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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.

## 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 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.

## The Matrix Exponential Function

### Positive Operator Semigroups (2017-01-01) 257: 43-53 , January 01, 2017

We continue our investigation of the asymptotic behavior of dynamical systems described by matrices, which was started in last chapter, now moving to the continuous time case. This means that we investigate the asymptotic properties of the matrix exponential function.

## Resource constrained scheduling with general truncated job-dependent learning effect

### Journal of Combinatorial Optimization (2017-02-01) 33: 626-644 , February 01, 2017

Scheduling with general truncated job-dependent learning effect and resource-dependent processing times is studied on a single machine. It is assumed that the job processing time is a function of the amount of resource allocated to the job, the general job-dependent learning effect and the job-dependent control parameter. For each version of the problem that differs in terms of the objective functions and the processing time functions, the optimal resource allocation is provided. Polynomial time algorithms are also developed to find the optimal schedule of several versions of the problem.

## SL(n) Invariant Valuations on Polytopes

### Discrete & Computational Geometry (2017-04-01) 57: 571-581 , April 01, 2017

A classification of $${\text {SL}}(n)$$ invariant valuations on the space of convex polytopes in $$\mathbb {R}^n$$ without any continuity assumptions is established. A corresponding result is obtained on the space of convex polytopes in $$\mathbb {R}^n$$ that contain the origin.

## Compact differences of weighted composition operators on the weighted Bergman spaces

### Journal of Inequalities and Applications (2017-01-03) 2017: 1-14 , January 03, 2017

In this paper, we consider the compact differences of weighted composition operators on the standard weighted Bergman spaces. Some necessary and sufficient conditions for the differences of weighted composition operators to be compact are given, which extends Moorhouse’s results in (J. Funct. Anal. 219:70-92, 2005).

## A Diophantine approach to the three and four exponentials conjectures

### The Ramanujan Journal (2017-01-01) 42: 199-221 , January 01, 2017

Some special cases of Wilkie’s conjecture are shown to be equivalent to real versions of the three and four exponentials conjectures. Wilkie’s conjecture is an open problem originating in model theory that concerns the density of algebraic points in sets defined using the exponential function; the latter conjectures concern the algebraic nature of values of the exponential function.

## Projection Operators

### Convex Analysis and Monotone Operator Theory in Hilbert Spaces (2017-01-01): 535-560 , January 01, 2017

Let *C* be a nonempty closed convex subset of
$$\mathcal{H}$$
. The projection *P*_{C}*x* of a point
$$x \in \mathcal{H}$$
onto *C* is characterized by (see Theorem
3.16
) or, equivalently, by (see Proposition
6.47
) *x* − *P*_{C}*x* ∈ *N*_{C}(*P*_{C}*x*). In this chapter, we investigate further the properties of projectors and provide a variety of examples.

## Asymptotics for the finite-time ruin probability of a risk model with a general counting process

### Japan Journal of Industrial and Applied Mathematics (2017-04-01) 34: 243-252 , April 01, 2017

This paper considers a general risk model. In this risk model there does not need independent or dependent assumption on the inter-arrival times. When the claim sizes have the WUOD structure, the asymptotics of the finite-time ruin probability of this risk model have been given. The obtained results extend and improve some existed results.