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## External automorphisms of ultraproducts of finite models

### Archive for Mathematical Logic (2012-05-01) 51: 433-441 , May 01, 2012

Let
$${\fancyscript{L}}$$
be a finite first-order language and
$${\langle{\fancyscript{M}_n} \,|\, {n < \omega}\rangle}$$
be a sequence of finite
$${\fancyscript{L}}$$
-models containing models of arbitrarily large finite cardinality. If the intersection of less than continuum-many dense open subsets of Cantor Space ^{ω}2 is non-empty, then there is a non-principal ultrafilter
$${\fancyscript{U}}$$
over *ω* such that the corresponding ultraproduct
$${\prod_\fancyscript{U}\fancyscript{M}_n}$$
has an automorphism that is not induced by an element of
$${\prod_{n<\omega}{\rm Aut}(\fancyscript{M}_n)}$$
.

## On the convergence of augmented Lagrangian methods for nonlinear semidefinite programming

### Journal of Global Optimization (2012-11-01) 54: 599-618 , November 01, 2012

In this paper, we present new convergence properties of the augmented Lagrangian method for nonlinear semidefinite programs (NSDP). Convergence to the approximately global solutions and optimal values of NSDP is first established for a basic augmented Lagrangian scheme under mild conditions, without requiring the boundedness condition of the multipliers. We then propose four modified augmented Lagrangian methods for NSDP based on different algorithmic strategies. We show that the same convergence of the proposed methods can be ensured under weaker conditions.

## Nonexpansive retracts and weak compatible pairs in metric Spaces

### Fixed Point Theory and Applications (2012-06-20) 2012: 1-10 , June 20, 2012

In this article, we introduce a new approach to common fixed point theory for a weak compatible pair. We first introduce the concepts of *R*-pair and *NR*-pair and establish some new common fixed point theorems for a weak compatible pair in hyperconvex metric spaces and uniformly convex metric spaces. We shall also establish the well-known De Marr's theorem for a family of weak compatible pairs in a hyperconvex metric space [0,1] and show by an example that it fails to hold in general hyperconvex metric spaces.

*Mathematics Subject Classification (2000)*: Primary 06F30, 46B20, 47E10.

## An efficient label setting/correcting shortest path algorithm

### Computational Optimization and Applications (2012-01-01) 51: 437-455 , January 01, 2012

We design a new label shortest path algorithm by applying the concept of a *pseudo permanent label*. This approach allows an algorithm to partition the set of nodes into two new sets: pseudo permanently labeled nodes and its complementary set. From this point of view, this new label method can be considered as a label setting method. Moreover, at least one node becomes permanently labeled when some nodes which belong to the set of pseudo permanently labeled nodes are scanned in each iteration of the algorithm. In the case of networks with non-negative length arcs it is easy to prove that this node has the minimum distance label among the non-pseudo permanently labeled nodes. On the other hand, it is not known during the computation which pseudo permanently labeled nodes are permanently labeled. Therefore, all distance labels are temporary and the algorithm becomes a label correcting method. Nevertheless, the proposed algorithm exhibits some nice features, such as: (1) the time bound for the running of the algorithm for a network with *n* nodes and *m* arcs is O(*nm*); (2) the number of node scan operations in the algorithm is less than the number of these operations in the previous label correcting algorithms as is observed in the computational experience; (3) the algorithm incorporates two new rules which allow easy detection of a negative cycle in the network; (4) the algorithm is quite simple and very easy to implement, and does not require sophisticated data structures; (5) the algorithm exhibits flexibility in the order in which the new pseudo permanently labeled nodes are scanned. The above features are possible through the application of the pseudo permanent label concept.

## The existence of strongly computable representations in the class of Boolean algebras

### Doklady Mathematics (2012-07-01) 86: 469-471 , July 01, 2012

## Degenerate scale problem in antiplane elasticity or Laplace equation for contour shapes of triangles or quadrilaterals

### Applied Mathematics and Mechanics (2012-04-01) 33: 525-538 , April 01, 2012

This paper provides several solutions to the degenerate scale for the shapes of triangles or quadrilaterals in an exterior boundary value problem (BVP) of the antiplane elasticity or the Laplace equation. The Schwarz-Christoffel mapping is used throughout. It is found that a complex potential with a simple form in the mapping plane satisfies the vanishing displacement condition (or *w*=0) along the boundary of the unit circle when the dimension *R* reaches its critical value 1. This means that the degenerate size in the physical plane is also achieved. The degenerate scales can be evaluated from the particular integrals depending on certain parameters in the mapping function. The numerical results of degenerate sizes for the shapes of triangles or quadrilaterals are provided.

## Front Matter - Einstein and the Changing Worldviews of Physics

### Einstein and the Changing Worldviews of Physics (2012-01-01): 12 , January 01, 2012

## Theorems about the Partition Function on Pages 189 and 182

### Ramanujan's Lost Notebook (2012-01-01): 181-204 , January 01, 2012

Each of these two isolated pages has a connection with Ramanujan’s famous paper in which he gives the first proofs of the congruences *p*(5*n*+4)≡0 (mod 5) and *p*(7*n*+5)≡0 (mod 7). One of Ramanujan’s proofs hinges upon the beautiful identity
$$\sum_{n=0}^{\infty}p(5n+4)q^n = 5 \frac{(q^5;q^5)_{\i}^5}{(q;q)_{\infty}^6}, \qquad |q|<1, $$
which is given on page 189. We provide a more detailed rendition of the proof given by Ramanujan, as well as a similarly beautiful identity yielding the congruence *p*(7*n*+5)≡0 (mod 7). On both pages, Ramanujan examines the more general partition function *p*_{r}(*n*) defined by
$$ \frac{1}{(q;q)_{\infty}^r}=\sum_{n=0}^{\infty}p_r(n)q^n, \qquad |q|<1. $$
In particular, he states new congruences for *p*_{r}(*n*).