Hilbert's Grand Hotel with a series twist

This paper presents a new twist on a familiar paradox, linking seemingly disparate ideas under one roof. Hilbert's Grand Hotel, a paradox which addresses infinite set comparisons is adapted and extended to incorporate ideas from calculus – namely infinite series. We present and resolve several variations, and invite the reader to explore his or her own variations.     

Mathematical model on malicious attacks in a mobile wireless network with clustering

A mathematical model has been formulated for the analysis of a wireless epidemic on a clustered heterogeneous network. The model introduces mobility into the epidemic framework assuming that the component nodes have a tendency to be attached with a frequently visited home cluster. This underlines the inherent regularity in the mobility pattern of mobile nodes in a wireless network. The analysis focuses primarily on features that arise because of the mobility considerations compared in the larger scenario formed by the epidemic aspects. A result on the invariance of the home cluster populations with respect to time provides an important view-point of the long-term behavior of the system. The analysis also focuses on obtaining a basic threshold condition that guides the epidemic behavior of the system. Analytical as well as numerical results have also been obtained to establish the asymptotic behavior of the connected components of the network, and that of the whole network when the underlying graph turns out to be irreducible. Applications to proximity based attacks and to scenarios with high cluster density have also been outlined.     

Algebraic attacks on a class of stream ciphers with unknown output function

Attacks on linear feedback shift register (LFSR) based cryptosystems typically assume that all the system details except the initial state of the LFSR are known. We address the problem assuming that the nonlinear output function is also unknown and frame the problem as one of a multivariate interpolation. The solution to this problem yields a system that produces an output identical to that of the original system with some other initial state. The attack needs to observe M bits of data and has complexity O(M ^ω) where ${M = \sum_{i=0}^{d} C(n, i)}$ is the number of monomials of degree at most d in n variables, n being the state size and d the degree of the output function. When the output function has annihilators of degree e < d then with O(D) bits of data one can reconstruct parts of the keystream where ${D = \sum_{i=0}^{e} C(n, i)}$ .     

Homoclinic twist bifurcations with ℤ2 symmetry

Summary We analyze bifurcations occurring in the vicinity of a homoclinic twist point for a generic two-parameter family of ℤ₂ equivariant ODEs in four dimensions. The results are compared with numerical results for a system of two coupled Josephson junctions with pure capacitive load.     

Geometric modeling and validation of twist drills with a generic point profile

Traditionally, twist drills with a few specific point geometry, such as planar, conical, cylindrical, ellipsoidal or hyperboloidal, have been designed and adapted for specific applications. Using CAD, the point geometry can be given a generic definition which will enhance the freedom to design drills with different point profiles and optimize them for multiple objectives. Such a definition can also be used for several downstream applications. This paper presents a methodology to model the twist drills with generic point geometry using NURBS. To begin with, a detailed basic model for a fluted twist drill with sectional geometry made up of arcs and straight lines has been presented in terms of bi-parametric surface patches. The coordinates of cutting lips and chisel edge of the drill have been obtained as solution to a surface-curve intersection problem using optimization algorithm. Subsequently, the model has been generalized by employing NURBS to represent the curves whereby the cutting edges and angles can be altered simply by changing the control points or their respective weights. Using this methodology, the generic definitions of the conventional angles on the drill point have been derived and presented. The proposed model has been illustrated in MATLAB environment and validated experimentally for a conical and an arbitrary point geometry. The experiments show a good conformity with the theoretical evaluations.     

On an effective model describing a layered periodic elastic medium with slide contacts on the interfaces

Effective models are derived for layered periodic elastic media'with slide contacts on all interfaces. In the case where each period consists of n layers with different plate velocities, the effective model has n phases. These models are investigated for typical media. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210, 1994, pp. 192–212. Translated by L. A. Molotkov.     

On the commutator length of a Dehn twist

We show that on a nonorientable surface of genus at least 7 any power of a Dehn twist is equal to a single commutator in the mapping class group and the same is true, under additional assumptions, for the twist subgroup, and also for the extended mapping class group of an orientable surface of genus at least 3.     

The optimal conical twist of a delta wing

The problem of reducing the drag of a wing at a specified lift in a supersonic flow is investigated. A solution for a delta wing is obtained in a simplified formulation of the optimization problem and a theoretical analysis. It is shown that the optimal conical wing is formed by elements of elliptical cones and planes. Numerical modelling of the flow of a non-viscous non-heat-conducting gas past the wing is performed, and the results of the theoretical analysis and direct optimization are compared. ©2012     

Sharp Sobolev inequalities in the presence of a twist

Let be a smooth compact Riemannian manifold of dimension . Let also be a smooth symmetrical positive -tensor field in . By the Sobolev embedding theorem, we can write that there exist such that for any ,

where is the standard Sobolev space of functions in with one derivative in . We investigate in this paper the value of the sharp in the equation above, the validity of the corresponding sharp inequality, and the existence of extremal functions for the saturated version of the sharp inequality.

     

Reflections and calculations on a prey-predator-patch problem

This paper is concerned with models for the interaction of plants, herbivores and their predators. We concentrate on situations in which local colonies of herbivores either over-exploit their host plant or are driven to extinction by predators. Starting from a complicated structured model, in which the local prey and predator density within patches is taken into account, we use time scale arguments to derive a three dimensional system of ordinary differential equations. The simplified system is analysed and the existence of multiple stable steady states is demonstrated.     

On data complexity of distinguishing attacks versus message recovery attacks on stream ciphers

We revisit the different approaches used in the literature to estimate the data complexity of distinguishing attacks on stream ciphers and analyze their inter-relationships. In the process, we formally argue which approach is applicable (or not applicable) in what scenario. To our knowledge, this is the first kind of such an exposition. We also perform a rigorous statistical analysis of the message recovery attack that exploits a distinguisher and show that in practice there is a significant gap between the data complexities of a message recovery attack and the underlying distinguishing attack. This gap is not necessarily determined by a constant factor as a function of the false positive and negative rate, as one would expect. Rather this gap is also a function of the number of samples of the distinguishing attack. We perform a case study on RC4 stream cipher to demonstrate that the typical complexities for message recovery attack inferred in the literature are but under-estimates and the actual estimates are quite larger.     

Practical key recovery attacks on FlexAEAD

Designs, Codes and Cryptography - FlexAEAD&nbsp;&nbsp;is a block cipher candidate submitted to the NIST Lightweight Cryptography standardization project, based on repeated application of an...     

Reflections as a challenge

Reflections on mathematics-based actions and practices bring an ethical dimension to the notion of reflection, and this is the aspect I consider and develop in this essay. I elaborate on the notion of reflection by addressing eight different issues. (1)The necessity of reflection emerges from the observation that mathematics-based actions do not have any intrinsic link to progress by virtue of being mathematics-based. Such actions can be as complex and as questionable as any other actions. (2) Although reflections, from this perspective, are believed to be necessary, one could cite afunctionality of non-reflection. For example, non-reflection enables the school mathematics tradition to continue to ensure that the future labour force has particular competencies in the right measures to match the social order for which they are destined. (3) Reflections often presupposespecificity, as they include general as well as specific reconsiderations with respect to some knowledge, actions and practices. (4) I usecollectivity of reflections to refer to the observation that ethical considerations can be facilitated through interaction and communication. Often this presupposes that challenging questions be formulated in order to open up the ethical dimension with respect to mathematics in action. (5) Reflections presuppose directedness and involvement, and this brings me to analyse theintentionality of reflections. (6) Reflections can address very many different issues, which leads me to recognise thediversity of reflections. (7) It is easy to ignore or to obstruct reflections, and when reflections emerge, they can easily be eliminated from an educational context. We should never ignore thefragility of reflections. (8) This brings me to recognise theuncertainty of reflection. Reflections cannot rely on any solid foundation. Still, I find that reflections are necessary.     

Reflections on a New Approach to Gittins Indexation

Traditional approaches to stochastic resource allocation problems (including the classical multi-armed bandit problems) have usually made use of dynamic programming (DP) methodology, perhaps buttressed by further ad hoc arguments. While such approaches seem ‘natural’ they have usually proved technically very difficult. Bertsimas and Niño-Mora have recently given a radically new account of many important results in this area which relate to Gittins indices. The key to their approach is in the characterisation of the region of achievable performance. The optimisation problems of interest are then solved as linear programs over this region. Here we exploit elements within the Bertsimas and Niño-Mora framework (in particular, its capacity to give formulae for the total return of a given policy in closed form) to obtain (i) a simple dynamic programming proof of the optimality of Gittins index policies and (ii) a range of index-based suboptimality bounds for general policies for a variety of stochastic models for resource allocation.     

Visibility with Multiple Reflections

We show that the region lit by a point light source inside a simple n -gon after at most k reflections off the boundary has combinatorial complexity O(n ^2k ) , for any k≥ 1 . A lower bound of Ω ((n/k-Θ(1)) ^2k ) is also established which matches the upper bound for any fixed k . A simple near-optimal algorithm for computing the illuminated region is presented, which runs in O(n ^2k log n) time and O(n ^2k ) space for k>1 , and in O(n ² log ² n) time and O(n ² ) space for k=1 . Received March 14, 1996, and in revised form December 22, 1997, and January 5, 1998.     

Cover attacks for elliptic curves with cofactor two

For cryptographic applications, in order to avoid a reduction of the discrete logarithm problem via the Chinese remainder theorem, one usually considers elliptic curves over finite fields whose order is a prime times a small so-called cofactor c. It is, however, possible to attack specific curves with this property via dedicated attacks. Particularly, if an elliptic curve \(E/\mathbb {F}_{q^n}\) is given, one might try to use the idea of cover attacks to reduce the problem to the corresponding problem in the Jacobian of a curve of genus \(g \ge n\) over \(\mathbb {F}_q\). In the given situation, the only attack so far which follows this idea is the GHS attack, this attack requires that the cofactor c is divisible by 4 as otherwise the genus of the resulting curve is too large. We present an algorithm for finding genus 3 hyperelliptic covers for the case \(c=2\). The construction works in odd characteristic and the resulting cover map has degree 3. As an application, two explicit examples of elliptic curves whose order are respectively 2 times a 149-bit prime and 2 times a 256-bit prime vulnerable to the attack are given.