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H~introduced the concept of continuous convergence of a sequence of continuous functions around 1920 (el. [7]). CARATH~ODORY subsequently used this concept in connection with normal families of holomorphie functions in [4], where he also states: ,,Mein Vorsehlag geht nun dahin, jedesmal, wo es vorteilhaft ist --und es ist, wie ich glaube, mit ganz wenigen Ausnahmen immer vorteilhaft --, den Begriff tier gleichm/~l~igen Konvergenz (i.e. compact convergence) in der Funktionentheorie durch den Begriff der ,stetigen Konvergenz' zu ersetzen, den H. HAH~ vor einigen Jahren in die Mathematik eingefiihrt hat und dessen Handhabung unvergle;chlieh einfacher ist." In 1955 H. SCHAEFER generalized the notion of continuous convergence to spaces of maps from a topological space to a mctrie space (cf. [8]) and obtained a characterization of this kind of convergence by means of pointwise convergence and the (local) "oscillation" of functions. He defined a normal family N as a set of maps such that every infinite subset of 2q contains a continuously convergent subsequence. He then showed that a normal family on a compact space :s a compact set of continuous functions (relative to uniform convergence) with the exception of a countable number of elements. This result is closely related to the "Ascoli theorem" we prove in section 3 in a much more general context.In this paper we show first of all that at least the following advantage is obtained when continuous convergence is used rather than compact convergence: Continuous convergence on a collection of continuous maps from one topological space to another is the coarsest "admissible convergence structure" without any further assumptions on the topologies of the two spaces. It is well-known that, in general, no such "admissible" topology exists (el. e.g. [1]).Consequently, continuous convergence will not be a topology, in general, but it always is a convergence structure (Limitierung, el. [6]).We shall make free use of the notations, results and the terminology of [6]. F(X) will always denote the set of all filters on the set X. In addition, we shall use the following conventions: If / is a function on X and ~-E F (X), then / (~-) is the filter generated by the sets ] (F), 2' E ~'. The symbol [{F}] will be used for the filter generated by the filter basis {F}; when {F} reduces to {{x}} for somex E X, then [{{x}}] will be denoted by ~; ~ is the ultrafilter of all subsets of X

Ising computing provides a new computing paradigm for many hard combinatorial optimization problems. Ising computing essentially tries to solve the quadratic unconstrained binary optimization problem, which is also described by the Ising spin glass model and is also the basis for so-called Quantum Annealing computers. In this work, we propose a novel General Purpose Graphics Processing Unit (GPGPU) solver for the balanced min-cut graph partitioning problem, which has many applications in the area of design automation and others. Ising model solvers for the balanced min-cut partitioning problem have been proposed in the past. However, they have rarely been demonstrated in existing quantum computers for many meaningful problem sizes. One difficulty is the fact that the balancing constraint in the balanced min-cut problem can result in a complete graph in the Ising model, which makes each local update a global update. Such global update from each GPU thread will diminish the efficiency of GPU computing, which favors many localized memory accesses for each thread. To mitigate this problem, we propose an novel Global Decoupled Ising (GDI) model and the corresponding annealing algorithm, in which the local update is still preserved to maintain the efficiency. As a result, the new Ising solver essentially eliminates the need for the fully connected graph and will use a more efficient method to track and update global balance without sacrificing cut quality. Experimental results show that the proposed Ising-based mincut partitioning method outperforms the state of art partitioning tool, METIS, on G-set graph benchmarks in terms of partitioning quality with similar CPU/GPU times.

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