How many atoms can be defined by boxes?

We study the functionb(n, d), the maximal number of atoms defined byn d-dimensional boxes, i.e. parallelopipeds in thed-dimensional Euclidean space with sides parallel to the coordinate axes. We characterize extremal interval families definingb(n, 1)=2n-1 atoms and we show thatb(n, 2)=2n ²-6n+7. We prove that for everyd, exists and . Moreover, we obtainb*(3)=8/9.     

How many normal measures can ℵ carry?

Relative to the existence of a supercompact cardinal with a measurable cardinal above it, we show that it is consistent for ?₁ to be regular and for ? to be measurable and to carry precisely τ normal measures, where τ ≥ ? is any regular cardinal. This extends the work of [2], in which the analogous result was obtained for ?_{ω +1} using the same hypotheses (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

How good can polynomial interpolation on the sphere be?

This paper explores the quality of polynomial interpolation approximations over the sphere S ^r−1⊂R ^r in the uniform norm, principally for r=3. Reimer [17] has shown there exist fundamental systems for which the norm ‖Λ _n ‖ of the interpolation operator Λ _n , considered as a map from C(S ^r−1) to C(S ^r−1), is bounded by d _n , where d _n is the dimension of the space of all spherical polynomials of degree at most n. Another bound is d _n ^1/2(λ_avg/λ_min)^1/2, where λ_avg and λ_min are the average and minimum eigenvalues of a matrix G determined by the fundamental system of interpolation points. For r=3 these bounds are (n+1)² and (n+1)(λ_avg/λ_min)^1/2, respectively. In a different direction, recent work by Sloan and Womersley [24] has shown that for r=3 and under a mild regularity assumption, the norm of the hyperinterpolation operator (which needs more point values than interpolation) is bounded by O(n ^1/2), which is optimal among all linear projections. How much can the gap between interpolation and hyperinterpolation be closed?

For interpolation the quality of the polynomial interpolant is critically dependent on the choice of interpolation points. Empirical evidence in this paper suggests that for points obtained by maximizing λ_min, the growth in ‖Λ _n ‖ is approximately n+1 for n<30. This choice of points also has the effect of reducing the condition number of the linear system to be solved for the interpolation weights. Choosing the points to minimize the norm directly produces fundamental systems for which the norm grows approximately as 0.7n+1.8 for n<30. On the other hand, ‘minimum energy points’, obtained by minimizing the potential energy of a set of (n+1)² points on S ², turn out empirically to be very bad as interpolation points.

This paper also presents numerical results on uniform errors for approximating a set of test functions, by both interpolation and hyperinterpolation, as well as by non-polynomial interpolation with certain global basis functions.

     

How artificial rain can be produced? A mathematical model

A non-linear mathematical model for rain making from water vapor in the atmosphere is proposed and analyzed. The model considers the process of artificial rain by introducing two kinds of aerosol particles conducive to nucleation of cloud droplets and formation of rain drops. The model analysis shows that, for uninterrupted rain, the water vapor in the atmosphere must be formed continuously with the required rate of rainfall. It is shown further that the intensity of rainfall increases as the concentrations of externally introduced aerosols, as well as the density of water vapor in the atmosphere, increase. Numerical simulation is also performed to see the effect of various parameters on the process of artificial rain making leading to rainfall.     

How many quasiplatonic surfaces?

We show that the number of isomorphism classes of quasiplatonic Riemann surfaces of genus ≦ g has a growth of type The number of non-isomorphic regular dessins of genus ≦ g has the same growth type. Received: 8 February 2005; revised: 2 May 2005     

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 can histograms be useful for introducing continuous probability distributions?

ZDM – Mathematics Education - The teaching of probability has changed a great deal since the end of the last century. The development of technologies is indeed part of this evolution. In...     

How many different cascades on a surface can have coinciding hyperbolic attractors?

It is shown that the number of essentially nonconjugate (i.e., not being iterations of topologically conjugate) diffeomorphisms of a surface having homeomorphic one-dimensional hyperbolic attractors can be arbitrarily large, provided that the genus of the surface is large enough. A lower bound for this number depending on the surface genus is given. The corresponding result for pseudo-Anosov homeomorphisms is stated.     

How far can relaxation functions be increasing in viscoelastic problems?

It is known that when we add a viscoelastic damping to a frictional damping acting in the domain we might lose the property of exponential stability of the system. Moreover, a necessary condition for a system to be sub-exponentially stable is that the kernel itself must be sub-exponentially decaying to zero. Having this in mind, a natural question to be asked is that of when this necessary condition is also sufficient. We prove that this is the case for a fairly large class of kernels.     

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 to find many counterfeit coins?

We propose an algorithm for findingm defective coins, that uses at most + 15m weighings on a balance scale, wheren is the number of all coins.     

How many lottery tickets to buy?

We investigate the optimal number of tickets an expected-utility-maximizing individual who participate in a lottery will buy. We show that the expected utility is not always unimodal in the number of tickets. We also show that a risk-averse individual will not buy more than one ticket. (S)he will be willing to own a share in several tickets only if that share is sufficiently small.     

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 large can a hereditarily separable or hereditarily Lindelöf space be?

The main result of this paper is that ifV satisfies GCH andω ₁<λ<μ are arbitrary regular cardinals, then in some cardinal preserving forcing extensionW ofV we have λ^<λ=λ=2 ^N 0,μ=2 ^λ and there are a hereditarily separableX ?2 ^λ with |X|= \(2^{\aleph _0 } \) =μ and a hereditarily LindelöfY ?2 ^μ withw(Y)= \(2^{2^\aleph 0} \) =μ. So far similar results have only been obtained under the assumption of CH.