Alice’s adventures in inverse tan land – mathematical argument, language and proof

2019 ◽  
Vol 103 (558) ◽  
pp. 388-400
Author(s):  
Paul Glaister
Keyword(s):  
Left Hand ◽  
Mathematical Argument ◽  
Image Position ◽  
Head Of Department

Andrew Palfreyman’s article [1] reminds us of the result (1) $${\rm{ta}}{{\rm{n}}^{{\rm{ - 1}}}}{\rm{ + ta}}{{\rm{n}}^{{\rm{ - 1}}}}\,2{\rm{ + ta}}{{\rm{n}}^{{\rm{ - 1}}}}{\rm{ 3 = }}\,\pi {\rm{, }}$$ having been set the challenge of finding the value of the left-hand side by his head of department at the start of a departmental meeting.

scholarly journals A Fourth Century Head in Central Museum, Athens

1895 ◽  
Vol 15 ◽  
pp. 194-201 ◽  
Author(s):  
E. F. Benson
Keyword(s):  
Fourth Century ◽  
Present Condition ◽  
Left Hand ◽  
Right Hand ◽  
Image Position ◽  
The Right

There is among the fourth century works in the Central Museum at Athens a head found at Laurium. It is made of Parian marble but it has been completely discoloured by slag or refuse from the lead mines, and is now quite black. In its present condition it is quite impossible to obtain a satisfactory photograph of it, and the reproduction given of it in the figure is from a cast.It has been published, as far as I am aware, only in M. Kavvadias' catalogue. There it is described as a head of the Lykeian Apollo. This identification rests solely on a passage of Lucian, who mentions a statue of the Lykeian Apollo in the gymnasium at Athens.He says of it ( 7)—It will be seen from a glance at the photograph that the grounds for this identification are very slender. The left hand with the bow does not exist, and the only reason for supposing therefore that this is a head of the Lykeian Apollo consists in the fact that the right hand of the statue rests on the head. This in itself seems insufficient and, among other reasons, it is I think rendered impossible by the phrase For the hand is not idly resting, it is not a tired hand; the posture of the fingers is firm and energetic.

An inequality relating means

1944 ◽  
Vol 40 (3) ◽  
pp. 253-255
Author(s):  
J. Bronowski
Keyword(s):  
Arithmetic Mean ◽  
Arithmetic Means ◽  
Geometric Means ◽  
Left Hand ◽  
Right Hand ◽  
Image Position ◽  
The Right

1. Let a, b be positive constants; and let y1, y2, …, yn be real exponents, not all equal, having arithmetic mean y defined by(here, and in what follows, the summation ∑ extends over the values i = 1, 2, …, n). Then it is clear thatsince the right-hand sides are the geometric means of the positive numbers whose arithmetic means stand on the left-hand sides. I know of no results, however, which relate the ratios and and I have had occasion recently to require such results. This note gives an inequality between these ratios, subject to certain restrictions on a and b.

scholarly journals I.—Elimination in the case of Equality of Fractions whose Numerators and Denominators are Linear Functions of the Variables.

1906 ◽  
Vol 45 (1) ◽  
pp. 1-7
Author(s):  
Thomas Muir
Keyword(s):  
Linear Functions ◽  
Short Paper ◽  
Left Hand ◽  
Image Position ◽  
Factor Α

(1) It is well known that if equations of the type referred to in the title be dealt with like ordinary quadrics, the eliminant obtained is marred by association with an irrelevant factor. Thus, to take the simplest case, viz.the left-hand member of which contains the irrelevant factor |α2β2|, being readily shown to be equal toThe object of the present short paper is to draw attention to other modes of procedure, and to formulate the results for n variables.

Ibycus; Stesichorus; Alcman (P. Oxy. 2735, 2618, 2737)

1971 ◽  
Vol 17 ◽  
pp. 89-98
Author(s):  
Denys Page
Keyword(s):  
Left Hand ◽  
Image Position ◽  
At Home

In PCPS N.S. XV (1969), 69 ff. I defined the triadic structure of P. Oxy. 2735 fr. I and suggested that the contents of this and other fragments would seem more at home in Ibycus than in Stesichorus. It was already clear that not all the fragments of this papyrus come from the same poem, but I had not yet noticed that the number of poems represented is probably at least three, and that there are now further reasons for preferring Ibycus as author.The following two fragments belong to poems different from each other and from fr. 1:(a) P. Oxy. 2735 fr. II(i) The tradic structure (top of column)(a), (b), (c) with paragraphos mark stanza-ends according to the three schemes described immediately below.Though the left-hand margin is nowhere preserved, I take it as self-evident that these are the beginnings of lines. It is likely, in Ibycus or Stesichorus, that eighteen lines will cover at least about two-thirds, perhaps the whole, of a triad; and in fact it seems obvious that strophe, antistrophe, and epode are represented here, although it is not quite certain where the stanzas end.There are three choices:(a) 1–6 = ant., 7–11 = ep., 12–17 = str., 18 ff. = ant.(b) 1–5 = ant., 6–10 = ep., 11–16 = str., 17 ff. = ant.(c) 1–5 = str., 6–16 = ant., 17 ff. = ep.

A note on a condition satisfied by certain random walks

1977 ◽  
Vol 14 (04) ◽  
pp. 843-849 ◽  
Author(s):  
R. A. Doney
Keyword(s):  
Final Section ◽  
Domain Of Attraction ◽  
Partial Result ◽  
Stable Law ◽  
Left Hand ◽  
Common Distribution ◽  
Common Distribution Function ◽  
Image Position ◽  
The Right ◽  
Necessary And Sufficient

The problem considered is to elucidate under what circumstances the condition holds, where and Xi are independent and have common distribution function F. The main result is that if F has zero mean, and (*) holds with F belongs to the domain of attraction of a completely asymmetric stable law of parameter 1/γ. The cases are also treated. (The case cannot arise in these circumstances.) A partial result is also given for the case when and the right-hand tail is ‘asymptotically larger’ than the left-hand tail. For 0 < γ < 1, (*) is known to be a necessary and sufficient condition for the arc-sine theorem to hold for Nn , the number of positive terms in (S 1, S 2, …, Sn ). In the final section we point out that in the case γ = 1 a limit theorem of a rather peculiar type can hold for Nn.

The Gibbs Phenomenon for Taylor Means and for [F, Dn] Means

1960 ◽  
Vol 12 ◽  
pp. 660-673 ◽  
Author(s):  
Chester L. Miracle
Keyword(s):  
Gibbs Phenomenon ◽  
Left Hand ◽  
Right Hand ◽  
Image Position

The Gibbs phenomenon may be described, quite generally, as follows. Let a sequence {fn(x)} (n = 0, 1, 2, … ,) converge to a function f(x) for x in the interval x0 < x < x0+ h. We say that {fn(x)} displays the Gibbs phenomenon in a right-hand neighbourhood of the point X0, ifA similar definition holds for a left-hand neighbourhood. If {fn(x)} displays the Gibbs phenomenon at both sides of x0, we say simply that {fn(x)} displays Gibbs phenomenon at the point X0.

On a Theorem of Herstein

1969 ◽  
Vol 21 ◽  
pp. 1348-1353 ◽  
Author(s):  
M. Chacron
Keyword(s):  
Division Ring ◽  
Monic Polynomial ◽  
Constant Term ◽  
Multiplicative Semigroup ◽  
Simple Ring ◽  
Left Hand ◽  
Ring Of Integers ◽  
Image Position

Throughout this paper, Z is the ring of integers, ƒ*(t) (ƒ(t)) is an integer monic (co-monic) polynomial in the indeterminate t (i.e., each coefficient of ƒ* (ƒ) is in Z and its highest (lowest) coefficient is 1 (5, p. 121, Definition) and M* (M) is the multiplicative semigroup of all integer monic (co-monic) polynomials ƒ* (ƒ) having no constant term. In (3, Theorem 2), Herstein proved that if R is a division ring with centre C such that1then R = C. In this paper we seek a generalization of Herstein's result to semi-simple rings. We also study the following condition:(1)*Our results are quite complete for a semi-simple ring R in which there exists a bound for the codegree ofƒ (ƒ*) (i.e., the degree of the lowest monomial of ƒ(ƒ*)) appearing in the left-hand side of (1) ((1)*).

A New Fragment of Greek Music in Cairo

1931 ◽  
Vol 51 (1) ◽  
pp. 91-100 ◽  
Author(s):  
J. F. Mountford
Keyword(s):  
Extreme Right ◽  
Bottom Edge ◽  
Left Hand ◽  
Right Hand ◽  
Formed Part ◽  
Image Position ◽  
The Right

The papyrus numbered 59533 in the Catalogue Général of the Cairo Museum is a mere scrap (13 cm. × 12 cm.), on one side of which is written a fragmentary text with suprascript musical signs:The writing is along the fibres, that is to say, on the recto if the scrap originally formed part of a roll; the verso is blank; and the papyrus has been folded horizontally. The right-hand and the top seem to be the original edges of a sheet or roll; the left-hand is clearly defective, and though the bottom edge may be the original edge of a sheet, it is not the bottom of a roll. The writing is carried to the extreme right-hand edge, without a margin. Below the text there is some scribbling which seems to have no particular significance and is probably to be taken as a probatio pennae rather than as a signature.

scholarly journals CONSISTENCY PROOF OF A FRAGMENT OF PV WITH SUBSTITUTION IN BOUNDED ARITHMETIC

2018 ◽  
Vol 83 (3) ◽  
pp. 1063-1090
Author(s):  
YORIYUKI YAMAGATA
Keyword(s):  
Propositional Logic ◽  
The Other ◽  
Bounded Arithmetic ◽  
Consistency Proof ◽  
Left Hand ◽  
Right Hand ◽  
Other Hand ◽  
Image Position

AbstractThis article presents a proof that Buss’s $S_2^2$ can prove the consistency of a fragment of Cook and Urquhart’s PV from which induction has been removed but substitution has been retained. This result improves Beckmann’s result, which proves the consistency of such a system without substitution in bounded arithmetic $S_2^1$.Our proof relies on the notion of “computation” of the terms of PV. In our work, we first prove that, in the system under consideration, if an equation is proved and either its left- or right-hand side is computed, then there is a corresponding computation for its right- or left-hand side, respectively. By carefully computing the bound of the size of the computation, the proof of this theorem inside a bounded arithmetic is obtained, from which the consistency of the system is readily proven.This result apparently implies the separation of bounded arithmetic because Buss and Ignjatović stated that it is not possible to prove the consistency of a fragment of PV without induction but with substitution in Buss’s $S_2^1$. However, their proof actually shows that it is not possible to prove the consistency of the system, which is obtained by the addition of propositional logic and other axioms to a system such as ours. On the other hand, the system that we have considered is strictly equational, which is a property on which our proof relies.

A Relation between Permanents and Determinants

1899 ◽  
Vol 22 ◽  
pp. 134-136 ◽  
Author(s):  
Thomas Muir
Keyword(s):  
Symmetric Function ◽  
Left Hand ◽  
Image Position

1. If all the negative terms of the determinant ∣ a1b2c3 … ∣ be changed in sign, we obtain a symmetric function, dealt with by Borchardt and Cayley, known as a Permanent and denoted byThe more important elementary properties of such functions are given in a paper published in the Proc. Roy. Soc. Edin., xi. pp. 409–418. As might be expected, relations are found to exist between them and determinants, an important instance being the theorem of § 7 of the said paper. Another theorem, not hitherto noted, deserves now to be put on record.2. For the case of the 2nd order it isthe truth of it being self-evident.For the case of the 3rd order it iswhich is easily verified by observing that the coefficients of a1, a2, a3 in the expression on the left-hand side are respectivelyand that by the previous case each of these vanishes.

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