Riccati Differential Equation for Hypergeometric Differential Equation
Rigid analytic spaces
Rings of constants of four-variable Lotka-Volterra systems
Lotka-Volterra systems appear in population biology, plasma physics, laser physics and derivation theory, among many others. We determine the rings of constants of four-variable Lotka-Volterra derivations with four parameters C 1, C 2, C 3, C 4 ∈ k, where k is a field of characteristic zero. Thus, we give a full description of polynomial first integrals of the respective systems of differential equations.
Rings of constants of generic 4D Lotka-Volterra systems
We show that the rings of constants of generic four-variable Lotka-Volterra derivations are finitely generated polynomial rings. We explicitly determine these rings, and we give a description of all polynomial first integrals of their corresponding systems of differential equations. Besides, we characterize cofactors of Darboux polynomials of arbitrary four-variable Lotka-Volterra systems. These cofactors are linear forms with coefficients in the set of nonnegative integers. Lotka-Volterra systems...
Rings of integers of type .
Rings of Real-Valued Continuous Functions. II.
Ringtopologien auf nichtalgebraischen Körpern.
Ringtopologien auf Z und Q.
Ritt's Second Theorem in arbitrary characteristic.
Root arrangements of hyperbolic polynomial-like functions.
A real polynomial P of degree n in one real variable is hyperbolic if its roots are all real. A real-valued function P is called a hyperbolic polynomial-like function (HPLF) of degree n if it has n real zeros and P(n) vanishes nowhere. Denote by xk(i) the roots of P(i), k = 1, ..., n-i, i = 0, ..., n-1. Then in the absence of any equality of the formxi(j) = xk(i) (1)one has∀i < j xk(i) < xk(j) < xk+j-i(i) (2)(the Rolle theorem). For n ≥ 4 (resp....
Root continuity theorems in valued fields.
Root continuity theorems in valued fields. II.
Roots of Unity over Large Algebraic Fields.
Round Quadratic Forms.