Ginzburg–Landau equations on Riemann surfaces of higher genus

We study the Ginzburg–Landau equations on Riemann surfaces of arbitrary genus. In particular, we

• –construct explicitly the (local moduli space of gauge-equivalent) solutions in the neighborhood of the constant curvature ones;
• –classify holomorphic structures on line bundles arising as solutions to the equations in terms of the degree, the Abel–Jacobi map, and symmetric products of the surface;
• –determine the form of the energy and identify when it is below the energy of the constant curvature (normal) solutions.

     

What is the Mathematics in Mathematics Education?

In this paper I tackle the question What is the mathematics in mathematics education? By providing three different frames for the word mathematics.

• 1.Frame 1: Mathematics as an abstract body of knowledge/ideas, the organization of that into systems and structures, and a set of methods for reaching conclusions.

• 2.Frame 2: Mathematics as contextual, ever present, as a lens or language to make sense of the world.

• 3.Frame 3: Mathematics as a verb (not a noun), a human activity, part of one’s identity.

After introducing the frames and examining their distinction and their overlap, I discuss their implication with respect to student-centered classroom, context, and culture.     

Fundamental polytopes of metric trees via parallel connections of matroids

We tackle the problem of a combinatorial classification of finite metric spaces via their fundamental polytopes, as suggested by Vershik (2010).In this paper we consider a hyperplane arrangement associated to every split pseudometric and, for tree-like metrics, we study the combinatorics of its underlying matroid.

• •We give explicit formulas for the face numbers of fundamental polytopes and Lipschitz polytopes of all tree-like metrics.
• •We characterize the metric trees for which the fundamental polytope is simplicial.

     

Forcing axioms via ground model interpretations

We study principles of the form: if a name σ is forced to have a certain property φ, then there is a ground model filter g such that ${\sigma }^g$ satisfies φ. We prove a general correspondence connecting these name principles to forcing axioms. Special cases of the main theorem are:

• •Any forcing axiom can be expressed as a name principle. For instance, $\mathsf{PFA}$ is equivalent to:

  • A principle for rank 1 names (equivalently, nice names) for subsets of ${\omega }_1$.

  • A principle for rank 2 names for sets of reals.
• •λ-bounded forcing axioms are equivalent to name principles. Bagaria's characterisation of $\mathsf{BFA}$ via generic absoluteness is a corollary.

We further systematically study name principles where φ is a notion of largeness for subsets of ${\omega }_1$ (such as being unbounded, stationary or in the club filter) and corresponding forcing axioms.     

Axial transport in dry ball mills

Ball mills are used for grinding of rocks, cement clinker and limestone from 10 to 100 mm feed sizes down to sub-millimetre product. They are typically rotating cylinders with diameters from 3 to 6 m and lengths from 6 to 12 m. The flow of particulate solids within these mills can be modelled using the discrete element method (DEM). Typically, such modelling is done for short durations of a few mill revolutions and either in two dimensions or using thin three-dimensional slices through the center of the mill with periodic boundary conditions in the axial direction. This facilitates an understanding of the radial motion of the charge, estimation of power draw and of liner wear, but it cannot provide information about axial transport within the mill. In this paper, we examine the axial transport in dry ball mills. This requires simulation of the entire mill and the full volume of the charge for significant periods of time (thousands of revolutions). We use a simple model for grate discharge that allows prediction of the time varying axial distribution of different particle sizes within a discharging ball mill. The distributions of sub-grate size ‘fines’ is shown to satisfy a one-dimensional diffusion equation with the diffusion coefficient decreasing with grate size. A pulse test, where a single mass of fines in injected at the feed end, is able to quantify the residence time distribution of the fines.     

Compactness and the axiom of choice

In the absence of the axiom of choice four versions of compactness (A-, B-, C-, and D-compactness) are investigated. Typical results:

1. C-compact spaces form the epireflective hull in Haus of A-compact completely regular spaces.
2. Equivalent are:
3. the axiom of choice,
4. A-compactness = D-compactness,
5. B-compactness = D-compactness,
6. C-compactness = D-compactness and complete regularity,
7. products of spaces with finite topologies are A-compact,
8. products of A-compact spaces are A-compact,
9. products of D-compact spaces are D-compact,
10. powers X ^k of 2-point discrete spaces are D-compact,
11. finite products of D-compact spaces are D-compact,
12. finite coproducts of D-compact spaces are D-compact,
13. D-compact Hausdorff spaces form an epireflective subcategory of Haus,
14. spaces with finite topologies are D-compact.

1. Equivalent are:
2. the Boolean prime ideal theorem,
3. A-compactness = B-compactness,
4. A-compactness and complete regularity = C-compactness,
5. products of spaces with finite underlying sets are A-compact,
6. products of A-compact Hausdorff spaces are A-compact,
7. powers X ^k of 2-point discrete spaces are A-compact,
8. A-compact Hausdorff spaces form an epireflective subcategory of Haus.

1. Equivalent are:
2. either the axiom of choice holds or every ultrafilter is fixed,
3. products of B-compact spaces are B-compact.

1. Equivalent are:
2. Dedekind-finite sets are finite,
3. every set carries some D-compact Hausdorff topology,
4. every T ₁-space has a T ₁-D-compactification,
5. Alexandroff-compactifications of discrete spaces and D-compact.

     

Mean-field stochastic control with elephant memory in finite and infinite time horizon

ABSTRACT

Our purpose of this paper is to study stochastic control problems for systems driven by mean-field stochastic differential equations with elephant memory, in the sense that the system (like the elephants) never forgets its history. We study both the finite horizon case and the infinite time horizon case.

• In the finite horizon case, results about existence and uniqueness of solutions of such a system are given. Moreover, we prove sufficient as well as necessary stochastic maximum principles for the optimal control of such systems. We apply our results to solve a mean-field linear quadratic control problem.

• For infinite horizon, we derive sufficient and necessary maximum principles.

  As an illustration, we solve an optimal consumption problem from a cash flow modelled by an elephant memory mean-field system.

     

Multidimensional symmetric stable processes

This paper surveys recent remarkable progress in the study of potential theory for symmetric stable processes. It also contains new results on the two-sided estimates for Green functions, Poisson kernels and Martin kernels of discontinuous symmetric α-stable process in boundedC ^1,1 open sets. The new results give explicit information on how the comparing constants depend on parameter α and consequently recover the Green function and Poisson kernel estimates for Brownian motion by passing α ↑ 2. In addition to these new estimates, this paper surveys recent progress in the study of notions of harmonicity, integral representation of harmonic functions, boundary Harnack inequality, conditional gauge and intrinsic ultracontractivity for symmetric stable processes. Here is a table of contents.

1. Introduction
2. Green function and Poisson kernel estimates

1. Estimates on balls
2. Estimates on boundedC ^1,1 domains
3. Estimates on boundedC ^1,1 open sets

1. Harmonic functions and integral representation
2. Two notions of harmonicity
3. Martin kernel and Martin boundary
4. Integral representation and uniqueness
5. Boundary Harnack principle
6. Conditional process and its limiting behavior
7. Conditional gauge and intrinsic ultracontractivity

     

Derivationen und Automorphismen von Algebren,in Denen Die Gleichung X2=0 Nur Trivial Lösbar Ist

Let A be a finite-dimensional algebra over a (commutative) field K of characteristic O, assume that x∈A and x²=0 implies x=0. We shall prove among others:

- The derivations and automorphisms of A are semisimple.

- If K is algebraically closed, then Der A=0 and |Aut A|<∞.

- If K=?, then Aut A (and hence Der A) is compact.

     

Immersionen mit paralleler zweiter Fundamentalform: Beispiele und Nicht-Beispiele

1. We show that symmetric R-spaces can. be imbedded in euclidean space with parallel second fundamental tensor (Dα=0).
2. We give a restrictive necessary condition for totally geodesic submanifolds of the Grassmannian to be the Gauss image of an immersion with Dα=0, c.f. [9].

     

Curvature Dimension Inequalities and Subelliptic Heat Kernel Gradient Bounds on Contact Manifolds

We study curvature dimension inequalities for the sub-Laplacian on contact Riemannian manifolds. This new curvature dimension condition is then used to obtain:

• Geometric conditions ensuring the compactness of the underlying manifold (Bonnet–Myers type results);
• Volume estimates of metric balls;
• Gradient bounds and stochastic completeness for the heat semigroup generated by the sub-Laplacian;
• Spectral gap estimates.

     

Planning systems and artificial intelligence

Traditional O.R. systems are compared with problem solving in Artificial Intelligence via Expert Systems. The discussion centers on explicit knowledge representation. The general aspects are illustrated by two planning systems:

1. LESP 2: A learning system for inspection plan generation
2. IDA: A system for finding functions and solutions in construction

     

Remarks on star countable discrete closed spaces

In this paper, we prove the following statements:

1. There exists a Tychonoff star countable discrete closed, pseudocompact space having a regular-closed subspace which is not star countable.
2. Every separable space can be embedded into an absolutely star countable discrete closed space as a closed subspace.
3. Assuming $2^{\aleph _0 } = 2^{\aleph _1 } $ , there exists a normal absolutely star countable discrete closed space having a regular-closed subspace which is not star countable.

     

The shortest augmenting path method for solving assignment problems — Motivation and computational experience

In this paper we discuss the shortest augmenting path method for solving assignment problems in the following respect:

we introduce this basic concept using matching theory

we present several efficient labeling techniques for constructing shortest augmenting paths

we show the relationship of this approach to several classical assignment algorithms

we present extensive computational experience for complete problems, and

we show how postoptimal analysis can be performed using this approach and naturally leads to a new, highly efficient hybrid approach for solving large-scale dense assignment problems

     

A new proof of a theorem of Dembowski

It was shown by P.Dembowski [1;Satz 3] that any finite semiaffine plane(=FSAP) is of the types:

1. A finite affine plane,
2. A finite projective plane with one line and all its points except one deleted,
3. A finite projective plane with one point deleted,
4. A finite projective plane.

This was established by using the results obtained for natural parallelisms of incidence structures. The purpose of this note is to give a new proof based on purely combinatorial arguments.     

Some results on entire functions of finite lower order

Letf(z) be an entire function of order λ and of finite lower order μ. If the zeros off(z) accumulate in the vicinity of a finite number of rays, then

1. λ is finite;
2. for every arbitrary numberk ₁>1, there existsk ₂>1 such thatT(k ₁ r,f)≤k ₂ T(r,f) for allr≥r ₀. Applying the above results, we prove that iff(z) is extremal for Yang's inequalityp=g/2, then
3. every deficient values off(z) is also its asymptotic value;
4. every asymptotic value off(z) is also its deficient value;
5. λ=μ;
6. $\sum\limits_{a \ne \infty } {\delta (a,f) \leqslant 1 - k(\mu ).} $

     

Analysis of performance of an iron-bath reactor using computational fluid dynamics

The performance of an iron-bath reactor has been studied using a comprehensive numerical model that combines a computational fluid dynamics approach for the gas phase and a heat and mass balance model for the bath. The model calculates:

• •coal, ore, flux and oxygen consumption;
• •post-combustion ratio (PCR);
• •heat-transfer efficiency (HTE);
• •off-gas temperature and composition;
• •heat transfer and chemical reactions between gas and iron and slag droplets; and
• •heat transfer between gas and bath, refractories and lance.

The model was validated with data reported by the Nippon Steel Corporation for a 100 t pilot plant, and the calculated and measured data are in good agreement. Modelling results showed that the dominant mechanisms of heat transfer from the gas to the bath are radiation to the slag surface and convection heat transfer to droplets.     

Maximal independent, analytic sets in Abelian polish groups

The paper deals with the questions:

1. whether a topological module admits maximal linearly independent subsets that are analytic
2. whether an Abelian topological group admits maximal independent subsets that are analytic
3. whether a topological field extension admits transcendence bases that are analytic.