Sequential theories and infinite distributivity in the lattice of chapters

1989 ◽  
Vol 54 (1) ◽  
pp. 190-206 ◽  
Author(s):  
Alan S. Stern
Keyword(s):  
Elementary Proof ◽  
Infinite Distributivity ◽  
Distributive Law ◽  
Image Position

AbstractWe introduce a notion of complexity for interpretations, which is used to prove some new results about interpretations of sequential theories. In particular, we give a new, elementary proof of Pudlák's theorem that sequential theories are connected. We also demonstrate a counterexample to the infinitary distributive lawin the lattice of chapters, in which the chapters a and bi are compact. (Counterexamples in which a is not compact have been found previously.)

scholarly journals An Elementary Proof of the Theorems of Cauchy and Mayer

1935 ◽  
Vol 4 (3) ◽  
pp. 112-117
Author(s):  
A. J. Macintyre ◽  
R. Wilson
Keyword(s):  
Differential Equation ◽  
Existence Theorem ◽  
Elementary Proof ◽  
Theoretical Discussion ◽  
Practical Advantage ◽  
Total Differential ◽  
Method Of Solution ◽  
The Subject ◽  
Image Position ◽  
Natural Way

Attention has recently been drawn to the obscurity of the usual presentations of Mayer's method of solution of the total differential equationThis method has the practical advantage that only a single integration is required, but its theoretical discussion is usually based on the validity of some other method of solution. Mayer's method gives a result even when the equation (1) is not integrable, but this cannot of course be a solution. An examination of the conditions under which the result is actually an integral of equation (1) leads to a proof of the existence theorem for (1) which is related to Mayer's method of solution in a natural way, and which moreover appears to be novel and of value in the presentation of the subject.

An elementary proof and an extension of Thas' theorem on k-arcs

1989 ◽  
Vol 105 (3) ◽  
pp. 459-462 ◽  
Author(s):  
Hitoshi Kaneta ◽  
Tatsuya Maruta
Keyword(s):  
Finite Field ◽  
Projective Space ◽  
Elementary Proof ◽  
Rational Curve ◽  
Image Position

Let q be the finite field of q elements. Denote by Sr q the projective space of dimension r over q. In Sr,q, where r ≥ 2, a k-arc is defined (see [4]) as a set of k points such that no j + 2 lie in a Sj,q, for j = 1,2,…, r−1. (For a k-arc with k > r, this last condition holds for all j when it holds for j = r−1.) A rational curve Cn of order n in Sr,q, is the set

scholarly journals A proof of the “Theorem of the Means.”

1943 ◽  
Vol 33 ◽  
pp. 17-18
Author(s):  
C. E. Walsh
Keyword(s):  
Elementary Proof ◽  
Image Position ◽  
Proof By Induction

Numerous proofs have been given of this familiar theorem, which states that if a1, a2, …, an are positive, and not all equal, thenThe following is an elementary proof by induction, which I have not seen used before. It is, of course, not claimed to be novel, and not likely to be so.

scholarly journals An Euler-type volume identity

1999 ◽  
Vol 59 (3) ◽  
pp. 495-508
Author(s):  
Kevin Callahan ◽  
Kathy Hann
Keyword(s):  
Elementary Proof ◽  
Three Dimensional ◽  
Convex Polytope ◽  
Image Position

In this paper we present an elementary proof of a congruence by subtraction relation. In order to prove congruence by subtraction, we produce a dissection relating equal sub-polytopes. An immediate consequence of this relation is an Euler-type volume identity in ℝ3 which appeared in the Unsolved Problems section of the December 1996 MAA Monthly.This Euler-type volume identity relates the volumes of subsets of a polytope called wedges that correspond to its faces, edges, and vertices. A wedge consists of the inward normal chords of the polytope emanating from a face, vertex, or edge. This identity is stated in the theorem below.Euler Volume Theorem. For any three dimensional convex polytope PThis identity follows immediately from

A lattice of interpretability types of theories

1977 ◽  
Vol 42 (2) ◽  
pp. 297-305 ◽  
Author(s):  
Jan Mycielski
Keyword(s):  
Order Theory ◽  
Partial Ordering ◽  
First Order ◽  
First Order Theory ◽  
Relative Consistency ◽  
Infinite Distributivity ◽  
Closed Fields ◽  
Image Position ◽  
Interpretability Type ◽  
Consistency Proofs

We consider first-order logic only. A theory S will be called locally interpretable in a theory T if every theorem of S is interpretable in T. If S is locally interpretable in T and T is consistent then S is consistent. Most known relative consistency proofs can be viewed as local interpretations. The classic examples are the cartesian interpretation of the elementary theorems of Euclidean n-dimensional geometry into the first-order theory of real closed fields, the interpretation of the arithmetic of integers (rational numbers) into the arithmetic of positive integers, the interpretation of ZF + (V = L) into ZF, the interpretation of analysis into ZFC, relative consistency proofs by forcing, etc. Those interpretations are global. Under fairly general conditions local interpretability implies global interpretability; see Remarks (7), (8), and (9) below.We define the type (interpretability type) of a theory S to be the class of all theories T such that S is locally interpretable in T and vice versa. There happen to be such types and they are partially ordered by the relation of local interpretability. This partial ordering is of lattice type and has the following form:The lattice is distributive and complete and satisfies the infinite distributivity law of Brouwerian lattices:We do not know if the dual lawis true. We will show that the lattice is algebraic and that its compact elements form a sublattice and are precisely the types of finitely axiomatizable theories, and several other facts.

scholarly journals A note on the zeros of the Bessel function

1939 ◽  
Vol 6 (1) ◽  
pp. 17-18 ◽  
Author(s):  
R. Wilson
Keyword(s):  
Bessel Function ◽  
Infinite Number ◽  
Elementary Proof ◽  
Real Parameter ◽  
Image Position

This note contains an elementary proof of the well-known result that every solution of Besse's equationwith a real parameter n has an infinite number of real positive zeros.

On a Theorem of Rav Concerning Egyptian Fractions

1975 ◽  
Vol 18 (1) ◽  
pp. 155-156 ◽  
Author(s):  
William A. Webb
Keyword(s):  
Positive Integer ◽  
Elementary Proof ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Egyptian Fractions ◽  
Image Position ◽  
Complete Bibliography ◽  
Necessary And Sufficient

Problems involving Egyptian fractions (rationals whose numerator is 1 and whose denominator is a positive integer) have been extensively studied. (See [1] for a more complete bibliography). Some of the most interesting questions, many still unsolved, concern the solvability ofwhere k is fixed.In [2] Rav proved necessary and sufficient conditions for the solvabilty of the above equation, as a consequence of some other theorems which are rather complicated in their proofs. In this note we give a short, elementary proof of this theorem, and at the same time generalize it slightly.

XLVII.—Non-Associative Arithmetics

1949 ◽  
Vol 62 (4) ◽  
pp. 442-453 ◽  
Author(s):  
I. M. H. Etherington
Keyword(s):  
Infinite Class ◽  
Algebraic Systems ◽  
Unique Product ◽  
Number Systems ◽  
Distributive Law ◽  
Image Position ◽  
Ordinary Number

Introduction and SummaryThe systems of “partitive numbers” introduced in this paper differ from ordinary number systems in being subject to non-associative addition. They are intended primarily to serve as the indices of powers in algebraic systems having non-associative multiplication, or as the coefficients of multiples in systems with non-associative addition, but are defined more generally than is probably necessary for these purposes. They are essentially the same as root-trees (Setzbäume) with non-branching knots other than terminal knots ignored, with operations of addition and multiplication defined.Partitive numbers are of two kinds, partitioned cardinals and partitioned serials, defined respectively as the partition-types of repeatedly partitioned classes and series. For each kind, multiplication is binary (i.e. any ordered pair has a unique product) and associative. Addition is in general a free operation (i.e. the summands are not limited to two, and indeed, assuming the multiplicative axiom, may form an infinite class or series); but it is non-associative, which means that for example a + b + c (involving one operation of addition) is distinguished from (a + b) + c and a + (b + c) (involving two operations). A one-sided distributive law is obeyed:Partitioned cardinals are commutative in addition.

On two integral inequalities

1977 ◽  
Vol 77 (3-4) ◽  
pp. 325-328 ◽  
Author(s):  
E. T. Copson
Keyword(s):  
Elementary Proof ◽  
Integral Inequalities ◽  
Image Position

SynopsisIn 1932, Hardy and Littlewood [1] proved the inequalityThe constant 4 is best possible; equality occurs when f(x) = A Y(Bx), wherey(x) = e−½x sin (x sin y−y) (y = ⅓π), (x ≧ o)and A and B (>0) are constants. In [2], three proofs are given. The inequality has also been discussed in [3, 4]. A very elementary proof in which the function Y(x) emerges naturally is given in this paper.

scholarly journals NONEXISTENCE OF A CIRCULANT EXPANDER FAMILY

2010 ◽  
Vol 83 (1) ◽  
pp. 87-95
Author(s):  
KA HIN LEUNG ◽  
VINH NGUYEN ◽  
WASIN SO
Keyword(s):  
Regular Graph ◽  
Elementary Proof ◽  
Explicit Construction ◽  
Simple Graph ◽  
Regular Graphs ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Largest Eigenvalue ◽  
Image Position ◽  
Second Largest Eigenvalue

AbstractThe expansion constant of a simple graph G of order n is defined as where $E(S, \overline {S})$ denotes the set of edges in G between the vertex subset S and its complement $\overline {S}$. An expander family is a sequence {Gi} of d-regular graphs of increasing order such that h(Gi)>ϵ for some fixed ϵ>0. Existence of such families is known in the literature, but explicit construction is nontrivial. A folklore theorem states that there is no expander family of circulant graphs only. In this note, we provide an elementary proof of this fact by first estimating the second largest eigenvalue of a circulant graph, and then employing Cheeger’s inequalities where G is a d-regular graph and λ2(G) denotes the second largest eigenvalue of G. Moreover, the associated equality cases are discussed.

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