How often can a finite group be realized as a Galois group over a field?

Let G be any finite group and any class of fields. By we denote the minimal number of realizations of G as a Galois group over some field from the class . For G abelian and the class of algebraic extensions of ℚ we give an explicit formula for . Similarly we treat the case of an abelian p-group G and the class which is conjectured to be the class of all fields of characteristic ≠p for which the Galois group of the maximal p-extension is finitely generated. For non-abelian groups G we offer a variety of sporadic results. Received: 27 October 1998 / Revised version: 3 February 1999     

How large dimension guarantees a given angle?

We study the following two problems: (1) Given $n\ge 2$ and $0\le \alpha \le 180^\circ $ , how large Hausdorff dimension can a compact set $A\subset \mathbb{R }^n$ have if $A$ does not contain three points that form an angle $\alpha $ ? (2) Given $\alpha $ and $\delta $ , how large Hausdorff dimension can a subset $A$ of a Euclidean space have if $A$ does not contain three points that form an angle in the $\delta $ -neighborhood of $\alpha $ ? An interesting phenomenon is that different angles show different behaviour in the above problems. Apart from the clearly special extreme angles $0$ and $180^\circ $ , the angles $60^\circ , 90^\circ $ and $120^\circ $ also play special role in problem (2): the maximal dimension is smaller for these angles than for the other angles. In problem (1) the angle $90^\circ $ seems to behave differently from other angles.     

How big can the circuits of a bridge of a maximal circuit be?

IfCE(G) is a maximum cardinality cocircuit of a 2-connected graphG, then no other maximum cocircuit is contained in one and the same block ofG-C. The analogous conjecture for real representable matroids would have important applications to classifying convex bodies with a certain Helly type property.     

How often is a polygon bounded by three sides?

LetL _n be the set of lines (no two parallel) determining ann-sided bounded faceF in the Euclidean plane. We show that the number,f(L _n), of triples fromL _n that determine a triangle containingF satisfies and these bounds are best. This result is generalized tod-dimensional Euclidean space (without the claim that the upper bound is attainable).     

How many integer homogeneous polynomials at small coprime integers have value of a univariate polynomial?

Let p(y) = p _m y ^m + p _m?1 y ^m?1 + ?+ p ₀ ?? $ \mathbb{Z} $ [y] be a polynomial of degree m > 0 in an integer variable. We estimate the number of times it equals some homogeneous polynomial in two variables with integer coefficients, degree at most n, and Euclidean norm at most N evaluated at a pair of small coprime integers (we count this number with the occurring multiplicities). For pairs of coprime integers of absolute value at most $ H<N/\sqrt{n} $ , this estimate is ?? _n,p (H)N ^n+1/m + O(N ^n+1/m?1 H ³ + N ⁿ H ²), where ?? _n,p (H) does not depend on N.     

How many consumer levels can survive in a chemotactic food chain?

We investigate the effect and the impact of predator-prey interactions, diffusivity and chemotaxis on the ability of survival of multiple consumer levels in a predator-prey microbial food chain. We aim at answering the question of how many consumer levels can survive from a dynamical system point of view. To solve this standing issue on food-chain length, first we construct a chemotactic food chain model. A priori bounds of the steady state populations are obtained. Then under certain sufficient conditions combining the effect of conversion efficiency, diffusivity and chemotaxis parameters, we derive the co-survival of all consumer levels, thus obtaining the food chain length of our model. Numerical simulations not only confirm our theoretical results, but also demonstrate the impact of conversion efficiency, diffusivity and chemotaxis behavior on the survival and stability of various consumer levels.      

How regular can the boundary of a quadratic Siegel disk be?

In the family of quadratic polynomials with an irrationally indifferent fixed point, we show the existence of Siegel disks with a fine control on the degree of regularity of the linearizing map on their boundary. A general theorem is stated and proved. As a particular case, we show that in the quadratic family, there are Siegel disks whose boundaries are but not Jordan curves.

     

How often are two permutations comparable?

Two permutations of are comparable in the Bruhat order if one is closer, in a natural way, to the identity permutation, , than the other. We show that the number of comparable pairs is of order at most, and at least. For the related weak order, the corresponding bounds are and , where . In light of numerical experiments, we conjecture that for each order the upper bound is qualitatively close to the actual number of comparable pairs.

     

How strange can an attractor for a dynamical system in a 3-manifold look?

The aim of this paper is to characterise those compact subsets K of 3-manifolds M that are (stable and not necessarily global) attractors for some flow on M. We will show that it is the topology of M−K, rather than that of K, the one that plays a relevant role in this problem.A necessary and sufficient condition for a set K to be an attractor is that it must be an “almost tame” subset of M in a sense made precise under the equivalent notions of “weakly tame” and “tamely embedded up to shape”, defined in the paper. These are complemented by a further equivalent condition, “algebraic tameness”, which has the advantage of being checkable by explicit computation.A final section of the paper is devoted to a partial analysis of the same question when one replaces flows by discrete dynamical systems.     

How Fast can a Subharmonic Function Decay Along a Ray?

Much is known about the connection between the growth and decayof subharmonic functions. The results indicate that there isa general principle: asubharmonic function cannot decay ‘toofast’ relative to its growth.Three theorems are provedwhich, together with work previously published elsewhere, givea fairly complete account of how this principle works out fora subharmonic function having extremal decay along a ray. 1991Mathematics Subject Classification: 30D20, 31A05.     

How best to recover a function from its inaccurately given spectrum?

Consider the problem of optimal recovery of a function and its derivatives on the line from the Fourier transform of the function known approximately on a set of finite measure. We find an optimal recovery method and an optimal set on which we must measure the Fourier transform with given error.     

How many data clusters are in the Galaxy data set?

Advances in Data Analysis and Classification - In model-based clustering, the Galaxy data set is often used as a benchmark data set to study the performance of different modeling approaches. Aitkin...     

The ?ojasiewicz exponent of a set of weighted homogeneous ideals

We give an expression for the ?ojasiewicz exponent of a set of ideals which are pieces of a weighted homogeneous filtration. We also study the application of this formula to the computation of the ?ojasiewicz exponent of the gradient of a semi-weighted homogeneous function (Cⁿ,0)→(C,0) with an isolated singularity at the origin.     

How fast can Maker win in fair biased games?

We study $(a:a)$ Maker–Breaker games played on the edge set of the complete graph on $n$ vertices. In the following four games — perfect matching game, Hamilton cycle game, star factor game and path factor game, our goal is to determine the least number of moves which Maker needs in order to win these games. Moreover, for all games except for the star factor game, we show how first player can win in the strong version of these games.     

How likely is Buffon's ring toss to intersect a planar Cantor set?

In Bateman and Volberg (2008) [1], it was shown that the n-th partial 1/4 Cantor in the plane set decays in Favard length no faster than $C\frac{\log n}{n}$. In Bond and Volberg (2008) [2], the so-called circular Favard length of the same set is studied, and the same estimate is shown to persist when the circle has radius $r?Cn$. By considering characteristic functions, the result of Bond and Volberg (2008) [2] naturally leads to a conjecture which (if true) would imply the sharpness of the $L\log \log L$ boundedness of the circular maximal operator proved by Seeger, Tao and Wright (2005) [3].