A new explicit solution method for the diffraction through a slit – Part 2

In the paper [N. Gorenflo, A new explicit solution method for the diffraction through a slit, ZAMP 53 (2002), 877–886] the problem of diffraction through a slit in a screen has been considered for arbitrary Dirichlet data, prescribed in the slit, and under the assumption that the normal derivative of the diffracted wave vanishes on the screen itself. For this problem certain functions with the following properties have been constructed: Each function f is defined on the whole of R and on the screen the values f(x), |x|  ≥  1, are the Dirichlet data of the diffracted wave which takes on the Dirichlet data f(x), |x|  ≤  1, in the slit. The problem of expanding arbitrary Dirichlet data, prescribed in the slit, into a series of functions of the considered form has been addressed, but not solved in a satisfactory way (only the application of the Gram-Schmidt orthogonalization process to such functions has been proposed). In this continuation of the aforementioned paper we choose the remaining degrees of freedom in the earlier given representations of such functions in a certain way. The resulting concrete functions can be expressed by Hankel functions and explicitly given coefficients. We suggest the expansion of arbitrary Dirichlet data, prescribed in the slit, into a series of these functions, here the expansion coefficients can be expressed explicitly by certain moments of the expanded data. Using this expansion, the diffracted wave can be expressed in an explicit form. In the future it should be examined whether similar techniques as those which are presented in the present paper can be used to solve other canonical diffraction problems, inclusively vectorial diffraction problems.     

The Limit Distribution of the Largest Interpoint Distance from a Symmetric Kotz Sample

Generalizing recent work of P. C. Matthews and A. L. Rukhin (Ann. Appl. Probab.3(1993), 454–466), we obtain the limit law of the largest interpoint Euclidean distance for a spherically symmetric multivariate sample of the Kotz distribution. While going through the proof, some errors in the reasoning given by Matthews and Rukhin are pointed out and corrected.     

Zerlegung total gerichteter Graphen in Kreise

The following statements are valid:The complete directed graph ¯K_n, n1 (mod 2p), is decomposable into directed 2p-cycles.The complete directed bipartite graph ¯K_m,n is decomposable into 2p-cycles if p is a divisor of m and np.If p is a prime, then this condition is necessary, too.The complete directed graph ¯K_n, n12, is decomposable into 6-cycles if and only if 6     

Multi-Faithful Spanning Trees of Infinite Graphs

For an end and a tree T of a graph G we denote respectively by m() and m _T () the maximum numbers of pairwise disjoint rays of G and T belonging to , and we define tm() := min{m _T(): T is a spanning tree of G}. In this paper we give partial answers — affirmative and negative ones — to the general problem of determining if, for a function f mapping every end of G to a cardinal f() such that tm() f() m(), there exists a spanning tree T of G such that m _T () = f() for every end of G.     

Why the theorem of Scheffé should be rather called a theorem of Riesz

In 1947 Henry Scheffé published a result which afterwards became known as Scheffé’s theorem, stating that the distributions of a sequence (f _n ) of densities, which converge almost everywhere to a density f, converge uniformly to the distribution of f. But almost 20 years earlier Frigyes Riesz proved a sufficient condition for convergence in the p-th mean (p ≥ 1), wherefrom the Scheffé theorem is just a special case.     

Another Look at the Erdős–Hajnal–Pósa Results on Partitioning Edges of the Rado Graph

Dedicated to the memory of Paul Erdős Erdős, Hajnal and Pósa exhibited in [1] a partition (U,D) of the edges of the Rado graph which is a counterexample to . They also obtained that if every vertex of a graph has either in or in the complement of finite degree then . We will characterize all graphs so that . Received October 29, 1999 RID="†" ID="†" Supported by NSERC of Canada Grant #691325.     

Crouzeix–Raviart boundary elements

This paper establishes a foundation of non-conforming boundary elements. We present a discrete weak formulation of hypersingular integral operator equations that uses Crouzeix–Raviart elements for the approximation. The cases of closed and open polyhedral surfaces are dealt with. We prove that, for shape regular elements, this non-conforming boundary element method converges and that the usual convergence rates of conforming elements are achieved. Key ingredient of the analysis is a discrete Poincaré–Friedrichs inequality in fractional order Sobolev spaces. A numerical experiment confirms the predicted convergence of Crouzeix–Raviart boundary elements. Norbert Heuer is supported by Fondecyt-Chile under grant no. 1080044. F.-J. Sayas is partially supported by MEC-FEDER Project MTM2007-63204 and Gobierno de Aragón (Grupo Consolidado PDIE).     

Regularisierte Faltung von Distributionen. Teil 1: Zur Berechnung von Fundamentallösungen

Zusammenfassung Für 2 Distributionen, deren Faltung nicht existiert, wird—mehrdeutig—eine bezüglich eines Differentialoperators regularisierte Faltung definiert.An 4 Beispielen wird die Anwendbarkeit der regularisierten Faltung bei der Berechnung von Fundamentallösungen faktorisierbarer Differentialoperatoren gezeigt. Dabei wurde die Fundamentallösung in Beispiel 3.2. erstmalig ohne Überlegungen physikalischer Art hergeleitet. Beispiel 3.3. verallgemeinert bekannte Ergebnisse, wobei diese als Spezialfälle erscheinen (Bemerkung 3) oder als fehlerhaft nachgewiesen werden (Bemerkung 4). Die Fundamentallösungen der Beispiele 3.1 und 3.4 scheinen in dieser Form neu zu sein.Schließlich folgen einige einfache Sätze, mittels derer Fundamentallösungen von iterierten Differentialoperatoren angegeben werden können. Sie werden im 2. Teil der Arbeit (Eine Tabelle von Fundamentallösungen) angewendet. Literaturhinweise finden sich am Ende von Teil 2.

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Summary For two distributions (whose convolution does not necessarily exist) we define a (manyvalued) convolution regularized with respect to a differential operator.We illustrate by four examples the usefulness of the concept of a regularized convolution for computing fundamental solutions of factorisable differential operators. The fundamental solution of example 3.2 is derived for the first time without considerations from physics. Example 3.3 generalizes well-known results, which either appear as special cases (remark 3) or are proved to contain errors (remark 4). The fundamental solutions of examples 3.1 and 3.4 seem to be new in this form.Finally we give some simple propositions concerning fundamental solutions of iterated differential operators. They will be used in part 2 of this paper (a table of fundamental solutions)The references are given at the end of part 2.

     

How to grow it? Strategies of mathematical development presented by the example of enumerating certain set partitions

Mathematische Semesterberichte - We describe in the form of a dialogue a development of various reflections on the combinatorics of set partitions; among the topics we pursue are...     

On vibrations in non-linear,forced, friction-excited systems

To accelerate the development of non-linear, friction-excited systems, i.e. automotive friction brakes, novel test procedures are being developed. These include the study of the system's forced vibrations. However, there is a lack of knowledge concerning feasible test procedures, the assessment of the gathered data, and the possible vibration phenomena. In this context, the contribution at hand discusses vibrations in non-linear, forced, friction-excited systems. In the pre-flutter regime proposed criteria for identification of parameter regions in which limit cycle oscillations can occur are challenged. In the post-flutter regime frequency spectra for different excitation conditions are studied. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)