Disjointly Additive Operators and Modular Spaces

1990 ◽  
Vol 42 (3) ◽  
pp. 508-519
Author(s):  
Iwo Labuda
Keyword(s):  
Additive Functional ◽  
Typical Result ◽  
Additive Functionals ◽  
Modular Spaces ◽  
The Subject ◽  
Image Position

By now the literature concerning the representation of disjointly additive functionals and operators is quite extensive. A few entries on the subject are [6, 7, 8, 11, 20, 21]. In [7, 8, 17] further references can be found, in [7] the “prehistory” of the subject is also discussed.To quote a typical result, we may take a 1967 theorem of Drewnowski and Orlicz ([6] Th. 3.2, [17] 12.4) which asserts that, under proper assumptions, an abstract modular (= disjointly countably additive functional) p on a “substantial“ subspace D of L° can be realized by the formula .

scholarly journals Reserves of Motor-Vehicle Insurance in Finland

1962 ◽  
Vol 2 (1) ◽  
pp. 161-173 ◽  
Author(s):  
Teivo Pentikäinen
Keyword(s):  
Motor Vehicle ◽  
Insurance Companies ◽  
Accurate Calculation ◽  
Short Term ◽  
The Subject ◽  
Image Position ◽  
Premium Income ◽  
The Cost ◽  
Motor Vehicle Insurance ◽  
Cluster Points

The Ministry of Social Affairs, which acts i.a. as the supervising office in Finland, has given instructions regarding the normal reserves of insurance companies. A summary of these and some comments are given here as far as they concern motor-vehicle insurance. The instructions as far as they concern the subject referred to in the following in the items 2-6, 9 and 10, were compiled by a committee, presided over by Mr. I. Ketola, M. Sc, which availed itself of the experience of several Finnish insurance companies.In order to give a review of the system as a whole many items, which are mathematically trivial and well-known, are briefly explained.The conventional principle of “pro rata parte temporis” is followed, which leads to the well-known reserve where P is the premium income of the company. This provides that the days when the premiums fall due are approximately equally distributed over the year (which can be checked from the premium sums of the different months in the book-keeping) or at least have no cluster points in the second half of the year and that the cost of the collecting of premiums is not less than 0.2 P. A more accurate calculation takes into account i.a. temporary short term policies etc.In casu-reserve. All unpaid claims (except those mentioned later) due to accidents which occured before the end of the account year, are listed and rated one by one. Doubtful cases, e.g. where the cause of the accident is still under litigation, are calculated in accordance with the “worst” alternative.

scholarly journals A Generalization of Cauchy' s Double Alternant

1964 ◽  
Vol 7 (2) ◽  
pp. 273-278 ◽  
Author(s):  
David Carlson ◽  
Chandler Davis
Keyword(s):  
The Subject ◽  
Image Position

The subject of alternants and alternating functions was widely studied during the last century (cf. Muir [6]). One of the best-known alternants is actually a double alternant (rows and columns) defined by Cauchy [2] in 1841. Cauchy's result may be stated as follows:If D = [dpq] p, q = l, …, n, where dpq = (xp+yq)-1, then1.

On Additive Operators

1971 ◽  
Vol 23 (3) ◽  
pp. 468-480 ◽  
Author(s):  
N. A. Friedman ◽  
A. E. Tong
Keyword(s):  
Banach Space ◽  
Compact Hausdorff Space ◽  
Weak Topology ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Additive Functional ◽  
Representation Theorems ◽  
Image Position ◽  
Vector Valued ◽  
Kernel Representation

Representation theorems for additive functional have been obtained in [2, 4; 6-8; 10-13]. Our aim in this paper is to study the representation of additive operators.Let S be a compact Hausdorff space and let C(S) be the space of real-valued continuous functions defined on S. Let X be an arbitrary Banach space and let T be an additive operator (see § 2) mapping C(S) into X. We will show (see Lemma 3.4) that additive operators may be represented in terms of a family of “measures” {μh} which take their values in X**. If X is weakly sequentially complete, then {μh} can be shown to take their values in X and are vector-valued measures (i.e., countably additive in the norm) (see Lemma 3.7). And, if X* is separable in the weak-* topology, T may be represented in terms of a kernel representation satisfying the Carathéordory conditions (see [9; 11; §4]):

scholarly journals On the Inversion of the Gauss Transformation

1957 ◽  
Vol 9 ◽  
pp. 459-464 ◽  
Author(s):  
P. G. Rooney
Keyword(s):  
Differential Operator ◽  
Recent Work ◽  
Suitable Definition ◽  
Inversion Theory ◽  
The Subject ◽  
Definition Of ◽  
Image Position ◽  
Gauss Transformation ◽  
Operational Methods

The inversion theory of the Gauss transformation has been the subject of recent work by several authors. If the transformation is defined by1.1,then operational methods indicate that,under a suitable definition of the differential operator.

scholarly journals An Elementary Proof of the Theorems of Cauchy and Mayer

1935 ◽  
Vol 4 (3) ◽  
pp. 112-117
Author(s):  
A. J. Macintyre ◽  
R. Wilson
Keyword(s):  
Differential Equation ◽  
Existence Theorem ◽  
Elementary Proof ◽  
Theoretical Discussion ◽  
Practical Advantage ◽  
Total Differential ◽  
Method Of Solution ◽  
The Subject ◽  
Image Position ◽  
Natural Way

Attention has recently been drawn to the obscurity of the usual presentations of Mayer's method of solution of the total differential equationThis method has the practical advantage that only a single integration is required, but its theoretical discussion is usually based on the validity of some other method of solution. Mayer's method gives a result even when the equation (1) is not integrable, but this cannot of course be a solution. An examination of the conditions under which the result is actually an integral of equation (1) leads to a proof of the existence theorem for (1) which is related to Mayer's method of solution in a natural way, and which moreover appears to be novel and of value in the presentation of the subject.

scholarly journals ADDITIVE FUNCTIONAL INEQUALITIES AND DERIVATIONS ON HILBERT C*-MODULES

2013 ◽  
Vol 55 (2) ◽  
pp. 341-348 ◽  
Author(s):  
FRIDOUN MORADLOU
Keyword(s):  
Banach Spaces ◽  
Functional Inequality ◽  
Additive Functional ◽  
Functional Inequalities ◽  
Ulam Stability ◽  
Formula Group ◽  
Image Position

AbstractIn this paper we investigate the following functional inequality $ \begin{eqnarray*} \| f(x-y-z) - f(x-y+z) + f(y) +f(z)\| \leq \|f(x+y-z) - f(x)\| \end{eqnarray*}$ in Banach spaces, and employing the above inequality we prove the generalized Hyers–Ulam stability of derivations in Hilbert C*-modules.

scholarly journals Limit theorems for additive functionals of continuous time random walks

2020 ◽  
pp. 1-22
Author(s):  
Yuri Kondratiev ◽  
Yuliya Mishura ◽  
Georgiy Shevchenko
Keyword(s):  
Continuous Time ◽  
Limit Theorems ◽  
Domain Of Attraction ◽  
Weak Limit ◽  
Additive Functional ◽  
Stable Law ◽  
Additive Functionals ◽  
Lévy Motion ◽  
Continuous Time Random Walks ◽  
Poisson Shot Noise

Abstract For a continuous-time random walk X = {X t , t ⩾ 0} (in general non-Markov), we study the asymptotic behaviour, as t → ∞, of the normalized additive functional $c_t\int _0^{t} f(X_s)\,{\rm d}s$ , t⩾ 0. Similarly to the Markov situation, assuming that the distribution of jumps of X belongs to the domain of attraction to α-stable law with α > 1, we establish the convergence to the local time at zero of an α-stable Lévy motion. We further study a situation where X is delayed by a random environment given by the Poisson shot-noise potential: $\Lambda (x,\gamma )= {\rm e}^{-\sum _{y\in \gamma } \phi (x-y)},$ where $\phi \colon \mathbb R\to [0,\infty )$ is a bounded function decaying sufficiently fast, and γ is a homogeneous Poisson point process, independent of X. We find that in this case the weak limit has both ‘quenched’ component depending on Λ, and a component, where Λ is ‘averaged’.

Comparative Bonuses under Whole Life and Endowment Assurances

1907 ◽  
Vol 41 (3) ◽  
pp. 273-304
Author(s):  
Hermann Julius Rietschel
Keyword(s):  
Special Regard ◽  
Interesting Discussion ◽  
The Subject ◽  
Image Position ◽  
To Receive

It is now some years since a paper was presented to the Institute dealing with the comparative bonus-earning powers of whole life and endowment assurances. In view of the importance of the subject I have ventured to submit this paper, which I hope may lead to a useful and interesting discussion. The matter has been so ably dealt with by the late Mr. Sunderland, Mr. Lidstone and Mr. Chatham that I have the greatest hesitation in presenting my own views, and I must therefore beg the Members of the Institute to receive this paper in the same kindly spirit which they have always manifested towards new contributors.My object in the paper has been to show the relative bonuses which should be allotted to whole life and endowment assurance policies having special regard to(1) The expenses of the office;(2) Variations in the rate of interest earned; and(3) Mortality.In Table A are given the office premiums which have been employed in this investigation. They are based on the H[M] 4 per-cent Table, with a loading for a £1 per-cent compound reversionary bonus, together with 10 per-cent and a constant of 3s. per-cent for expenses and commission—

The Measure Spectrum of a Uniform Algebra and Subharmonicity

1982 ◽  
Vol 34 (3) ◽  
pp. 673-685
Author(s):  
Donna Kumagai
Keyword(s):  
Probability Measure ◽  
Maximal Ideal ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Uniform Algebra ◽  
Ideal Space ◽  
Representing Measure ◽  
Uniform Algebras ◽  
The Subject ◽  
Image Position

Let A be a uniform algebra on a compact Hausdorff space X. The spectrum, or the maximal ideal space, MA, of A is given byWe define the measure spectrum, SA, of A bySA is the set of all representing measures on X for all Φ ∈ MA. (A representing measure for Φ ∈ MA is a probability measure μ on X satisfyingThe concept of representing measure continues to be an effective tool in the study of uniform algebras. See for example [12, Chapters 2 and 3], [5, pp. 15-22] and [3]. Most of the known results on the subject of representing measures, however, concern measures associated with a single homomorphism.

The Meromorphisms of an Elliptic Function-Field

1936 ◽  
Vol 32 (2) ◽  
pp. 212-215 ◽  
Author(s):  
H. Davenport
Keyword(s):  
Finite Field ◽  
Elliptic Function ◽  
Function Field ◽  
Unit Element ◽  
Number Of Solutions ◽  
Complicated Case ◽  
Zero Divisors ◽  
The Subject ◽  
Image Position ◽  
The Given

1. Hasse's second proof of the truth of the analogue of Riemann's hypothesis for the congruence zeta-function of an elliptic function-field over a finite field is based on the consideration of the normalized meromorphisms of such a field. The meromorphisms form a ring of characteristic 0 with a unit element and no zero divisors, and have as a subring the natural multiplications n (n = 0, ± 1, …). Two questions concerning the nature of meromorphisms were left open, first whether they are commutative, and secondly whether every meromorphism μ satisfies an algebraic equation with rational integers n0, … not all zero. I have proved that except in the case (which is equivalent to |N−q|=2 √q, where N is the number of solutions of the Weierstrassian equation in the given finite field of q elements), both these results are true. This proof, of which I give an account in this paper, suggested to Hasse a simpler treatment of the subject, which throws still more light on the nature of meromorphisms. Consequently I only give my proof in full in the case in which the given finite field is the mod p field, and indicate briefly in § 4 how it generalizes to the more complicated case.

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