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In modern information technology, research on block cipher security is imperative. Concerning the ultra lightweight block cipher PICO, there has been only one study focused on recovering its complete master key, with a large search space of 2 64 , and no fault analysis yet. This paper proposes a new fault analysis approach, combining differential fault and algebraic equation techniques. It achieved the recovery of PICO's entire master key with 40 faults in an average time of 0.57 h. S-box decomposition was utilized to optimize our approach, reducing the time by a remarkable 75.83% under the identical 40-fault condition. Furthermore, PICO's complete master key could be recovered with 28 faults in an average time of 0.78 h, indicating a significant 2 37 reduction in its search space compared to the previous study. This marks the first fault analysis on PICO. Compared to conventional fault analysis methods DFA (differential fault analysis) and AFA (algebraic fault analysis), our approach outperforms in recovering PICO's entire master key, highlighting the cruciality of key expansion complexity in block cipher security. Therefore, our approach could serve to recover master keys of block ciphers with comparably complicated key expansions, and production of more secure block ciphers could result. [ABSTRACT FROM AUTHOR]

We study the following problem: given a regular symmetric form (linear functional) v, find all the regular symmetric forms u which satisfy the equation x 3 u = λ x v , λ ∈ C − { 0 }. We give the second-order recurrence relation of the orthogonal polynomial sequence with respect to u. Moreover, in the case where v is a semi-classical form of class s, we show that u is semi-classical and its class is analyzed in term of the class of v. An example is highlighted. [ABSTRACT FROM AUTHOR]

We consider the family of the irreducible algebraic equations of the Bour's minimal surfaces for integers m = 2, 3, 4 in the three dimensional Euclidean space, and obtain its degrees in cartesian coordinates, classes in inhomogeneous coordinates. We also reveal its integral free representations. Moreover, we visualise the equations of the surface family. [ABSTRACT FROM AUTHOR]

Algebraic persistent fault analysis (APFA), which combines algebraic analysis with persistent fault attacks, brings new challenges to the security of lightweight block ciphers and has received widespread attention since its introduction. Threshold Implementation (TI) is one of the most widely used countermeasures for side channel attacks. Inspired by this method, the SKINNY block cipher adopts the S_box decomposition to reduce the number of variables in the set of algebraic equations and the number of Conjunctive Normal Form (CNF) equations in this paper, thus speeding up the algebraic persistent fault analysis and reducing the number of fault ciphertexts. In our study, we firstly establish algebraic equations for full-round faulty encryption, and then analyze the relationship between the number of fault ciphertexts required and the solving time in different scenarios (decomposed S_boxes and original S_box). By comparing the two sets of experimental results, the success rate and the efficiency of the attack are greatly improved by using S_box decomposition. In this paper, We can recover the master key in a minimum of 2000s using 11 pairs of plaintext and fault ciphertext, while the key recovery cannot be done in effective time using the original S_box expression equations. At the same time, we apply S_box decomposition to another kind of algebraic persistent fault analysis, and the experimental results show that using S_box decomposition can effectively reduce the solving time and solving success rate under the same conditions. [ABSTRACT FROM AUTHOR]

Recently, Hao and Li [Fully coupled forward-backward SDEs involving the value function. Nonlocal Hamilton–Jacobi–Bellman equations, ESAIM: Control Optim, Calc. Var.22 (2016) 519–538] studied a new kind of forward-backward stochastic differential equations (FBSDEs), namely the fully coupled FBSDEs involving the value function in the case where the diffusion coefficient σ in forward stochastic differential equations depends on control, but does not depend on z. In our paper, we generalize their work to the case where σ depends on both control and z , which is called the general fully coupled FBSDEs involving the value function. The existence and uniqueness theorem of this kind of equations under suitable assumptions is proved. After obtaining the dynamic programming principle for the value function W , we prove that the value function W is the minimum viscosity solution of the related nonlocal Hamilton–Jacobi–Bellman equation combined with an algebraic equation. [ABSTRACT FROM AUTHOR]

It is known that the immediate basins of attraction of zeros of a complex polynomial f(z) with simple zeros under the iteration of family of iterations, i.e., Schröder's formula of the second kind (König's formula or the Basic Family) with the convergence degree m ≥ 2, converge to the Voronoi diagram of the zeros as m → ∞ . In a previous work, we consider a region, i.e., the intersection of the regions surrounded by the circles of Apollonius for zeros and show that for each zero, this region included in the immediate basin of attraction of the zero converges to the Voronoi cell of the zero as m → ∞ . Here, for a polynomial f(z) with multiple zeros, by extending our previous work, we show how the immediate basins of attraction of the family to the rational function f (z) / f ′ (z) , whose zeros are simple, converge to the Voronoi diagram. Honorato has shown a similar result, but our approach is more direct and quantitative. Numerical examples illustrate the relationship between the regions above, the immediate basins of attraction and the Voronoi cells, and show that the region above monotonically enlarges to the Voronoi cell. [ABSTRACT FROM AUTHOR]

The numerical solution of systems of linear ordinary differential equations with nonlocal nonlinear conditions depending on the values of the desired function at intermediate points is investigated. Conditions for the existence of a solution to the problem under consideration are given. For the numerical solution, an approach is proposed that reduces the problem to two auxiliary linear systems of differential equations with linear conditions and to a single nonlinear algebraic system with a dimension depending only on the number of given intermediate points. The proposed approach is illustrated by solving two problems. The auxiliary problems are solved analytically in one of them and numerically in the other. [ABSTRACT FROM AUTHOR]

Schröder's iterative formula of the second kind (S2 formula) for finding zeros of a function f(z) is a generalization of Newton's formula to an arbitrary order m of convergence. For iterative formulae, convergence regions of initial values to zeros in the complex plane z are essential. From numerical experiments, it is suggested that as order m of the S2 formula grows, the complicated fractal structure of the boundary of convergence regions gradually diminishes. We propose a method of estimating the convergence regions with the circles of Apollonius to verify this result for polynomials f(z) with simple zeros. We indeed show that as m grows, each region surrounded by the circles of Apollonius monotonically enlarges to the Voronoi cell of a zero of f(z). Numerical examples illustrate convergence regions for several values of m and some polynomials. [ABSTRACT FROM AUTHOR]

This manuscript is concerning to investigate numerical solutions for different classes including parabolic, elliptic and hyperbolic partial differential equations of arbitrary order (PDEs). The proposed technique depends on some operational matrices of fractional order differentiation and integration. To compute the mentioned operational matrices, we apply shifted Jacobi poly- nomials in two dimension. Thank to these matrices, we convert the PDE under consideration to an algebraic equation which is can be easily solved for unknown coefficient matrix required for the numerical solution. The proposed method is very efficient and need no discretization of the data for the proposed PDE. The approximate solution obtain via this method is highly accurate and the computation is easy. The proposed method is supported by solving various examples from well known articles. [ABSTRACT FROM AUTHOR]

Real‐time estimation techniques are presented to estimate solar irradiance and photovoltaic (PV) module temperature simultaneously from maximum power point condition. An algebraic equation which is function of PV output voltage and current measurements is utilised to estimate solar radiation. A non‐linear model‐based technique of immersion and invariance is employed to derive an update low for module temperature estimation. It is shown that the estimated temperature globally asymptotically converges, where its accuracy depends on solar radiation estimator. The proposed algorithms are accurate, fast, and easy to implement at optimal operating conditions of PV modules. [ABSTRACT FROM AUTHOR]

This paper deals with Mellin–Barnes integral, representing solution to thenth degree algebraic equation. We prove convergence of this integral in the boundary points of its convergence domain. The cases of tetranomial and pentanomial equations are completely studied. [ABSTRACT FROM PUBLISHER]

We consider a general reduced algebraic equation of degree n with complex coefficients. The solution to this equation, a multifunction, is called a general algebraic function. In the coefficient space we consider the discriminant set ∇ of the equation and choose in its complement the maximal polydisk domain D containing the origin. We describe the monodromy of the general algebraic function in a neighborhood of D. In particular, we prove that ∇ intersects the boundary ∂D along n real algebraic surfaces $S^{(j)} $ of dimension n − 2. Furthermore, every branch y( x) of the general algebraic function ramifies in D only along the pair of surfaces $S^{(j)} $ and $S^{(j - 1)} $. [ABSTRACT FROM AUTHOR]

□ We present a new approach to carry out transient analysis for Markov modulated fluid flows. The system of partial differential equations for the time-dependent distributions is transformed into a system of ordinary differential equations by Laplace transform. This system of ordinary differential equations is solved by means of decomposing its coefficient matrix into two parts, one with eigenvalues with negative real parts and the other with positive real parts. The results we obtained are equivalent, but in simpler form, to those of Ahn and Ramaswami, which are derived by probabilistic arguments. [ABSTRACT FROM PUBLISHER]

Resultants are important special functions used to describe nonlinear phenomena. The resultant $$R_{r_1 \ldots r_n }$$ determines a consistency condition for a system of n homogeneous polynomials of degrees r, ..., r in n variables in precisely the same way as the determinant does for a system of linear equations. Unfortunately, there is a lack of convenient formulas for resultants in the case of a large number of variables. In this paper we use Cauchy contour integrals to obtain a polynomial formula for resultants, which is expected to be useful in applications. [ABSTRACT FROM AUTHOR]

Abstract: Rimming flow of a non-Newtonian fluid on the inner surface of a horizontal rotating cylinder is investigated. Simple lubrication theory is applied since the Reynolds number is small and liquid film is thin. For the steady-state flow of a power-law fluid the mathematical model reduces to a simple algebraic equation regarding the thickness of the liquid film. The qualitative analysis of this equation is carried out and the existence of two possible solutions is rigorously proved. Based on this qualitative analysis, different regimes of the rimming flow are defined and analyzed analytically. For the particular case, when the flow index in a power-law constitutive equation is equal to 1/2, the problem reduces to the fourth order algebraic equation which is solved analytically by Ferrari method. [Copyright &y& Elsevier]

Abstract: The present paper illustrates an iterative numerical method to solve nonlinear equations of the form f(x)=0, especially those containing the partial and non partial involvement of transcendental terms. Comparative analysis shows that the present method is faster than Newton–Raphson method, hybrid iteration method, new hybrid iteration method and others. Cost is also found to be minimum than these methods. The beauty in our method can be seen because of the optimization in important effecting factors, i.e. lesser number of iteration steps, lesser number of functional evaluations and lesser value of absolute error in final as well as in individual step as compared to the other methods. This work also demonstrates the higher order convergence of the present method as compared to others without going to the computation of second derivative. [Copyright &y& Elsevier]

Abstract: A piecewise truncated quadratic speed trajectory is proposed to mimic the unknown speed trajectory between point detectors. The basis functions of the new method consist of quadratic and constant functions of time. The constant functions, corresponding to upper and lower speed bounds, are determined using the maximum likelihood estimates of highest and lowest speeds that have been historically observed within a time interval. The purpose of setting a lower (upper) speed bound for simulating vehicle speed trajectory is to mimic a low (high) average speed during transition flow and congestion, and to restrict a quadratic speed trajectory to be within a realistic speed range, respectively. It was found that travel time estimation using different approaches is similar during free-flow conditions but significantly different during transition flow and congestion conditions. Using the actual travel time obtained from field experiment, the new method yields more accurate travel time estimation than other trajectory-based methods. Compared to travel time estimation using speed and density information, the new method only needs speed measurements, and therefore, it is more robust and easier to implement in practice than density-based methods. Computational implementation of the new trajectory method is tractable and can be done very efficiently, making it suitable for on-line real time travel time estimation. [Copyright &y& Elsevier]

In this paper, fast Fourier transform and complex analysis are used to analyze the dynamic response of slabs on a viscoelastic foundation caused by a moving harmonic load. Critical speed and resonance frequency of the slab to a moving harmonic load are obtained analytically. It is proved that there exists a bifurcation in critical speed. One branch of Critical speed increases as load frequency increases, while the other branch of critical speed decreases as load frequency increases. There are two critical speeds when the load frequency is low, but only one critical speed exists when the load frequency is high. A parametric study is also performed to study the effect of load speed, load frequency, material properties of the slab and the damping coefficient on dynamic response. It is found that the damping coefficient has significant influence on dynamic response. For small damping, the maximum response of the slab increases with increased load speed and frequency. However, for large damping, the maximum response of the slab decreases with increased load speed and frequency. [ABSTRACT FROM AUTHOR]

We propose a method for constructing systems of polynomial equations that define submanifolds of degenerate binary forms of an arbitrary degeneracy degree. [ABSTRACT FROM AUTHOR]

Abstract: The present paper considers the existence of continuous roots of algebraic equations with coefficients being continuous functions defined on compact Hausdorff spaces. For a compact Hausdorff space X, denotes the Banach algebra of all continuous complex-valued functions on X with the sup norm . The algebra is said to be algebraically closed if each monic algebraic equation with coefficients has a root in . First we study a topological characterization of a first-countable compact (connected) Hausdorff space X such that is algebraically closed. The result has been obtained by Countryman Jr, Hatori–Miura and Miura–Niijima and we provide a simple proof for metrizable spaces. Also we consider continuous approximate roots of the equation with respect to z, where , and provide a topological characterization of compact Hausdorff space X with such that the above equation has an approximate root in for each , in terms of the first Čech cohomology of X. [Copyright &y& Elsevier]

Abstract: In this paper, a new iteration scheme is proposed to solve the roots of an algebraic equation f(x)=0. Given an initial guess, x 0, the roots of the equation can be obtained using the following iteration scheme:This iteration scheme has unique convergence characteristics different from the well-known Newton’s method. It is shown that this iteration method has cubic local convergence in the neighborhood of the root. Using this scheme, real or complex roots for specific algebraic equations can be found. Because there are two iteration directions, for a given initial guess, two solutions can be found for certain algebraic equations with multiple roots. Examples are presented and compared with other methods. [Copyright &y& Elsevier]

We obtain an integral formula for a solution to a general algebraic equation. In this formula the integrand is an elementary function and integration is carried out over an interval. The advantage of this formula over the well-known Mellin formula is that the integral has a broader convergence domain. This circumstance makes it possible to describe the monodromy of a solution for trinomial equations. [ABSTRACT FROM AUTHOR]

In this paper we introduce a new algorithm based on Adomian method to solve algebraic equations. [Copyright &y& Elsevier]

We further develop our recent direct method for the optimized effective potential (OEP) in density functional theory (DFT) [Yang and Wu, Phys. Rev. Lett.89, 143002 (2002)]. First, we show that the stationary condition in our optimization approach leads to a proper nonlinear algebraic equation for the OEP in a finite basis set, which differs from other finite basis set approaches. Then by constructing an approximate second derivative matrix of the energy functional in conjunction with the use of the Newton method, we significantly accelerate the convergence of the iterative optimization for OEP. Enhancement of the method is made in using the Tikhonov regularization method for the inversion of the second derivative matrix when it is singular or nearly singular and the direct inversion in the iterative space. It is shown that under a fixed stepsize condition, the optimization approach is equivalent to the self-consistent solution to the nonlinear algebraic equation for OEP. Because the approximate second derivatives are easy to compute and the iteration numbers are small now, the computation costs of OEP become comparable to that of regular DFT calculations as shown by calculations of some molecules, small and larger ones. We show how to find balanced results between energies and potentials when choosing a basis set for potentials. [ABSTRACT FROM AUTHOR]

The Weibull model is frequently used to describe non-linear microbial survival due to its simplicity and flexibility. An essential part of developing a predictive microbiological approach is to accurately estimate survival parameters. In this study, an explicit integral representation of the Weibull model was derived from its original differential form. Calculated microbial survival curves under the same conditions were identical when both forms of the equation were used. The integral equation was able to be simplified into an explicit algebraic equation under specific processing conditions that consisted of a come-up time (CUT) followed by a lethal agent having a constant magnitude. These conditions are found in microbial survival tests in which currently used methods ignore or incorporate CUT using an empirical approach to estimate microbial survival parameters. Using the proposed method, microbial survival parameters were accurately estimated by fitting the obtained explicit algebraic equation to microbial survival data, whereas the conventional method resulted in inaccurate parameter estimation. • An explicit integral form of the Weibull model was derived from its differential form. • The integral can be simplified to an algebraic equation under conditions with CUTs. • The algebraic equation can be used to accurately estimate the survival parameters. [ABSTRACT FROM AUTHOR]