A discrete chain of degrees of index sets

1972 ◽  
Vol 37 (1) ◽  
pp. 139-149 ◽  
Author(s):  
Louise Hay
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Local Information ◽  
Truth Table ◽  
Partial Ordering ◽  
Index Set ◽  
Recursively Enumerable ◽  
Index Sets ◽  
The One ◽  
Standard Enumeration

Let {Wi} be a standard enumeration of all recursively enumerable (r.e.) sets, and for any class A of r.e. sets, let θA denote the index set of A = {n ∣ Wn ∈ A}. (Clearly, .) In [1], the index sets of nonempty finite classes of finite sets were classified under one-one reducibility into an increasing sequence {Ym}, 0 ≤ m < ∞. In this paper we examine further properties of this sequence within the partial ordering of one-one degrees of index sets. The main results are as follows: (1) For each m, Ym < Ym + 1 and < Ym + 1; (2) Ym is incomparable to ; (3) Ym + 1 and ; are immediate successors (among index sets) of Ym and m; (4) the pair (Ym + 1, ) is a “least upper bound” for the pair (Ym, ) in the sense that any successor of both Ym and is ≥ Ym + 1or; (5) the pair (Ym, ) is a “greatest lower bound” for the pair (Ym + 1, ) in the sense that any predecessor of both Ym + 1 and is ≤ Ym or . Since and all Ym are in the bounded truth-table degree of K, this yields some local information about the one-one degrees of index sets which are “at the bottom” in the one-one ordering of index sets.

A noninitial segment of index sets

1974 ◽  
Vol 39 (2) ◽  
pp. 209-224 ◽  
Author(s):  
Louise Hay
Keyword(s):  
Partial Order ◽  
Initial Segment ◽  
Finite Sets ◽  
Index Set ◽  
Difference Hierarchy ◽  
Recursively Enumerable ◽  
Index Sets ◽  
The Difference ◽  
The One ◽  
Standard Enumeration

Let {Wk}k ≥ 0 be a standard enumeration of all recursively enumerable (r.e.) sets. If A is any class of r.e. sets, let θA denote the index set of A, i.e., θA = {k ∣ Wk ∈ A}. The one-one degrees of index sets form a partial order ℐ which is a proper subordering of the partial order of all one-one degrees. Denote by ⌀ the one-one degree of the empty set, and, if b is the one-one degree of θB, denote by the one-one degree of . Let . Let {Ym}m≥0 be the sequence of index sets of nonempty finite classes of finite sets (classified in [5] and independently, in [2]) and denote by am the one-one degree of Ym. As shown in [2], these degrees are complete at each level of the difference hierarchy generated by the r.e. sets. It was proved in [3] that, for each m ≥ 0,(a) am+1 and ām+1 are incomparable immediate successors of am and ām, and(b) .For m = 0, since Y0 = θ{⌀}, it follows from (a) that(c) .Hence it follows that(d) {⌀, , ao, ā0, a1, ā1 is an initial segment of ℐ.

Index sets of finite classes of recursively enumerable sets

1969 ◽  
Vol 34 (1) ◽  
pp. 39-44 ◽  
Author(s):  
Louise Hay
Keyword(s):  
Index Set ◽  
Recursive Functions ◽  
Recursively Enumerable ◽  
Index Sets ◽  
Recursively Enumerable Sets ◽  
Partial Recursive Functions ◽  
Recursive Isomorphism ◽  
Standard Enumeration

Let q0, q1,… be a standard enumeration of all partial recursive functions of one variable. For each i, let wi = range qi and for any recursively enumerable (r.e.) set α, let θα = {n | wn = α}. If A is a class of r.e. sets, let θA = the index set of A = {n | wn ∈ A}. It is the purpose of this paper to classify the possible recursive isomorphism types of index sets of finite classes of r.e. sets. The main theorem will also provide an answer to the question left open in [2] concerning the possible double isomorphism types of pairs (θα, θβ) where α ⊂ β.

scholarly journals The number of olfactory stimuli that humans can discriminate is still unknown

eLife ◽  
2015 ◽  
Vol 4 ◽  
Author(s):  
Richard C Gerkin ◽  
Jason B Castro
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Olfactory Stimuli ◽  
The One ◽  
Reported Data

It was recently proposed (<xref ref-type="bibr" rid="bib2">Bushdid et al., 2014</xref>) that humans can discriminate between at least a trillion olfactory stimuli. Here we show that this claim is the result of a fragile estimation framework capable of producing nearly any result from the reported data, including values tens of orders of magnitude larger or smaller than the one originally reported in (<xref ref-type="bibr" rid="bib2">Bushdid et al., 2014</xref>). Additionally, the formula used to derive this estimate is well-known to provide an upper bound, not a lower bound as reported. That is to say, the actual claim supported by the calculation is in fact that humans can discriminate at most one trillion olfactory stimuli. We conclude that there is no evidence for the original claim.

ESTIMATE OF THE HAUSDORFF MEASURE OF THE SIERPINSKI TRIANGLE

Fractals ◽  
2009 ◽  
Vol 17 (02) ◽  
pp. 137-148
Author(s):  
PÉTER MÓRA
Keyword(s):  
Lower Bound ◽  
Hausdorff Dimension ◽  
Hausdorff Measure ◽  
Upper Bound ◽  
Open Problem ◽  
Dimensional Hausdorff Measure ◽  
Sierpinski Triangle ◽  
The One

It is well-known that the Hausdorff dimension of the Sierpinski triangle Λ is s = log 3/ log 2. However, it is a long standing open problem to compute the s-dimensional Hausdorff measure of Λ denoted by [Formula: see text]. In the literature the best existing estimate is [Formula: see text] In this paper we improve significantly the lower bound. We also give an upper bound which is weaker than the one above but everybody can check it easily. Namely, we prove that [Formula: see text] holds.

Classifications of generalized index sets of open classes

1978 ◽  
Vol 43 (4) ◽  
pp. 694-714 ◽  
Author(s):  
Nancy Johnson
Keyword(s):  
Finite Collection ◽  
Finite Sets ◽  
Index Set ◽  
Recursively Enumerable ◽  
Ershov Hierarchy ◽  
Index Sets ◽  
Finite Set ◽  
The Difference ◽  
Original Classification

The Rice-Shapiro Theorem [4] says that the index set of a class of recursively enumerable (r.e.) sets is r.e. if and only if consists of all sets which extend an element of a canonically enumerable sequence of finite sets. If an index of a difference of r.e. (d.r.e.) sets is defined to be the pair of indices of the r.e. sets of which it is the difference, then the following generalization due to Hay [3] is obtained: The index set of a class of d.r.e. sets is d.r.e. if and only if is empty or consists of all sets which extend a single fixed finite set. In that paper Hay also classifies index sets of classes consisting of d.r.e. sets which extend one of a finite collection of finite sets. These sets turn out to be finite Boolean combinations of r.e. sets. The question then arises “What about the classification of the index set of a class consisting of d.r.e. sets which extend an element of a canonically enumerable sequence of finite sets?” The results in this paper come from an attempt to answer this question.Since classes of sets which are Boolean combinations of r.e. sets form a hierarchy (the finite Ershov hierarchy, see Ershov [1]) with the r.e. and d.r.e. sets respectively levels 1 and 2 of this hierarchy, we may define index sets of classes of level n sets. If is a class of level n sets which extend some element of a canonically enumerable sequence of finite sets and if we let co-, then we extend the original classification question to the classification of the index sets of the classes and co-.Now if the sequence of finite sets enumerates only finitely many sets or if only finitely many of the finite sets are minimal under inclusion, then it is a routine computation to verify that the index sets of and co- are in the finite Ershov hierarchy. Thus we are interested in the case in which infinitely many of the sequence of finite sets are minimal under inclusion. However if the infinite sequence is fairly simple, for instance{0}, {1}, {2}, … then the r.e. index set of co- is Σ20-complete as well as the index sets of and co- for all levels n > 2. Since the finite Ershov hierarchy does not exhaust ⊿20 there is a lot of “room” between these two extreme cases.

scholarly journals Sur les solutions friables de l'équation a+b=c

2013 ◽  
Vol 154 (3) ◽  
pp. 439-463 ◽  
Author(s):  
SARY DRAPPEAU
Keyword(s):  
Lower Bound ◽  
Asymptotic Behaviour ◽  
Recent Work ◽  
Upper Bound ◽  
Smooth Function ◽  
Riemann Hypothesis ◽  
Smooth Solutions ◽  
Compactly Supported ◽  
The One

AbstractIn a recent paper [5], Lagarias and Soundararajan study the y-smooth solutions to the equation a+b=c. Conditionally under the Generalised Riemann Hypothesis, they obtain an estimate for the number of those solutions weighted by a compactly supported smooth function, as well as a lower bound for the number of bounded unweighted solutions. In this paper, we prove a more precise conditional estimate for the number of weighted solutions that is valid when y is relatively large with respect to x, so as to connect our estimate with the one obtained by La Bretèche and Granville in a recent work [2]. We also prove, conditionally under the Generalised Riemann Hypothesis, the conjectured upper bound for the number of bounded unweighted solutions, thus obtaining its exact asymptotic behaviour.

Rice Theorems For D.R.E. Sets

1975 ◽  
Vol 27 (2) ◽  
pp. 352-365 ◽  
Author(s):  
Louise Hay
Keyword(s):  
Finite Sets ◽  
Index Set ◽  
Recursively Enumerable ◽  
Index Sets

Two of the basic theorems in the classification of index sets of classes of recursively enumerable (r.e.) sets are the following:(i) The index set of a class C of r.e. sets is recursive if and only if C is empty or contains all r.e. sets; and(ii) the index set of a class C or r.e. sets is recursively enumerable if and only if C is empty or consists of all r.e. sets which extend some element of a canonically enumerable class of finite sets.The first theorem is due to Rice [7, p. 364, Corollary B]. The second was conjectured by Rice [7, p. 361] and proved independently by McNaughton, Shapiro, and Myhill [6].

scholarly journals Number of Directions Determined by a Set in $$\mathbb {F}_{q}^{2}$$ and Growth in $$\mathrm {Aff}(\mathbb {F}_{q})$$

2021 ◽  
Author(s):  
Daniele Dona
Keyword(s):  
Lower Bound ◽  
Finite Field ◽  
Upper Bound ◽  
Finite Plane ◽  
Structural Theorem ◽  
Prime Fields ◽  
The One

AbstractWe prove that a set A of at most q non-collinear points in the finite plane $$\mathbb {F}_{q}^{2}$$ F q 2 spans more than $${|A|}/\!{\sqrt{q}}$$ | A | / q directions: this is based on a lower bound by Fancsali et al. which we prove again together with a different upper bound than the one given therein. Then, following the procedure used by Rudnev and Shkredov, we prove a new structural theorem about slowly growing sets in $$\mathrm {Aff}(\mathbb {F}_{q})$$ Aff ( F q ) for any finite field $$\mathbb {F}_{q}$$ F q , generalizing the analogous results by Helfgott, Murphy, and Rudnev and Shkredov over prime fields.

scholarly journals The Degree/Diameter Problem in Maximal Planar Bipartite graphs

10.37236/4468 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Cristina Dalfó ◽  
Clemens Huemer ◽  
Julián Salas
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Planar Graphs ◽  
Maximum Degree ◽  
Bipartite Graphs ◽  
Upper Bounds ◽  
The One

The $(\Delta,D)$ (degree/diameter) problem consists of finding the largest possible number of vertices $n$ among all the graphs with maximum degree $\Delta$ and diameter $D$. We consider the $(\Delta,D)$ problem for maximal planar bipartite graphs, that is, simple planar graphs in which every face is a quadrangle. We obtain that for the $(\Delta,2)$ problem, the number of vertices is $n=\Delta+2$; and for the $(\Delta,3)$ problem, $n= 3\Delta-1$ if $\Delta$ is odd and $n= 3\Delta-2$ if $\Delta$ is even. Then, we prove that, for the general case of the $(\Delta,D)$ problem, an upper bound on $n$ is approximately $3(2D+1)(\Delta-2)^{\lfloor D/2\rfloor}$, and another one is $C(\Delta-2)^{\lfloor D/2\rfloor}$ if $\Delta\geq D$ and $C$ is a sufficiently large constant. Our upper bounds improve for our kind of graphs the one given by Fellows, Hell and Seyffarth for general planar graphs. We also give a lower bound on $n$ for maximal planar bipartite graphs, which is approximately $(\Delta-2)^{k}$ if $D=2k$, and $3(\Delta-3)^k$ if $D=2k+1$, for $\Delta$ and $D$ sufficiently large in both cases.

scholarly journals On Ensuring Multilayer Wirability by Stretching Layouts

VLSI Design ◽  
1998 ◽  
Vol 7 (4) ◽  
pp. 365-383
Author(s):  
Teofilo F. Gonzalez ◽  
Si-Qing Zheng
Keyword(s):  
Dynamic Programming ◽  
Lower Bound ◽  
Upper Bound ◽  
Area Increase ◽  
Different Types ◽  
Np Complete ◽  
Layout Area ◽  
The One

Every knock-knee layout is four-layer wirable. However, there are knock-knee layouts that cannot be wired in less than four layers. While it is easy to determine whether a knock-knee layout is one-layer wirable or two-layer wirable, the problem of determining three-layer wirability of knock-knee layouts is NP-complete. A knock-knee layout may be stretched vertically (horizontally) by introducing empty rows (columns) so that it can be wired in fewer than four layers. In this paper we discuss two different types of stretching schemes. It is known that under these two stretching schemes, any knock-knee layout is three-layer wirable by stretching it up to (4/3) of the knock-knee layout area (upper bound). We show that there are knock-knee layouts that when stretched and wired in three layers under scheme I (II) require at least 1.2 (1.07563) of the original layout area. Our lower bound for the area increase factor can be used to guide the search for effective stretching-based dynamic programming three-layer wiring algorithms similar to the one presented in [8].

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