··· are independent random variables and . For each k, EXk = 0 and , while is a non-increasing sequence of positive numbers. Then, for any ε > 0 and any positive integers n and m (n < m), The well-known Kolmogorov inequality is the particular case ck = 1, all k, and n = 1 of (1). It is the object of the present note to produce an extended version of (1) where no moment conditions need be satisfied. This provides a useful general bound and illuminates the role of certain standard techniques in the study of the almost sure behaviour of sums of independent random variables.

The nilpotence degree of quantum Lie nilpotent algebras

2018 ◽  
Vol 28 (06) ◽  
pp. 1119-1128
Author(s):  
Elena Kireeva ◽  
Vladimir Shchigolev
Keyword(s):  
Associative Algebra ◽  
Invertible Element ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Ground Field ◽  
Commutator Formula ◽  
Quantum Analog ◽  
Nilpotent Algebras

We consider the quantum analog of the Lie commutator [Formula: see text] for an invertible element [Formula: see text] of the ground field and prove lower and upper bounds for the nilpotence degree of an associative algebra satisfying an identity of the form [Formula: see text].

Some formulae for the G -function

1963 ◽  
Vol 59 (2) ◽  
pp. 347-350 ◽  
Author(s):  
R. K. Saxena
Keyword(s):  
General Result ◽  
Present Note ◽  
The Other ◽  
Positive Integers

The object of the present note is to evaluate some integrals involving Meijer's G-function, in which the argument of the G-function contains a factor where m and n are positive integers and t is the variable of integration. Two different forms of the general result have been obtained, one for m > n and the other for m < n. The value of the corresponding integral when m = n is also obtained. For the definition, properties and the behaviour of the G-function, see (2), §§ 5·3, 5·31 and (5), § 18.

An Extension of the Hájek-Rényi inequality for the case without moment conditions

1968 ◽  
Vol 5 (02) ◽  
pp. 481-483
Author(s):  
C. C. Heyde
Keyword(s):  
Present Note ◽  
Random Variables ◽  
Independent Random Variables ◽  
Extended Version ◽  
Moment Conditions ◽  
General Bound ◽  
Image Position ◽  
Positive Integers ◽  
Standard Techniques

In the paper [1], Hájek and Rényi established an inequality which they formulated in the following way: X 1,X 2,··· are independent random variables and . For each k, EXk = 0 and , while is a non-increasing sequence of positive numbers. Then, for any ε &gt; 0 and any positive integers n and m (n &lt; m), The well-known Kolmogorov inequality is the particular case ck = 1, all k, and n = 1 of (1). It is the object of the present note to produce an extended version of (1) where no moment conditions need be satisfied. This provides a useful general bound and illuminates the role of certain standard techniques in the study of the almost sure behaviour of sums of independent random variables.

X-ray Structure Refinement and Vibrational Spectroscopy of Ca8Gd2 (PO4)6 O2

2013 ◽  
Vol 12 (10) ◽  
pp. 719-726
Author(s):  
R. Ayadi ◽  
Mohamed Boujelbene ◽  
T. Mhiri
Keyword(s):  
Solid State ◽  
General Formula ◽  
Vibrational Spectroscopy ◽  
Powder Diffraction ◽  
Infrared Spectra ◽  
Solid State Reaction ◽  
Structure Refinement ◽  
Unit Cell ◽  
Cell Group ◽  
X Ray

The present paper is interested in the study of compounds from the apatite family with the general formula Ca10 (PO4)6A2. It particularly brings to light the exploitation of the distinctive stereochemistries of two Ca positions in apatite. In fact, Gd-Bearing oxyapatiteCa8 Gd2 (PO4)6O2 has been synthesized by solid state reaction and characterized by X-ray powder diffraction. The site occupancies of substituents is0.3333 in Gd and 0.3333 for Ca in the Ca(1) position and 0. 5 for Gd in the Ca (2) position.  Besides, the observed frequencies in the Raman and infrared spectra were explained and discussed on the basis of unit-cell group analyses.

Language, procedures, and the non-perceptual origin of natural number concepts

2016 ◽  
Author(s):  
David Barner
Keyword(s):  
General Framework ◽  
Ad Hoc ◽  
Building Blocks ◽  
Number Words ◽  
Linguistic Markers ◽  
Logical Foundations ◽  
Large Numbers ◽  
Perceptual Representations ◽  
Positive Integers ◽  
Perceptual Systems

Perceptual representations – e.g., of objects or approximate magnitudes –are often invoked as building blocks that children combine with linguisticsymbols when they acquire the positive integers. Systems of numericalperception are either assumed to contain the logical foundations ofarithmetic innately, or to supply the basis for their induction. Here Ipropose an alternative to this general framework, and argue that theintegers are not learned from perceptual systems, but instead arise toexplain perception as part of language acquisition. Drawing oncross-linguistic data and developmental data, I show that small numbers(1-4) and large numbers (~5+) arise both historically and in individualchildren via entirely distinct mechanisms, constituting independentlearning problems, neither of which begins with perceptual building blocks.Specifically, I propose that children begin by learning small numbers(i.e., *one, two, three*) using the same logical resources that supportother linguistic markers of number (e.g., singular, plural). Several yearslater, children discover the logic of counting by inferring the logicalrelations between larger number words from their roles in blind countingprocedures, and only incidentally associate number words with perception ofapproximate magnitudes, in an *ad hoc* and highly malleable fashion.Counting provides a form of explanation for perception but is not causallyderived from perceptual systems.

Post-quantum digital signature schemes: setting a hidden group with two-dimensional cyclicity

2020 ◽  
Vol 4 ◽  
pp. 75-82
Author(s):  
D.Yu. Guryanov ◽  
◽  
D.N. Moldovyan ◽  
A. A. Moldovyan ◽  
Keyword(s):  
Associative Algebra ◽  
Digital Signature ◽  
Quantum Signature ◽  
Commutative Group ◽  
Two Dimensional ◽  
Signature Scheme ◽  
Signature Schemes ◽  
Fixed Element ◽  
Commutative Associative Algebra ◽  
Quantum Digital Signature

For the construction of post-quantum digital signature schemes that satisfy the strengthened criterion of resistance to quantum attacks, an algebraic carrier is proposed that allows one to define a hidden commutative group with two-dimensional cyclicity. Formulas are obtained that describe the set of elements that are permutable with a given fixed element. A post-quantum signature scheme based on the considered finite non-commutative associative algebra is described.

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