Pointless metric spaces

1990 ◽  
Vol 55 (1) ◽  
pp. 207-219 ◽  
Author(s):  
Giangiacomo Gerla
Keyword(s):  
Lower Bound ◽  
Metric Space ◽  
Upper Bound ◽  
Metric Spaces ◽  
Space Theory ◽  
Research Projects ◽  
Natural Numbers ◽  
Real Numbers ◽  
Euclidean Metric

In the last years several research projects have been motivated by the problem of constructing the usual geometrical spaces by admitting “regions” and “inclusion” between regions as primitives and by defining the points as suitable sequences or classes of regions (for references see [2]).In this paper we propose and examine a system of axioms for the pointless space theory in which “regions”, “inclusion”, “distance” and “diameter” are assumed as primitives and the concept of point is derived. Such a system extends a system proposed by K. Weihrauch and U. Schreiber in [5].In the sequel R and N denote the set of real numbers and the set of natural numbers, and E is a Euclidean metric space. Moreover, if X is a subset of R, then ⋁X is the least upper bound and ⋀X the greatest lower bound of X.

The Heine-Borel theorem in extended basic logic

1949 ◽  
Vol 14 (1) ◽  
pp. 9-15 ◽  
Author(s):  
Frederic B. Fitch
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper Bound ◽  
Fundamental Theorem ◽  
Upper Bounds ◽  
Basic Logic ◽  
Real Numbers ◽  
Consistent Theory

A demonstrably consistent theory of real numbers has been outlined by the writer in An extension of basic logic1 (hereafter referred to as EBL). This theory deals with non-negative real numbers, but it could be easily modified to deal with negative real numbers also. It was shown that the theory was adequate for proving a form of the fundamental theorem on least upper bounds and greatest lower bounds. More precisely, the following results were obtained in the terminology of EBL: If С is a class of U-reals and is completely represented in Κ′ and if some U-real is an upper bound of С, then there is a U-real which is a least upper bound of С. If D is a class of (U-reals and is completely represented in Κ′, then there is a U-real which is a greatest lower bound of D.

scholarly journals TESTING EMBEDDABILITY BETWEEN METRIC SPACES

2009 ◽  
Vol 20 (02) ◽  
pp. 313-329
Author(s):  
CHING-LUEH CHANG ◽  
YUH-DAUH LYUU ◽  
YEN-WU TI
Keyword(s):  
Metric Space ◽  
Upper Bound ◽  
Metric Spaces ◽  
Graph Isomorphism ◽  
Upper And Lower Bounds ◽  
Wrong Answer ◽  
Probability Of Error ◽  
Finite Metric Space ◽  
Finite Metric Spaces ◽  
Special Case

Let L ≥ 1, ε > 0 be real numbers, (M, d) be a finite metric space and (N, ρ) be a metric space. A query to a metric space consists of a pair of points and asks for the distance between these points. We study the number of queries to metric spaces (M, d) and (N, ρ) needed to decide whether (M, d) is L-bilipschitz embeddable into (N, ρ) or ∊-far from being L-bilipschitz embeddable into N, ρ). When (M, d) is ∊-far from being L-bilipschitz embeddable into (N, ρ), we allow an o(1) probability of error (i.e., returning the wrong answer "L-bilipschitz embeddable"). However, no error is allowed when (M, d) is L-bilipschitz embeddable into (N, ρ). That is, algorithms with only one-sided errors are studied in this paper. When |M| ≤ |N| are both finite, we give an upper bound of [Formula: see text] on the number of queries for determining with one-sided error whether (M, d) is L-bilipschitz embeddable into (N, ρ) or ∊-far from being L-bilipschitz embeddable into (N, ρ). For the special case of finite |M| = |N|, the above upper bound evaluates to [Formula: see text]. We also prove a lower bound of Ω(|N|3/2) for the special case when |M| = |N| are finite and L = 1, which coincides with testing isometry between finite metric spaces. For finite |M| = |N|, the upper and lower bounds thus match up to a multiplicative factor of at most [Formula: see text], which depends only sublogarithmically in |N|. We also investigate the case when (N, ρ) is not necessarily finite. Our results are based on techniques developed in an earlier work on testing graph isomorphism.

Decomposition Space Theory and the Bing Shrinking Criterion

2021 ◽  
pp. 62-76
Author(s):  
Christopher W. Davis ◽  
Boldizsár Kalmár ◽  
Min Hoon Kim ◽  
Henrik Rüping
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Embedding Theorem ◽  
Space Theory ◽  
Compact Metric Space ◽  
Decomposition Spaces ◽  
Compact Metric Spaces ◽  
Space Decomposition ◽  
Decomposition Space ◽  
Continuous Decomposition

‘Decomposition Space Theory and the Bing Shrinking Criterion’ gives a proof of the central Bing shrinking criterion and then provides an introduction to the key notions of the field of decomposition space theory. The chapter begins by proving the Bing shrinking criterion, which characterizes when a given map between compact metric spaces is approximable by homeomorphisms. Next, it develops the elements of the theory of decomposition spaces. A key fact is that a decomposition space associated with an upper semi-continuous decomposition of a compact metric space is again a compact metric space. Decomposition spaces are key in the proof of the disc embedding theorem.

OPTIMAL LOWER BOUND ON THE SUPREMAL STRICT p-NEGATIVE TYPE OF A FINITE METRIC SPACE

2009 ◽  
Vol 80 (3) ◽  
pp. 486-497 ◽  
Author(s):  
ANTHONY WESTON
Keyword(s):  
Lower Bound ◽  
Metric Space ◽  
Triangle Inequality ◽  
Metric Spaces ◽  
Elementary Approach ◽  
Negative Type ◽  
Finite Metric Space ◽  
Image Position ◽  
Finite Metric Spaces ◽  
Optimal Lower Bound

AbstractDetermining meaningful lower bounds on the supremal strict p-negative type of classes of finite metric spaces is a difficult nonlinear problem. In this paper we use an elementary approach to obtain the following result: given a finite metric space (X,d) there is a constant ζ>0, dependent only on n=∣X∣ and the scaled diameter 𝔇=(diamX)/min{d(x,y)∣x⁄=y} of X (which we may assume is >1), such that (X,d) has p-negative type for all p∈[0,ζ] and strict p-negative type for all p∈[0,ζ). In fact, we obtain A consideration of basic examples shows that our value of ζ is optimal provided that 𝔇≤2. In other words, for each 𝔇∈(1,2] and natural number n≥3, there exists an n-point metric space of scaled diameter 𝔇 whose supremal strict p-negative type is exactly ζ. The results of this paper hold more generally for all finite semi-metric spaces since the triangle inequality is not used in any of the proofs. Moreover, ζ is always optimal in the case of finite semi-metric spaces.

scholarly journals On a new generalization of Alzer's inequality

2000 ◽  
Vol 23 (12) ◽  
pp. 815-818 ◽  
Author(s):  
Feng Qi ◽  
Lokenath Debnath
Keyword(s):  
Lower Bound ◽  
Large Class ◽  
Mean Value ◽  
Mathematical Induction ◽  
Mean Value Theorem ◽  
Natural Numbers ◽  
Real Numbers ◽  
Positive Real

Let{an}n=1∞be an increasing sequence of positive real numbers. Under certain conditions of this sequence we use the mathematical induction and the Cauchy mean-value theorem to prove the following inequality:anan+m≤((1/n)∑i=1nair(1/(n+m))∑i=1n+mair)1/r, wherenandmare natural numbers andris a positive number. The lower bound is best possible. This inequality generalizes the Alzer's inequality (1993) in a new direction. It is shown that the above inequality holds for a large class of positive, increasing and logarithmically concave sequences.

scholarly journals Diophantine Equations Relating Sums and Products of Positive Integers: Computation-Aided Study of Parametric Solutions, Bounds, and Distinct-Term Solutions

Mathematics ◽  
2021 ◽  
Vol 9 (21) ◽  
pp. 2779
Author(s):  
Petr Karlovsky
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Diophantine Equations ◽  
Natural Numbers ◽  
Parametric Solutions ◽  
Special Cases ◽  
Positive Integers

Diophantine equations ∏i=1nxi=F∑i=1nxi with xi,F∈ℤ+ associate the products and sums of n natural numbers. Only special cases have been studied so far. Here, we provide new parametric solutions depending on F and the divisors of F or F2. One of these solutions shows that the equation of any degree with any F is solvable. For n = 2, exactly two solutions exist if and only if F is a prime. These solutions are (2F,2F) and (F + 1, F(F + 1)). We generalize an upper bound for the sum of solution terms from n = 3 established by Crilly and Fletcher in 2015 to any n to be F+1F+n−1 and determine a lower bound to be nnFn−1. Confining the solutions to n-tuples consisting of distinct terms, equations of the 4th degree with any F are solvable but equations of the 5th to 9th degree are not. An upper bound for the sum of terms of distinct-term solutions is conjectured to be F+1F+n−2n−1!/2+1/n−2!. The conjecture is supported by computation, which also indicates that the upper bound equals the largest sum of solution terms if and only if F=n+k−2n−2!−1, k∈ℤ+. Computation provides further insights into the relationships between F and the sum of terms of distinct-term solutions.

scholarly journals Generalized F-Contractions on Product of Metric Spaces

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1040
Author(s):  
Nicolae Adrian Secelean ◽  
Mi Zhou
Keyword(s):  
Fixed Point ◽  
Metric Space ◽  
Product Space ◽  
Metric Spaces ◽  
Natural Numbers ◽  
Positive Integers

Our purpose in this paper is to extend the fixed point results of a ψ F -contraction introduced by Secelean N.A. and Wardowski D. ( ψ F -Contractions: Not Necessarily Nonexpansive Picard Operators, Results. Math.70(3), 415–431 (2016)) defined on a metric space X into itself to the case of mapping defined on the product space X I , where I is a set of positive integers (natural numbers). Some improvements to the conditions imposed on function F and space X are provided. An illustrative example is given.

scholarly journals Increasing subsequences of random walks

2016 ◽  
Vol 163 (1) ◽  
pp. 173-185 ◽  
Author(s):  
OMER ANGEL ◽  
RICHÁRD BALKA ◽  
YUVAL PERES
Keyword(s):  
Asymptotic Behavior ◽  
Random Walk ◽  
Lower Bound ◽  
Upper Bound ◽  
Simple Random Walk ◽  
Real Numbers ◽  
Record Times ◽  
One Dimensional ◽  
Expected Length ◽  
Longest Increasing Subsequence

AbstractGiven a sequence of n real numbers {Si}i⩽n, we consider the longest weakly increasing subsequence, namely i1 < i2 < . . . < iL with Sik ⩽ Sik+1 and L maximal. When the elements Si are i.i.d. uniform random variables, Vershik and Kerov, and Logan and Shepp proved that ${\mathbb E} L=(2+o(1)) \sqrt{n}$.We consider the case when {Si}i⩽n is a random walk on ℝ with increments of mean zero and finite (positive) variance. In this case, it is well known (e.g., using record times) that the length of the longest increasing subsequence satisfies ${\mathbb E} L\geq c\sqrt{n}$. Our main result is an upper bound ${\mathbb E} L\leq n^{1/2 + o(1)}$, establishing the leading asymptotic behavior. If {Si}i⩽n is a simple random walk on ℤ, we improve the lower bound by showing that ${\mathbb E} L \geq c\sqrt{n} \log{n}$.We also show that if {Si} is a simple random walk in ℤ2, then there is a subsequence of {Si}i⩽n of expected length at least cn1/3 that is increasing in each coordinate. The above one-dimensional result yields an upper bound of n1/2+o(1). The problem of determining the correct exponent remains open.

Non-homogeneous binary cubic forms

1951 ◽  
Vol 47 (3) ◽  
pp. 457-460 ◽  
Author(s):  
R. P. Bambah
Keyword(s):  
Lower Bound ◽  
Finite Volume ◽  
Upper Bound ◽  
The Other ◽  
Real Numbers ◽  
Cubic Form ◽  
Homogeneous Form ◽  
Other Hand ◽  
Image Position ◽  
Cubic Forms

1. Let f(x1, x2, …, xn) be a homogeneous form with real coefficients in n variables x1, x2, …, xn. Let a1, a2, …, an be n real numbers. Define mf(a1, …, an) to be the lower bound of | f(x1 + a1, …, xn + an) | for integers x1, …, xn. Let mf be the upper bound of mf(a1, …, an) for all choices of a1, …, an. For many forms f it is known that there exist estimates for mf in terms of the invariants alone of f. On the other hand, it follows from a theorem of Macbeath* that no such estimates exist if the regionhas a finite volume. However, for such forms there may be simple estimates for mf dependent on the coefficients of f; for example, Chalk has conjectured that:If f(x,y) is reduced binary cubic form with negative discriminant, then for any real a, b there exist integers x, y such that

scholarly journals Upper Limit Superior and Lower Limit Inferior of Soft Sequences

2018 ◽  
Vol 7 (4.7) ◽  
pp. 306
Author(s):  
Ahmed Hasan Hameed ◽  
Ekhlas Annon Mousa ◽  
Abdulsattar Abdullah hamad
Keyword(s):  
Lower Bound ◽  
Lower Limit ◽  
Upper Bound ◽  
Subject Classification ◽  
Real Numbers ◽  
Soft Sets ◽  
Upper Limit

In this paper we have provided some of evidence work of the authors R.N.Hasan, O.A Tantawy in 2016 [1] thy given proof of the concept between two bound soft sets & subsets of soft elements real numbers also was concluded an upper bound and lower bound by using two sequences of soft element real numbers which is   In this paper, supposed to extend R.N.Hasan, O.A Tantawy work but here we are given new notion and proof for upper limit superior and lower limit inferior with two sequences and subsequences for the conclude new proof after recalled that, which is upper limit superior and lower limit inferior By this we have proved above new two theorems and one proposition & strengthen the example.AMS Subject Classification: 06D72, 40A05, 54A40  

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