scholarly journals Multipole expansion of integral powers of cosine theta

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
E. O. Jobunga ◽  
O. S. Okeyo
Keyword(s):  
Legendre Polynomial ◽  
Legendre Polynomials ◽  
Coulomb Repulsion ◽  
Multipole Expansion ◽  
The Other ◽  
Polynomial Basis ◽  
Angular Function ◽  
Multipole Expansions ◽  
Spatially Varying ◽  
Special Case

AbstractLegendre polynomials form the basis for multipole expansion of spatially varying functions. The technique allows for decomposition of the function into two separate parts with one depending on the radial coordinates only and the other depending on the angular variables. In this work, the angular function $$\cos ^k \theta$$ cos k θ is expanded in the Legendre polynomial basis and the algorithm for determining the corresponding coefficients of the Legendre polynomials is generated. This expansion together with the algorithm can be generalized to any case in which a dot product of any two vectors appears. Two alternative multipole expansions for the electron–electron Coulomb repulsion term are obtained. It is shown that the conventional multipole expansion of the Coulomb repulsion term is a special case for one of the expansions generated in this work.

scholarly journals The game of Double-Silver on intervals

2001 ◽  
Vol 26 (8) ◽  
pp. 485-496 ◽  
Author(s):  
Gerald A. Heuer
Keyword(s):  
Measure Zero ◽  
Optimal Strategies ◽  
The Other ◽  
Parameter Plane ◽  
Left Endpoint ◽  
Relative Placement ◽  
Special Case

Silverman's game on intervals was analyzed in a special case by Evans, and later more extensively by Heuer and Leopold-Wildburger, who found that optimal strategies exist (and gave them) quite generally when the intervals have no endpoints in common. They exist in about half the parameter plane when the intervals have a left endpoint or a right endpoint, but not both, in common, and (as Evans had earlier found) exist only on a set of measure zero in this plane if the intervals are identical. The game of Double-Silver, where each player has its own threshold and penalty, is examined. There are several combinations of conditions on relative placement of the intervals, the thresholds and penalties under which optimal strategies exist and are found. The indications are that in the other cases no optimal strategies exist.

scholarly journals Smooth Wavelet Approximations of Truncated Legendre Polynomials via the Jacobi Theta Function

2014 ◽  
Vol 2014 ◽  
pp. 1-24 ◽  
Author(s):  
David W. Pravica ◽  
Njinasoa Randriampiry ◽  
Michael J. Spurr
Keyword(s):  
Fourier Transform ◽  
Theta Function ◽  
Legendre Polynomial ◽  
Legendre Polynomials ◽  
Inverse Fourier Transform ◽  
Recursion Relations ◽  
Jacobi Theta Function ◽  
Convergence Results ◽  
The Family ◽  
Reciprocal Symmetry

The family ofnth orderq-Legendre polynomials are introduced. They are shown to be obtainable from the Jacobi theta function and to satisfy recursion relations and multiplicatively advanced differential equations (MADEs) that are analogues of the recursion relations and ODEs satisfied by thenth degree Legendre polynomials. Thenth orderq-Legendre polynomials are shown to have vanishingkth moments for0≤k<n, as does thenth degree truncated Legendre polynomial. Convergence results are obtained, approximations are given, a reciprocal symmetry is shown, and nearly orthonormal frames are constructed. Conditions are given under which a MADE remains a MADE under inverse Fourier transform. This is used to construct new wavelets as solutions of MADEs.

Silent Victims Revisited: The Special Case of Domestic Violence

PEDIATRICS ◽  
1995 ◽  
Vol 96 (3) ◽  
pp. 511-513
Author(s):  
Barry Zuckerman ◽  
Marilyn Augustyn ◽  
Betsy McAlister Groves ◽  
Steven Parker
Keyword(s):  
Domestic Violence ◽  
Community Violence ◽  
Clinical Experience ◽  
Behavioral Problems ◽  
The Other ◽  
Media Attention ◽  
Witness Violence ◽  
Primary Focus ◽  
Special Case ◽  
At Home

In a commentary published previously, we communicated our concern regarding the plight of children who witness violence.1 Research suggests that children who witness violence suffer significant psychologic and behavioral problems that interfere with their ability to function in school, at home, and with peers. The primary focus of that commentary was children who witnessed community violence. Our ongoing clinical experience, heightened by media attention on domestic violence, including the O.J. Simpson case, leads us to revisit silent victims with a sole focus on those children who witness domestic violence. Domestic violence is a particularly devastating event for a child who, in the presence of danger, typically turns to a parent for protection and for whom there is no comfort or security if one parent is the perpetrator of violence, and the other is a terrified victim.

Existence of Periodic Solution for Equation of Motion of Simple Beams With Harmonically Variable Length

1995 ◽  
Author(s):  
Ebrahim Esmailzadeh ◽  
Gholamreza Nakhaie-Jazar ◽  
Bahman Mehri
Keyword(s):  
Periodic Solution ◽  
Fixed Point ◽  
Nonlinear Function ◽  
Axial Direction ◽  
Equation Of Motion ◽  
The Other ◽  
Sufficient Condition ◽  
Vibrating String ◽  
The Stability ◽  
Special Case

Abstract The transverse vibrating motion of a simple beam with one end fixed while driven harmonically along its axial direction from the other end is investigated. For a special case of zero value for the rigidity of the beam, the system reduces to that of a vibrating string with the corresponding equation of its motion. The sufficient condition for the periodic solution of the beam is then derived by means of the Green’s function and Schauder’s fixed point theorem. The criteria for the stability of the system is well defined and the condition for which the performance of the beam behaves as a nonlinear function is stated.

scholarly journals Can you reach for the planets or grasp at the stars? – Modified noun, verb, or preposition constituents in idiom processing

2019 ◽  
Author(s):  
Eva Smolka ◽  
Carsten Eulitz
Keyword(s):  
Sentence Type ◽  
The Other ◽  
Psycholinguistic Research ◽  
Idiom Comprehension ◽  
Figurative Meaning ◽  
Idiomatic Phrase ◽  
Idiom Processing ◽  
Special Case ◽  
Idiomatic Meaning ◽  
Control Sentence

Idioms are a special case of multi-word expressions in that their meaning cannot be compositionally constructed from the meaning of the single constituents. The question of how the idiomatic meaning is assembled remains an unsettled issue in psycholinguistic research. The present study examines whether the figurative meaning of an idiom is recognized if critical idiomatic constituents, such as the noun, verb, or preposition, are modified. In three paraphrase experiments, participants saw (a) the canonical idiomatic phrase (e.g., She reached for the stars), (b) the idiomatic phrase with a modified constituent (e.g., She reached/grasped for/at the stars/planets), or (c) a matched literal control sentence (e.g., She reached for the sweets) and rated on a scale from 1 (not at all) - 7 (completely) how strongly the sentence reflected the meaning of a paraphrase of the idiom (e.g., She has always aspired to unattainable goals). Sentence type and constituent type strongly affected paraphrase ratings with highest ratings for canonical idiomatic phrases, lowest ratings for control sentences, and ratings in between for idioms with modified constituents. Further, idioms with modified verbs were rated higher in matching the figurative meaning than idioms with modified prepositions or nouns. Overall these findings indicate that the figurative meaning was assembled in spite of the modifications. We conclude that idioms are not fully ‘semantically fixed’ but allow for some flexibility in the processing of idioms. Modified constituents that activate meanings similar to those of the canonical constituents will co-activate the figurative meaning of the idiom together with the other idiomatic constituents. We discuss psycholinguistic models on idiom comprehension.

scholarly journals Optical axes of catadioptric systems including visual, Purkinje and other nonvisual systems of a heterocentric astigmatic eye

2010 ◽  
Vol 69 (3) ◽  
Author(s):  
W. F. Harris
Keyword(s):  
Visual System ◽  
Optical Axis ◽  
The Other ◽  
Optical Systems ◽  
Numerical Examples ◽  
Straight Line ◽  
Optical Axes ◽  
Catadioptric Systems ◽  
Reflecting Surfaces ◽  
Special Case

For a dioptric system with elements which may be heterocentric and astigmatic an optical axis has been defined to be a straight line along which a ray both enters and emerges from the system.  Previous work shows that the dioptric system may or may not have an optical axis and that, if it does have one, then that optical axis may or may not be unique.  Formulae were derived for the locations of any optical axes.  The purpose of this paper is to extend those results to allow for reflecting surfaces in the system in addition to refracting elements.  Thus the paper locates any optical axes in catadioptric systems (including dioptric systems as a special case).  The reflecting surfaces may be astigmatic and decentred or tilted.  The theory is illustrated by means of numerical examples.  The locations of the optical axes are calculated for seven optical systems associated with a particular heterocentric astigmatic model eye.  The optical systems are the visual system, the four Purkinje systems and two other nonvisual systems of the eye.  The Purkinje systems each have an infinity of optical axes whereas the other nonvisual systems, and the visual system, each have a unique optical axis. (S Afr Optom 2010 69(3) 152-160)

EU demonstration project for separate discharge and treatment of urine, faeces and greywater – Part I: Results

2007 ◽  
Vol 56 (5) ◽  
pp. 239-249 ◽  
Author(s):  
A. Peter-Fröhlich ◽  
L. Pawlowski ◽  
A. Bonhomme ◽  
M. Oldenburg
Keyword(s):  
Life Cycle Assessment ◽  
Life Cycle ◽  
Cost Analysis ◽  
Nutrient Recycling ◽  
Residential Area ◽  
Water Saving ◽  
Demonstration Project ◽  
The Other ◽  
Energy Reduction ◽  
Special Case

The main aim of this EU demonstration project was to test two new sanitation concepts to determine if these concepts are more sustainable, compared with the conventional sanitation system, particularly with regard to nutrient recycling. Two different sanitation concepts were tested. One concept comprised the use of gravity separation toilets, the other used vacuum separation toilets. Results from a life-cycle-assessment investigation show that the new sanitation concepts are more sustainable. A cost analysis for an existing residential area did not prove lower costs for the new sanitation concepts in this special case. The experience from this demonstration project shows that prior to a widespread use of the new sanitation concepts, several improvements have to be made. One important issue is the improvement of separation toilets. Since nutrient recycling, water saving and reuse as well energy reduction become more and more important, further research should be undertaken in this field.

Trans‐European Party Groupings: Emergence of New and Alignment of Old Parties in the Light of the Direct Elections to the European Parliament

1979 ◽  
Vol 14 (4) ◽  
pp. 455-478 ◽  
Author(s):  
Paul‐H. Claeys ◽  
Nicole Loeb‐Mayer
Keyword(s):  
Political Parties ◽  
Ad Hoc ◽  
European Parliament ◽  
The Other ◽  
European Level ◽  
Or Groups ◽  
Direct Elections ◽  
Special Case ◽  
Domestic Level

TWO QUESTIONS ARISE WHEN CONSIDERING THE CHANGES that might be brought about by direct elections and by developments in the new European Parliament. One concerns institutionalized cooperation between political parties. To what extent can the three existing European party federations – Socialist, Christian Democrat, Liberal – be considered as a step towards the formation of genuine European political parties? Are they anything more than alignments of traditional parties coordinating their action at European level? The other question is related to parties or groups which have not until now created close-knit ad hoc structures. A special case is that of the Communist parties, which have not organized specific links at Communit level. Another problem is raised by non-traditional parties and groups that have in most cases little or no parlia mentary representation at either national or European level. Will some of them take advantage of the European sphere of action to make more impression than they have been able to do at domestic level, in cooperation with similarly oriented partners in other member countries?

LEGENDRE POLYNOMIALS AS THE NORMALIZATION OF PHOTON-SUBTRACTED SQUEEZED STATES

2009 ◽  
Vol 24 (20) ◽  
pp. 1597-1603 ◽  
Author(s):  
HONG-YI FAN ◽  
LI-YUN HU ◽  
XUE-XIANG XU
Keyword(s):  
Legendre Polynomial ◽  
Legendre Polynomials ◽  
Hermite Polynomial ◽  
Vacuum State ◽  
Squeezed States ◽  
Squeezed State ◽  
Normalization Factor ◽  
Squeezed Vacuum State ◽  
Squeezed Vacuum ◽  
Excitation State

By converting the photon-subtracted squeezed state (PSSS) to a squeezed Hermite-polynomial excitation state we find that the normalization factor of PSSS is an m-order Legendre polynomial of the squeezing parameter, where m is the number of subtracted photons. Some new relations about the Legendre polynomials are obtained by this analysis. We also show that the PSSS can also be treated as a Hermite-polynomial excitation on squeezed vacuum state.

Zero Knowledge Proofs

2018 ◽  
pp. 111-123 ◽  
Author(s):  
Kannan Balasubramanian ◽  
Mala K.
Keyword(s):  
Real World ◽  
The Other ◽  
Interactive Proof ◽  
Zero Knowledge ◽  
Practical Applications ◽  
Proof Of Knowledge ◽  
Real World Applications ◽  
Special Case ◽  
Zero Knowledge Proof

Zero knowledge protocols provide a way of proving that a statement is true without revealing anything other than the correctness of the claim. Zero knowledge protocols have practical applications in cryptography and are used in many applications. While some applications only exist on a specification level, a direction of research has produced real-world applications. Zero knowledge protocols, also referred to as zero knowledge proofs, are a type of protocol in which one party, called the prover, tries to convince the other party, called the verifier, that a given statement is true. Sometimes the statement is that the prover possesses a particular piece of information. This is a special case of zero knowledge protocol called a zero-knowledge proof of knowledge. Formally, a zero-knowledge proof is a type of interactive proof.

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