scholarly journals Presenting the Symmetrical Group with Transpositions

1994 ◽  
Vol 168 (2) ◽  
pp. 521-524 ◽  
Author(s):  
L. Solomon
Keyword(s):  

In a recent article F. Hund has treated the problem of the specific heat of the hydrogen molecule on the basis of the wave mechanics. The total number of rotational states are divided due to the homopolar character of the molecule into two groups, to the one of which belong wave functions symmetrical in the two nuclei, and to the other wave functions which are antisymmetrical in the nuclei. Hund has suggested that the presence of both groups in hydrogen may be accounted for by assuming that the nuclei possess a spin, in which case transitions between symmetrical or between antisymmetrical states will have their usual intensity but transitions between symmetrical and antisymmetrical states will be very weak, of the order of the coupling of the nuclear spins. He then writes the following expression for the rotational specific heat, C r /R = σ 2 d 2 / d σ 2 log Q, Q = β [1 + 5 e -6σ + 9 e -20σ + ...] + 3 e -2σ + 7 e -12σ + 11 e -30σ +...., (1) where σ = h 2 /8π 2 I k T and β is the ratio of the weights of the symmetrical group of states to the antisymmetrical group. Hund has found that he obtains a close agreement between (1) and the observed specific heat curve only when β has about the value 2, that is when the symmetrical states have twice the weight of the antisymmetrical. He further obtains for this case I = 1·54 × 10 -41 gm. cm. 2 , the moment of inertia of the H 2 molecule.


1914 ◽  
Vol 34 ◽  
pp. 76-88
Author(s):  
W. R. Lethaby

The Square Pedestals.—In some notes on the sculpture from the Artemision at the British Museum, printed in the last volume of this Journal (p. 87), I suggested that the fragment No. 1201 most probably belonged to a relief representing either Herakles in the Garden of the Hesperides or Herakles and the Hydra. Subsequent examination and the attempt to make a restoration from the given data have made me sure that the former was the subject of the sculpture. Only this would account for the quiet action of the left hand of Herakles and for the closely associated female figure. If this were indeed the subject, how could its normal elements be arranged so as to suit the conditions of the square pedestal having a vertical joint in the centre, and making proper use of the existing fragment of which Fig. 1 is a rough sketch? This question I have tried to answer. The fragment is now fixed in the side of a built-up pedestal close to its left-hand angle, but there is nothing which settles this position and it is a practically impossible one, for there is not room left in which to complete the figure of Herakles. If, however, we shift the piece to the right hand half of the pedestal, and sketch in the completion of the two figures, we at once see how perfectly the tree and serpent would occupy the centre of the composition and leave the left-hand space for the two other watching maidens—the whole making a symmetrical group.


1999 ◽  
Vol 6 (29) ◽  
Author(s):  
Søren Riis

<p>It is shown that any sequence  psi_n of tautologies which expresses the<br />validity of a fixed combinatorial principle either is "easy" i.e. has polynomial<br />size tree-resolution proofs or is "difficult" i.e requires exponential<br />size tree-resolution proofs. It is shown that the class of tautologies which<br />are hard (for tree-resolution) is identical to the class of tautologies which<br />are based on combinatorial principles which are violated for infinite sets.<br />Actually it is shown that the gap-phenomena is valid for tautologies based<br />on infinite mathematical theories (i.e. not just based on a single proposition).<br />We clarify the link between translating combinatorial principles (or<br />more general statements from predicate logic) and the recent idea of using<br /> the symmetrical group to generate problems of propositional logic.<br />Finally, we show that it is undecidable whether a sequence  psi_n (of the<br />kind we consider) has polynomial size tree-resolution proofs or requires<br />exponential size tree-resolution proofs. Also we show that the degree of<br />the polynomial in the polynomial size (in case it exists) is non-recursive,<br />but semi-decidable.</p><p>Keywords: Logical aspects of Complexity, Propositional proof complexity,<br />Resolution proofs.</p><p> </p>


2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Takehiro Yamashita ◽  
Hiroto Terasaki ◽  
Ryo Asaoka ◽  
Naoya Yoshihara ◽  
Naoko Kakiuchi ◽  
...  

AbstractThe purpose of this study was to compare the incidences of macular irregularities of elementary school (ES) and junior high school (JHS) students. This was a prospective cross-sectional observational study of 122 right eyes of 122 ES students (8–9 years) and 173 right eyes of 173 JHS students (12–13 years). Vertical cross-sectional images of the macula were obtained by optical coherence tomography. The eyes were classified based on the vertical symmetry of the posterior pole, and then sub-classified as convex-, flat-, concave-, or dome-shaped based on the direction of the curvature of the retinal pigment epithelium. One hundred and two eyes (83.6%) were placed in the symmetrical group in the ES students and 149 eyes (86.1%) in the JHS students. Twenty eyes (16.4%) were placed in the asymmetric groups in the ES students and 24 eyes (13.9%) in the JHS students. In symmetrical group, 78 and 118 eyes had a convex-shape (76.4 and 79.2%), followed by 22 and 29 eyes of dome-shaped group (21.6 and 19.4%) in ES and JHS students respectively. Because the incidences of the posterior pole shapes were not significantly different between the groups, it is likely that the macular irregularities develop before the age of ES.


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