On the solution of certain algebraic equations

This note deals with the problem of determining the roots of simple algebrac equations by constructing polynomial equations that have the same roots.     

On the algebraic integrals of the dynamical equations

We bring to the reader’s attention a translation of Levi-Civita’s work “Sugli integrali algebrici delle equazioni dinamiche” (1896).     

On the equations defining irreducible algebraic groups

Abstract

We consider five known problems concerning positive laws in groups. One of them has a counterexample, the others are open in general. It is shown that three of the problems are equivalent and all of them have a positive solution in the class of locally graded groups.     

On differential algebraic equations with discontinuities

Summary The dynamic features of systems in mathematical process simulation are often studied by looking at the response of the system to discontinuities in the input variables (Sprung-Antwort-Verhalten). A more detailed analysis ([2]) shows that for systems in the form of differential algebraic equations (even with index 1), which frequently occur e.g. in chemical engineering, a solution to the problem need not exist.In this paper we derive necessary and sufficient conditions for such systems to guarantee the solvability of the problem (Theorem 1). Further, a simple algorithm is stated (Theorem 3), which is suitable for numerical computation.

---

Zusammenfassung Die Untersuchung der dynamischen Eigenschaften von Systemen mit Hilfe der mathematischen Prozeßsimulation geschieht häufig durch die Betrachtung des Systemverhaltens bei sprungartiger Veränderung der Eingangsgrößen (Sprung-Antwort-Verhalten). Eine genauere Analyse ([2]) zeigt, daß für Systeme in Form von Differential-Algebraischen Gleichungssystemen (sogar mit Index 1), welche häufig auftreten z.B. in der chemischen Verfahrenstechnik, eine Lösung dieser Aufgabe nicht existieren muß.In dieser Arbeit leiten wir für Systeme vom Index 1 notwendige und hinreichende Bedingungen her, welche die Lösbarkeit des Problems garantieren (Theorem 1). Ferner wird ein einfacher Algorithmus zur numerischen Berechnung beschrieben (Theorem 3).

     

On algebraic differential equations with knotted trajectories

It is proven that a system of three autonomous differential equations with polynomials (of degree no lower than three) for right sides can have a knotted curve as its solution.Translated from Matematicheskie Zametki, Vol. 11, No. 6, pp. 645–650, June, 1972.     

On the solution of algebraic equations by the decomposition method

The decomposition method (G. Adomian, “Stochastic Systems,” Academic Press, New York, 1983) developed to solve nonlinear stochastic differential equations has recently been generalized to nonlinear (and/or) stochastic partial differential equations, systems of equations, and delay equations and applied to diverse applications. As pointed out previously (see reference above) the methodology is an operator method which can be used for nondifferential operators as well. Extension has also been made to algebraic equations involving real or complex coefficients. This paper deals specifically with quadratic, cubic, and general higher-order polynomial equations and negative, or nonintegral powers, and random algebraic equations. Further work on this general subject appears elsewhere (G. Adomian, “Stochastic Systems II,” Academic Press, New York, in press).     

Normwise, mixed and componentwise condition numbers of nonsymmetric algebraic Riccati equations

This paper is devoted to the perturbation analysis for nonsymmetric algebraic Riccati equations. The upper bounds for the normwise, mixed and componentwise condition numbers are presented. The results are illustrated by numerical examples.     

On meromorphic solutions of generalized algebraic differential equations

Summary In this paper we investigate the rate of growth of meromorphic functions f which are solutions of certain algebraic differential equation whose coefficients a(z) are arbitrary meromorphic functions. By method based on Nevanlinna's theory of meromorphic functions, it has been shown that if the zeros and poles of f satisfy the condition N(r, f′/f)=S(r, f′/f) then the ratio T(r, f′/f)/(T(r, a(z)), as r → ∞ outside a set of r values of finite measure, is bounded for at least one of the coefficients a(z). The content of an invited address delivered by the author on March 27, 1971 to the 683th meeting of the American Mathematical Society of the University of Illinois at Chicago Circle, Chicago, Illinois, U.S.A. Entrato in Redazione il 16 novembre 1970.     

On the algebraic properties of solutions of inhomogeneous hypergeometric equations

Generalized hypergeometric differential equations of arbitrary order are considered. Necessary and sufficient conditions for the algebraic independence of solutions of collections of such equations, as well as of their values at algebraic points, are obtained.     

On the growth of entire solutions of algebraic differential equations

This is the opening memorial plenary lecture on memorial meeting in honor of Professor Shlomo Strelitz on Conference of Differential Equations and Complex Analysis, University of Haifa, Israel, December 2000. Shlomo Strelitz: 7.1.1923–27.9.1999, teached at Vilnius University 1946–1973, professor 1967–1973.__________Published in Lietuvos Matematikos Rinkinys, Vol. 45, No. 1, pp. 57–63, January–March, 2005.     

On Diophantine equations over the ring of all algebraic integers

For each odd prime q an integer NH_q (NH₃ = ?1, NH₅ = ?1, NH₇ = 97, NH₁₁ = ?243, …) is defined as the norm from L to $\text{Q}$ of the Heilbronn sum H_q = Tr_I^{$\text{Q}$(ζ)}(ζ), where ζ is a primitive q²th root of unity and L ?- $\text{Q}$(ζ) the subfield of degree q. Various properties are proved relating the congruence properties of H_q and NH_q modulo p (p ≠ q prime) to the Fermat quotient $\text{(p}{}^{\mathrm{q\; ?\; 1}}\text{? 1)}\text{q}\text{(}\text{mod}\text{q)}$; in particular, it is shown that NH_q is even iff 2^{q ? 1} ≡ 1 (mod q²).     

On a method to transform algebraic differential equations into universal equations

The paper introduces an algorithm which transforms homogeneous algebraic differential equations into universal differential equations (in the sense of L. A. Rubel) havingC ⁿ (ℝ)-solutions. By applications of the algorithm to different initial equations some new universal differential equations are found, and all the known equations due to R. J. Duffin are rediscovered with this method. Assuming weak conditions one can find Cⁿ(ℝ)-solutionsy of the differential equation close to any continuous function such that 1, with 0 ≤k ₁ <k ₂ < .... <k _s ≤n are linearly independent over the field of real algebraic numbers at the rational points q1,...,qs.     

On the absolute continuity of the roots of some algebraic equations

We prove that for every polynomial of degreem≤3, with coefficients depending smoothly on a real parametert, it is possible to select am-tuple of roots absolutely continuous int.

---

Sunto Proviamo che per ogni polinomio di gradom≤3, i cui coefficienti dipendano con sufficiente regolarità da un parametro realet, si possono selezionarem radici (indipendenti) che sono assolutamente continue int.

     

On solving sparse algebraic equations over finite fields

A system of algebraic equations over a finite field is called sparse if each equation depends on a small number of variables. In this paper new deterministic algorithms for solving such equations are presented. The mathematical expectation of their running time is estimated. These estimates are at present the best theoretical bounds on the complexity of solving average instances of the above problem. In characteristic 2 the estimates are significantly lower the worst case bounds provided by SAT solvers.     

On solving equations of algebraic sum of equal powers

It is well known that a system of equations of sum of equal powers can be converted to an algebraic equation of higher degree via Newton's identities. This is the Viete-Newton theorem. This work reports the generalizations of the Viete-Newton theorem to a system of equations of algebraic sum of equal powers. By exploiting some facts from algebra and combinatorics, it is shown that a system of equations of algebraic sum of equal powers can be converted in a closed form to two algebraic equations, whose degree sum equals the number of unknowns of the system of equations of algebraic sum of equal powers.     

On the order of growth of solutions of algebraic differential equations

Assume thatf is an integer transcendental solution of the differential equationP _n (z, f, f′)=P _n−1(z, f, f′, ... f ^(p)), whereP _n andP _n−1 are polynomials in all variables, the degree ofP _n with respect tof andf′ is equal ton, and the degree ofP _n−1 with respect tof, f′, ... f ^(p) is at mostn−1. We prove that the order ρ of growth off satisfies the relation 1/2≤ρ<∞. We also prove that if ρ=1/2, then, for a certain real ν, in the domain {z: ν<argz<ν+2π}/E _*, whereE _* is a certain set of disks with finite sum of radii, the estimate lnf(z)=z ^1/2 (β+o(1)), β∈C, holds forz=re ^iϕ,r≥r(ϕ)≥0. Furthermore, on the ray {z: argz=ν}, the following relation is true: ln‖f(re ^iν)‖=o(r ^1/2),r→+∞,r>0, , where Δ is a certain set on the semiaxisr>0 with mes Δ<∞. “L'vivs'ka Politekhnika” University, Lvov. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 1, pp. 69–77, January, 1999.