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 Characterizations and Identities for Isosceles Triangular Numbers

2021 ◽  
Vol 14 (2) ◽  
pp. 380-395
Author(s):  
Jiramate Punpim ◽  
Somphong Jitman
Keyword(s):  
Positive Integer ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Triangular Numbers ◽  
Triangular Number ◽  
Recursive Formulas ◽  
Necessary And Sufficient ◽  
Positive Integers ◽  
Special Case

Triangular numbers have been of interest and continuously studied due to their beautiful representations, nice properties, and various links with other figurate numbers. For positive integers n and l, the nth l-isosceles triangular number is a generalization of triangular numbers defined to be the arithmetic sum of the formT(n, l) = 1 + (1 + l) + (1 + 2l) + · · · + (1 + (n − 1)l).In this paper, we focus on characterizations and identities for isosceles triangular numbers as well as their links with other figurate numbers. Recursive formulas for constructions of isosceles triangular numbers are given together with necessary and sufficient conditions for a positive integer to be a sum of isosceles triangular  numbers. Various identities for isosceles triangular numbers are established. Results on triangular numbers can be viewed as a special case.

On Nilpotent Products of Cyclic Groups. II

1961 ◽  
Vol 13 ◽  
pp. 557-568 ◽  
Author(s):  
Ruth Rebekka Struik
Keyword(s):  
Normal Subgroup ◽  
Finite Number ◽  
Free Product ◽  
Multiplication Table ◽  
Unique Representation ◽  
Cyclic Groups ◽  
Positive Integers

In a previous paper (18), G = F/Fn was studied for F a free product of a finite number of cyclic groups, and Fn the normal subgroup generated by commutators of weight n. In that paper the following cases were completely treated:(a) F a free product of cyclic groups of order pαi, p a prime, αi positive integers, and n = 4, 5, … , p + 1.(b) F a free product of cyclic groups of order 2αi, and n = 4.In this paper, the following case is completely treated:(c) F a free product of cyclic groups of order pαi p a prime, αi positive integers, and n = p + 2.(Note that n = 2 is well known, and n — 3 was studied by Golovin (2).) By ‘'completely treated” is meant: a unique representation of elements of the group is given, and the order of the group is indicated. In the case of n = 4, a multiplication table was given.

Several weighted sum formulas of multiple zeta values

2017 ◽  
Vol 13 (09) ◽  
pp. 2253-2264 ◽  
Author(s):  
Minking Eie ◽  
Wen-Chin Liaw ◽  
Yao Lin Ong
Keyword(s):  
Real Number ◽  
Weighted Sum ◽  
Multiple Zeta Values ◽  
Explicit Evaluation ◽  
Multiple Zeta ◽  
Zeta Values ◽  
Number Formula ◽  
Positive Integers ◽  
Special Case

For a real number [Formula: see text] and positive integers [Formula: see text] and [Formula: see text] with [Formula: see text], we evaluate the sum of multiple zeta values [Formula: see text] explicitly in terms of [Formula: see text] and [Formula: see text]. The special case [Formula: see text] gives an evaluation of [Formula: see text]. An explicit evaluation of the multiple zeta-star value [Formula: see text] is also obtained, as well as some applications to evaluation of multiple zeta values with even arguments.

scholarly journals A finiteness theorem on symplectic singularities

2016 ◽  
Vol 152 (6) ◽  
pp. 1225-1236 ◽  
Author(s):  
Yoshinori Namikawa
Keyword(s):  
Finite Number ◽  
Contact Structure ◽  
Finiteness Theorem ◽  
Maximal Weight ◽  
Fixed Dimension ◽  
Positive Integers ◽  
Boundedness Result ◽  
Symplectic Varieties

An affine symplectic singularity$X$with a good$\mathbf{C}^{\ast }$-action is called a conical symplectic variety. In this paper we prove the following theorem. For fixed positive integers$N$and$d$, there are only a finite number of conical symplectic varieties of dimension$2d$with maximal weights$N$, up to an isomorphism. To prove the main theorem, we first relate a conical symplectic variety with a log Fano Kawamata log terminal (klt) pair, which has a contact structure. By the boundedness result for log Fano klt pairs with fixed Cartier index, we prove that conical symplectic varieties of a fixed dimension and with a fixed maximal weight form a bounded family. Next we prove the rigidity of conical symplectic varieties by using Poisson deformations.

scholarly journals Detachments Preserving Local Edge-Connectivity of Graphs

1999 ◽  
Vol 6 (35) ◽  
Author(s):  
Tibor Jordán ◽  
Zoltán Szigeti
Keyword(s):  
Edge Connectivity ◽  
Common Generalization ◽  
Connectivity Augmentation ◽  
Local Edge ◽  
Positive Integers ◽  
Special Case

Let G = (V +s,E) be a graph and let S = (d1, ..., dp) be a set of positive integers with<br />Sum dj = d(s). An S-detachment splits s into a set of p independent vertices s1, ..., sp with<br />d(sj) = dj, 1 <= j <= p. Given a requirement function r(u, v) on pairs of vertices of V , an<br />S-detachment is called r-admissible if the detached graph G' satisfies lambda_G' (x, y) >= r(x, y)<br />for every pair x, y in V . Here lambda_H(u, v) denotes the local edge-connectivity between u and v<br />in graph H.<br />We prove that an r-admissible S-detachment exists if and only if (a) lambda_G(x, y) >= r(x, y),<br />and (b) lambda_G−s(x, y) >= r(x, y) − Sum |dj/2| hold for every x, y in V .<br />The special case of this characterization when r(x, y) = lambda_G(x, y) for each pair in V was conjectured by B. Fleiner. Our result is a common generalization of a theorem of W. Mader on edge splittings preserving local edge-connectivity and a result of B. Fleiner on detachments preserving global edge-connectivity. Other corollaries include previous results of L. Lov´asz and C.J.St.A. Nash-Williams on edge splittings and detachments, respectively. As a new application, we extend a theorem of A. Frank on local edge-connectivity augmentation to the case when stars of given degrees are added.

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.

The Bracket Function and Complementary Sets of Integers

1969 ◽  
Vol 21 ◽  
pp. 6-27 ◽  
Author(s):  
Aviezri S. Fraenkel
Keyword(s):  
Irrational Numbers ◽  
Image Position ◽  
Positive Integers ◽  
Special Case

The following result is well known (as usual, [x]denotes the integral part of x):(A) Let α and β be positive irrational numbers satisfying1Then the sets [nα], [nβ], n= 1, 2, …, are complementary with respect to the set of all positive integers]see, e.g. (1; 2; 4; 5; 6; 7; 8; 10; 13; 14; 15; 16). In some of these references the result, or a special case thereof, is mentioned in connection with Wythoff's game, with or without proof. It appears that Beatty (4) was the originator of the problem.The theorem has a converse, and the following holds:(B) Let α and β be positive. The sets [nα] and [nβ], n = 1, 2, …, are complementary with respect to the set of all positive integers if and only if α and β are irrational, and (1) holds.

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

scholarly journals Juggler's Exclusion Process

2012 ◽  
Vol 49 (01) ◽  
pp. 266-279
Author(s):  
Lasse Leskelä ◽  
Harri Varpanen
Keyword(s):  
Markov Process ◽  
Gibbs Measure ◽  
Equilibrium Distribution ◽  
Exclusion Process ◽  
Jump Height ◽  
The Family ◽  
System Of Particles ◽  
Unit Speed ◽  
Positive Integers ◽  
Special Case

Juggler's exclusion process describes a system of particles on the positive integers where particles drift down to zero at unit speed. After a particle hits zero, it jumps into a randomly chosen unoccupied site. We model the system as a set-valued Markov process and show that the process is ergodic if the family of jump height distributions is uniformly integrable. In a special case where the particles jump according to a set-avoiding memoryless distribution, the process reaches its equilibrium in finite nonrandom time, and the equilibrium distribution can be represented as a Gibbs measure conforming to a linear gravitational potential.

Weight-Equitable Subdivision of Red and Blue Points in the Plane

2018 ◽  
Vol 28 (01) ◽  
pp. 39-56 ◽  
Author(s):  
Jude Buot ◽  
Mikio Kano
Keyword(s):  
General Position ◽  
Sufficient Conditions ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Total Weight ◽  
Blue Point ◽  
Disjoint Sets ◽  
Necessary And Sufficient ◽  
Positive Integers ◽  
Special Case

Let [Formula: see text] and [Formula: see text] be two disjoint sets of red points and blue points, respectively, in the plane in general position. Assign a weight [Formula: see text] to each red point and a weight [Formula: see text] to each blue point, where [Formula: see text] and [Formula: see text] are positive integers. Define the weight of a region in the plane as the sum of the weights of red and blue points in it. We give necessary and sufficient conditions for the existence of a line that bisects the weight of the plane whenever the total weight [Formula: see text] is [Formula: see text], for some integer [Formula: see text]. Moreover, we look closely into the special case where [Formula: see text] and [Formula: see text] since this case is important to generate a weight-equitable subdivision of the plane. Among other results, we show that for any configuration of [Formula: see text] with total weight [Formula: see text], for some integer [Formula: see text] and odd integer [Formula: see text], the plane can be subdivided into [Formula: see text] convex regions of weight [Formula: see text] if and only if [Formula: see text]. Using the proofs of the main result, we also give a polynomial time algorithm in finding a weight-equitable subdivision in the plane.

Sign in / Sign up

Export Citation Format

Share Document