On parallelepipeds of minimal volume containing a convex symmetric body in ℝn

1991 ◽  
Vol 109 (1) ◽  
pp. 125-148 ◽  
Author(s):  
A. Pelczynski ◽  
S. J. Szarek
Keyword(s):  
Local Minimum ◽  
The Body ◽  
Symmetric Body ◽  
Alternative Proof ◽  
Euclidean Ball ◽  
Maximal Volume ◽  
Minimal Volume ◽  
Convex Symmetric Body ◽  
Unit Euclidean Ball ◽  
Image Position

AbstractGiven a convex symmetric body C ⊂ ℝn we put a(C) = |C| sup |P|−1 where the supremum extends over all parallelepipeds containing C and |A| denotes the volume of a set A ⊂ ℝn. Let an = inf {a(C): C ⊂ ℝn}. We show thatwhich slightly improves the estimate due to Dvoretzky and Rogers [6]. In every dimension n we construct a convex symmetric polytope Wn such that the unit Euclidean ball is the ellipsoid of maximal volume inscribed into Wn and the volume of every parallelepiped containing Wn is greater thanfor large n which shows ‘the limit’ to the Dvoretzky Rogers method for bounding an below. We present an alternative proof of the result of I. K. Babenko [l] that . We show that , and that a local minimum of the function C→a(C) for C ⊂ ℝn is attained only at an equiframed convex body (that is, a body such that every point of its boundary belongs to a parallelepiped of minimal volume containing the body).

Readers in the Underworld: Lucretius, De Rerum Natura 3.912–1075

2004 ◽  
Vol 94 ◽  
pp. 27-46
Author(s):  
Tobias Reinhardt
Keyword(s):  
The Body ◽  
Fear Of Death ◽  
De Rerum Natura ◽  
Image Position

Readers have always acknowledged the comparatively clear macrostructure of De rerum natura 3. It begins with a prooemium in which is described the terrifying impact which the fear of death has on human lives, as well as the fact that Epicurus has provided a cure against this fear, namely his physical doctrines (1–93). Particular attention is paid to fears of an afterlife in which we have to suffer pain and grief in the underworld; cf., for instance, the programmatic lines 3.37–40 (translation by Ferguson Smith, which will be used throughout):This prooemium is followed by a long passage (94–829) in which Lucretius explains the basics of Epicurean psychology and tries to show that the soul is (like the body) material and hence mortal; this last point is driven home with particular force in II. 417–829 where Lucretius lists twenty-five proofs for the mortality of the soul.

The physical basis of mollusk shell chiral coiling

2021 ◽  
Vol 118 (48) ◽  
pp. e2109210118
Author(s):  
Régis Chirat ◽  
Alain Goriely ◽  
Derek E. Moulton
Keyword(s):  
Mathematical Model ◽  
The Body ◽  
Symmetric Body ◽  
Model Organisms ◽  
Physical Interaction ◽  
Mechanical Forces ◽  
Hard Shell ◽  
Left Right Asymmetry ◽  
Development And Evolution ◽  
Mollusk Shell

Snails are model organisms for studying the genetic, molecular, and developmental bases of left–right asymmetry in Bilateria. However, the development of their typical helicospiral shell, present for the last 540 million years in environments as different as the abyss or our gardens, remains poorly understood. Conversely, ammonites typically have a bilaterally symmetric, planispiraly coiled shell, with only 1% of 3,000 genera displaying either a helicospiral or a meandering asymmetric shell. A comparative analysis suggests that the development of chiral shells in these mollusks is different and that, unlike snails, ammonites with asymmetric shells probably had a bilaterally symmetric body diagnostic of cephalopods. We propose a mathematical model for the growth of shells, taking into account the physical interaction during development between the soft mollusk body and its hard shell. Our model shows that a growth mismatch between the secreted shell tube and a bilaterally symmetric body in ammonites can generate mechanical forces that are balanced by a twist of the body, breaking shell symmetry. In gastropods, where a twist is intrinsic to the body, the same model predicts that helicospiral shells are the most likely shell forms. Our model explains a large diversity of forms and shows that, although molluscan shells are incrementally secreted at their opening, the path followed by the shell edge and the resulting form are partly governed by the mechanics of the body inside the shell, a perspective that explains many aspects of their development and evolution.

scholarly journals VI.—The Body-Temperature of Fishes and other Marine Animals.

1908 ◽  
Vol 28 ◽  
pp. 66-84 ◽  
Author(s):  
Sutherland Simpson
Keyword(s):  
Body Temperature ◽  
Temperature Difference ◽  
Great Majority ◽  
The Body ◽  
Shore Crab ◽  
First Year ◽  
Marine Animals ◽  
Mean Temperature ◽  
The Mean ◽  
Image Position

SUMMARYThe body-temperature of the following fishes, crustaceans, and echinoderms has been examined and compared with the temperature of the water in which they live:—Cod-fish (Gadus morrhua), ling (Molva vulgaris), torsk (Brosmius brosme), coal-fish or saithe (Gadus virens), haddock (Gadus œgelfinus), flounder (Pleuronectes flesus), smelt (Osmerus eperlanus), dog-fish (Scyllium catulus), shore crab (Carcinus mœnas), edible crab (Cancer pagurus), lobster (Homarus vulgaris), sea-urchin (Echinus esculentus), and starfish (Asterias rubens). The minimum, maximum, and mean temperature difference for each species are given in the following table:—The excess of temperature is most evident in the larger specimens. This is well shown in the case of the coal-fish, where in the adult it was 0°·7 C., and in the great majority (11 out of 12) of the young of the first year, 0°·0 C. The body-weight and the conditions under which the fish are captured probably form the most important factors in determining the temperature difference.In 14 codfish, where the rectal, blood, and muscle temperatures were recorded in the same individual, it was found to be highest in the muscle and lowest in the rectum, the mean temperature difference being 0°·46 C. for the muscle, 0°·41 C for the blood, and 0°·36 C. for the rectum.

scholarly journals The Chaetotaxy of the Pupa of Stegomyia fasciata

1920 ◽  
Vol 10 (2) ◽  
pp. 161-169 ◽  
Author(s):  
J. W. S. Macfie
Keyword(s):  
The Body ◽  
Posterior Angle ◽  
Image Position ◽  
The Right ◽  
Two Sides

The pupa is bilaterally symmetrical, that is, setae occur in similar situations on each side of the body, so that it will suffice to describe the arrangement on one side only. The setae on the two sides of the same pupa, however, often vary as regards their sub-divisions, and similar variations occur between different individuals; as an example, in Table I are shown some of the variations that were found in ten pupae taken at random. An examination of a larger number would have revealed a wider range. As a rule, a seta which is sometimes single, sometimes divided, is longer when single. For example, in one pupa the seta at the posterior angle ofthe seventh segment was single on the right side, double on the left; the former measuring 266μ, and the latter only 159μ in length. This fact is not specifically mentioned in the descriptions which follow, but should be understood.

On the number of interior peaks of solutions to a non-autonomous singularly perturbed Neumann problem

2009 ◽  
Vol 139 (2) ◽  
pp. 427-448 ◽  
Author(s):  
Chunyi Zhao
Keyword(s):  
Positive Integer ◽  
Neumann Problem ◽  
Smooth Function ◽  
Local Minimum ◽  
Singularly Perturbed ◽  
Minimum Point ◽  
Local Minimum Point ◽  
Image Position

We study the following non-autonomous singularly perturbed Neumann problem:where the index p is subcritical and a(x) is a positive smooth function in . We show that, given ε small enough, there exists a K(ε) such that, for any positive integer K ≤ K(ε), there always exists a solution with K interior peaks concentrating at a strict sth-order local minimum point of a.

scholarly journals Two Enumerative Proofs of an Identity of Jacobi

1966 ◽  
Vol 15 (1) ◽  
pp. 67-71 ◽  
Author(s):  
C. Sudler
Keyword(s):  
Alternative Proof ◽  
Ferrers Graph ◽  
Image Position ◽  
Durfee Square

In (7), Wright gives an enumerative proof of an identity algebraically equivalent to that of Jacobi, namelyHere, and in the sequel, products run from 1 to oo and sums from - oo to oo unless otherwise indicated. We give here a simplified version of his argument by working directly with (1), the substitution leading to equation (3) of his paper being omitted. We then supply an alternative proof of (1) by means of a generalisation of the Durfee square concept utilising the rectangle of dimensions v by v + r for fixed r and maximal v contained in the Ferrers graph of a partition.

scholarly journals Irrationality proofs à la Hermite

2011 ◽  
Vol 95 (534) ◽  
pp. 407-413
Author(s):  
Li Zhou
Keyword(s):  
Recurrence Relation ◽  
Alternative Proof ◽  
Image Position

In [1] Niven used the integralto give a well-known proof of the irrationality of π. Recently Zhou and Markov [2] used a recurrence relation satisfied by this integral to present an alternative proof which may be more direct than Niven's.Niven did not cite any references in [1] and thus the origin or Hn seems rather mysterious and ingenious. However if we heed Abel's advice to ‘study the masters’, we find that Hn emerged much more naturally from the great works of Lambert [3] and Hermite [4].

Proportionality of organ development in osteodystrophic dwarf beef cattle

1960 ◽  
Vol 55 (3) ◽  
pp. 351-358 ◽  
Author(s):  
E. S. E. Hafez ◽  
E. H. Rupnow
Keyword(s):  
Abdominal Cavity ◽  
Live Weight ◽  
The Body ◽  
Muscle Area ◽  
Eye Muscle ◽  
Pituitary Glands ◽  
The Difference ◽  
Image Position ◽  
And Control ◽  
Dwarf Males

Sixteen osteodystrophic dwarf cattle and ten controls of comparable age were slaughtered. The components of the body and eviscerated carcass were weighed and measured. At birth the dwarfs were thick and blocky. At the time of slaughter a bulging forehead was common but not always extreme and not always present. The symptoms of dwarfism became increasingly pronounced with age, due to retarded growth. The dwarfs had shorter thoracic cavity, abdominal cavity, body, loin, hind leg, arm bone and forearm bone than the controls. No explanation can be given for the difference. However, the dwarfs were hydrocephalic and had significantly lighter adrenal and pituitary glands than the control animals. The dwarf animals had more blood, heavier feet, less abdominal fat, smaller loin ‘eye muscle’ area at the 12th rib and a less deep loin ‘eye muscle’. The dwarf females had a lighter rumen (with and without contents) and large intestines (without content) as a percentage of live weight than the controls and dwarf males. There was no difference in palatability of the meat or percentage of wholesale cuts from the dwarf and control animals except for percentage of plate. The following three ratios were disproportionate in the dwarfs as compared with the controls:

The auditor Thaumasius in the Vita Plotini

1993 ◽  
Vol 113 ◽  
pp. 157-160
Author(s):  
Richard Lim
Keyword(s):  
The Body ◽  
Philosophical Discourse ◽  
Lecture Room ◽  
Image Position

In his Vita Plotini, Porphyry recounts a colourful episode which, for a brief moment, brings to life the dynamics within the lecture room of Plotinus in Rome. The author explains how he was in the habit of posing questions to Plotinus frequently and persistently while his teacher was conducting his philosophical discourse before a mixed body of listeners. On one occasion, such an exchange between the two over the issue of the connexion between the soul and the body continued intermittently over a period of some three days, with the following outcome (Porph. V. Plot. xiii 12-15):

scholarly journals Existence and Concentration of Positive Solutions for Nonlinear Kirchhoff-Type Problems with a General Critical Nonlinearity

2018 ◽  
Vol 61 (4) ◽  
pp. 1023-1040 ◽  
Author(s):  
Jianjun Zhang ◽  
David G. Costa ◽  
João Marcos do Ó
Keyword(s):  
Positive Solutions ◽  
Local Minimum ◽  
Type Equation ◽  
Bound State ◽  
State Solution ◽  
Critical Nonlinearity ◽  
Bound State Solution ◽  
Kirchhoff Type ◽  
Image Position ◽  
Kirchhoff Type Problems

AbstractWe are concerned with the following Kirchhoff-type equation$$ - \varepsilon ^2M\left( {\varepsilon ^{2 - N}\int_{{\open R}^N} {\vert \nabla u \vert^2{\rm d}x} } \right)\Delta u + V(x)u = f(u),\quad x \in {{\open R}^N},\quad N{\rm \ges }2,$$whereM ∈ C(ℝ+, ℝ+),V ∈ C(ℝN, ℝ+) andf(s) is of critical growth. In this paper, we construct a localized bound state solution concentrating at a local minimum ofVasε → 0 under certain conditions onf(s),MandV. In particular, the monotonicity off(s)/sand the Ambrosetti–Rabinowitz condition are not required.

Sign in / Sign up

Export Citation Format

Share Document