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## Grundwissen Differenzialrechnung

### (Hoch)Schulmathematik (2017-01-01): 113-150 , January 01, 2017

### Zusammenfassung

In diesem Kapitel konzentrieren wir uns auf das, was in der Schule oftmals etwas zu kurz kommt: Eine präzise Einführung des Begriffes der ″Ableitung einer Funktion″. Zusammen mit dem Integralbegriff ist dies wohl die bedeutsamste Anwendung des Grenzwert-Konzepts in der Mathematik schlechthin. Auf Dinge wie z.B. Berechnung von Extrem- und Wendepunkten gehen wir hier nicht näher ein – dies wird im Matheunterricht ja auch bis zum Umfallen eingeübt. Allerdings stellen wir ausführlich die Ableitungsregeln bereit, die wir in Kapitel 8 benötigen werden.

## General Distortion Theorem for Univalent Functions with Quasiconformal Extension

### Complex Analysis and Operator Theory (2017-10-01) 11: 1491-1501 , October 01, 2017

One of the long-standing problems in the quasiconformal theory is finding sharp distortion bounds for *k*-quasiconformal maps for arbitrary
$$k <1$$
. We provide a general distortion theorem for univalent functions in arbitrary quasiconformal disks with *k*-quasiconformal extensions to
$$\mathbb {C}$$
giving a universal power bound. Generically, this power cannot be strengthened.

## Strong convergence of an extragradient-type algorithm for the multiple-sets split equality problem

### Journal of Inequalities and Applications (2017-02-28) 2017: 1-11 , February 28, 2017

This paper introduces a new extragradient-type method to solve the multiple-sets split equality problem (MSSEP). Under some suitable conditions, the strong convergence of an algorithm can be verified in the infinite-dimensional Hilbert spaces. Moreover, several numerical results are given to show the effectiveness of our algorithm.

## A Method and an Algorithm to Reconstruct the Spatial Structure of Current Density Vectors in Magnetocardiography

### Cybernetics and Systems Analysis (2017-05-01) 53: 485-494 , May 01, 2017

The spatial distribution of parameter values (measured at points of the plane of observation) of the magnetic field of the human heart is associated with the distribution of current density vectors in the plane parallel to the plane of measurements and secant relative to the heart. The inverse problem is solved with the help of the Fourier integral transform. The algorithm is modeled based on real data of magnetometric investigations of the human heart.

## ESL-SELO: A robust image denoising algorithm with penalty

### Acta Mathematicae Applicatae Sinica, English Series (2017-07-01) 33: 753-770 , July 01, 2017

Robust image recovery methods have been attracted more and more attention in recent decades for its good property of tolerating system errors or measuring noise. In this paper, we propose a new robust method (ESL-SELO) to recover nosing image, which combine exponential loss function and seamless-L0 (SELO) penalty function to guarantee both accuracy and robustness of the estimator. Theoretical result showed that our method has a local optimal solution and good asymptotic properties. Finally, we compare our method with other methods in simulation which shows better robustness and takes much less time.

## Analytic knots, satellites and the 4-ball genus

### Mathematische Zeitschrift (2017-06-01) 286: 263-290 , June 01, 2017

Call a smooth knot (or smooth link) in the unit sphere in
$$\mathbb {C}^2$$
analytic (respectively, smoothly analytic) if it bounds a complex curve (respectively, a smooth complex curve) in the complex ball. Let *K* be a smoothly analytic knot. For a small tubular neighbourhood of *K* we give a sharp lower bound for the 4-ball genus of analytic links *L* contained in it.

## Acyclic Chromatic Index of Triangle-free 1-Planar Graphs

### Graphs and Combinatorics (2017-07-01) 33: 859-868 , July 01, 2017

An acyclic edge coloring of a graph *G* is a proper edge coloring such that every cycle is colored with at least three colors. The acyclic chromatic index
$$\chi _{a}'(G)$$
of a graph *G* is the least number of colors in an acyclic edge coloring of *G*. It was conjectured that
$$\chi '_{a}(G)\le {\varDelta }(G) + 2$$
for any simple graph *G* with maximum degree
$${\varDelta }(G)$$
. A graph is *1-planar* if it can be drawn on the plane such that every edge is crossed by at most one other edge. In this paper, we show that every triangle-free 1-planar graph *G* has an acyclic edge coloring with
$${\varDelta }(G) + 16$$
colors.

## Well-Posedness of the Green–Lindsay Variational Problem of Dynamic Thermoelasticity

### Journal of Mathematical Sciences (2017-10-01) 226: 11-27 , October 01, 2017

On the basis of the Green–Lindsay initial-boundary-value problem of thermoelasticity, we formulate the corresponding variational problem in terms of displacements and temperature. Sufficient conditions of regularity of the initial data of the problem and the uniqueness of its solution are established from the energy equation of the variational problem. To prove the existence of the generalized solution (and simultaneously, as the first step to a well-justified procedure for finding its approximation), we use the method of Galerkin semidiscretization with respect to the spatial variables and show that the limit of the sequence of its approximations is the solution of the Green–Lindsay variational problem.

## Reihen

### Höhere Mathematik für Naturwissenschaftler und Ingenieure (2017-01-01): 191-264 , January 01, 2017

### Zusammenfassung

Im vorangegangenen Kapitel wurden TAYLOR-Polynome, d.h. Summen aus endlich vielen Potenzfunktionen, als Mittel zur Approximation von hinreichend oft differenzierbaren Funktionen behandelt. Im folgenden Kapitel sollen nun Summen mit ”unendlich” vielen Summanden betrachtet werden. Solche Summen werden auch Reihen genannt.

## A Note on Generalized Lagrangians of Non-uniform Hypergraphs

### Order (2017-03-01) 34: 9-21 , March 01, 2017

Set
$A\subset {\mathbb N}$
is less than
$B\subset {\mathbb N}$
in the *colex ordering* if *m**a**x*(*A*△*B*)∈*B*. In 1980’s, Frankl and Füredi conjectured that the *r*-uniform graph with *m* edges consisting of the first *m* sets of
${\mathbb N}^{(r)}$
in the colex ordering has the largest Lagrangian among all *r*-uniform graphs with *m* edges. A result of Motzkin and Straus implies that this conjecture is true for *r*=2. This conjecture seems to be challenging even for *r*=3. For a hypergraph *H*=(*V*,*E*), the set *T*(*H*)={|*e*|:*e*∈*E*} is called the *edge type* of *H*. In this paper, we study non-uniform hypergraphs and define *L*(*H*) a generalized Lagrangian of a non-uniform hypergraph *H* in which edges of different types have different weights. We study the following two questions: 1. Let *H* be a hypergraph with *m* edges and edge type *T*. Let *C*_{m,T} denote the hypergraph with edge type *T* and *m* edges formed by taking the first *m* sets with cardinality in *T* in the colex ordering. Does *L*(*H*)≤*L*(*C*_{m,T}) hold? If *T*={*r*}, then this question is the question by Frankl and Füredi. 2. Given a hypergraph *H*, find a minimum subhypergraph *G* of *H* such that *L*(*G*) = *L*(*H*). A result of Motzkin and Straus gave a complete answer to both questions if *H* is a graph. In this paper, we give a complete answer to both questions for {1,2}-hypergraphs. Regarding the first question, we give a result for {1,*r*_{1},*r*_{2},…,*r*_{l}}-hypergraph. We also show the connection between the generalized Lagrangian of {1,*r*_{1},*r*_{2},⋯ ,*r*_{l}}-hypergraphs and {*r*_{1},*r*_{2},⋯ ,*r*_{l}}-hypergraphs concerning the second question.