scholarly journals Banach limit in convexity and geometric means for convex bodies

2016 ◽  
Vol 23 (0) ◽  
pp. 41-51 ◽  
Author(s):  
Liran Rotem
Keyword(s):  
Convex Bodies ◽  
Banach Limit ◽  
Geometric Means

Algebraically inspired results on convex functions and bodies

2016 ◽  
Vol 18 (06) ◽  
pp. 1650027 ◽  
Author(s):  
Liran Rotem
Keyword(s):  
Convex Functions ◽  
Final Section ◽  
Polar Body ◽  
Convex Bodies ◽  
Type Inequality ◽  
Dual Function ◽  
Minkowski Addition ◽  
Geometric Means ◽  
Better Than

We show how algebraic identities, inequalities and constructions, which hold for numbers or matrices, often have analogs in the geometric classes of convex bodies or convex functions. By letting the polar body [Formula: see text] or the dual function [Formula: see text] play the role of the inverses “[Formula: see text]” and “[Formula: see text]”, we are able to conjecture many new results, which often turn out to be correct. As one example, we prove that for every convex function [Formula: see text] one has [Formula: see text] where [Formula: see text]. We also prove several corollaries of this identity, including a Santal type inequality and a contribution to the theory of summands. We proceed to discuss the analogous identity for convex bodies, where an unexpected distinction appears between the classical Minkowski addition and the more modern 2-addition. In the final section of the paper we consider the harmonic and geometric means of convex bodies and convex functions, and discuss their concavity properties. Once again, we find that in some problems the 2-addition of convex bodies behaves even better than the Minkowski addition.

Estimation of geographical variation of cancer risks in drinking water in Japan

2006 ◽  
Vol 6 (2) ◽  
pp. 31-37
Author(s):  
K. Ohno ◽  
E. Kadota ◽  
Y. Kondo ◽  
T. Kamei ◽  
Y. Magara
Keyword(s):  
Geographical Variation ◽  
Statistical Data ◽  
Raw Water ◽  
Cancer Risks ◽  
Geometric Means ◽  
Toxicological Effects ◽  
Total Cancer ◽  
Big Cities ◽  
The Relationship ◽  
Plausible Reason

The cancer risks posed by ten substances in raw and purified water were estimated for each municipality in Japan to compare risks between raw and purified water, and inter-municipality. Water concentrations were estimated by use of statistical data. Assigning cancer unit risks to each substance and applying the assumption of additive toxicological effects to multiple carcinogens, total cancer risks of the waters were estimated. As a result, the geometric means of total cancer risks in raw and purified water were 1.16×10−5 and 2.18×10−5, respectively. In raw water, the contribution ratio of arsenic to total cancer risk accounted for 97%. In purified water, that of four trihalomethanes (THMs) accounted for 54%. The increase of total cancer risks in purified water was due to THMs. In regard to the geographical variation, the relationship between population size and total cancer risks were investigated. The result was that there were higher cancer risks in the big cities with the population more than a million both in raw and purified water. One plausible reason for the higher risks in purified water in the big cities is a larger chlorination dose due to the huge water supply areas. The reason for the increase in raw water remained unclear.

Excess entropy of the systems of molecules differing in size and shape

1983 ◽  
Vol 48 (1) ◽  
pp. 192-198 ◽  
Author(s):  
Tomáš Boublík
Keyword(s):  
Equation Of State ◽  
Hard Spheres ◽  
Excess Entropy ◽  
Convex Bodies ◽  
Pure Substance ◽  
Liquid Equilibrium ◽  
Simple Rules ◽  
Different Shapes ◽  
The Mean

The excess entropy of mixing of mixtures of hard spheres and spherocylinders is determined from an equation of state of hard convex bodies. The obtained dependence of excess entropy on composition was used to find the accuracy of determining ΔSE from relations employed for the correlation and prediction of vapour-liquid equilibrium. Simple rules were proposed for establishing the mean parameter of nonsphericity for mixtures of hard bodies of different shapes allowing to describe the P-V-T behaviour of solutions in terms of the equation of state fo pure substance. The determination of ΔSE by means of these rules is discussed.

scholarly journals Some Hermite–Hadamard type integral inequalities for convex functions defined on convex bodies in ℝn\mathbb{R}^{n}

2020 ◽  
Vol 26 (1) ◽  
pp. 67-77 ◽  
Author(s):  
Silvestru Sever Dragomir
Keyword(s):  
Euclidean Space ◽  
Convex Functions ◽  
Convex Bodies ◽  
Integral Inequalities ◽  
Divergence Theorem ◽  
Several Variables ◽  
Type Integral ◽  
Bounded Convex

AbstractIn this paper, by the use of the divergence theorem, we establish some integral inequalities of Hermite–Hadamard type for convex functions of several variables defined on closed and bounded convex bodies in the Euclidean space {\mathbb{R}^{n}} for any {n\geq 2}.

scholarly journals General methods of convergence and summability

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Francisco Javier García-Pacheco ◽  
Ramazan Kama ◽  
María del Carmen Listán-García
Keyword(s):  
Geometric Structure ◽  
Banach Limit ◽  
Limit Operator ◽  
Finite Dimensional ◽  
Free Ultrafilter ◽  
General Method

AbstractThis paper is on general methods of convergence and summability. We first present the general method of convergence described by free filters of $\mathbb{N} $ N and study the space of convergence associated with the filter. We notice that $c(X)$ c ( X ) is always a space of convergence associated with a filter (the Frechet filter); that if X is finite dimensional, then $\ell _{\infty }(X)$ ℓ ∞ ( X ) is a space of convergence associated with any free ultrafilter of $\mathbb{N} $ N ; and that if X is not complete, then $\ell _{\infty }(X)$ ℓ ∞ ( X ) is never the space of convergence associated with any free filter of $\mathbb{N} $ N . Afterwards, we define a new general method of convergence inspired by the Banach limit convergence, that is, described through operators of norm 1 which are an extension of the limit operator. We prove that $\ell _{\infty }(X)$ ℓ ∞ ( X ) is always a space of convergence through a certain class of such operators; that if X is reflexive and 1-injective, then $c(X)$ c ( X ) is a space of convergence through a certain class of such operators; and that if X is not complete, then $c(X)$ c ( X ) is never the space of convergence through any class of such operators. In the meantime, we study the geometric structure of the set $\mathcal{HB}(\lim ):= \{T\in \mathcal{B} (\ell _{\infty }(X),X): T|_{c(X)}= \lim \text{ and }\|T\|=1\}$ HB ( lim ) : = { T ∈ B ( ℓ ∞ ( X ) , X ) : T | c ( X ) = lim  and  ∥ T ∥ = 1 } and prove that $\mathcal{HB}(\lim )$ HB ( lim ) is a face of $\mathsf{B} _{\mathcal{L}_{X}^{0}}$ B L X 0 if X has the Bade property, where $\mathcal{L}_{X}^{0}:= \{ T\in \mathcal{B} (\ell _{\infty }(X),X): c_{0}(X) \subseteq \ker (T) \} $ L X 0 : = { T ∈ B ( ℓ ∞ ( X ) , X ) : c 0 ( X ) ⊆ ker ( T ) } . Finally, we study the multipliers associated with series for the above methods of convergence.

Microbiological Quality of Fresh Blue Crabmeat, Clams and Oysters

1983 ◽  
Vol 46 (11) ◽  
pp. 978-981 ◽  
Author(s):  
B. A. WENTZ ◽  
A. P. DURAN ◽  
A. SWARTZENTRUBER ◽  
A. H. SCHWAB ◽  
R. B. READ
Keyword(s):  
Escherichia Coli ◽  
Staphylococcus Aureus ◽  
Microbiological Quality ◽  
Fecal Coliforms ◽  
Geometric Means ◽  
Colony Forming Units ◽  
Plate Counts ◽  
The Mean ◽  
Eastern Oysters

The microbiological quality of fresh blue crabmeat, soft- and hardshell clams and shucked Eastern oysters was determined at the retail (crabmeat, oysters) and wholesale (clams) levels. Geometric means of aerobic plate counts incubated at 35°C were: blue crabmeat 140,000 colony-forming units (CFU)/g, hardshell clams, 950 CFU/g, softshell clams 680 CFU/g and shucked Eastern oysters 390,000 CFU/g. Coliform geometric means ranged from 3,6/100 g for hardshell clams to 21/g for blue crabmeat. Means for fecal coliforms or Escherichia coli ranged from <3/100 g for clams to 27/100 g for oysters, The mean Staphylococcus aureus count in blue crabmeat was 10/g.

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