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Polynomials and degrees of maps in real normed algebras

Takis Sakkalis (2020)

Communications in Mathematics

Let 𝒜 be the algebra of quaternions ℍ or octonions 𝕆 . In this manuscript an elementary proof is given, based on ideas of Cauchy and D’Alembert, of the fact that an ordinary polynomial f ( t ) ∈ 𝒜 [ t ] has a root in 𝒜 . As a consequence, the Jacobian determinant | J ( f ) | is always non-negative in 𝒜 . Moreover, using the idea of the topological degree we show that a regular polynomial g ( t ) over 𝒜 has also a root in 𝒜 . Finally, utilizing multiplication ( * ) in 𝒜 , we prove various results on the topological degree of products...

Polynomials, sign patterns and Descartes' rule of signs

Vladimir Petrov Kostov (2019)

Mathematica Bohemica

By Descartes’ rule of signs, a real degree d polynomial P with all nonvanishing coefficients with c sign changes and p sign preservations in the sequence of its coefficients ( c + p = d ) has pos ≤ c positive and ¬ ≤ p negative roots, where pos ≡ c ( mod 2 ) and ¬ ≡ p ( mod 2 ) . For 1 ≤ d ≤ 3 , for every possible choice of the sequence of signs of coefficients of P (called sign pattern) and for every pair ( pos , neg ) satisfying these conditions there exists a polynomial P with exactly pos positive and exactly ¬ negative roots (all of them simple). For d ≥ 4 this is not...

Positively homogeneous functions and the Łojasiewicz gradient inequality

Alain Haraux (2005)

Annales Polonici Mathematici

It is quite natural to conjecture that a positively homogeneous function with degree d ≥ 2 on ℝ N satisfies the Łojasiewicz gradient inequality with exponent θ = 1/d without any need for an analyticity assumption. We show that this property is true under some additional hypotheses, but not always, even for N = 2.

Positivity and contractivity in the dynamics of clusters’ splitting with derivative of fractional order

Emile Franc Doungmo Goufo, Stella Mugisha (2015)

Open Mathematics

Classical models of clusters’ fission have failed to fully explain strange phenomenons like the phenomenon of shattering (Ziff et al., 1987) and the sudden appearance of infinitely many particles in some systems with initial finite particles number. Furthermore, the bounded perturbation theorem presented in (Pazy, 1983) is not in general true in solution operators theory for models of fractional order γ (with 0 < γ ≤ 1). In this article, we introduce and study a model that can be understood as...

Positivity and stabilization of fractional 2D linear systems described by the Roesser model

Tadeusz Kaczorek, Krzysztof Rogowski (2010)

International Journal of Applied Mathematics and Computer Science

A new class of fractional 2D linear discrete-time systems is introduced. The fractional difference definition is applied to each dimension of a 2D Roesser model. Solutions of these systems are derived using a 2D Z-transform. The classical Cayley-Hamilton theorem is extended to 2D fractional systems described by the Roesser model. Necessary and sufficient conditions for the positivity and stabilization by the state-feedback of fractional 2D linear systems are established. A procedure for the computation...

Currently displaying 3221 – 3240 of 4562