Diophantine equations as a context for technology-enhanced training in conjecturing and proving

Solving indeterminate algebraic equations in integers is a classic topic in the mathematics curricula across grades. At the undergraduate level, the study of solutions of non-linear equations of this kind can be motivated by the use of technology. This article shows how the unity of geometric contextualization and spreadsheet-based amplification of this topic can provide a discovery experience for prospective secondary teachers and information technology students. Such experience can be extended to include a transition from a computationally driven conjecturing to a formal proof based on a number of simple yet useful techniques.

Dynamic modelling of long-term care decisions

This paper describes and analyzes research on the dynamics of long-term care and the policy relevance of identifying the sources of persistence in caregiving arrangements (including the effect of dynamics on parameter estimates, implications for family welfare, parent welfare, child welfare, and cost of government programs). We discuss sources and causes of observed persistence in caregiving arrangements including inertia/state dependence (confounded by unobserved heterogeneity) and costs of changing caregivers. We comment on causes of dynamics including learning/human capital accumulation; burnout; and game-playing. We suggest how to deal with endogenous geography; dynamics in discrete and continuous choices; and equilibrium issues (multiple equilibria, dynamic equilibria). We also present an overview of commonly used longitudinal data sets and evaluate their relative advantages/disadvantages. We also discuss other data issues related to noisy measures of wealth and family structure. Finally, we suggest some methods to handle econometric problems such as endogeneous geography. © 2014 Springer Science+Business Media New York.

Two dynamic discrete choice estimation problems and simulation method solutions

This paper considers two problems that frequently arise in dynamic discrete choice problems but have not received much attention with regard to simulation methods. The first problem is how to simulate unbiased simulators of probabilities conditional on past history. The second is simulating a discrete transition probability model when the underlying dependent variable is really continuous. Both methods work well relative to reasonable alternatives in the application discussed. However, in both cases, for this application, simpler methods also provide reasonably good results.

Approximate solutions to stochastic dynamic programs

This paper examines the properties of various approximation methods for solving stochastic dynamic programs in structural estimation problems. The problem addressed is evaluating the expected value of the maximum of available choices. The paper shows that approximating this by the maximum of expected values frequently has poor properties. It also shows that choosing a convenient distributional assumptions for the errors and then solving exactly conditional on the distributional assumption leads to small approximation errors even if the distribution is misspecified. © 1997 Cambridge University Press.

Cryptanalysis of block ciphers with overdefined systems of equations

Several recently proposed ciphers, for example Rijndael and Serpent, are built with layers of small S-boxes interconnected by linear key-dependent layers. Their security relies on the fact, that the classical methods of cryptanalysis (e.g. linear or differential attacks) are based on probabilistic characteristics, which makes their security grow exponentially with the number of rounds N r r. In this paper we study the security of such ciphers under an additional hypothesis: the S-box can be described by an overdefined system of algebraic equations (true with probability 1). We show that this is true for both Serpent (due to a small size of S-boxes) and Rijndael (due to unexpected algebraic properties). We study general methods known for solving overdefined systems of equations, such as XL from Eurocrypt’00, and show their inefficiency. Then we introduce a new method called XSL that uses the sparsity of the equations and their specific structure. The XSL attack uses only relations true with probability 1, and thus the security does not have to grow exponentially in the number of rounds. XSL has a parameter P, and from our estimations is seems that P should be a constant or grow very slowly with the number of rounds. The XSL attack would then be polynomial (or subexponential) in N r> , with a huge constant that is double-exponential in the size of the S-box. The exact complexity of such attacks is not known due to the redundant equations. Though the presented version of the XSL attack always gives always more than the exhaustive search for Rijndael, it seems to (marginally) break 256-bit Serpent. We suggest a new criterion for design of S-boxes in block ciphers: they should not be describable by a system of polynomial equations that is too small or too overdefined.

A dynamic Web agent for verifying the security and integrity of a Web site's contents

To harness safe operation of Web-based systems in Web environments, we propose an SSPA (Server-based SHA-1 Page-digest Algorithm) to verify the integrity of Web contents before the server issues an HTTP response to a user request. In addition to standard security measures, our Java implementation of the SSPA, which is called the Dynamic Security Surveillance Agent (DSSA), provides further security in terms of content integrity to Web-based systems. Its function is to prevent the display of Web contents that have been altered through the malicious acts of attackers and intruders on client machines. This is to protect the reputation of organisations from cyber-attacks and to ensure the safe operation of Web systems by dynamically monitoring the integrity of a Web site's content on demand. We discuss our findings in terms of the applicability and practicality of the proposed system. We also discuss its time metrics, specifically in relation to its computational overhead at the Web server, as well as the overall latency from the clients' point of view, using different Internet access methods. The SSPA, our DSSA implementation, some experimental results and related work are all discussed

Amplitude equations and asymptotic expansions for multi-scale problems

In this paper we introduce a new technique to obtain the slow-motion dynamics in nonequilibrium and singularly perturbed problems characterized by multiple scales. Our method is based on a straightforward asymptotic reduction of the order of the governing differential equation and leads to amplitude equations that describe the slowly-varying envelope variation of a uniformly valid asymptotic expansion. This may constitute a simpler and in certain cases a more general approach toward the derivation of asymptotic expansions, compared to other mainstream methods such as the method of Multiple Scales or Matched Asymptotic expansions because of its relation with the Renormalization Group. We illustrate our method with a number of singularly perturbed problems for ordinary and partial differential equations and recover certain results from the literature as special cases. © 2010 - IOS Press and the authors. All rights reserved.

Examples illustrating the use of renormalization techniques for singularly perturbed differential equations

With nine examples, we seek to illustrate the utility of the Renormalization Group approach as a unification of other asymptotic and perturbation methods.

Reduction of amplitude equations by the renormalization group approach

This article elucidates and analyzes the fundamental underlying structure of the renormalization group (RG) approach as it applies to the solution of any differential equation involving multiple scales. The amplitude equation derived through the elimination of secular terms arising from a naive perturbation expansion of the solution to these equations by the RG approach is reduced to an algebraic equation which is expressed in terms of the Thiele semi-invariants or cumulants of the eliminant sequence { Zi } i=1 . Its use is illustrated through the solution of both linear and nonlinear perturbation problems and certain results from the literature are recovered as special cases. The fundamental structure that emerges from the application of the RG approach is not the amplitude equation but the aforementioned algebraic equation. © 2008 The American Physical Society.

The renormalization group and the implicit function theorem for amplitude equations

This article lays down the foundations of the renormalization group (RG) approach for differential equations characterized by multiple scales. The renormalization of constants through an elimination process and the subsequent derivation of the amplitude equation [Chen, Phys. Rev. E 54, 376 (1996)] are given a rigorous but not abstract mathematical form whose justification is based on the implicit function theorem. Developing the theoretical framework that underlies the RG approach leads to a systematization of the renormalization process and to the derivation of explicit closed-form expressions for the amplitude equations that can be carried out with symbolic computation for both linear and nonlinear scalar differential equations and first order systems but independently of their particular forms. Certain nonlinear singular perturbation problems are considered that illustrate the formalism and recover well-known results from the literature as special cases. © 2008 American Institute of Physics.

Secular series and renormalization group for amplitude equations

We have developed a technique that circumvents the process of elimination of secular terms and reproduces the uniformly valid approximations, amplitude equations, and first integrals. The technique is based on a rearrangement of secular terms and their grouping into the secular series that multiplies the constants of the asymptotic expansion. We illustrate the technique by deriving amplitude equations for standard nonlinear oscillator and boundary-layer problems. © 2008 The American Physical Society.

Improvement of a dynamic accumulator at ICICS 07 and its application in multi-user keyword-based retrieval on encrypted data

A dynamic accumulator is an algorithm, which merges a large set of elements into a constant-size value such that for an element accumulated, there is a witness confirming that the element was included into the value, with a property that accumulated elements can be dynamically added and deleted into/from the original set. Recently Wang et al. presented a dynamic accumulator for batch updates at ICICS 2007. However, their construction suffers from two serious problems. We analyze them and propose a way to repair their scheme. We use the accumulator to construct a new scheme for common secure indices with conjunctive keyword-based retrieval.

Dynamic self-organization phenomena in complex ionized gas systems : new paradigms and technological aspects

An overview of dynamic self-organization phenomena in complex ionized gas systems, associated physical phenomena, and industrial applications is presented. The most recent experimental, theoretical, and modeling efforts to understand the growth mechanisms and dynamics of nano- and micron-sized particles, as well as the unique properties of the plasma-particle systems (colloidal, or complex plasmas) and the associated physical phenomena are reviewed and the major technological applications of micro- and nanoparticles are discussed. Until recently, such particles were considered mostly as a potential hazard for the microelectronic manufacturing and significant efforts were applied to remove them from the processing volume or suppress the gas-phase coagulation. Nowadays, fine clusters and particulates find numerous challenging applications in fundamental science as well as in nanotechnology and other leading high-tech industries.

Keyword field-free conjunctive keyword searches on encrypted data and extension for dynamic groups

We consider the following problem: a user stores encrypted documents on an untrusted server, and wishes to retrieve all documents containing some keywords without any loss of data confidentiality. Conjunctive keyword searches on encrypted data have been studied by numerous researchers over the past few years, and all existing schemes use keyword fields as compulsory information. This however is impractical for many applications. In this paper, we propose a scheme of keyword field-free conjunctive keyword searches on encrypted data, which affirmatively answers an open problem asked by Golle et al. at ACNS 2004. Furthermore, the proposed scheme is extended to the dynamic group setting. Security analysis of our constructions is given in the paper.

Lower partial moment and information entropy-based dynamic electricity purchasing strategy

In the electricity market environment, load-serving entities (LSEs) will inevitably face risks in purchasing electricity because there are a plethora of uncertainties involved. To maximize profits and minimize risks, LSEs need to develop an optimal strategy to reasonably allocate the purchased electricity amount in different electricity markets such as the spot market, bilateral contract market, and options market. Because risks originate from uncertainties, an approach is presented to address the risk evaluation problem by the combined use of the lower partial moment and information entropy (LPME). The lower partial moment is used to measure the amount and probability of the loss, whereas the information entropy is used to represent the uncertainty of the loss. Electricity purchasing is a repeated procedure; therefore, the model presented represents a dynamic strategy. Under the chance-constrained programming framework, the developed optimization model minimizes the risk of the electricity purchasing portfolio in different markets because the actual profit of the LSE concerned is not less than the specified target under a required confidence level. Then, the particle swarm optimization (PSO) algorithm is employed to solve the optimization model. Finally, a sample example is used to illustrate the basic features of the developed model and method.