On the Polygons Inscribed in one Conic and Circumscribed to Another

1924 ◽  
Vol 22 (2) ◽  
pp. 163-166
Author(s):  
W. Burnside
Keyword(s):  
Difference Equation ◽  
Present Note ◽  
Elliptic Functions ◽  
Point Of View ◽  
Image Position

This problem is in general treated in connection with the division of the periods of the elliptic functions. It is the object of the present note to shew, from a purely algebraical point of view, that the condition of closing of the polygon depends solely on a difference equation of the form

scholarly journals The Esscher Premium Principle: A Criticism

1981 ◽  
Vol 12 (1) ◽  
pp. 77-78 ◽  
Author(s):  
Benjamin Zehnwirth
Keyword(s):  
Loss Function ◽  
Utility Theory ◽  
Random Quantity ◽  
Present Note ◽  
Random Variable ◽  
Point Of View ◽  
The Other ◽  
Important Distinction ◽  
Image Position ◽  
Insight Into

The Esscher premium principle has recently had some exposure, namely, with the works of Bühlmann (1980) and Gerber (1980).Bühlmann (1980) devised the principle and coined the name for it within the framework of utility theory and risk exchange. Geruber (1980), on the other hand, gives further insight into the principle by studying it within the realm of forecasting in much the same spirit as credibility theorists forecast premiums. However, there is an important distinction: the choice of loss function.The present note sets out to criticize this relatively embryonic principle using decision theoretic arguments and indicates that the Esscher premium is essentially a small perturbation of the well established linearized credibility premium Bühlmann (1970).Let H denote the Esscher premium principle with loading h > o. That is, if X is an observable random variable and Y is a parameter (a risk or a random quantity) to be forecasted then the Esscher premium is given byThat is, H(Y ∣ X) is the Bayes decision rule for estimating Y given the data X relative to the loss functionwhere a is the estimate of Y, and of course the loading h is greater than zero.Now, for the clincher. This loss function is nonsensical from the point of view of estimation. It indicates a loss (or error) to the forecaster that is essentially the antithesis of relative loss.

On chains of two-two relations and the theory of elliptic functions

1926 ◽  
Vol 23 (1) ◽  
pp. 92-102 ◽  
Author(s):  
H. F. Baker
Keyword(s):  
Present Note ◽  
Elliptic Functions ◽  
Reduced Form ◽  
A Value ◽  
Image Position

1. The relationin which we suppose each of a, b, c, a2 − c2 to be different from zero, leads, from a value θ, to two values of φ, which we may denote by θ1 and θ−1. Each of these, put in place of θ, leads, beside the value θ of φ, to another value of φ, say, respectively, θ2 and θ−2. If θ2 or θ−2 be put for θ, the same relation leads to two values of φ, say, θ1 and θ3 or θ−1 and θ−3, respectively. And so on. It may happen that θn is the same as θ, in which case also θ−n is the same as θ; this we may express by saying that the relation is closed, or that there is closure, after n links. It is the object of the present note to express in reduced form, in terms of a, b, c, the condition that this may be so. Evidently, if n = pq, the condition of closure after n links is satisfied when the condition for closure after p links (or also after q links) is satisfied. But there is a condition for closure after n links which is not satisfied for any less number of links; this is the condition which we call the reduced or proper condition for closure after n links, and it is this which we seek to express.

The elliptic products of Jacobi and the theory of linear congruences

1926 ◽  
Vol 23 (4) ◽  
pp. 337-355
Author(s):  
P. A. MacMahon
Keyword(s):  
Elliptic Functions ◽  
Linear Congruences ◽  
Image Position ◽  
Theory Of Numbers

In the application of Elliptic Functions to the Theory of Numbers the two formulae of Jacobiare of great importance.

scholarly journals On certain infinite integrals involving Struve functions and parabolic cylinder functions

1946 ◽  
Vol 7 (4) ◽  
pp. 171-173 ◽  
Author(s):  
S. C. Mitra
Keyword(s):  
Present Note ◽  
Parabolic Cylinder ◽  
Parabolic Cylinder Functions ◽  
Struve Functions ◽  
Image Position

The object of the present note is to obtain a number of infinite integrals involving Struve functions and parabolic cylinder functions. 1. G. N. Watson(1) has proved thatFrom (1)follows provided that the integral is convergent and term-by-term integration is permissible. A great many interesting particular cases of (2) are easily deducible: the following will be used in this paper.

Anomalous Behavior of Heterogeneous Materials at Microwaves Frequencies: Introduction to Fractional Derivatives in Electromagnetism

1990 ◽  
Vol 189 ◽  
Author(s):  
F. Heliodore ◽  
D. Cottevieille ◽  
A. Le Mehaute
Keyword(s):  
Electromagnetic Wave ◽  
Physical Meaning ◽  
Fractional Derivatives ◽  
Heterogeneous Media ◽  
Present Note ◽  
Heterogeneous Materials ◽  
Anomalous Behavior ◽  
Point Of View ◽  
Unique Point ◽  
Validity Range

ABSTRACTThe present note introduces new trends in electromagnetic spectroscopy in complex media.When an electromagnetic wave propagates in heterogeneous media, some questions arise about both physical meaning and validity range of the traditional analysis. The aim of our advanced research is related to the generalisation of Maxwell's equations able todescribe both homogeneous and heterogeneous media from an unique point of view.

scholarly journals Some Remarks on a Recent Paper by Borch

1961 ◽  
Vol 1 (5) ◽  
pp. 265-272 ◽  
Author(s):  
Paul Markham Kahn
Keyword(s):  
Distribution Function ◽  
Random Variable ◽  
Fixed Number ◽  
Point Of View ◽  
Optimum Amount ◽  
Measurable Transformation ◽  
The Difference ◽  
Image Position ◽  
Stop Loss ◽  
Condition B

In his recent paper, “An Attempt to Determine the Optimum Amount of Stop Loss Reinsurance”, presented to the XVIth International Congress of Actuaries, Dr. Karl Borch considers the problem of minimizing the variance of the total claims borne by the ceding insurer. Adopting this variance as a measure of risk, he considers as the most efficient reinsurance scheme that one which serves to minimize this variance. If x represents the amount of total claims with distribution function F (x), he considers a reinsurance scheme as a transformation of F (x). Attacking his problem from a different point of view, we restate and prove it for a set of transformations apparently wider than that which he allows.The process of reinsurance substitutes for the amount of total claims x a transformed value Tx as the liability of the ceding insurer, and hence a reinsurance scheme may be described by the associated transformation T of the random variable x representing the amount of total claims, rather than by a transformation of its distribution as discussed by Borch. Let us define an admissible transformation as a Lebesgue-measurable transformation T such thatwhere c is a fixed number between o and m = E (x). Condition (a) implies that the insurer will never bear an amount greater than the actual total claims. In condition (b), c represents the reinsurance premium, assumed fixed, and is equal to the expected value of the difference between the total amount of claims x and the total retained amount of claims Tx borne by the insurer.

The Representation of (C, k) Summable Series in Fourier Form

1978 ◽  
Vol 21 (2) ◽  
pp. 149-158 ◽  
Author(s):  
G. E. Cross
Keyword(s):  
Trigonometric Series ◽  
Point Of View ◽  
Perron Integral ◽  
Image Position

Several non-absolutely convergent integrals have been defined which generalize the Perron integral. The most significant of these integrals from the point of view of application to trigonometric series are the Pn- and pn-integrals of R. D. James [10] and [11]. The theorems relating the Pn -integral to trigonometric series state essentially that if the series1.1

Sophocles, ‘Electra’ 1205–10

1973 ◽  
Vol 19 ◽  
pp. 45-46
Author(s):  
R. D. Dawe
Keyword(s):  
Present Note ◽  
Greek Tragedy ◽  
Image Position

The attribution of lines to different speakers in Greek tragedy is a matter on which MSS have notoriously little authority. As for Electra itself, there are at least three places where the name of the heroine has been incorrectly added in some or all MSS. In my Studies in the Text of Sophocles, I, 198, I list these places and suggest that the same error has happened at a fourth place, viz. 1323. The purpose of the present note is to suggest that at El. 1205–10 the same mistake has happened yet again.The situation is that Electra is holding the urn which she falsely believes to contain the ashes of her dead brother, Orestes. But Orestes is alive, and before her at this very moment. He is trying to persuade her to give up the urn. If the text before us had been preserved in a MS devoid of ascriptions to speakers, no one would have been so perverse as to do what all MSS and editors do in fact do, namely attribute the words οὔ φημ᾿ ἐάσειν to Orestes.

A Singular Perturbation Problem and A Neutral Differential-Difference Equation

1974 ◽  
Vol 17 (1) ◽  
pp. 77-83
Author(s):  
Edward Moore
Keyword(s):  
Difference Equation ◽  
Taylor Series Expansion ◽  
Linear Difference Equation ◽  
Constant Matrix ◽  
Perturbation Problem ◽  
Constant Coefficients ◽  
Explicit Formulae ◽  
Differential Difference Equation ◽  
The Difference ◽  
Image Position

Vasil’eva, [2], demonstrates a close connection between the explicit formulae for solutions to the linear difference equation with constant coefficients(1.1)where z is an n-vector, A an n×n constant matrix, τ>0, and a corresponding differential equation with constant coefficients(1.2)(1.2) is obtained from (1.1) by replacing the difference z(t—τ) by the first two terms of its Taylor Series expansion, combined with a suitable rearrangement of the terms.

scholarly journals On general curves lying on a quadric

1927 ◽  
Vol 1 (1) ◽  
pp. 19-30 ◽  
Author(s):  
H. F. Baker
Keyword(s):  
Present Note ◽  
Rational Curves ◽  
Stationary Points ◽  
Image Position

Introduction. The present note, though in continuation of the preceding one dealing with rational curves, is written so as to be independent of this. It is concerned to prove that if a curve of order n, and genus p, with k cusps, or stationary points, lying on a quadric, Ω, in space of any number of dimensions, is such that itself, its tangents, its osculating planes, … , and finally its osculating (h – 1)-folds, all lie on the quadric Ω, then the number of its osculating h-folds which lie on the quadric isTwo proofs of this result are given, in §§ 4 and 5.

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