scholarly journals III. Memoir on the theory of the partitions of numbers. —Part V. Partitions in two-dimensional space

1912 ◽  
Vol 211 (471-483) ◽  
pp. 75-110 ◽  
Keyword(s):  
Complete Lattice ◽  
Dimensional Space ◽  
Two Dimensions ◽  
Algebraic Expression ◽  
Two Dimensional ◽  
Vertical Column ◽  
Order Of Magnitude ◽  
The Subject ◽  
First Time ◽  
Made In

In previous papers I have broached the question of the two-dimensional partitions of numbers—or, say, the partitions in a plane—without, however, having succeeded in establishing certain conjectured formulas of enumeration. The parts of such partitions are placed at the nodes of a complete, or of an incomplete, lattice in two dimensions, in such wise that descending order of magnitude is in evidence in each horizontal row of nodes and in each vertical column. No decided advance was made in regard to the complete lattice, and the question of the incomplete lattice is considered for the first time in the present paper. I return to the subject because I am now able to throw a considerable amount of fresh light upon the problem, and have succeeded in overcoming most of the difficulties which surround it. In fact, I am now able to show how the generating functions may be constructed in respect of any lattice, complete or incomplete, in forms which are free from redundant terms. I have not succeeded, so far, in giving a general algebraic expression to the functions, but, in the case of the complete lattice, I have shown that an assumption as to form, consistent with all results that have been arrived at in particular cases, leads at once to the expression that has been for so long the conjectured result. For the complete lattice of two rows, and for the incomplete lattice of two rows, the results have been obtained without any assumption in regard to form, and must be regarded as rigidly established.

scholarly journals Memoir on the theory of the partitions of numbers. Part V.—Partitions in two-dimensional space

1911 ◽  
Vol 85 (578) ◽  
pp. 304-305 ◽  
Keyword(s):  
Generating Function ◽  
Dimensional Space ◽  
Two Dimensional ◽  
Order Of Magnitude

This paper is principally concerned with the enumerations of the partitions of numbers in two-dimensional space. Such partitions are such that the parts are placed at the points or nodes of a lattice, rectangular in shape, and . . . . . . . . . . . . having given numbers of nodes in the sides, and the parts of the numbers partitioned are placed at the nodes in such wise that descending order of magnitude in each row of parts is from left to right and in each column of parts from top to bottom. If the nodes in the sides be m, n, respectively, and the part magnitude be restricted so as to be not greater than l , the generating function which gives by the coefficients of x w in the ascending expansion the number of such partitions of the number w has been long conjectured by the author to have the expression ( l + 1) ( l + 2 ) . . . ( l + m ) ( l + 2) . . . (l + m + 1) . . . . . . ( l + n ) . . . (l + m + n - 1)/(1) (2) . . . ( m ) (2) . . . ( m + 1) . . . . . . ( n ) . . . ( m + n - 1) where ( s ) denotes 1— x s ; but previous attempts to establish this had, except in certain particular cases, resulted in failure.

Developing an Understanding of Organisational Knowledge Creation: A Review Framework

2017 ◽  
Vol 16 (02) ◽  
pp. 1750020 ◽  
Author(s):  
Pamela Chidiogo Izunwanne
Keyword(s):  
Knowledge Creation ◽  
Empirical Studies ◽  
Two Dimensions ◽  
Extensive Review ◽  
Extant Literature ◽  
The Past ◽  
Organisational Knowledge ◽  
The Subject ◽  
Made In

The purpose of this paper is to develop an understanding of the dimensions of organisational knowledge creation. To this end, the literature from numerous bodies of research is reviewed. Over the past 20 years, the literature on organisational knowledge creation has significantly grown and advanced. Many new and conflicting terminologies have evolved to shed more light on the subject. As a result, there are overlaps and conflicts in the literature. The framework suggested in this paper is therefore considered to be a useful means to structure further research on the subject. This paper presents an extensive review of the organisational knowledge creation literature as well as an overview of existing empirical studies. The framework that is presented in this paper emerged from the review and maps out two dimensions of organisational knowledge creation. Although there are many other possible dimensions that could potentially be included in the framework, this study focuses on the dimensions that are believed to reflect more obvious distinctions made in the extant literature on organisational knowledge creation. Furthermore, limiting the review to only two dimensions is believed to provide more coherence.

scholarly journals Analytic materials

2016 ◽  
Vol 472 (2195) ◽  
pp. 20160613 ◽  
Author(s):  
Graeme W. Milton
Keyword(s):  
Connected Domain ◽  
Free Field ◽  
Exact Results ◽  
Two Dimensional ◽  
Simply Connected Domain ◽  
Maximum Value ◽  
Two Dimensional Materials ◽  
The Subject ◽  
Simply Connected ◽  
Made In

The theory of inhomogeneous analytic materials is developed. These are materials where the coefficients entering the equations involve analytic functions. Three types of analytic materials are identified. The first two types involve an integer p . If p takes its maximum value, then we have a complete analytic material. Otherwise, it is incomplete analytic material of rank p . For two-dimensional materials, further progress can be made in the identification of analytic materials by using the well-known fact that a 90 ° rotation applied to a divergence-free field in a simply connected domain yields a curl-free field, and this can then be expressed as the gradient of a potential. Other exact results for the fields in inhomogeneous media are reviewed. Also reviewed is the subject of metamaterials, as these materials provide a way of realizing desirable coefficients in the equations.

scholarly journals Helium-like ions in d-dimensions: analyticity and generalized ground state Majorana solutions

2021 ◽  
Author(s):  
Adrian Mauricio Escobar ◽  
Horacio Olivares-Pilón ◽  
Norberto Aquino ◽  
Salvador Antonio Cruz-Jimenez
Keyword(s):  
Ground State Energy ◽  
Ground State ◽  
State Energy ◽  
Dimensional Space ◽  
Algebraic Function ◽  
Approximate Solutions ◽  
Algebraic Expression ◽  
Three Body ◽  
First Time ◽  
Analytical Expressions

Abstract Non-relativistic Helium-like ions (−e, −e, Ze) with static nucleus in a d−dimensional space (d > 1) are considered. Assuming r−1Coulomb interactions, a 2-parametric correlated Hylleraas-type trial function is used to calculate the ground state energy of the system in the domain Z ≤ 10. For odd d = 3, 5, the variational energy is given by a rational algebraic function of the variational parameters whilst for even d = 2, 4 it is shown for the first time that it corresponds to a more complicated non-algebraic expression. This twofold analyticity will hold for any d. It allows us to construct reasonably accurate approximate solutions for the ground state energy E0(Z, d) in the form of compact analytical expressions. We call them generalized Majorana solutions. They reproduce the first leading terms in the celebrated 1Z expansion, and serve as generating functions for certain correlation-dependent properties. The (first) critical charge Zc vs d and the Shannon entropy S(d)r vs Z are also calculated within the present variational approach. In the light of these results, for the physically important case d = 3 a more general 3-parametric correlated Hylleraas-type trial is used to compute the finite mass effects in the Majorana solution for a three-body Coulomb system with arbitrary charges and masses. It admits a straightforward generalization to any d as well. Concrete results for the systems e− e− e+, H+2 and H− are indicated explicitly. Our variational analytical results are in excellent agreement with the exact numerical values reported in the literature.

Bronze Swords in Northern Europe: a reconsideration of Sprockhoff's Griffzungenschwerter

1953 ◽  
Vol 18 (2) ◽  
pp. 129-147
Author(s):  
J. D. Cowen
Keyword(s):  
Northern Europe ◽  
Central Theme ◽  
The Past ◽  
The North ◽  
Intimate Knowledge ◽  
The Subject ◽  
First Time ◽  
Made In

It is just over twenty years since Professor Ernst Sprockhoff published his classic study of bronze swords in Northern Europe, and a review of the situation as it presents itself today, surveyed from a point well outside the limits of the Nordic area, may not be out of place.The ground covered in this fine work had already in part been traversed by Sophus Müller and Gustav Kossinna; but in the process it had become a field of battle where the bitterest partisan spirit had all too recently been displayed, and might all too easily have been re-aroused. It is not the least part of our debt to Sprockhoff that he refused to treat his material on controversial lines, and confined himself to a presentation so objective that it immediately became possible, for the first time for many years, once more to discuss the subject in a sane and cool manner. Thus, adding much that was new and solely his own, he set down in plain, precise terms the whole of the evidence relating to the history, development, and chronology of the flange-hilted bronze swords of the North.Of this structure the main fabric, without any doubt, stands firm. The central theme, based on a large number of closed finds, and supported by an intimate knowledge of the material, need fear no criticism. Yet some aspects at least of the relations between the Nordic world and other parts of Europe call for re-examination, and the work of the past two decades enables some adjustments to be made. In fairness to Sprockhoff it should be stated quite clearly, at the outset, that the most important of these adjustments have been either made possible, or actually anticipated, by his own work in related fields since 1931.

scholarly journals Thermodynamics of Acoustic Black Holes in Two Dimensions

2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
Baocheng Zhang
Keyword(s):  
Fluid Dynamics ◽  
Black Holes ◽  
Black Hole ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Schwarzschild Black Hole ◽  
First Law Of Thermodynamics ◽  
Entropic Gravity ◽  
Acoustic Black Holes ◽  
Made In

It is well-known that the thermal Hawking-like radiation can be emitted from the acoustic horizon, but the thermodynamic-like understanding for acoustic black holes was rarely made. In this paper, we will show that the kinematic connection can lead to the dynamic connection at the horizon between the fluid and gravitational models in two dimensions, which implies that there exists the thermodynamic-like description for acoustic black holes. Then, we discuss the first law of thermodynamics for the acoustic black hole via an intriguing connection between the gravitational-like dynamics of the acoustic horizon and thermodynamics. We obtain a universal form for the entropy of acoustic black holes, which has an interpretation similar to the entropic gravity. We also discuss the specific heat and find that the derivative of the velocity of background fluid can be regarded as a novel acoustic analogue of the two-dimensional dilaton potential, which interprets why the two-dimensional fluid dynamics can be connected to the gravitational dynamics but it is difficult for four-dimensional case. In particular, when a constraint is added for the fluid, the analogue of a Schwarzschild black hole can be realized.

A Hill-determinant approach to symmetric double-well potentials in two dimensions

1995 ◽  
Vol 73 (9-10) ◽  
pp. 632-637 ◽  
Author(s):  
M. R. M. Witwit ◽  
J. P. Killingbeck
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Dimensional Space ◽  
Energy Levels ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Degenerate Eigenvalues ◽  
Double Well Potential ◽  
Range Of Values

Energy levels of the Schrödinger equation for a double-well potential V(x,y;Z2,λ) = −Z2[x2 + y2] + λ[axxx4 + 2axyx2y2 + ayyy4] in two-dimensional space are calculated, using a Hill-determinant approach for several eigenstates and a range of values of λ and Z2. Special emphasis is placed on the larger values of Z2, for which the eigenvalues for different states have almost degenerate eigenvalues.

scholarly journals XVII. Bakerian lecture.─On the grand currents of atmospheric circulation

1892 ◽  
Vol 183 ◽  
pp. 653-684 ◽  
Keyword(s):  
Atmospheric Circulation ◽  
Important Paper ◽  
Trade Winds ◽  
The World ◽  
True Theory ◽  
The Subject ◽  
First Time ◽  
Special Notice ◽  
Made In

In the early times of the Boyal Society (a little more than 200 years ago) a spirit of inquiry and of speculation as to the causes of the Trade Winds arose among its members. The papers which we may presume to have first brought the subject into special notice in the Society, and which were published in the * Transactions,’ offered views which, in the light of subsequent knowledge and theory, show themselves as being untenable, and in part even grotesque. But those papers were soon followed by, and probably had an effect in leading to, a much more important paper by the eminent astronomer Edmund Halley; and this was followed 49 years later by one, more important still, by George Hadley, in which we may with confidence judge that a substantially true theory of a large part of the system of Atmospheric Circulation in its grandest and most dominant conditions was for the first time offered to the world through the pages of the ‘ Philosophical Transactions.’ Further speculations on the subject and advances in our knowledge of it have been made in later times and have been brought into notice in various ways. I believe that I have myself arrived at some improved considerations which are to a large extent trustworthy and go far towards completing the true theory of the grand currents of atmospheric circulation, and I entertain the ambition to have my views placed on record by this Society—the Society in which the subject had its most important beginnings.

Evidence of Intermittent Type Fluctuation of Target Residue in Relativistic Nuclear Collisions at a Few Gev/n

2003 ◽  
Vol 12 (03) ◽  
pp. 407-419 ◽  
Author(s):  
Dipak Ghosh ◽  
Argha Deb ◽  
Swarnapratim Bhattacharyya ◽  
Jayita Ghosh Rupa Das ◽  
Sima Mukherjee
Keyword(s):  
Fractal Dimension ◽  
Energy Dependence ◽  
Dimensional Space ◽  
Azimuthal Angle ◽  
Emission Angle ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Nuclear Collisions ◽  
Angle Space ◽  
Relativistic Nuclear Collisions

This paper presents evidence of intermittency in target fragmentation in 28Si-AgBr interactions at 14.5 AGeV in emission angle space and azimuthal angle space separately and also in two-dimensional (cos θ,ϕ) space. From the intermittency exponent, the anomalous fractal dimension dq is calculated and the variation of dq with the order q is investigated. The authors also compare the results with those obtained from 16O-AgBr interactions at 60 AGeV and at 2.1 AGeV and 32S-AgBr interactions at 200 AGeV to study the energy dependence of intermittent type of fluctuations in emission angle and azimuthal angle space separately. Comparing the intermittency exponents in one- and two-dimensions it is further seen that intermittency exponent has a higher value in two-dimensional space.

scholarly journals A survey of two-dimensional graph layout techniques for information visualisation

2012 ◽  
Vol 12 (3-4) ◽  
pp. 324-357 ◽  
Author(s):  
Helen Gibson ◽  
Joe Faith ◽  
Paul Vickers
Keyword(s):  
Graph Drawing ◽  
Dimensional Space ◽  
Two Dimensional ◽  
Graph Layout ◽  
Information Visualisation ◽  
Detailed Assessment ◽  
Node Attributes ◽  
Multi Level ◽  
Dimensional Graph ◽  
Made In

Many algorithms for graph layout have been devised over the last 30 years spanning both the graph drawing and information visualisation communities. This article first reviews the advances made in the field of graph drawing that have then often been applied by the information visualisation community. There then follows a discussion of a range of techniques developed specifically for graph visualisations. Graph drawing algorithms are categorised into the following approaches: force-directed layouts, the use of dimension reduction in graph layout and computational improvements including multi-level techniques. Methods developed specifically for graph visualisation often make use of node-attributes and are categorised based on whether the attributes are used to introduce constraints to the layout, provide a clustered view or define an explicit representation in two-dimensional space. The similarities and distinctions between these techniques are examined and the aim is to provide a detailed assessment of currently available graph layout techniques, specifically how they can be used by visualisation practitioners, and to motivate further research in the area.

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