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## Singular Integrals on Product Homogeneous Groups

### Integral Equations and Operator Theory (2013-05-01) 76: 55-79 , May 01, 2013

We consider singular integral operators with rough kernels on the product space of homogeneous groups. We prove *L*^{p} boundedness of them for
$${p \in (1,\infty)}$$
under a sharp integrability condition of the kernels.

## Nonself-similar flow with a shock wave reflected from the center of symmetry and new self-similar solutions with two reflected shocks

### Computational Mathematics and Mathematical Physics (2013-03-01) 53: 350-368 , March 01, 2013

In some problems concerning cylindrically and spherically symmetric unsteady ideal (inviscid and nonheat-conducting) gas flows at the axis and center of symmetry (hereafter, at the center of symmetry), the gas density vanishes and the speed of sound becomes infinite starting at some time. This situation occurs in the problem of a shock wave reflecting from the center of symmetry. For an ideal gas with constant heat capacities and their ratio γ (adiabatic exponent), the solution of this problem near the reflection point is self-similar with a self-similarity exponent determined in the course of the solution construction. Assuming that γ on the reflected shock wave decreases, if this decrease exceeds a threshold value, the flow changes substantially. Assuming that the type of the solution remains unchanged for such γ, self-similarity is preserved if a piston starts expanding from the center of symmetry at the reflection time preceded by a finite-intensity reflected shock wave propagating at the speed of sound. To answer some questions arising in this formulation, specifically, to find the solution in the absence of the piston, the evolution of a close-to-self-similar solution calculated by the method of characteristics is traced. The required modification of the method of characteristics and the results obtained with it are described. The numerical results reveal a number of unexpected features. As a result, new self-similar solutions are constructed in which two (rather than one) shock waves reflect from the center of symmetry in the absence of the piston.

## Die Bolyai und die Belletristik

### János Bolyai (2013-01-01) 16: 245-251 , January 01, 2013

### Zusammenfassung

Im gesellschaftlichen Leben Ungarns zählte das Erscheinen der beiden Bolyai nicht gerade zu den alltäglichen Ereignissen. Wir müssen uns aufrichtig eingestehen, dass es im Land bis zur Zeit der Bolyai nicht allzu viele herausragende und weltberühmte Gelehrte gab. Die allgemeine Einstellung in Ungarn, die eher die Literatur und die Dichtkunst begünstigte, schenkte den Wissenschaftlern kaum Beachtung.

## Einleitung

### Geometrische Denkweisen beim Lösen von PISA-Aufgaben (2013-01-01) 3: 1-4 , January 01, 2013

### Zusammenfassung

Geometrische Denkweisen durchdringen die gesamte Mathematik, denn mathematisches Denken bedient sich oft geometrischer Stötzen. Immer, wenn es darum geht, visuell dargebotene Informationen aufzunehmen, zu analysieren, zu speichern und mit ihnen in der Vorstellung zu operieren, sind dabei geometrische Denkweisen von grundlegender Bedeutung.

## Mutation on knots and Whitney’s 2-isomorphism theorem

### Acta Mathematica Sinica, English Series (2013-06-01) 29: 1219-1230 , June 01, 2013

Whitney’s 2-switching theorem states that any two embeddings of a 2-connected planar graph in *S*^{2} can be connected via a sequence of simple operations, named 2-switching. In this paper, we obtain two operations on planar graphs from the view point of knot theory, which we will term “twisting” and “2-switching” respectively. With the twisting operation, we give a pure geometrical proof of Whitney’s 2-switching theorem. As an application, we obtain some relationships between two knots which correspond to the same signed planar graph. Besides, we also give a necessary and sufficient condition to test whether a pair of reduced alternating diagrams are mutants of each other by their signed planar graphs.

## A Partial Manuscript on Fourier and Laplace Transforms

### Ramanujan's Lost Notebook (2013-01-01): 285-305 , January 01, 2013

Most of the results in the partial manuscript on integral transforms discussed in this chapter are classical. However, the partial manuscript contains one of the highlights of the book, a beautiful new transformation formula involving the logarithmic derivative of the gamma function. An extremely clever device used to prove this transformation formula harkens back to Ramanujan’s paper, *New expressions for Riemann’s functions*
$$\xi(s)\,and\,\Xi(s) $$

## Back Matter - Stability and Bifurcation Theory for Non-Autonomous Differential Equations

### Stability and Bifurcation Theory for Non-Autonomous Differential Equations (2013-01-01): 2065 , January 01, 2013

## Geometric Analysis of the Bergman Kernel and Metric

### Geometric Analysis of the Bergman Kernel and Metric (2013-01-01): 268 , January 01, 2013

## PyCX: a Python-based simulation code repository for complex systems education

### Complex Adaptive Systems Modeling (2013-03-13) 1: 1-10 , March 13, 2013

We introduce PyCX, an online repository of simple, crude, easy-to-understand sample codes for various complex systems simulation, including iterative maps, cellular automata, dynamical networks and agent-based models. All the sample codes were written in plain Python, a general-purpose programming language widely used in industry as well as in academia, so that students can gain practical skills for both complex systems simulation and computer programming simultaneously. The core philosophy of PyCX is on the simplicity, readability, generalizability and pedagogical values of simulation codes. PyCX has been used in instructions of complex systems modeling at several places with successful outcomes.

## Solutions of perturbed p-Laplacian equation with critical nonlinearity and magnetic fields

### Boundary Value Problems (2013-11-05) 2013: 1-11 , November 05, 2013

In this paper, we consider a perturbed *p*-Laplacian equation with criticalnonlinearity and magnetic fields on
. By using the variational method, we establish theexistence of nontrivial solutions of the least energy.

*MSC:* 35B33, 35J60, 35J65.