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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]

Nowadays, with the development of the Internet of Things and information security technologies, lightweight block ciphers are gradually being widely used. As a side-channel attack method, algebraic fault analysis has received attention from experts and scholars since its introduction. The familiar nonlinear operation in lightweight block ciphers is the S-box substitution, and the performance index of the S-box directly determines the security strength of the cipher. To further improve the efficiency of algebraic fault analysis, this paper proposes a method to rewrite the algebraic equations of S-box substitution by decomposing the original cubic S-boxes into two quadratic S-boxes. The results show that this method is significantly effective compared to the original method in GIFT-64 and SKINNY-64, especially in the SKINNY-64 block cipher, where the average solving time is reduced by several hundred times in the best case with the same samples. At the same time, our best results show that s single-bit fault injection is enough to recover the master key of SKINNY-64. In addition, the PRESENT-64 block cipher is also studied in this paper, and the results show that the method can also improve efficiency significantly. When the location of the single-bit fault is unknown, using the S-box decomposition method for SKINNY-64 can also significantly improve the solving success rate, reduce the number of faults, and speed up the solving. [ABSTRACT FROM AUTHOR]

In reforming the college physical education curriculum, we can introduce algebraic equations to solve the optimization problem of college physical education curriculum training. The article expounds on the principle of algebraic equations based on consulting many documents. At the same time, we apply it to physical education teaching. This method dramatically improves the numerical behavior of the violation correction method in traditional curriculum education and training. At the same time, the technique is simple, practical, and efficient. Finally, a numerical comparison is made between this method, the traditional augmentation method, and it’s default correction method.

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 264, 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 237 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.

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]

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.

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]

This paper is devoted to solve a class of differential equation with simultaneously combining variable coefficients and variable delays namely variable-delay differential equations (VDDEs). For this purpose, a numerical method is proposed in which the unknown function and its derivative are approximated with the basis of interpolating Multiquadric radial basis functions (MQRBFs) at arbitrary collocation points. According to the existing mechanism, the synchronization problem is recast to a system of algebraic equations. In the other hand, the proposed method provides a very adjustable framework for approximation according to the discretization and due to a board range of arbitrary nodes. Finally, some illustrative examples are given to verify the validity and applicability of the new technique. [ABSTRACT FROM AUTHOR]

This study aims to determine the effect of Photomath mobile application on the achievement and attitude of learning algebraic equations for lower secondary school students in Malaysia. Photomath is a free mobile application, for both ios and android platforms, which uses computer vision and image understanding technology to scan math equations using the mobile camera to subsequently provide worked solutions. The application recognises equations of both printed and handwritten and provides step-by-step calculations and graphical solutions. In this study, data were collected from 33 lower secondary students through pre and post-test and questionnaire. Data from pre-test and post-test were used to investigate the performance of students after the application of Photomath, whereas the questionnaire was used to identify the attitude of students towards the application of Photomath in learning algebraic equations, regarding belief and readiness aspects. Different mean scores for pre-test and post-test were tested, and hypothesis testing using a t-test was used. Data from the questionnaire were analysed using descriptive statistics. This study has found that the application of Photomath significantly enhances the learning of algebraic equations among Form Two students. However, the participants' belief and readiness towards Photomath are still at a moderate level. [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]

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]

In this note we recall (see also [8]) the structure of all recurrent sequences which satisfy a fixed recurrence relation, with entries in a perfect field. As a consequence of these considerations we give a reasonable proof for the known result that the Hadamard product of two recurrent sequences is also a recurrent sequence.

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]

We discuss the representability almost everywhere (a.e.) in C of an irreducible algebraic function as the Cauchy transform of a signed measure supported on a finite number of compact semi-analytic curves and a finite number of isolated points. This brings us to the study of trajectories of the particular family of quadratic differentials A( z − a)( z − b)×( z − c) d z . More precisely, we give a necessary and sufficient condition on the complex numbers a and b for these quadratic differentials to have finite critical trajectories. We also discuss all possible configurations of critical graphs. [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]

In this paper we compare different methods of solving algebraic equations based on resolvent polynomial equations. These methods were considered by several authors, including the second author of this article. Due to the fact that no demonstration of convergence was made for these methods and therefore the speed of convergence was not determined, this study offers some important data in terms of effectiveness. [ABSTRACT FROM AUTHOR]

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]

This paper explored algebraic features of nonlinear parts in Serpent encryption algorithm and offered an 11-round Serpent-128 impossible differential algebraic attack through utilizing the method in constructing S-box algebraic equations. The new method analyzed block 11-round Serpent with 2127 selected plaintexts and 2129 bytes memory space at the same time of giving a new design principle of S-box anti-algebraic attack. [ABSTRACT FROM AUTHOR]

In this paper we establish a relationship between two approaches to the solution of algebraic fifth-degree equations, namely, the Hermite-Kronecker method (based on the modular elliptic equation) and the Mellin method (based on hypergeometric series). [ABSTRACT FROM AUTHOR]

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]

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]

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]

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]

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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]