scholarly journals Note on the Inequality Theorem that mx m–1( x – 1)> xm – 1> m( x – 1) unless when 0< m<1, when mx m–1( x – 1)< xm – 1< m( x – 1), where x is any positive quantity other than unity

1901 ◽  
Vol 20 ◽  
pp. 29
Author(s):  
H. S. Carslaw
Keyword(s):  
Positive Quantity ◽  
Inequality Theorem

The generalized Laguerre inequalities and functions in the Laguerre-Pólya class

2013 ◽  
Vol 11 (9) ◽  
Author(s):  
George Csordas ◽  
Anna Vishnyakova
Keyword(s):  
Entire Function ◽  
Open Problem ◽  
A Priori ◽  
Sufficient Conditions ◽  
Class Definition ◽  
Lindelöf Theorem ◽  
Order And Type ◽  
Geometric Result ◽  
Real Entire Function ◽  
Inequality Theorem

AbstractThe principal goal of this paper is to show that the various sufficient conditions for a real entire function, φ(x), to belong to the Laguerre-Pólya class (Definition 1.1), expressed in terms of Laguerre-type inequalities, do not require the a priori assumptions about the order and type of φ(x). The proof of the main theorem (Theorem 2.3) involving the generalized real Laguerre inequalities, is based on a beautiful geometric result, the Borel-Carathédodory Inequality (Theorem 2.1), and on a deep theorem of Lindelöf (Theorem 2.2). In case of the complex Laguerre inequalities (Theorem 3.2), the proof is sketched for it requires a slightly more delicate analysis. Section 3 concludes with some other cognate results, an open problem and a conjecture which is based on Cardon’s recent, ingenious extension of the Laguerre-type inequalities.

scholarly journals XIX. —Consideration of the objections raised against the geometrical representation of the square roots of negative quantities

1829 ◽  
Vol 119 ◽  
pp. 241-254 ◽  
Keyword(s):  
The Other ◽  
Square Root ◽  
Square Roots ◽  
The Third ◽  
Oblique Direction ◽  
Real Nature ◽  
Positive Quantity ◽  
Algebraic Operations ◽  
The One ◽  
Straight Lines

Some years ago my attention was drawn to those algebraic quantities, which are commonly called impossible roots or imaginary quantities: it appeared extraordinary, that mathematicians should be able by means of these quan­tities to pursue their investigations, both in pure and mixed mathematics, and to arrive at results which agree with the results obtained by other independent processes; and yet that the real nature of these quantities should be entirely unknown, and even their real existence denied. One thing was evident re­specting them; that they were quantities capable of undergoing algebraic operations analogous to the operations performed on what are called possible quantities, and of producing correct results: thus it was manifest, that the operations of algebra were more comprehensive than the definitions and funda­mental principles; that is, that they extended to a class of quantities, viz. those commonly called impossible roots, to which the definitions and funda­mental principles were inapplicable. It seemed probable, therefore, that there was a deficiency in the definitions and fundamental principles of algebra ; and that other definitions and fundamental principles might be discovered of a more comprehensive nature, which would extend to every class of quantities to which the operations of algebra were applicable; that is, both to possible and impossible quantities, as they are called. I was induced therefore to examine into the nature of algebraic operations, with a view, if possible, of arriving at these general definitions and fundamental principles: and I found, that, by considering algebra merely as applied to geometry, such principles and definitions might be obtained. The fundamental principles and definitions which I arrived at were these: that all straight lines drawn in a given plane from a given point, in any direction whatever, are capable of being algebra­ically represented, both in length and direction; that the addition of such lines (when estimated both in length and direction) must be performed in the same manner as composition of motion in dynamics; and that four such lines are proportionals, -both in length and direction, when they are proportionals in length, and the fourth is inclined to the third at the same angle that the second is to the first. From these principles I deduced, that, if a line drawn in any given direction be assumed as a positive quantity, and consequently its oppo­site, a negative quantity, a line drawn at right angles to the positive or nega­tive direction will be the square root of a negative quantity, and a line drawn in an oblique direction will be the sum of two quantities, the one either posi­tive or negative, and the other, the square root of a negative quantity.

Consideration of the objections raised against the geometrical repre­sentation of the square roots of negative quantities

1833 ◽  
Vol 2 ◽  
pp. 371-372
Keyword(s):  
A Priori ◽  
The Other ◽  
Square Root ◽  
Square Roots ◽  
Negative Direction ◽  
The Third ◽  
Oblique Direction ◽  
Positive Quantity ◽  
Algebraic Operations ◽  
The One

It has always appeared a paradox in mathematics, that by em­ploying what are called imaginary or impossible quantities, and sub­jecting them to the same algebraic operations as those which are performed on quantities that are real and possible, the results ob­tained should always prove perfectly correct. The author inferring from this fact, that the operations of algebra are of a more compre­hensive nature than its definitions and fundamental principles, was led to inquire what extension might be given to these definitions and principles, so as to render them strictly applicable to quantities of every description, whether real or imaginary. This deficiency, he conceives, may be supplied by having recourse to certain geometrical considerations. By taking into account the directions as well as the lengths of lines drawn in a given plane, from a given point, the ad­dition of such lines may admit of being performed in the same man­ner as the composition of motions in dynamics; and four such lines may be regarded as proportional, both in length and direction, when they are proportionals in length, and, when also the fourth is inclined to the third at the same angle that the second is to the first. From this principle he deduces, that if a line drawn in any given di­rection be assumed as a positive quantity, and consequently its op­posite a negative quantity, a line drawn at right angles to the posi­tive or negative direction will be represented by the square root of a negative quantity ; and a line drawn in an oblique direction will be represented by the sum of two quantities, the one either positive or negative, and the other the square root of a negative quantity. On this subject, the author published a treatise in April 1828; since which period several objections have been made to this hypothesis. The purpose of the present paper is to answer these objections. The first of these is, that impossible roots should be considered merely as the indications of some impossible condition, which the pro­position that has given rise to them involves; and that they have in fact no real or absolute existence. To this it is replied by the author, that although such a statement may be true in some cases, it is by no means necessarily so in all; and that these quantities re­semble in this respect fractional and negative roots, which, whenever they are excluded by the nature of the question, are indeed signs of impossibility, but yet in other cases are admitted to be real and significant quantities. We have therefore no stronger reasons, à priori , for denying the real existence of what are called impossible roots, because they are in some cases the signs of impossibility, than we should have for refusing that character to fractional or negative roots on similar grounds.

scholarly journals TACHYONS IN ROBERTSON–WALKER COSMOLOGY

1998 ◽  
Vol 07 (02) ◽  
pp. 279-298 ◽  
Author(s):  
ROMAN TOMASCHITZ
Keyword(s):  
Life Time ◽  
Momentum Conservation ◽  
Communication Process ◽  
Expansion Factor ◽  
Negatively Curved ◽  
Positive Quantity ◽  
Space Expansion ◽  
Three Space ◽  
Galactic Background ◽  
Faster Than Light

Superluminal signal transfer is studied in the context of a preferred cosmic frame of reference provided by the galactic background. The receding galaxies constitute a frame of absolute rest, in which the energy of tachyons (faster-than-light particles) is unambi-guously defined as a positive quantity. The causality violation which arises in relativistic tachyonic theories is avoided. We define interactions of particles and tachyons in terms of elastic head-on collisions and energy-momentum conservation. To compare the theory developed with existing relativistic theories, tachyons are studied at first in a Minkowski universe, and the causality of a superluminal communication process is analyzed. Then we discuss the dynamics of tachyons in a Robertson–Walker universe with linear expansion factor and negatively curved three-space. We point out the consequences that the space expansion has on tachyons, like a finite life-time in the frame of absolute rest, and multiple images in the rest frames of moving observers.

scholarly journals Change in Semigraph Energy Due to Edge Deletion and its Relation with Distance Energy

2019 ◽  
Vol 8 (2S10) ◽  
pp. 873-875
Keyword(s):  
Adjacency Matrix ◽  
Singular Values ◽  
Edge Deletion ◽  
Graph Energy ◽  
Inequality Theorem

If there is an adjacency matrix A, the sum total of the singular values of A is known as the graph energy. We can find the change in energy of a graph by removing the edges using the inequality theorem on singular values. In this paper we discuss about the change in semigraph energy due to deletion of edges and its relation with distance energy

scholarly journals Strength Characteristics of Recycled Aggregate Concrete by Ann

2020 ◽  
Vol 9 (3) ◽  
pp. 1210-1214
Keyword(s):  
Compressive Strength ◽  
Power Plants ◽  
Building Materials ◽  
Neural System ◽  
Recycled Aggregate Concrete ◽  
Recycled Aggregate ◽  
Percentage Error ◽  
Ann Model ◽  
Positive Quantity ◽  
Ultimate Objective

Development Practices is the Key to the Next Generation for having a progressively imperative and better work concerning Engineering Perspectives. Various sorts of research have been done previously and are being done in the present on building materials extensively used for Constructions. With the ultimate objective to shield the future and proportion the trademark resources, various examinations have been coordinated over some vague period on reactions and wastes leaving undertakings, fantastically warm power plants, to facilitate the use of wastes thusly reusing them and screen the normal resources which are comprehensively using being developed practices[1-5]. A positive quantity of mortar and cement paste from the authentic concrete stays connected to stone particles in the recycled combination when demolished concrete is crushed [11,15]. The adhered mortar presence at the surface of an overwhelmed concrete mixture usually degrades the great of the recycled mixture and therefore the fresh and hardened residences of concrete crafted from it compared to herbal aggregates. As per the investigation, the compressive strength of cement was anticipated utilizing artificial neural system models Firstly, to prepare the ANN model to anticipate the compressive strength of RAC, The predicted compressive strength was contrasted and the exploratory compressive strength and correlation are carried out[12-14]. Training and testing of the ANN model are done utilizing compressive strength results of RAC collected from literature, the practical values obtained are used to validate the ANN model. Then the percentage error between the experiment and predicted compressive strength is obtained

scholarly journals VI. On the general theory integration

1905 ◽  
Vol 204 (372-386) ◽  
pp. 221-252 ◽  
Keyword(s):  
Arbitrary Point ◽  
Continuous Functions ◽  
Common Value ◽  
Theory Integration ◽  
Positive Quantity ◽  
The Common ◽  
Definition Of ◽  
Special Case ◽  
Ordinary Integral ◽  
Definite Limit

Riemann was the first to consider the theory of integration of non-continuous functions. As is well known, his definition of the integral of a function between the limits a and b is as follows:— Divide the segment ( a, b ) into any finite number of intervals, each less, say, than a positive quantity, or norm d ; take the product of each such interval by the value of the function at any point of that interval, and form the sum of all these products; if this sum has a limit, when d is indefinitely diminished which is independent of the mode of division into intervals, and of the choice of the points in those intervals at which the values of the function are considered, this limit is called the integral of the function from a to b . The most convenient mode, however, of defining a Riemann (that is an ordinary) integral of a function, is due to Darboux; it is based on the introduction of upper and lower integrals (intégrale par excès, par défaut: oberes, unteres Integral). The definitions of these are as follows:— It may be shown that, if the interval ( a, b ) be divided as before, and the sum of the products taken as before, but with this difference, that instead of the value of the function at an arbitrary point of the part, the upper (lower) limit of the values of the function in the part be taken and multiplied by the length of the corresponding part, these summations have, whatever he the type of function, each of them a definite limit, independent of the mode of division and the mode in which d approaches the value zero. This limit is called the upper (lower) integral of the function. In the special case in which these two limits agree, the common value is called the integral the function .

scholarly journals Extensions of the Heisenberg-Weyl inequality

1986 ◽  
Vol 9 (1) ◽  
pp. 185-192 ◽  
Author(s):  
H. P. Heinig ◽  
M. Smith
Keyword(s):  
Entropy Inequality ◽  
Optimal Form ◽  
One Dimensional ◽  
Higher Dimensional ◽  
Weyl Inequality ◽  
Uncertainty Inequality ◽  
Weighted Entropy ◽  
Inequality Theorem

In this paper a number of generalizations of the classical Heisenberg-Weyl uncertainty inequality are given. We prove then-dimensional Hirschman entropy inequality (Theorem 2.1) from the optimal form of the Hausdorff-Young theorem and deduce a higher dimensional uncertainty inequality (Theorem 2.2). From a general weighted form of the Hausdorff-Young theorem, a one-dimensional weighted entropy inequality is proved and some weighted forms of the Heisenberg-Weyl inequalities are given.

An inequality for real positive functions

1952 ◽  
Vol 48 (1) ◽  
pp. 93-105 ◽  
Author(s):  
W. K. Hayman
Keyword(s):  
Complex Variable ◽  
Positive Quantity ◽  
Similar Quantity ◽  
Image Position ◽  
Positive Functions

In the theory of functions of a complex variable a positive quantity g(r), denned in terms of a circle |z| = r, can often be estimated by means of another similar quantity f(R) denned in terms of a circle |z| = R with R > r. Inequalities of the formarise in this connexion. Putting R = r + h we have for every h > 0.

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