The concept ‘model’ and its potential role in mathematics education

Under the general term ‘mathematical models’ is now subsumed a sufficiently diversified collection of distinct types of models to merit categorization and an attempt at a unified definition. In this study a broad ‘working definition’ for the concept ‘model’ is proposed; basic types of models are identified and illustrated, and the impact of conscious instruction of models on mathematics education is analysed.     

On a certain set of Lerch’s zeta-functions and their derivatives∗

Lithuanian Mathematical Journal - For 0 < α ≤ 1 and 0 < λ ≤ 1, let L(α, λ, s) denote the associated Lerch zeta-function, which is defined as $$...     

Chains and Trees: ‘Strong’–‘Weak’ Order in Job Scheduling

We present a summary of recent NP-hardness and polynomial time solvability results for the distinction between strong and weak precedence for chains and trees in scheduling. We distinguish between chains and proper trees which are not chains, and demonstrate that the strong-weak precedence distinction for chains is not inclusive with regards to NP-hardness, and conjecture that the same holds for strong-weak tree precedence. The objective is to show that different interpretations for chain and tree order relations in scheduling might have far reaching computational implications.     

Determinants and Symmetries in ‘Yetter—Drinfeld’ Categories

A Yetter—Drinfeld category over a Hopf algebra H with a bijective antipode, is equipped with a braiding which may be symmetric for some of its subcategories (e.g. when H is a triangular Hopf algebra). We prove that under an additional condition (which we term the u-condition) such symmetric subcategories completely resemble the category of vector spaces over a field k, with the ordinary flip map. Consequently, when Char k=0, one can define well behaving exterior algebras and non-commutative determinant functions.     

Preconditioners and their analyses for edge element saddle-point systems arising from time-harmonic Maxwell’s equations

Numerical Algorithms - We derive and propose a family of new preconditioners for the saddle-point systems arising from the edge element discretization of the time-harmonic Maxwell’s equations...     

Metric kinematic planes and their representation by ��-local systems

Planes with a regularly operating commutative group and with reflections on the lines through a distinguished point o, such that the 3-reflection theorem is valid for any three lines through o, are characterized algebraically as subplanes of locally euclidean planes. A slight weakening of the conditions yields to locally pseudo euclidean planes.     

Hill’s potentials in weighted Sobolev spaces and their spectral gaps

We describe a new, short proof of some facts relating the gap lengths of the spectrum of a potential q of Hill’s equation, −y′′ + qy = λy, to its regularity. For example, a real potential is in a weighted Gevrey-Sobolev space if and only if its gap lengths, γ _n , belong to a similarly weighted sequence space. An extension of this result to complex potentials is proven as well. We also recover Trubowitz results about analytic potentials. The proof essentially employs the implicit function theorem.     

Unique determination of the convolutions of measures in Rm,m⩾2, by their restriction to a set

Restrictions are indicated on a complex-valued measure in the spaceR ^m ,m 2., under which the n-fold convolution ^n*,n 2, is uniquely determined by its values on any semispacex ₁<r, rR.Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 50, pp. 86–90, 1988.     

Simplicial complexes and Macaulay’s inverse systems

Let Δ be a simplicial complex on V = {x ₁, . . . , x _n }, with Stanley–Reisner ideal ${I_{\Delta}\subseteq R=k[x_1,\ldots, x_n]}Let Δ be a simplicial complex on V = {x ₁, . . . , x _n }, with Stanley–Reisner ideal I_D í R=k[x₁,?, x_n]{I_{\Delta}\subseteq R=k[x_1,\ldots, x_n]} . The goal of this paper is to investigate the class of artinian algebras A=A(D,a₁,?,a_n) = R/(I_D,x₁^a₁,?,x_n^a_n){A=A(\Delta,a_1,\ldots,a_n)= R/(I_{\Delta},x_1^{a_1},\ldots,x_n^{a_n})} , where each a _i ≥ 2. By utilizing the technique of Macaulay’s inverse systems, we can explicitly describe the socle of A in terms of Δ. As a consequence, we determine the simplicial complexes, that we will call levelable, for which there exists a tuple (a ₁, . . . , a _n ) such that A(Δ, a ₁, . . . , a _n ) is a level algebra.     

A new example of ‘independence’ and ‘white noise’

Summary We examine the notion of free independence according to Voiculescu. This form of independence is used for defining free white noise or process with stationary and freely independent increments. We prove a general limit theorem giving the combinatorics of infinitely freely divisible states and thus of free white noises with the help of admissible partitions. We realize the free analogues of the Wiener process and of the Poisson process as processes on the full Fock space ofL ² (—).     

‘New mathematics’ and teaching

Inside the scientific world it is not always understood that the mood of mathematics, which is a product and a part of culture, can change with time. This is partly why many have been surprised by the coming of the so‐called new mathematics.

In the truly creative mathematical mind two opposite tendencies coexist: the logical and the imaginative. Apparently it seems that new mathematics can be reduced to a purely logical machinery. In fact it contains as much imaginative contributions as classical mathematics. But it is difficult to show simultaneously the logical sequence of propositions and the clumsy progression of research itself. Mathematical exposition does not always follow the ‘ most natural slopes’ of the mind. Unfamiliar presentations often give an impression of ‘ abstraction ‘, more familiar ones an impression of concreteness ‘.

So it appears that difficulties with new mathematics are mostly of psychological origin. Misuses of it can easily raise up intolerance reactions and emotional blocks. Perhaps insisting upon the fact that, here as elsewhere, it is important to be able to guess, to realize that intuition and imagination are essential, could help to make new mathematics better understood, more useful and more able to be considered as a unifing element among sciences.     

Brownian motion with killing and reflection and the ‘‘hot–spots’’ problem

We investigate the hot–spots property for the survival time probability of Brownian motion with killing and reflection in planar convex domains whose boundary consists of two curves, one of which is an arc of a circle, intersecting at acute angles. This leads to the hot–spots property for the mixed Dirichlet–Neumann eigenvalue problem in the domain with Neumann conditions on one of the curves and Dirichlet conditions on the other.Supported in part by NSF Grant # 9700585-DMSSupported in part by NSF Grant # 0203961-DMS     

Normal systems over ANR’s, rigid embeddings and nonseparable absorbing sets

Most of results of Bestvina and Mogilski [Characterizing certain incomplete infinite-di-mensional absolute retracts. Michigan Math. J., 33, 291-313 (1986)] on strong Z-sets in ANR's and absorbing sets is generalized to nonseparable case. It is shown that if an ANR X is locally homotopy dense embeddable in infinite-dimensional Hilbert manifolds and w(U ) = w(X ) (where "w"is the topological weight) for each open nonempty subset U of X , then X itself is homotopy dense embeddable in a Hilbert manifold. It is also demonstrated that whenever X is an AR, its weak product W (X, ) = {(x n ) ∞ n=1 ∈ X ω : x n = for almost all n} is homeomorphic to a pre-Hilbert space E with E ～ = ΣE. An intrinsic characterization of manifolds modelled on such pre-Hilbert spaces is given.     

Cryptanalytic results on ‘Dual CRT’ and ‘Common Prime’ RSA

In this paper we study weaknesses of two variants of RSA: Dual RSA and Common Prime RSA. Several schemes under the framework of Dual RSA have been proposed by Sun et al. (IEEE Trans Inf Theory 53(8):2922–2933, 2007). We here concentrate on the Dual CRT-RSA scheme and present certain range of parameters where it is insecure. As a corollary of our work, we prove that the Dual Generalized Rebalanced-RSA (Scheme III of Sun et al.) can be efficiently broken for a significant region where the scheme has been claimed to be secure. Next we consider the Common Prime RSA as proposed by Wiener (IEEE Trans. Inf. Theory 36:553–558, 1990). We present new range of parameters in Common Prime RSA where it is not secure. We use lattice based techniques for the attacks.     

On Levitin–Polyak well-posedness and stability in set optimization

Positivity - In this paper, Levitin–Polyak (in short LP) well-posedness in the set and scalar sense are defined for a set optimization problem and a relationship between them is found....