The Problem of Relativity in reference to several bodies

1926 ◽  
Vol 23 (3) ◽  
pp. 191-197
Author(s):  
R. Hargreaves
Keyword(s):  
Relative Motion ◽  
Point Of View ◽  
Image Position ◽  
Primary Form

§ 1. If the kinetic potential for the relative motion of two masses is written with an added constant asa close connexion with the relativity quadratic appears. The latter is in factwhere a modification of the primary formwhich shows an unaltered determinant. The condition in respect to the determinant, suggested, I believe, by Schwarzschild, is one which to me appears to give the most significant form to the results. From the dynamical standpoint we may regard it as imposing a counterpoise in the inertia coefficients to the modification introduced by the potential; or from a geometrical point of view we may regard it as minimizing the departure from the normal use of coordinates. An illuminating example of the loss of meaning that accompanies transformation in which this condition is disregarded is furnished by the isotropic form which is sometimes given to Einstein's quadratic.

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

scholarly journals The Magnetic Fields of Pulsars

1974 ◽  
Vol 64 ◽  
pp. 187-187
Author(s):  
D. M. Sedrakian
Keyword(s):  
Magnetic Field ◽  
Angular Velocity ◽  
Magnetic Fields ◽  
Relative Motion ◽  
Kinematic Viscosity ◽  
Charged Particles ◽  
Normal Fluid ◽  
Inertia Effects ◽  
Generation Mechanisms ◽  
Image Position

Two generation mechanisms of magnetic fields in pulsars are considered.If the temperature of a star is more than 108K, the star consists of a normal fluid of neutrons, protons and electrons. Because the angular velocity of pulsars is not constant dω/dt ≠0, inertia effects can occur, and generate magnetic fields through the relative motion of charged particles with different masses. The kinematic viscosity of electrons is 30 times larger than that of protons; hence electrons move with the crust, but the proton-neutron fluid will move relative to the electrons. The magnetic momentum can be calculated by the following formula where Meff = Mp + Mn(Nn/Np), R = radius of the star, σ = conductivity. For typical neutron stars we have dω/dt~ 10-8 s-2, R~106 cm, σ~1029 s-1 and we get a magnetic field of the order of 1010 G.

Formulas for Entraining Velocity in Lubricated Line Contacts

2002 ◽  
Vol 124 (4) ◽  
pp. 856-858
Author(s):  
Enrico Ciulli
Keyword(s):  
Relative Motion ◽  
Physical Meaning ◽  
Point Of View ◽  
Lubricated Contacts ◽  
Line Contacts ◽  
Two Bodies

The knowledge of the entraining velocity is necessary for the investigation of lubricated contacts. The entraining velocity is the average of the surface velocities of the two bodies in contact relative to the contact itself; its estimation can be actually not always immediate. In this work the general case of two pairing cylindrical surfaces in planar relative motion is analyzed from a kinematical point of view. Formulas for the evaluation of the entraining velocity are presented that are directly applicable to any case of connected members of a mechanism. The physical meaning of the terms of the proposed formulas is also briefly investigated from a lubrication point of view.

scholarly journals Minimum Variance Reinsurance

1962 ◽  
Vol 2 (2) ◽  
pp. 257-260 ◽  
Author(s):  
Stefan Vajda
Keyword(s):  
Minimum Variance ◽  
Point Of View ◽  
Stieltjes Integral ◽  
Optimum Amount ◽  
Natural Conditions ◽  
Excess Loss ◽  
Total Claim ◽  
Image Position ◽  
Stop Loss ◽  
Elegant Proof

In a paper entitled “An Attempt to Determine the Optimum Amount of Stop Loss Reinsurance” (XVIth Int. Congr. Act. Bruxelles 1960) Karl Borch has shown that, if the reinsurance premium is given, the smallest variance of the cedent's payments is obtained by a stop-loss reinsurance contract. Paul Markham Kahn, in “Some Remarks on a Recent Paper by Borch”, a paper read to the 1961 Astin Colloquium, has given an elegant proof of this theorem which appears to apply also to cases not considered by Borch. In this paper we study the problem from the reinsurer's point of view and it will be seen that, under natural conditions which are also used in the proof of the Borch-Kahn theorem, the minimum variance of the reinsurer's payments is obtained by a quota contract. This focusses attention on a peculiar opposition of interests of the two partners of a reinsurance contract. However, we do not enter any further into the investigation of a possible resolution of this conflict.We study a problem concerning the division of risk between a cedent and his reinsurer. The risk may refer to a whole portfolio (in which case one might consider a Stop-Loss contract), or to a single contract (when an Excess-Loss contract is a possibility). We shall here use the nomenclature of a portfolio reinsurance.Let it be assumed that a function F(x) is known which gives the probability of a total claim not exceeding x. We have then in Stieltjes integral notationThe two partners to a reinsurance arrangement agree that the reinsurer reimburses m(x).x out of a claim of x, where m(x) is a continuous and differentiate function of x and o ≤ m(x) ≤ 1.

Questions about quantifiers

1984 ◽  
Vol 49 (2) ◽  
pp. 443-466 ◽  
Author(s):  
Johan van Benthem
Keyword(s):  
Natural Language ◽  
Negative Polarity ◽  
Point Of View ◽  
Methodological Issues ◽  
Mathematical Structures ◽  
Fixed Model ◽  
Semantic Treatment ◽  
The Subject ◽  
Image Position ◽  
Quantifier Phrases

The importance of the logical ‘generalized quantifiers’ (Mostowski [1957]) for the semantics of natural language was brought out clearly in Barwise & Cooper [1981]. Basically, the idea is that a quantifier phrase QA (such as “all women”, “most children”, “no men”) refers to a set of sets of individuals, viz. those B for which (QA)B holds. Thus, e.g., given a fixed model with universe E,where ⟦A⟧ is the set of individuals forming the extension of the predicate “A” in the model. This point of view permits an elegant and uniform semantic treatment of the subject-predicate form that pervades natural language.Such denotations of quantifier phrases exhibit familiar mathematical structures. Thus, for instance, all A produces filters, and no A produces ideals. The denotation of most A is neither; but it is still monotone, in the sense of being closed under supersets. Mere closure under subsets occurs too; witness a quantifier phrase like few A. These mathematical structures are at present being used in organizing linguistic observations and formulating hypotheses about them. In addition to the already mentioned paper of Barwise & Cooper, an interesting example is Zwarts [1981], containing applications to the phenomena of “negative polarity” and “conjunction reduction”. In the course of the latter investigation, several methodological issues of a wider logical interest arose, and these have inspired the present paper.In order to present these issues, let us shift the above perspective, placing the emphasis on quantifier expressions per se (“all”, “most”, “no”, “some”, etcetera), viewed as denoting relations Q between sets of individuals.

scholarly journals On the spectral theory of a tapered rod

2002 ◽  
Vol 132 (3) ◽  
pp. 729-764 ◽  
Author(s):  
C. A. Stuart
Keyword(s):  
Function Space ◽  
Spectral Theory ◽  
Spectral Properties ◽  
Boundary Value ◽  
Point Of View ◽  
Point Boundary ◽  
Cross Sectional ◽  
Mechanical Interpretation ◽  
The One ◽  
Image Position

This paper establishes the main features of the spectral theory for the singular two-point boundary-value problem and which models the buckling of a rod whose cross-sectional area decays to zero at one end. The degree of tapering is related to the rate at which the coefficient A tends to zero as s approaches 0. We say that there is tapering of order p ≥ 0 when A ∈ C([0, 1]) with A(s) > 0 for s ∈ (0, 1] and there is a constant L ∈ (0, ∞) such that lims→0A(s)/sp = L. A rigorous spectral theory involves relating (1)−(3) to the spectrum of a linear operator in a function space and then investigating the spectrum of that operator. We do this in two different (but, as we show, equivalent) settings, each of which is natural from a certain point of view. The main conclusion is that the spectral properties of the problem for tapering of order p = 2 are very different from what occurs for p < 2. For p = 2, there is a non-trivial essential spectrum and possibly no eigenvalues, whereas for p < 2, the whole spectrum consists of a sequence of simple eigenvalues. Establishing the details of this spectral theory is an important step in the study of the corresponding nonlinear model. The first function space that we choose is the one best suited to the mechanical interpretation of the problem and the one that is used for treating the nonlinear problem. However, we relate this formulation in a precise way to the usual L2 setting that is most common when dealing with boundary-value problems.

scholarly journals REPRESENTING CONJUNCTIVE DEDUCTIONS BY DISJUNCTIVE DEDUCTIONS

2016 ◽  
Vol 10 (1) ◽  
pp. 145-157
Author(s):  
KOSTA DOŠEN ◽  
ZORAN PETRIĆ
Keyword(s):  
Point Of View ◽  
The Other ◽  
Theoretical Point ◽  
Single Object ◽  
Finite Products ◽  
Image Position ◽  
Propositional Letter

AbstractA skeleton of the category with finite coproducts${\cal D}$ freely generated by a single object has a subcategory isomorphic to a skeleton of the category with finite products ${\cal C}$ freely generated by a countable set of objects. As a consequence, we obtain that ${\cal D}$ has a subcategory equivalent with ${\cal C}$. From a proof-theoretical point of view, this means that up to some identifications of formulae the deductions of pure conjunctive logic with a countable set of propositional letters can be represented by deductions in pure disjunctive logic with just one propositional letter. By taking opposite categories, one can replace coproduct by product, i.e., disjunction by conjunction, and the other way round, to obtain the dual results.

HANDMADE DENSITY SETS

2017 ◽  
Vol 82 (1) ◽  
pp. 208-223 ◽  
Author(s):  
GEMMA CAROTENUTO
Keyword(s):  
Set Theory ◽  
Real Line ◽  
Borel Measure ◽  
Point Of View ◽  
Topological Complexity ◽  
Descriptive Set Theory ◽  
Locally Finite ◽  
Finite Borel Measure ◽  
Image Position ◽  
Descriptive Set

AbstractGiven a metric space (X , d), equipped with a locally finite Borel measure, a measurable set $A \subseteq X$ is a density set if the points where A has density 1 are exactly the points of A. We study the topological complexity of the density sets of the real line with Lebesgue measure, with the tools—and from the point of view—of descriptive set theory. In this context a density set is always in $\Pi _3^0$. We single out a family of true $\Pi _3^0$ density sets, an example of true $\Sigma _2^0$ density set and finally one of true $\Pi _2^0$ density set.

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