Zeros of regular functions of quaternionic and octonionic variable: a division lemma and the camshaft effect

This works introduces several notions of subharmonicity for real-valued functions of one quaternionic variable. These notions are related to the theory of slice regular quaternionic functions introduced by Gentili and Struppa in 2006. The interesting properties of these new classes of functions are studied and applied to construct the analogs of Green's functions.

Reduction of everywhere convergent power series with respect to Gröbner bases

In their seminal paper [HILL99], Håstad, Impagliazzo, Levin and Luby show that a pseudorandom generator can be constructed from any one-way function. This plausibility result is one of the most fundamental theorems in cryptography and helps shape our understanding of hardness and randomness in the field. Unfortunately, the reduction of [HILL99] is not nearly as efficient nor as security preserving as one may desire. The main reason for the security deterioration is the blowup to the size of the input. In particular, given one-way functions on n bits one obtains by [HILL99] pseudorandom generators with seed length \(\cal{O}\)(n 8). Alternative constructions that are far more efficient exist when assuming the one-way function is of a certain restricted structure (e.g. a permutations or a regular function). Recently, Holenstein [Hol06] addressed a different type of restriction. It is demonstrated in [Hol06] that the blowup in the construction may be reduced when considering one-way functions that have exponential hardness. This result generalizes the original construction of [HILL99] and obtains a generator from any exponentially hard one-way function with a blowup of \(\cal{O}\)(n 5), and even \(\cal{O}\)(n 4 log2 n) if the security of the resulting pseudorandom generator is allowed to have weaker (yet super-polynomial) security.In this work we show a construction of a pseudorandom generator from any exponentially hard one-way function with a blowup of only \(\cal{O}\)(n 2) and respectively, only \(\cal{O}\)(n log2 n) if the security of the resulting pseudorandom generator is allowed to have only super-polynomial security. Our technique does not take the path of the original [HILL99] methodology, but rather follows by using the tools recently presented in [HHR05] (for the setting of regular one-way functions) and further developing them.KeywordsHash FunctionSeed LengthPseudorandom GeneratorPseudorandom FunctionInput LengthThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We introduce and study Hankel operators defined on the Hardy space of regular functions of a quaternionic variable. Theorems analogous to those of Nehari and Fefferman are proved.

Normal matrices over hermitian discrete valuation rings

In this chapter we prepare the ground for the proof of the Jung-Abhyankar theorem in section 2 and the study of quasiordinary power series in section 4 of chapter V. We assume that the reader is acquainted with the notion of power series over a field; in section 1, for the convenience of the reader, we give some background, introduce convergent power series over the real and complex numbers in section 2, and prove Weierstraß division and preparation theorem in section 3. The category of formal (resp. analytic) algebras—these are homomorphic images of formal (resp. convergent) rings of power series—is treated in section 4. The last section considers, in particular, the completion of an analytic algebra; these results are needed to prove the convergent case of the Jung-Abhyankar theorem.

Given a continuous boundary value on the boundary of a “simply closed surface” ∂S that encloses the whole Earth, a regular harmonic fictitious field V*(P) in the domain outside an inner sphere Ki that lies inside the Earth could be determined, and it is proved that V*(P) coincides with the Earth’s real field V(P) in the whole domain outside the Earth. Since in the domain outside the inner sphere Ki and the fictitious regular harmonic function V*(P) could be expressed as a uniformly convergent spherical harmonic series, it is concluded that the Earth’s potential field could be expressed as a uniformly convergent spherical harmonic expansion series in the whole domain outside the Earth.

In this manuscript, we are concerned with the problem of locating the zeros of some special quaternionic polynomials and regular functions with restricted coefficients; namely quaternionic coefficients whose real components or their moduli satisfy suitable inequalities. The obtained results for this subclass of quaternionic polynomials and regular functions produce extensions of the classical Enestr?om-Kakeya theorem and its various variants from complex to the quaternionic setting.

Published accounts of properties of characteristic functions of third-order, irregular boundary value problems and of expansions in infinite series of such functions have been given by D. Jackson and Hopkinst and by the author.T In these papers were considered only cases which may be regarded as elementary in the sense that the characteristic functions were elementary functions. The present paper deals with the characteristic functions and the associated series of the differential system (1) d3u/dx3 + [p3 + r(x)]u = 0, u(0) = u'(0) = U(7) = 0. We shall suppose throughout that r(x) is a convergent power series in x3, and that, when the radius of the circle of convergence of this series is less than 7r, r(x) is continuous in the interval 0 ? x < w. Many of the methods and results obtained in this paper (but not all of them) can be carried over immediately to the case in which the differential equation is the same as the present one but the boundary conditions are of the type in Part I of the 1927 paper. The first part of this paper is devoted to a study of the characteristic functions, the second part to necessary conditions for the convergence of a formal series of these functions arising from a given function f(x), and the third part to a statement of conditions sufficient to ensure the convergence to f(x) of the formal series forf(x), and to the proof of such convergence.

We consider two of the most fundamental theorems in cryptography. The first, due to Håstad et al. [SIAM J. Comput., 28 (1999), pp. 1364–1396] is that pseudorandom generators can be constructed from any one-way function. The second, due to Yao [Proceedings of the $23$rd Annual Symposium on Foundations of Computer Science (FOCS), 1982, pp. 80–91], states that the existence of weak one-way functions implies the existence of full-fledged one-way functions. These powerful plausibility results shape our understanding of hardness and randomness in cryptography, but unfortunately their proofs are not as tight (i.e., security preserving) as one may desire. This work revisits a technique that we call the randomized iterate, introduced by Goldreich, Krawczyk, and Luby [SIAM J. Comput., 22 (1993), pp. 1163–1175]. This technique was used by Goldreich, Krawczyk, and Luby [SIAM J. Comput., 22 (1993), pp. 1163–1175] to give a construction of pseudorandom generators from regular one-way functions. We simplify and strengthen this technique in order to obtain a similar construction, where the seed length of the resulting generators is as short as $\Theta(n \log n)$ (rather than $\Theta(n^3)$ achieved by Goldreich, Krawczyk, and Luby [SIAM J. Comput., 22 (1993), pp. 1163–1175]). Our technique has the potential of implying seed length $\Theta(n)$, and the only bottleneck for such a result are the parameters of current generators against bounded-space computations. We give a construction with similar parameters for security amplification of regular one-way functions. This improves upon the construction of Goldreich et al. [Proceedings of the $31$st Annual Symposium on Foundations of Computer Science, (FOCS), 1990, pp. 318–326] in that the construction does not need to “know" the regularity parameter of the functions (in terms of security, the two reductions are incomparable). In addition, we use the randomized iterate to show a construction of a pseudorandom generator based on an exponentially hard one-way function that has a seed length of only $\Theta(n^2)$. This improves a recent result of Holenstein [Proceedings of the Theory of Cryptography, Third Theory of Cryptography Conference (TCC), 2006] that shows a construction with seed length $\Theta(n^5)$ based on such one-way functions. Finally, we show that the randomized iterate may even be useful in the general context of Håstad et al. [SIAM J. Comput., 28 (1999), pp. 1364–1396]. In particular, we use the randomized iterate to replace the basic building block of the Håstad et al. [SIAM J. Comput., 28 (1999), pp. 1364–1396] construction. Interestingly, this modification improves efficiency by an $\Theta(n^2)$ factor and reduces the seed length to $\Theta(n^7)$ (which also implies improvement in the security of the construction).

Polynomial approximation of [formula omitted] functions by means of boundary values and applications: A survey

It has been shown by some researchers that in a problem of weighted least-square (WLS) design of finite-impulse response (FIR) filters, bulk of the design computation is concerned with the evaluation of the inverse of a matrix in order to solve a system of equations. In this paper, a new algorithm for the WLS design of FIR filters is presented, in which an iterative procedure is developed for the inversion of the matrix involved in the design. By imposing a mild constraint on the updation factor of the weighting function, the inverse of a matrix is expanded as a convergent power series. By investigating the properties of some of the matrices from the design formulation, a modified version of the series that converges rapidly is then proposed to evaluate the inverse in each iteration. It is shown that due to the fast convergence of the power series, one needs to evaluate only the first two or three terms of the series except during the initial stages of the iterations, implying that the conventional operation for matrix inversion is simplified significantly.

Mathematical modeling using noninteger‐order derivatives has gained rapid attention among scientists and researchers in recent years. This manuscript investigates the dynamics of rabies transmission in a predator–prey framework by formulating the problem as a system of nonlinear fractional initial value problems (FIVPs). Rabies modeling is important for understanding and controlling the spread of this fatal disease in humans and animals. To solve these complex equations, we employ the Laplace Residual Power Series (LRPS) method, which combines the Laplace transform with the Residual Power Series to efficiently produce rapidly convergent fractional power series solutions. Our results demonstrate that LRPS provides accurate and computationally efficient solutions, offering valuable insights into the effects of fractional dynamics on rabies spread. These findings highlight the potential of LRPS for studying complex epidemiological systems and improving disease management strategies.

converge to P(z) uniformly over an arbitrary closed region, K, exterior to the real axis. (ii) If P(z) is a Stieltjes power series with radius of convergence R>0, then an arbitrary infinite sequence of distinct approximants converges to P(z) uniformly over the circle Iz -=R-6, a > 0. (iii) If P(z) is a positive definite power series which is summable (Borel) to F(z), then Sk(k > 1) converges to F(z) uniformly over K. If the reciprocal of P(z) is summable (Borel) the same holds for k < 1. (iv) If P(z) is a Stieltjes series possessing minimal extensionsi D=-n Cif(-z)i for every n, and if the series EZiXo(1)ic-ilz-i-' is summable (Borel) to F(z), then in the Pade table for P(z), S-1 converges to F(z) uniformly over an arbitrary closed region exterior to the negative half of the real axis. As a corollary to (iii) we find that when P(z) is summable (Borel) and has a corresponding continued fraction, the latter is necessarily convergent to the Borel sum. The converse is not true. That is, the corresponding continued fraction may converge when P(z) is not summable (Borel). 2. The convergence of the diagonal files in the Pade table for a convergent positive definite power series. We recall that P(z) is a positive definite series if all the quadratic forms j= n= , 1, 2, * , are positive de-

This chapter treats the theory of ordinary differential equations, both linear and nonlinear.Sections 1–4 establish existence and uniqueness theorems for ordinary differential equations. The first section gives some examples of first-order equations, mostly nonlinear, to illustrate certain kinds of behavior of solutions. The second section shows, in the presence of continuity for a vectorvalued F satisfying a “Lipschitz condition,” that the first-order system y′ = F(t, y) has a unique local solution satisfying an initial condition y (t0) = y0. Since higher-order equations can always be reduced to first-order systems, these results address existence and uniqueness for nth-order equations as a special case. Section 3 shows that the solutions to a system depend well on the initial condition and on any parameters that are present in F. Section 4 applies these results to existence of integral curves for a vector field and to construction of coordinate systems from families of integral curves.Sections 5–8 concern linear systems. Section 5 shows that local solutions of linear systems may be extended to global solutions and that in the homogeneous case the vector space of global solutions has dimension equal to the size of the system. The method of variation of parameters reduces the solution of any linear system to the solution of a homogeneous linear system. Sections 6–7 identify explicit solutions to nth-order linear equations and first-order linear systems. The “Jordan canonical form” of a square matrix plays a role in the case of a system. Section 8 discusses power-series solutions to second-order homogeneous linear equations whose coefficients are given by convergent power series, as well as solutions that arise in the case of regular singular points. Two kinds of special functions are mentioned that result from this study-Legendre polynomials and Bessel functions.KeywordsOrdinary Differential EquationUniqueness TheoremDiagonal EntryLipschitz ConditionIntegral CurfThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.