Geometrical solutions of some quadratic equations with non-real roots

This note gives geometrical/graphical methods of finding solutions of the quadratic equation ax ² + bx + c = 0, a p 0, with non-real roots. Three different cases which give rise to non-real roots of the quadratic equation have been discussed. In case I a geometrical construction and its proof for finding the solutions of the quadratic equation ax ² + bx + c = 0, a p 0, when a,b,c ] R, the set of real numbers, are presented. Case II deals with the geometrical solutions of the quadratic equation ax ² + bx + c = 0, a p 0, when b ] R, the set of real numbers; and a,c ] C, the set of complex numbers. Finally, the solutions of the quadratic equation ax ² + bx + c = 0, a p 0, when a,c ] R, the set of real numbers, and b ] C, the set of complex numbers, are presented in case III.     

Vanishing Derivations and Co-Centralizing Generalized Derivations on Multilinear Polynomials in Prime Rings

Let R be a noncommutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, and f(x₁,…, x_n) be a multilinear polynomial over C, which is not central valued on R. Suppose that F and G are two generalized derivations of R and d is a nonzero derivation of R such that d(F(f(r))f(r) ? f(r)G(f(r))) = 0 for all r = (r₁,…, r_n) ∈ Rⁿ, then one of the following holds:

1. There exist a, p, q, c ∈ U and λ ∈C such that F(x) = ax + xp + λx, G(x) = px + xq and d(x) = [c, x] for all x ∈ R, with [c, a ? q] = 0 and f(x₁,…, x_n)² is central valued on R;

2. There exists a ∈ U such that F(x) = xa and G(x) = ax for all x ∈ R;

3. There exist a, b, c ∈ U and λ ∈C such that F(x) = λx + xa ? bx, G(x) = ax + xb and d(x) = [c, x] for all x ∈ R, with b + αc ∈ C for some α ∈C;

4. R satisfies s₄ and there exist a, b ∈ U and λ ∈C such that F(x) = λx + xa ? bx and G(x) = ax + xb for all x ∈ R;

5. There exist a′, b, c ∈ U and δ a derivation of R such that F(x) = a′x + xb ? δ(x), G(x) = bx + δ(x) and d(x) = [c, x] for all x ∈ R, with [c, a′] = 0 and f(x₁,…, x_n)² is central valued on R.

     

Greek ladders via linear algebra

Theon's ladder is an ancient method for easily approximating nth roots of a real number k. Previous work in this area has focused on modifying Theon's ladder to approximate roots of quadratic polynomials. We extend this work using techniques from linear algebra. We will show that a ladder associated to the quadratic polynomial ax ² + bx + c can be adjusted to approximate either root. Other situations such as quadratics with no real roots and corresponding matrices with complex eigenvalues are also addressed.     

Geometrical solution of some cubic equations with non-real roots

A general cubic equation ax ³ + bx ² + cx + d = 0 where a, b, c, d ∈R, a ≠ 0 has three roots with two possibilities—either all three roots are real or one root is real and the remaining two roots are imaginary. Dealing with the second possibility this paper attempts to give the geometrical locations of the imaginary roots of the equation under three different sets of conditions. These sets of conditions include: (i) the real root of the given cubic equation is given, (ii) the real part of an imaginary root is given, and (iii) the imaginary part of an imaginary root is given.     

Relations between roots and coefficients of cubic equations with one root negative the reciprocal of another

Under predetermined conditions on the roots and coefficients, necessary and sufficient conditions relating the coefficients of a given cubic equation x ³?+?ax ²?+?bx?+?c?=?0 can be established so that the roots possess desired properties. In this note, the condition for one root of a cubic equation to be the negative reciprocal of another one is obtained. Given that the coefficients a, b, c of the cubic equation are in arithmetical or geometrical progression, further conditions are deived for one root to be the negative reciprocal of another. These results provide useful means for checking calculated roots of cubic equations and could serve the needs of teachers and students of Mathematical Sciences in tertiary institutions when the solution of cubic equations are first studied.     

Dynamics of a family of piecewise-linear area-preserving plane maps III. Cantor set spectra

This paper studies the behavior under iteration of the maps T _ab (x,y) = (F _ab (x) ? y, x) of the plane ?², in which F _ab (x) = ax if x ≥ 0 and bx if x < 0. These maps are area-preserving homeomorphisms of ?² that map rays from the origin to rays from the origin. Orbits of the map correspond to solutions of the nonlinear difference equation x _n+2 = 1/2(a ? b)|x _n+1|+1/2(a+b)x _n+1 ? x _n . This difference equation can be rewritten in an eigenvalue form for a nonlinear difference operator of Schrödinger type ? x _n+2+2x _n+1 ? x _n +V _μ(x _n+1)x _n+1 = Ex _n+1, in which μ = (1/2)(a ? b) is fixed, and V _μ(x) = μ(sgn(x)) is an antisymmetric step function potential, and the energy E = 2 ? 1/2(a+b). We study the set Ω _SB of parameter values where the map T _ab has at least one bounded orbit, which correspond to l _∞-eigenfunctions of this difference operator. The paper shows that for transcendental μ the set Spec_∞[μ] of energy values E having a bounded solution is a Cantor set. Numerical simulations suggest the possibility that these Cantor sets have positive (one-dimensional) measure for all real values of μ.     

Hidden lessons: How a focus on slope-like properties of quadratic functions encouraged unexpected generalizations

This article presents secondary students’ generalizations about the connections between algebraic and graphical representations of quadratic functions, focusing specifically on the roles of the parameters a, b, and c in the general form of a quadratic function, y = ax² + bx + c. Students’ generalizations about these connections led to a surprising finding: two-thirds of the students interviewed identified the parameter a as the “slope” of the parabola. Analysis of qualitative data from interviews and classroom observations led to the development of three focusing phenomena in the classroom environment that inadvertently supported a focus on slope-like properties of quadratic functions: (a) the use of linear analogies, (b) the rise over run method, and (c) viewing a as dynamic rather than static.     

The Structure of Finite c-Local Rings

For a commutative ring R, assume that c is a nonzero element of Z(R) with the property that cZ(R) = {0}. A local ring R is called c-local if Z(R)² = {0, c}, Z(R)³ = {0}, and xZ(R) = {0} implies x ∈ {0, c}. For any finite c-local ring (R, 𝔪), it is proved that the ideal m has a minimal generating set which has a c-partition. The structure and classification up to isomorphism of all finite commutative c-local rings with order greater than 2⁵ are determined.     

On Quadratic Integral Polynomials with only Finitely Many Roots in any Commutative Finite-Dimensional Algebra

The monic quadratic polynomials f with integer coefficients such that each commutative finite-dimensional algebra over a field contains only finitely many roots of f are determined as the polynomials of the form f = X ² + (2m + 1)X + m ² + m, where ${m \in \mathbb{Z}}$ .     

AA-Rings

Abstract

Let R be a ring with identity such that R ⁺, the additive group of R, is torsion-free of finite rank (tffr). The ring R is called an E-ring if End(R ⁺) = {x ? ax : a ∈ R} and is called an A-ring if Aut(R ⁺) = {x ? ux : u ∈ U(R)}, where U(R) is the group of units of R. While E-rings have been studied for decades, the notion of A-rings was introduced only recently. We now introduce a weaker notion. The ring R, 1 ∈ R, is called an AA-ring if for each α ∈ Aut(R ⁺) there is some natural number n such that α ⁿ  ∈ {x ? ux : u ∈ U(R)}. We will find all tffr AA-rings with nilradical N(R) ≠ {0} and show that all tffr AA-rings with N(R) = {0} are actually E-rings. As a consequence of our results on AA-rings, we are able to prove that all tffr A-rings are indeed E-rings.     

Spreading Properties and Complex Dynamics for Monostable Reaction–Diffusion Equations

This paper is concerned with the study of the large-time behavior of the solutions u of a class of one-dimensional reaction–diffusion equations with monostable reaction terms f, including in particular the classical Fisher-KPP nonlinearities. The nonnegative initial data u ₀(x) are chiefly assumed to be exponentially bounded as x tends to + ∞ and separated away from the unstable steady state 0 as x tends to ? ∞. On the one hand, we give some conditions on u ₀ which guarantee that, for some λ > 0, the quantity c _λ = λ +f′(0)/λ is the asymptotic spreading speed, in the sense that lim  _t→+∞ u(t, ct) = 1 (the stable steady state) if c < c _λ and lim  _t→+∞ u(t, ct) = 0 if c > c _λ. These conditions are fulfilled in particular when u ₀(x) e ^λx is asymptotically periodic as x → + ∞. On the other hand, we also construct examples where the spreading speed is not uniquely determined. Namely, we show the existence of classes of initial conditions u ₀ for which the ω-limit set of u(t, ct + x) as t tends to + ∞ is equal to the whole interval [0, 1] for all x ∈ ? and for all speeds c belonging to a given interval (γ₁, γ₂) with large enough γ₁ < γ₂ ≤ + ∞.     

Dispersive Estimates of Solutions to the Wave Equation with a Potential in Dimensions n ≥ 4

We prove dispersive estimates for solutions to the wave equation with a real-valued potential V ∈ L ^∞(R ⁿ ), n ≥ 4, satisfying V(x) = O(?x?^?(n+1)/2?ε), ε > 0.     

Elementary matrix reduction over certain rings

We explore elementary matrix reduction over certain rings characterized by properties related to stable range. Let R be a commutative ring. We call R locally stable if aR+bR = R??x∈R such that R∕(a+bx)R has stable range 1. We study locally stable rings and prove that every locally stable Bézout ring is an elementary divisor ring. Many known results on domains are thereby generalized.     

Simplicity of Some Derivations of k[x,y]

If k is a field of characteristic zero, c ∈ k?0, and 1 ≤ s ≤ r are integers such that either r ? s + 1 divides s or s divides r ? s + 1, then it is shown that the derivation y ^r ? _x  + (xy ^s  + c)? _y of the polynomial ring k[x, y] is simple.     

Subsets with Generalized Derivations Having Nilpotent Values on Lie Ideals

Let R be a prime ring, with no nonzero nil right ideal, Q the two-sided Martindale quotient ring of R, F a generalized derivation of R, L a noncommutative Lie ideal of R, and b ∈ Q. If, for any u, w ∈ L, there exists n = n(u, w) ≥1 such that (F(uw) ? bwu)ⁿ = 0, then one of the following statements holds:

1. F = 0 and b = 0;

2. R ? M₂(K), the ring of 2 × 2 matrices over a field K, b² = 0, and F(x) = ?bx, for all x ∈ R.

     

A Class of Simple Derivations of k[x,y]

If k is a field of characteristic zero, c ∈ k?{0}, s, t ≥ 1, and r ≥ 0 are integers, then it is shown that the k-derivation y ^r ? _x  + (y ^s x ^t  + c)? _y of the polynomial algebra k[x, y] is simple.     

Transforming algebraic Riccati equations into unilateral quadratic matrix equations

The problem of reducing an algebraic Riccati equation XCX − AX − XD + B = 0 to a unilateral quadratic matrix equation (UQME) of the kind PX ² + QX + R = 0 is analyzed. New transformations are introduced which enable one to prove some theoretical and computational properties. In particular we show that the structure preserving doubling algorithm (SDA) of Anderson (Int J Control 28(2):295–306, 1978) is in fact the cyclic reduction algorithm of Hockney (J Assoc Comput Mach 12:95–113, 1965) and Buzbee et al. (SIAM J Numer Anal 7:627–656, 1970), applied to a suitable UQME. A new algorithm obtained by complementing our transformations with the shrink-and-shift technique of Ramaswami is presented. The new algorithm is accurate and much faster than SDA when applied to some examples concerning fluid queue models.