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Harmonic Functions (harmonic + function)
Selected AbstractsHarmonic functions on the real hyperbolic ball II Hardy-Sobolev and Lipschitz spacesMATHEMATISCHE NACHRICHTEN, Issue 1 2004Sandrine Grellier Abstract In this paper, we pursue the study of harmonic functions on the real hyperbolic ball started in [13]. Our focus here is on the theory of Hardy-Sobolev and Lipschitz spaces of these functions. We prove here that these spaces admit Fefferman-Stein like characterizations in terms of maximal and square functionals. We further prove that the hyperbolic harmonic extension of Lipschitz functions on the boundary extend into Lipschitz functions on the whole ball. In doing so, we exhibit differences of behaviour of derivatives of harmonic functions depending on the parity of the dimension of the ball and on the parity of the order of derivation. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] Brahms and the Principle of Destabilised BeginningsMUSIC ANALYSIS, Issue 1 2009Ryan Mcclelland ABSTRACT Despite the considerable research on moment-to-moment motivic development in Brahms's instrumental music, surprisingly few studies emphasise global thematic processes which involve transformations of initially destabilised thematic material. After placing Brahms's destabilised beginnings in the context of earlier nineteenth-century works, the article explores several pieces with destabilised beginnings in order to demonstrate Brahms's range of techniques for tonal and rhythmic-metric destabilisation, strategies used to maintain destabilisation at preliminary thematic returns and the relationships between destabilised beginnings and their eventual stabilised form. Tonal destabilisation subsumes several related and somewhat overlapping techniques, and the article pursues six which have particular relevance in Brahms's music: (1) stylistically marked chromaticism, (2) unusual dissonance treatment, (3) denial of harmonic or melodic cadence, (4) minimally stable diatonic harmonisation, (5) disjuncture between harmonic function and sonority type and (6) ambiguous establishment of key. The briefer consideration of rhythmic-metric destabilisation studies dissonance (1) at the level of metre and (2) at some level of hypermetre. The concluding section addresses stylistic and genre-specific constraints on destabilised beginnings as well as the implications of destabilised beginnings for the analysis of musical form, especially the distinction between rondo and modified sonata designs. [source] Large deviations of multiclass M/G/1 queues,THE CANADIAN JOURNAL OF STATISTICS, Issue 3 2009André Dabrowski Abstract Consider a multiclass M/G/1 queue where queued customers are served in their order of arrival at a rate which depends on the customer class. We model this system using a chain with states represented by a tree. Since the service time distribution depends on the customer class, the stationary distribution is not of product form so there is no simple expression for the stationary distribution. Nevertheless, we can find a harmonic function on this chain which provides information about the asymptotics of this stationary distribution. The associated h -transformation produces a change of measure that increases the arrival rate of customers and decreases the departure rate thus making large deviations common. The Canadian Journal of Statistics 37: 327,346; 2009 © 2009 Statistical Society of Canada Considérons une file d'attente M/G/1 multicatégorie où les consommateurs dans la file d'attente sont servis selon leur ordre d'arrivée à un taux dépendant de leur catégorie de consommateurs. Nous modélisons ce système en utilisant une chaîne où les états sont représentés à l'aide d'un arbre. Puisque la distribution du temps de service dépend du type de consommateurs, la distribution stationnaire ne peut pas s'écrire sous la forme d'un produit. Par conséquent, il n'existe pas d'expression simple pour représenter la distribution stationnaire. Cependant, nous pouvons obte-nir une transformation harmonique de cette chaîne contenant de l'information sur les propriétés asymptotiques de cette distribution stationnaire. La transformation- h associée conduit à un chan-gement de mesure qui augmente le taux d'arrivée et décroît le taux de service ce qui augmente la probabilité de grandes déviations. La revue canadienne de statistique 37: 327,346; 2009 © 2009 Société statistique du Canada [source] Application of the equivalent multipole moment method with polar translations to forward calculation of neuromagnetic fieldsELECTRONICS & COMMUNICATIONS IN JAPAN, Issue 4 2008Shoji Hamada Abstract This paper describes an application of the equivalent multipole moment method (EMMM) with polar translations to calculation of magnetic fields induced by a current dipole placed in a human head model. Although the EMMM is a conventional Laplacian field solver based on spherical harmonic functions, the polar translations enable it to treat eccentric and exclusive spheres in arbitrary arrangements. The head model is composed of seven spheres corresponding to skin, two eyeballs, skull, cerebral spinal fluid, gray matter, and white matter. The validity of the calculated magnetic fields and the magnetic flux linkages with a loop coil located near the model is successfully confirmed by the reciprocity theorem derived by Eaton. © 2008 Wiley Periodicals, Inc. Electron Comm Jpn, 91(4): 34,44, 2008; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecj.10079 [source] The annual cycle of heavy precipitation across the United Kingdom: a model based on extreme value statisticsINTERNATIONAL JOURNAL OF CLIMATOLOGY, Issue 12 2009D. Maraun Abstract The annual cycle of extreme 1-day precipitation events across the UK is investigated by developing a statistical model and fitting it to data from 689 rain gauges. A generalized extreme-value distribution (GEV) is fit to the time series of monthly maxima, across all months of the year simultaneously, by approximating the annual cycles of the location and scale parameters by harmonic functions, while keeping the shape parameter constant throughout the year. We average the shape parameter of neighbouring rain gauges to decrease parameter uncertainties, and also interpolate values of all model parameters to give complete coverage of the UK. The model reveals distinct spatial patterns for the estimated parameters. The annual mean of the location and scale parameter is highly correlated with orography. The annual cycle of the location parameter is strong in the northwest UK (peaking in late autumn or winter) and in East Anglia (where it peaks in late summer), and low in the Midlands. The annual cycle of the scale parameter exhibits a similar pattern with strongest amplitudes in East Anglia. The spatial patterns of the annual cycle phase suggest that they are linked to the dominance of frontal precipitation for generating extreme precipitation in the west and convective precipitation in the southeast of the UK. The shape parameter shows a gradient from positive values in the east to negative values in some areas of the west. We also estimate 10-year and 100-year return levels at each rain gauge, and interpolated across the UK. Copyright © 2008 Royal Meteorological Society [source] Numerical analysis of sleeve monopole in parallel-plate waveguideINTERNATIONAL JOURNAL OF RF AND MICROWAVE COMPUTER-AIDED ENGINEERING, Issue 2 2001Zhi Ning Chen Abstract A monopole with double sleeves, which consists of a resonant loading and a conventional sleeve monopole, is experimentally investigated. The loaded monopole is put vertically in a parallel-plate waveguide and driven by a coaxial feeder. The new structure exhibits a remarkably broad impedance bandwidth. In this paper, a modal expansion technique is used to numerically evaluate the impedance characteristics of the monopole by modeling the fields between the plates using cylindrical harmonic functions. A Fourier least-square integration is applied to finding the expansion coefficients by the boundary and continuity conditions. Prior to modeling the proposed sleeve monopole, the developed analysis scheme is examined for its convergence and accuracy. Calculated results are validated by the measurements. For the optimum design at 5.8 Ghz, we investigate the effects of the structure parameters on the impedance characteristics. © 2001 John Wiley & Sons, Inc. Int J RF and Microwave CAE 11: 86,98, 2001. [source] Some Riemann boundary value problems in Clifford analysisMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 3 2010Klaus Gürlebeck Abstract In this paper, we mainly study the Rm (m>0) Riemann boundary value problems for functions with values in a Clifford algebra C,(V3, 3). We prove a generalized Liouville-type theorem for harmonic functions and biharmonic functions by combining the growth behaviour estimates with the series expansions for k -monogenic functions. We obtain the result under only one growth condition at infinity by using the integral representation formulas for harmonic functions and biharmonic functions. By using the Plemelj formula and the integral representation formulas, a more generalized Liouville theorem for harmonic functions and biharmonic functions are presented. Combining the Plemelj formula and the integral representation formulas with the above generalized Liouville theorem, we prove that the Rm (m>0) Riemann boundary value problems for monogenic functions, harmonic functions and biharmonic functions are solvable. Explicit representation formulas of the solutions are given. Copyright © 2009 John Wiley & Sons, Ltd. [source] Equivalence of weak Dirichlet's principle, the method of weak solutions and Perron's method towards classical solutions of Dirichlet's problem for harmonic functionsMATHEMATISCHE NACHRICHTEN, Issue 4 2006Christian G. Simader Abstract For boundary data , , W1,2(G ) (where G , ,N is a bounded domain) it is an easy exercise to prove the existence of weak L2 -solutions to the Dirichlet problem ",u = 0 in G, u |,G = , |,G". By means of Weyl's Lemma it is readily seen that there is , , C,(G ), ,, = 0 and , = u a.e. in G . On the contrary it seems to be a complicated task even for this simple equation to prove continuity of , up to the boundary in a suitable domain if , , W1,2(G ) , C0(). The purpose of this paper is to present an elementary proof of that fact in (classical) Dirichlet domains. Here the method of weak solutions (resp. Dirichlet's principle) is equivalent to the classical approaches (Poincaré's "sweeping-out method", Perron's method). (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] Harmonic functions on the real hyperbolic ball II Hardy-Sobolev and Lipschitz spacesMATHEMATISCHE NACHRICHTEN, Issue 1 2004Sandrine Grellier Abstract In this paper, we pursue the study of harmonic functions on the real hyperbolic ball started in [13]. Our focus here is on the theory of Hardy-Sobolev and Lipschitz spaces of these functions. We prove here that these spaces admit Fefferman-Stein like characterizations in terms of maximal and square functionals. We further prove that the hyperbolic harmonic extension of Lipschitz functions on the boundary extend into Lipschitz functions on the whole ball. In doing so, we exhibit differences of behaviour of derivatives of harmonic functions depending on the parity of the dimension of the ball and on the parity of the order of derivation. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] Gaussian curvature estimates for the convex level sets of p -harmonic functionsCOMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 7 2010Xi-Nan Ma We give a positive lower bound for the Gaussian curvature of the convex level sets of p -harmonic functions with the Gaussian curvature of the boundary and the norm of the gradient on the boundary. Combining the deformation process, this estimate gives a new approach to studying the convexity of the level sets of the p -harmonic function. © 2010 Wiley Periodicals, Inc. [source] Sharp integral inequalities for harmonic functionsCOMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 1 2008Fengbo Hang Motivated by Carleman's proof of isoperimetric inequality in the plane, we study some sharp integral inequalities for harmonic functions on the upper half-space. We also derive the regularity for nonnegative solutions of the associated integral system and some Liouville-type theorems. © 2007 Wiley Periodicals, Inc. 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