scholarly journals Two criteria in Fermat infinite descent method

2017 ◽  
Vol 2 (4) ◽  
Author(s):  
Anatoly A. Grinberg
Keyword(s):  
Finite Number ◽  
Descent Method ◽  
Equation Solution ◽  
Preceding Step ◽  
Special Case ◽  
Infinite Descent ◽  
Fermat's Equation

A method infinite descent is traditionally used to proof the Fermat’s theorem for the special case of exponent n=4. At each step, the method sequentially generates a new Fermat’s   equation with one of the term being smaller than that in the preceding step. After a finite number of steps the term becomes less than one and this is taken as criterion of the insolvency of the original Fermat’s equation. We show that the power of factor 2, in even parameter of Pythagoras’ equation solution used in the proof, decreases by one at each step of the descent. As a result we arrive at an unsolvable equation. This is the second criterion for the descent method. Which of the two criteria is reached first depends on the parameters of the initial Pythagorean solutions chosen for the analysis.

scholarly journals C-Commutativity

1980 ◽  
Vol 30 (2) ◽  
pp. 252-255 ◽  
Author(s):  
T. Cheatham ◽  
E. Enochs
Keyword(s):  
Finite Number ◽  
Commutative Ring ◽  
Associative Ring ◽  
Nonzero Divisor ◽  
Special Case

AbstractAn associative ring R with identity is said to be c-commutative for c ∈ R if a, b ∈ R and ab = c implies ba = c. Taft has shown that if R is c-commutative where c is a central nonzero divisor]can be omitted. We show that in R[x] is h(x)-commutative for any h(x) ∈ R [x] then so is R with any finite number of (commuting) indeterminates adjoined. Examples adjoined. Examples are given to show that R [[x]] need not be c-commutative even if R[x] is, Finally, examples are given to answer Taft's question for the special case of a zero-commutative ring.

Solution of a Problem of L. Fuchs Concerning Finite Intersections of Pure Subgroups

1986 ◽  
Vol 38 (2) ◽  
pp. 304-327 ◽  
Author(s):  
R. Göbel ◽  
R. Vergohsen
Keyword(s):  
Finite Number ◽  
Abelian Groups ◽  
Natural Numbers ◽  
P 75 ◽  
Image Position ◽  
Special Case

L. Fuchs states in his book “Infinite Abelian Groups” [6, Vol. I, p. 134] the followingProblem 13. Find conditions on a subgroup of A to be the intersection of a finite number of pure (p-pure) subgroups of A.The answer to this problem will be given as a special case of our theorem below. In order to find a better setting of this problem recall that a subgroup S ⊆ E is p-pure if pnE ∩ S = pnS for all natural numbers. Then S is pure in E if S is p-pure for all primes p. This generalizes to pσ-isotype, a definition due to L. J. Kulikov, cf. [6, Vol. II, p. 75] and [11, pp. 61, 62]. If α is an ordinal, then S is pσ-isotype if

Server sharing with a limited number of service positions and symmetric queues

1987 ◽  
Vol 24 (04) ◽  
pp. 990-1000 ◽  
Author(s):  
Benjamin Avi-Itzhak ◽  
Shlomo Halfin
Keyword(s):  
Finite Number ◽  
Generating Functions ◽  
Delay Time ◽  
Time Distribution ◽  
Equilibrium Distribution ◽  
Laplace Transforms ◽  
Processor Sharing ◽  
Time Sharing ◽  
Time Variance ◽  
Special Case

A class of M/G/1 time-sharing queues with a finite number of service positions and unlimited waiting space is described. The equilibrium distribution of symmetric queues belonging to this class is invariant under arbitrary service-independent reordering of the customers at instants of arrivals and departures. The delay time distribution, in the special case of one service position where preempted customers join the end of the line, is provided in terms of Laplace transforms and generating functions. It is shown that placing preempted customers at the end of the line rather than at the beginning of the line results in a reduction of the delay time variance. Comparisons with the delay time variance of the case of unlimited number of service positions (processor sharing system) are presented.

scholarly journals Finitely-generated solutions of certain integral equations

1994 ◽  
Vol 37 (2) ◽  
pp. 325-345 ◽  
Author(s):  
D. Porter ◽  
D. S. G. Stirling
Keyword(s):  
Integral Equation ◽  
Finite Number ◽  
Recent Work ◽  
Integral Equations ◽  
Finite Rank ◽  
Integral Operators ◽  
Compact Operators ◽  
Particular Solutions ◽  
Singular Kernels ◽  
Special Case

Recent work has shown that the solutions of the second-kind integral equation arising from a difference kernel can be expressed in terms of two particular solutions of the equation. This paper establishes analogous results for a wider class of integral operators, which includes the special case of those arising from difference kernels, where the solution of the general case is generated by a finite number of particular cases. The generalisation is achieved by reducing the problem to one of finite rank. Certain non-compact operators, including those arising from Cauchy singular kernels, are amenable to this approach.

scholarly journals A prime decomposition symbol for a non-abelian central extension which is abelian over a bicyclic biquadratic field

1980 ◽  
Vol 79 ◽  
pp. 79-109 ◽  
Author(s):  
Yoshiomi Furuta
Keyword(s):  
Number Field ◽  
Rational Number ◽  
Central Extension ◽  
Descent Method ◽  
Prime Decomposition ◽  
Abelian Extensions ◽  
Residue Symbol ◽  
Special Case

In a previous paper [6] we had some criteria for the prime decomposition in certain non-abelian extensions over the rational number field Q, and as its special case we had a reciprocity of the biquadratic residue symbol. The reciprocity was obtained by using a descent method of the prime decomposition for a central extension over Q which is abelian over a biquadratic field In the present paper we study on the case over a biquadratic field in general. We define a symbol [d1, d2, p] which expresses the decomposition law of a rational prime p in a central extension mentioned above.

Server sharing with a limited number of service positions and symmetric queues

1987 ◽  
Vol 24 (4) ◽  
pp. 990-1000 ◽  
Author(s):  
Benjamin Avi-Itzhak ◽  
Shlomo Halfin
Keyword(s):  
Finite Number ◽  
Generating Functions ◽  
Delay Time ◽  
Time Distribution ◽  
Equilibrium Distribution ◽  
Laplace Transforms ◽  
Processor Sharing ◽  
Time Sharing ◽  
Time Variance ◽  
Special Case

A class of M/G/1 time-sharing queues with a finite number of service positions and unlimited waiting space is described. The equilibrium distribution of symmetric queues belonging to this class is invariant under arbitrary service-independent reordering of the customers at instants of arrivals and departures. The delay time distribution, in the special case of one service position where preempted customers join the end of the line, is provided in terms of Laplace transforms and generating functions. It is shown that placing preempted customers at the end of the line rather than at the beginning of the line results in a reduction of the delay time variance. Comparisons with the delay time variance of the case of unlimited number of service positions (processor sharing system) are presented.

scholarly journals Unions of a clique and a co-clique as star complements for non-main graph eigenvalues

2019 ◽  
Vol 35 ◽  
pp. 90-99 ◽  
Author(s):  
Zoran Stanic
Keyword(s):  
Finite Number ◽  
Numerical Data ◽  
Arbitrary Size ◽  
Graph Eigenvalues ◽  
Strongly Regular ◽  
Special Case

Graphs consisting of a clique and a co-clique, both of arbitrary size, are considered in the role of star complements for an arbitrary non-main eigenvalue. Among other results, the sign of such a eigenvalue is discussed, the neigbourhoods of star set vertices are described, and the parameters of all strongly regular extensions are determined. It is also proved that, unless in a specified special case, if the size of a co-clique is fixed then there is a finite number of possibilities for our star complement and the corresponding non-main eigenvalue. Numerical data on these possibilities is presented.

Cooperative games with additive multiple attributes

2021 ◽  
Vol 41 (1) ◽  
pp. 1135-1150
Author(s):  
Haitao Liu ◽  
Qiang Zhang
Keyword(s):  
Finite Number ◽  
Cooperative Game ◽  
Shapley Value ◽  
Cooperative Games ◽  
Polynomial Form ◽  
The Core ◽  
The Shapley Value ◽  
Efficient Resource ◽  
Multiple Attributes ◽  
Special Case

This paper studies cooperative games in which players have multiple attributes. Such games are applicable to situations in which each player has a finite number of independent additive attributes in cooperative games and the payoffs of coalitions are endogenous functions of these attributes. The additive attributes cooperative game, which is a special case of the multiattribute cooperative game, is studied with respect to the core, the conditions for existence and boundedness and methods of transformation regarding a general cooperative game. A coalitional polynomial form is also proposed to discuss the structure of coalition. Moreover, a Shapley-like solution called the efficient resource (ER) solution for additive attributes cooperative games is studied via the axiomatical method, and the ER solution of two additive attribute games with equivalent total resources coincides with the Shapley value. Finally, some examples of additive attribute games are given.

Interpretations of sets of conditions

1954 ◽  
Vol 19 (2) ◽  
pp. 97-102 ◽  
Author(s):  
W. V. Quine
Keyword(s):  
Finite Number ◽  
Predicate Calculus ◽  
Present Purpose ◽  
Positive Integers ◽  
Special Case ◽  
Empty Universe

The celebrated theorem of Löwenheim and Skolem tells us that every consistent set S of quantificational schemata (i.e., every set of well-formed formulas of the lower predicate calculus admitting of a true interpretation in some non-empty universe) admits of a true numerical interpretation (i.e., an interpretation of predicate letters such that all schemata of S come out true when the variables of quantification are construed as ranging over just the positive integers).Later literature goes farther, and shows how, given S, actually to produce a numerical interpretation which will fit S in case S is consistent. The general case is covered by Kleene (see Bibliography). The special case where S contains just one schema (or any finite number, since we can form their conjunction) had been dealt with by Hilbert and Bernays. Certain extensions, along lines not to be embarked on here, have been made by Kleene, Kreisel, Hasenjäger, and Wang.My present purpose is expository: to make the construction of the numerical interpretation, and the proof of its adequacy, more easily intelligible than they hitherto have been. The reasoning is mainly Kleene's, though closer in some ways to earlier reasoning of Gödel.

scholarly journals A Hilbert type inequality

2000 ◽  
Vol 31 (4) ◽  
pp. 311-316
Author(s):  
G. D. Handley ◽  
J. J. Koliha ◽  
J. Pecaric
Keyword(s):  
Finite Number ◽  
Type Inequality ◽  
Real Numbers ◽  
Special Case ◽  
Hilbert Type

In this paper we obtain a new inequality of Hilbert type for a finite number of nonnegative sequences of real numbers from which we can recover as a special case an inequality due to Pachpatte. We also obtain an integral variant of the inequality.

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