The determination of a function from its projections

1979 ◽  
Vol 85 (2) ◽  
pp. 351-355 ◽  
Author(s):  
K. J. Falconer
Keyword(s):  
Lebesgue Measure ◽  
Unit Vector ◽  
Fubini’S Theorem ◽  
Perpendicular Distance ◽  
Almost All ◽  
Dimensional Lebesgue Measure ◽  
Fubini's Theorem

Let H(t, θ) be the hyperplane in Rn (n ≥ 2) which is perpendicular to the unit vector θ and perpendicular distance t from the origin, that is H(t, θ) = {x ε Rn: x.θ = t} (Note that H(t, θ) and H(−t, −θ) are the same hyperplane.) If f(x)ε L1(Rn) we will denote by F(t, θ) the projection of f perpendicular to θ, that is the integral of f(x) over H(t, θ) with respect to (n − 1)-dimensional Lebesgue measure. By Fubini's Theorem, if f(x) ε L1 (Rn), F(t, θ) exists for almost all t for every θ.

Consistency conditions for a finite set of projections of a function

1979 ◽  
Vol 85 (1) ◽  
pp. 61-68 ◽  
Author(s):  
K. J. Falconer
Keyword(s):  
Lebesgue Measure ◽  
Convex Subset ◽  
Compact Convex ◽  
Unit Vector ◽  
Compact Convex Subset ◽  
Finite Collection ◽  
Finite Set ◽  
Almost All ◽  
Dimensional Lebesgue Measure ◽  
Unit Vectors

Let H(μ, θ) be the hyperplane in Rn (n ≥ 2) that is perpendicular to the unit vector 6 and perpendicular distance μ from the origin; that is, H(μ, θ) = (x ∈ Rn: x. θ = μ). (Note that H(μ, θ) and H(−μ, −θ) are the same hyperplanes.) Let X be a proper compact convex subset of Rm. If f(x) ∈ L1(X) we will denote by F(μ, θ) the projection of f perpendicular to θ; that is, the integral of f(x) over H(μ, θ) with respect to (n − 1)-dimensional Lebesgue measure. By Fubini's Theorem, if f(x) ∈ L1(X), F(μ, θ) exists for almost all μ for every θ. Our aim in this paper is, given a finite collection of unit vectors θ1, …, θN, to characterize the F(μ, θi) that are the projections of some function f(x) with support in X for 1 ≤ i ≤ N.

Decidability of the “almost all” theory of degrees

1972 ◽  
Vol 37 (3) ◽  
pp. 501-506 ◽  
Author(s):  
John Stillwell
Keyword(s):  
Lebesgue Measure ◽  
Measure Theory ◽  
Random Sequence ◽  
Recursion Theory ◽  
Sharp Contrast ◽  
Standard Theory ◽  
Simple Manner ◽  
Fubini’S Theorem ◽  
Almost All ◽  
Random Pair

Ever since Spector's brilliant application of measure theory to recursion theory in 1958 [6] it has been realized that measure theory promotes sweeping simplifications in the theory of degrees. Results previously thought to be pathological were shown by Spector, and later Sacks [4], [5], to hold for almost all degrees (“almost all” in the sense of Lebesgue measure), often with much simpler proofs. Good examples of this phenomenon are Spector's demonstration that almost all pairs of sets are of incomparable degree (as an immediate consequence of Fubini's theorem) and Sacks' exquisitely simple deduction from this result that almost every degree is the join of two incomparable degrees (for if a random sequence is decomposed into its even and odd parts, the result is a random pair).The present paper attempts to vindicate the feeling that almost all degrees behave in a simple manner by showing that if the quantifier in the theory of degrees with ′(jump), ∪ (join) and ∩ (meet) is taken to be (almost ∀a) instead of (∀a) then the theory is decidable. We are able to use ∩ because it will be shown that if t1, t2 are any terms built from degree variables a1, …, am with ′ and ∪ then t1 ∩ t2 exists for almost all a1, …, am. Thus the “almost all” theory presents a sharp contrast to the standard theory, where ∩ is not always defined (Kleene-Post [1]) and which is known to be undecidable (Lachlan [2]).

Function space topologies defined by sectional integrals and applications to an extremal problem

1980 ◽  
Vol 87 (1) ◽  
pp. 81-96 ◽  
Author(s):  
K. J. Falconer
Keyword(s):  
Extremal Problem ◽  
Final Section ◽  
Convex Set ◽  
Compact Convex ◽  
Unit Vector ◽  
Extremal Values ◽  
Almost All ◽  
Function Space Topologies ◽  
Dimensional Lebesgue Measure ◽  
Uniformly Bounded

Let H(t, θ) be the hyperplane in Rn (n ≥ 2) which is perpendicular to the unit vector θ, and distant t from the origin; that is H(t, θ) = {x ε Rn: x.θ = t}. (Note that H(t, θ) and H(−t, − θ) are the same hyperplane.) If f(x) εℒ1(Rn), we will denote the integral of f with respect to (n − 1)-dimensional Lebesgue measure over H(t, θ) by F(t, θ), termed the projection or sectional integral of f over H(t,θ). By Fubini's theorem, F(t, θ) exists for almost all t for any θ. Throughout this paper we will assume that f(x) has support in X, a compact convex subset of Rn. In Section 2 we examine some of the topologies that may be defined on functions on X in terms of the F(t, θ), and in the remainder of the paper we examine the extremal problem suggested by Croft (4), that of maximizing the integral of f over the set X with the constraint that the F(t, θ) are uniformly bounded above. We examine in particular how the extremal values depend on the convex set X. In the final section the extremal problem is related to a generalization of Bang's plank theorem and the theory of capacities, and several conjectures are proposed.

Population Change, Modernization and Welfare. By Joseph Spengler. Prentice-Hall Inc., Englewood Cliffs, N.J. 1974.

1977 ◽  
Vol 16 (2) ◽  
pp. 220-222
Author(s):  
Zeba A. Sathar
Keyword(s):  
Population Growth ◽  
Population Change ◽  
Population Control ◽  
Stable Condition ◽  
Developed Countries ◽  
Wide Field ◽  
Wide Scale ◽  
Socio Economic Variables ◽  
Almost All

The book covers a wide field, touching on almost all aspects of popula¬tion change on a world-wide scale. It discusses, using world and country data, the relationships between demographic and socio-economic variables, and elaborates on" their relative importance in the determination of population problems which confront the world as a whole and nations individually. Policies designed to alleviate these problems are discussed with an emphasis on those related to population control. The first chapter is entitled "Population Growth: Past and Prospective" and reviews the various parameters associated with population change in the past and in the future. It touches upon the concept of a stable population in order to show the elements which cause a population to change (i.e. remove it from its stable condition). The main elements of change, population growth, migration, mortality and natality are discussed individually. The chapter is concluded by a description of the main differences in these elements and other socio-economic conditions as they exist in the less-developed and developed countries.

scholarly journals Comparison on the Nutrient Plunder Capacity of Orychophragmus violaceus and Brassica napus L. Based on Electrophysiological Information

Horticulturae ◽  
2021 ◽  
Vol 7 (8) ◽  
pp. 206
Author(s):  
Cheng Zhang ◽  
Yue Su ◽  
Yanyou Wu ◽  
Haitao Li ◽  
Ying Zhou ◽  
...  
Keyword(s):  
Brassica Napus ◽  
Orychophragmus Violaceus ◽  
Nutrient Metabolism ◽  
Brassica Napus L ◽  
Adaptive Mechanisms ◽  
Capacitive Reactance ◽  
Time Determination ◽  
Almost All ◽  
First Time

The nutrient metabolism, growth and development of plants are strongly affected by its nutrient plunder, and plants have different adaptive mechanisms to low-nutrient environments. The electrophysiological activities involve almost all life processes of plants. In this study, the active transport flow of nutrient (NAF) and nutrient plunder capacity (NPC) of plants were defined based on leaf intrinsic impedance (IZ), capacitive reactance (IXc), inductive reactance (IXL) and capacitance (IC) to evaluate the nutrient plunder capacity of plants for the first time. The results indicate that Orychophragmus violaceus had higher (p < 0.01) NPC and IC and lower (p < 0.01) IR, IXc, IXL and IZ as compared to Brassica napus L., which supports a superior ion affinity and that it could be better adapted to low-nutrient environments. UAF and NPC of plants exhibited good correlations with crude protein, crude ash and water content, and precisely revealed the plunder capacity and adaptive strategies of plants to nutrients. The present work highlights that O. violaceus had superior NPC and ion affinity compared with B. napus, and provided a novel, rapid, reliable method based on the plant’s electrophysiological information for real-time determination of the nutrient plunder capacity of plants.

scholarly journals Cyanobacterial Toxins and Peptides in Lake Vegoritis, Greece

Toxins ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 394
Author(s):  
Sevasti-Kiriaki Zervou ◽  
Kimon Moschandreou ◽  
Aikaterina Paraskevopoulou ◽  
Christophoros Christophoridis ◽  
Elpida Grigoriadou ◽  
...  
Keyword(s):  
Public Health ◽  
Risk Assessment ◽  
Simultaneous Determination ◽  
Aquatic Ecosystems ◽  
Human Exposure ◽  
Bioactive Metabolites ◽  
Cyanobacterial Toxins ◽  
Major Threat ◽  
Almost All

Cyanotoxins (CTs) produced by cyanobacteria in surface freshwater are a major threat for public health and aquatic ecosystems. Cyanobacteria can also produce a wide variety of other understudied bioactive metabolites such as oligopeptides microginins (MGs), aeruginosins (AERs), aeruginosamides (AEGs) and anabaenopeptins (APs). This study reports on the co-occurrence of CTs and cyanopeptides (CPs) in Lake Vegoritis, Greece and presents their variant-specific profiles obtained during 3-years of monitoring (2018–2020). Fifteen CTs (cylindrospermopsin (CYN), anatoxin (ATX), nodularin (NOD), and 12 microcystins (MCs)) and ten CPs (3 APs, 4 MGs, 2 AERs and aeruginosamide (AEG A)) were targeted using an extended and validated LC-MS/MS protocol for the simultaneous determination of multi-class CTs and CPs. Results showed the presence of MCs (MC-LR, MC-RR, MC-YR, dmMC-LR, dmMC-RR, MC-HtyR, and MC-HilR) and CYN at concentrations of <1 μg/L, with MC-LR (79%) and CYN (71%) being the most frequently occurring. Anabaenopeptins B (AP B) and F (AP F) were detected in almost all samples and microginin T1 (MG T1) was the most abundant CP, reaching 47.0 μg/L. This is the first report of the co-occurrence of CTs and CPs in Lake Vegoritis, which is used for irrigation, fishing and recreational activities. The findings support the need for further investigations of the occurrence of CTs and the less studied cyanobacterial metabolites in lakes, to promote risk assessment with relevance to human exposure.

scholarly journals Estimation of the mean normal measure from flat sections

2008 ◽  
Vol 40 (01) ◽  
pp. 31-48
Author(s):  
Markus Kiderlen
Keyword(s):  
A Priori ◽  
The Other ◽  
Consistent Estimator ◽  
Random Set ◽  
Normal Measure ◽  
Mild Assumption ◽  
The Mean ◽  
Vertical Sections ◽  
Almost All

We discuss the determination of the mean normal measure of a stationary random set Z ⊂ ℝ d by taking measurements at the intersections of Z with k-dimensional planes. We show that mean normal measures of sections with vertical planes determine the mean normal measure of Z if k ≥ 3 or if k = 2 and an additional mild assumption holds. The mean normal measures of finitely many flat sections are not sufficient for this purpose. On the other hand, a discrete mean normal measure can be verified (i.e. an a priori guess can be confirmed or discarded) using mean normal measures of intersections with m suitably chosen planes when m ≥ ⌊d / k⌋ + 1. This even holds for almost all m-tuples of k-dimensional planes are viable for verification. A consistent estimator for the mean normal measure of Z, based on stereological measurements in vertical sections, is also presented.

scholarly journals On the application of Fubini's theorem in the integration of functiions of two variables in a measure space

2013 ◽  
Vol 1 (2) ◽  
Author(s):  
'Bankole V Akinremi ◽  
Ubong Sam Idiong ◽  
Bridjet Akintewe ◽  
Kayode Samuel Famuagun
Keyword(s):  
Measure Space ◽  
Fubini’S Theorem ◽  
Fubini's Theorem

The trajectory of the turning point is dense for almost all tent maps

1996 ◽  
Vol 16 (6) ◽  
pp. 1173-1183 ◽  
Author(s):  
Karen Brucks ◽  
Michal Misiurewicz
Keyword(s):  
Lebesgue Measure ◽  
Turning Point ◽  
Tent Map ◽  
Tent Maps ◽  
Almost All

AbstractWe prove that for almost every (with respect to the Lebesgue measure) a ∈ [√2, 2], the forward trajectory of the turning point of the tent map fa with slope a is dense in the interval of transitivity of fa.

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