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 Lower Bounds for the Cop Number when the Robber is Fast

2011 ◽  
Vol 20 (4) ◽  
pp. 617-621 ◽  
Author(s):  
ABBAS MEHRABIAN
Keyword(s):  
Graph Theory ◽  
Lower Bound ◽  
Lower Bounds ◽  
Regular Graph ◽  
Upper Bound ◽  
Upper Bounds ◽  
Cops And Robbers ◽  
Vertex Graph

We consider a variant of the Cops and Robbers game where the robber can movetedges at a time, and show that in this variant, the cop number of ad-regular graph with girth larger than 2t+2 is Ω(dt). By the known upper bounds on the order of cages, this implies that the cop number of a connectedn-vertex graph can be as large as Ω(n2/3) ift≥ 2, and Ω(n4/5) ift≥ 4. This improves the Ω($n^{\frac{t-3}{t-2}}$) lower bound of Frieze, Krivelevich and Loh (Variations on cops and robbers,J. Graph Theory, to appear) when 2 ≤t≤ 6. We also conjecture a general upper boundO(nt/t+1) for the cop number in this variant, generalizing Meyniel's conjecture.

scholarly journals The Complexity of Constructing Evolutionary Trees Using Experiments

2001 ◽  
Vol 8 (1) ◽  
Author(s):  
Gerth Stølting Brodal ◽  
Rolf Fagerberg ◽  
Christian N. S. Pedersen ◽  
Anna Östlin
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper Bound ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Optimal Time ◽  
Evolutionary Trees ◽  
Dynamic Algorithm ◽  
Running Time ◽  
Small Height

<p>We present tight upper and lower bounds for the problem of constructing evolutionary trees in the experiment model. We describe an algorithm which constructs an evolutionary tree of n species in time O(n d logd n) using at most n |d/2| (log2|d/2|−1 n + O(1)) experiments for d > 2, and<br />at most n(log n + O(1)) experiments for d = 2, where d is the degree of the tree. This improves the previous best upper bound by a factor Theta(log d). For d = 2 the previously best algorithm with running time O(n log n) had a bound of 4n log n on the number of experiments. By an explicit adversary argument, we show an <br />Omega(nd logd n) lower bound, matching our upper bounds and improving the previous best lower bound<br />by a factor Theta(logd n). Central to our algorithm is the construction and maintenance of separator trees of small height. We present how to maintain separator trees with height log n + O(1) under the insertion of new nodes in amortized time O(log n). Part of our dynamic algorithm is an algorithm for computing a centroid tree in optimal time O(n).</p><p>Keywords: Evolutionary trees, Experiment model, Separator trees, Centroid tree, Lower bounds</p>

scholarly journals Encoding Two-Dimensional Range Top-k Queries

Algorithmica ◽  
2021 ◽  
Author(s):  
Seungbum Jo ◽  
Rahul Lingala ◽  
Srinivasa Rao Satti
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper Bound ◽  
Total Order ◽  
Two Dimensional ◽  
Information Theoretic ◽  
Cartesian Tree ◽  
Dimensional Range

AbstractWe consider the problem of encoding two-dimensional arrays, whose elements come from a total order, for answering $${\text{Top-}}{k}$$ Top- k queries. The aim is to obtain encodings that use space close to the information-theoretic lower bound, which can be constructed efficiently. For an $$m \times n$$ m × n array, with $$m \le n$$ m ≤ n , we first propose an encoding for answering 1-sided $${\textsf {Top}}{\text {-}}k{}$$ Top - k queries, whose query range is restricted to $$[1 \dots m][1 \dots a]$$ [ 1 ⋯ m ] [ 1 ⋯ a ] , for $$1 \le a \le n$$ 1 ≤ a ≤ n . Next, we propose an encoding for answering for the general (4-sided) $${\textsf {Top}}{\text {-}}k{}$$ Top - k queries that takes $$(m\lg {{(k+1)n \atopwithdelims ()n}}+2nm(m-1)+o(n))$$ ( m lg ( k + 1 ) n n + 2 n m ( m - 1 ) + o ( n ) ) bits, which generalizes the joint Cartesian tree of Golin et al. [TCS 2016]. Compared with trivial $$O(nm\lg {n})$$ O ( n m lg n ) -bit encoding, our encoding takes less space when $$m = o(\lg {n})$$ m = o ( lg n ) . In addition to the upper bound results for the encodings, we also give lower bounds on encodings for answering 1 and 4-sided $${\textsf {Top}}{\text {-}}k{}$$ Top - k queries, which show that our upper bound results are almost optimal.

scholarly journals The Essential Dimension of Stacks of Parabolic Vector Bundles over Curves

2012 ◽  
Vol 10 (3) ◽  
pp. 455-488 ◽  
Author(s):  
Indranil Biswas ◽  
Ajneet Dhillon ◽  
Nicole Lemire
Keyword(s):  
Lower Bounds ◽  
Upper Bound ◽  
Vector Bundles ◽  
Upper Bounds ◽  
Essential Dimension ◽  
Parabolic Structure ◽  
Moduli Stack

AbstractWe find upper bounds on the essential dimension of the moduli stack of parabolic vector bundles over a curve. When there is no parabolic structure, we improve the known upper bound on the essential dimension of the usual moduli stack. Our calculations also give lower bounds on the essential dimension of the semistable locus inside the moduli stack of vector bundles of rank r and degree d without parabolic structure.

scholarly journals On the Rank of $p$-Schemes

10.37236/3097 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Fateme Raei Barandagh ◽  
Amir Rahnamai Barghi
Keyword(s):  
Lower Bound ◽  
Prime Number ◽  
Lower Bounds ◽  
Upper Bound ◽  
Upper And Lower Bounds ◽  
Minimum Rank

Let $n>1$ be an integer and $p$ be a prime number. Denote by $\mathfrak{C}_{p^n}$ the class of non-thin association $p$-schemes of degree $p^n$. A sharp upper and lower bounds on the rank of schemes in $\mathfrak{C}_{p^n}$ with a certain order of thin radical are obtained. Moreover, all schemes in this class whose rank are equal to the lower bound are characterized and some schemes in this class whose rank are equal to the upper bound are constructed. Finally, it is shown that the scheme with minimum rank in $\mathfrak{C}_{p^n}$ is unique up to isomorphism, and it is a fusion of any association $p$-schemes with degree $p^n$.

scholarly journals When is non-trivial estimation possible for graphons and stochastic block models?‡

2017 ◽  
Vol 7 (2) ◽  
pp. 169-181
Author(s):  
Audra McMillan ◽  
Adam Smith
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper Bound ◽  
Upper And Lower Bounds ◽  
Random Networks ◽  
Block Models

Abstract Block graphons (also called stochastic block models) are an important and widely studied class of models for random networks. We provide a lower bound on the accuracy of estimators for block graphons with a large number of blocks. We show that, given only the number $k$ of blocks and an upper bound $\rho$ on the values (connection probabilities) of the graphon, every estimator incurs error ${\it{\Omega}}\left(\min\left(\rho, \sqrt{\frac{\rho k^2}{n^2}}\right)\right)$ in the $\delta_2$ metric with constant probability for at least some graphons. In particular, our bound rules out any non-trivial estimation (that is, with $\delta_2$ error substantially less than $\rho$) when $k\geq n\sqrt{\rho}$. Combined with previous upper and lower bounds, our results characterize, up to logarithmic terms, the accuracy of graphon estimation in the $\delta_2$ metric. A similar lower bound to ours was obtained independently by Klopp et al.

A LOWER BOUND FOR ORDER-PRESERVING BROADCAST IN THE POSTAL MODEL

1993 ◽  
Vol 03 (04) ◽  
pp. 313-320 ◽  
Author(s):  
PHILIP D. MACKENZIE
Keyword(s):  
Communication Network ◽  
Lower Bound ◽  
Lower Bounds ◽  
Message Passing ◽  
Upper Bounds ◽  
Communication Latency ◽  
Multiple Messages ◽  
Postal Model ◽  
Parameter Ranges

In the postal model of message passing systems, the actual communication network between processors is abstracted by a single communication latency factor, which measures the inverse ratio of the time it takes for a processor to send a message and the time that passes until the recipient receives the message. In this paper we examine the problem of broadcasting multiple messages in an order-preserving fashion in the postal model. We prove lower bounds for all parameter ranges and show that these lower bounds are within a factor of seven of the best upper bounds. In some cases, our lower bounds show significant asymptotic improvements over the previous best lower bounds.

The Unrestricted Black-Box Complexity of Jump Functions

2016 ◽  
Vol 24 (4) ◽  
pp. 719-744 ◽  
Author(s):  
Maxim Buzdalov ◽  
Benjamin Doerr ◽  
Mikhail Kever
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper Bounds ◽  
Black Box ◽  
Problem Instance ◽  
Theory Approach ◽  
Function Classes ◽  
Fitness Value ◽  
Jump Function ◽  
Jump Sizes

We analyze the unrestricted black-box complexity of the Jump function classes for different jump sizes. For upper bounds, we present three algorithms for small, medium, and extreme jump sizes. We prove a matrix lower bound theorem which is capable of giving better lower bounds than the classic information theory approach. Using this theorem, we prove lower bounds that almost match the upper bounds. For the case of extreme jump functions, which apart from the optimum reveal only the middle fitness value(s), we use an additional lower bound argument to show that any black-box algorithm does not gain significant insight about the problem instance from the first [Formula: see text] fitness evaluations. This, together with our upper bound, shows that the black-box complexity of extreme jump functions is [Formula: see text].

scholarly journals A Computational Perspective on Network Coding

2010 ◽  
Vol 2010 ◽  
pp. 1-11
Author(s):  
Qin Guo ◽  
Mingxing Luo ◽  
Lixiang Li ◽  
Yixian Yang
Keyword(s):  
Graph Theory ◽  
Computational Complexity ◽  
Lower Bound ◽  
Network Coding ◽  
Upper Bound ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Multicast Networks ◽  
Computational Perspective ◽  
Source Rate

From the perspectives of graph theory and combinatorics theory we obtain some new upper bounds on the number of encoding nodes, which can characterize the coding complexity of the network coding, both in feasible acyclic and cyclic multicast networks. In contrast to previous work, during our analysis we first investigate the simple multicast network with source rateh=2, and thenh≥2. We find that for feasible acyclic multicast networks our upper bound is exactly the lower bound given by M. Langberg et al. in 2006. So the gap between their lower and upper bounds for feasible acyclic multicast networks does not exist. Based on the new upper bound, we improve the computational complexity given by M. Langberg et al. in 2009. Moreover, these results further support the feasibility of signatures for network coding.

scholarly journals Class numbers of real cyclotomic fields of composite conductor

2014 ◽  
Vol 17 (A) ◽  
pp. 404-417 ◽  
Author(s):  
John C. Miller
Keyword(s):  
Lower Bounds ◽  
Upper Bound ◽  
Upper Bounds ◽  
Class Numbers ◽  
Prime Ideals ◽  
Cyclotomic Fields ◽  
Class Field ◽  
Divisibility Properties ◽  
Composite Conductor ◽  
Hilbert Class Field

AbstractUntil recently, the ‘plus part’ of the class numbers of cyclotomic fields had only been determined for fields of root discriminant small enough to be treated by Odlyzko’s discriminant bounds.However, by finding lower bounds for sums over prime ideals of the Hilbert class field, we can now establish upper bounds for class numbers of fields of larger discriminant. This new analytic upper bound, together with algebraic arguments concerning the divisibility properties of class numbers, allows us to unconditionally determine the class numbers of many cyclotomic fields that had previously been untreatable by any known method.In this paper, we study in particular the cyclotomic fields of composite conductor.

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