Interactive generalizing of geometric configurations

The heuristic method of generalizing in the sense of theorem finding through conceptual generalization is a productive scheme of cognition which, in the field of synthetic elementary geometry, can be acquired and practised by the student in a direct way. With the computer as an interactive construction tool motivating and efficient possibilities arise to realize these methods.     

On topological and geometric (194) configurations

An $\left(n_k\right)$ configuration is a set of  $n$ points and  $n$ lines such that each point lies on  $k$ lines while each line contains  $k$ points. The configuration is geometric, topological, or combinatorial depending on whether lines are considered to be straight lines, pseudolines, or just combinatorial lines. The existence and enumeration of $\left(n_k\right)$ configurations for a given  $k$ has been subject to active research. A current front of research concerns geometric $\left(n_4\right)$ configurations: it is now known that geometric $\left(n_4\right)$ configurations exist for all  $n\ge 18$, apart from sporadic exceptional cases. In this paper, we settle by computational techniques the first open case of $\left(19_4\right)$ configurations: we obtain all topological $\left(19_4\right)$ configurations among which none are geometrically realizable.     

On the geometric form of free boundaries satisfying a bernoulli condition

In the x–y plane, let Ω denote an annular domain bounded by the simple closed curves ? and ?^*, such that the capacity potential U in Ω satisfies ∣ ▽ U∣ = 1 on ?. Assuming ?^* to be known, we obtain qualitative properties of ?. For example, the number of local maxima or minima of the y-co-ordinate relative to ? cannot exceed the corresponding number for ?^*.     

On the geometric form of solutions of a free boundary problem involving galvanization

In [6], T. I. Vogel studied a free boundary problem originating in the galvanization process. He showed that if the given boundary Γ* is starlike or convex, then so is the free boundary solution Γ. Our purpose is to generalize Vogel's second result by showing (under certain assumptions) that Γ cannot have more (local) maxima or minima (relative to a given direction) than Γ*; also that Γ cannot have more inflection points or greater total curvature than Γ*. The author has already proven analogous results for the Bernoulli free boundary problem in [1], [2] and [3].     

On the suprema of Bernoulli processes

In this note we announce the affirmative solution of the so-called Bernoulli Conjecture concerning the characterization of the sample boundedness of Bernoulli processes. We also present some applications and discuss related open problems.     

On the Bernoulli two-armed bandit problem

The paper is initially concerned with monotonic properties of the posterior success probabilities when the prior success probabilities are distributed according to an arbitrary joint distribution function (Bayesian approach). Next a dynamic programming model is proposed and monotonic properties of the optimal expected cumulative discounted reward are proved. Finally, optimality properties are given for the case when one prior success probability is known.     

On the permanent of random Bernoulli matrices

We show that the permanent of an n×n matrix with iid Bernoulli entries ±1 is of magnitude with probability 1−o(1). In particular, it is almost surely non-zero.     

On the symbolic calculus of Bernoulli convolutions

A probability measure onT whose symbolic calculus is minimal is constucted.     

On Bernoulli and Euler numbers

The main purpose of this paper is to investigate some basic relations (e.g., Voronoi's and Kummer's congruences) of Bernoulli and Euler numbers by manipulating Euler factors in a natural way.     

On some inequalities for the Bernoulli numbers

In modo elementare si migliorano sensibilmente alcune diseguaglianze recentemente stabilite da D. J. Leeming sui numeri di Bernoulli [3].     

On a formal analogue of the Bernoulli numbers

Using properties of the modular forms G_k, it is shown that G_K(z) = α_kω^2k where z is an integer in an imaginary quadratic field and α_k is algebraic and involves analogues of the Bernoulli numbers. A recursion formula (3) is given for these numbers.     

On convergence of configurations

The transition rule F of a cellular automaton may sometimes be regarded as a “rule of growth” of a crystal from a “seed” ω. A study is made of the iterates ω, Fω, F²ω,…. For certain growth rules F it is proved that when ω is “sufficiently large” the sequence F^pω “converges” to a rational polytope W. The limiting shape W depends only on F.