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 matrices can be spectrally balanced simultaneously?

We prove that any ? positive definite d × d matrices, M₁,...,M_?, of full rank, can be simultaneously spectrally balanced in the following sense: for any k < d such that ? ≤ \(\ell \leqslant \left\lfloor {\frac{{d - 1}}{{k - 1}}} \right\rfloor \), there exists a matrix A satisfying \(\frac{{{\lambda _1}\left( {{A^T}{M_i}A} \right)}}{{Tr\left( {{A^T}{M_i}A} \right)}} < \frac{1}{k}\) 1/k for all i, where λ₁(M) denotes the largest eigenvalue of a matrix M. This answers a question posed by Peres, Popov and Sousi ([PPS13]) and completes the picture described in that paper regarding sufficient conditions for transience of self-interacting random walks. Furthermore, in some cases we give quantitative bounds on the transience of such walks.     

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 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 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 can a cause-of-death reduction be compensated for by the population heterogeneity? A dynamic approach

In the context of widening socioeconomic inequalities in mortality, it has become crucially important to understand the impact of population heterogeneity and its evolution on mortality. In particular, recent developments in multi-population mortality have raised a number of questions, among which is the issue of evaluating cause-of-death reduction targets set by national and international institutions in the presence of heterogeneity. The aim of this paper is to show how the population dynamics framework contributes to addressing these issues, relying on English population data and cause-specific number of deaths by socioeconomic circumstances, over the period 1981–2015.The analysis of the data first highlights the complexity of recent demographic developments, characterized by significant compositional changes, with considerable variations according to the age class or cohort, along with a widening of socioeconomic inequalities. We then introduce a dynamic framework for studying the impact of composition changes on the mortality of the global population. In particular, we show how a cause-of-death reduction could be compensated for in the presence of heterogeneity, which could lead to misinterpretations when assessing public policies impacts and/or for the forecasting of future trends.     

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 can emphasizing mathematical modeling principles benefit students in a traditionally taught differential equations course?

This paper reports a mixed-methods, causal-comparative study of two instructional approaches in a differential equations course for engineering undergraduates. One classroom emphasized decontextualized techniques to solve canonical differential equations. The other classroom emphasized modeling principles to derive and interpret canonical differential equations as models of real world phenomena. Both classrooms were lecture based. This paper presents a brief review of both texts and compares a lesson on the same topic from each classroom in order to illustrate what it means to emphasize modeling principles. An ANCOVA was used to compare student performance on common final exam items while controlling for prior mathematics achievement. The main effect for instructional approach was statistically significant suggesting that a modeling perspective for teaching differential equations aided student learning. This quasi-experimental study corroborates previously reported recommendations for improving teaching and learning of differential equations.     

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.     

What can be done with PRA?

Let S⁺ denote system JR _° ² + AC _° ¹ in a classical second-order arithmetic, in which the induction rule is permitted to apply only to quantifier-free formulas and to _° ² -formulas not containing functional variables, while the convolution axiom is permitted to apply only to _° ¹ -formulas without functional variables. Also postulated is the closedness of the function class being examined, relative to primitive recursive operations. System, S⁺ turns out to be sufficiently rich: in it a theory of recursions and an elementary recursive analysis can be developed, a theorem on the continuity of effective operators and a theorem on cuteliminability from -deductions can be proved, and the usual analytic proofs of many number-theoretic theorems, including the prime distribution law, can be derived (with insignificant changes). (A formalization in S⁺ of the proof of Konig's lemma on paths in binary trees and of Godel's completeness theorem is described in the note.) On the other hand, the system admits of an interpretation in primitive recursive arithmetic (PRA). In particular, quantifier-free theorems in S⁺ are deducible in PRA, while theorems of form xyR(x,y) with a quantifier-free formula R have calculi R (x,(x)) with primitive recursive function , deducible in PRA. Thus, the suppressing part of operating constructive analysis can be developed already at the finite stages of the Shanin majorant hierarchy. In addition, a purely mechanical method exists for obtaining elementary number-theoretic proofs from many analytic proofs.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 60, pp. 93–102, 1976. Results announced September 4, 1975.     

A remark on “A nonlinear mathematical model of the corneal shape”

An analytical method using Taylor series is proposed to solve a nonlinear two-point boundary problem arising in corneal shape. The solution process makes it extremely easy to obtain a relatively accurate solution. The pencil-and-paper solution procedure can be extended to other boundary value problems.     

A Bus Driver Scheduling Problem: a?new mathematical model and a GRASP approximate solution

This paper addresses the problem of determining the best scheduling for Bus Drivers, a $\mathcal{NP}$ -hard problem consisting of finding the minimum number of drivers to cover a set of Pieces-Of-Work (POWs) subject to a variety of rules and regulations that must be enforced such as spreadover and working time. This problem is known in literature as Crew Scheduling Problem and, in particular in public transportation, it is designated as Bus Driver Scheduling Problem. We propose a new mathematical formulation of a Bus Driver Scheduling Problem under special constraints imposed by Italian transportation rules. Unfortunately, this model can only be usefully applied to small or medium size problem instances. For large instances, a Greedy Randomized Adaptive Search Procedure (GRASP) is proposed. Results are reported for a set of real-word problems and comparison is made with an exact method. Moreover, we report a comparison of the computational results obtained with our GRASP procedure with the results obtained by Huisman et al. (Transp. Sci. 39(4):491?C502, 2005).     

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