A kinematic formula for integral invariant of degree 4 in real space form

In this paper,kinematic formulae in a real space form are investigated.A kinematic formula for homogeneous polynomial of degree 4 on the second fundamental forms in a real space form is obtained.The formula obtained is a concrete form of the result of Howard and the analogue of the known formula of Chen and Zhou.     

B-Construction and C-Construction

B-construction is a way of obtaining a graded algebra from the triple consisting of an additive category, an object, and an autoequivalence, while C-construction is a way of obtaining an algebra (without unity) from the pair consisting of an additive category and a set of objects. In this article, we study and compare three important classes of algebras in noncommutative algebraic geometry and representation theory of finite dimensional algebras, namely, quantum polynomial algebras, preprojetive algebras and trivial extensions, via these constructions.     

Skew Derivations and Engel Conditions

It is known that for a nonzero derivation d of a prime ring R, if a nonzero ideal I of R satisfies the Engel-type identity [[…[[d(x ^{k ₀} ), x ^{k ₁} ], x ^{k ₂} ],…], x ^{k _n} ], then R is commutative. Here we extend this result to a skew derivation of R for a Lie ideal I, which has an immediate corollary that replaces d by an automorphism of R. A related result in two variables is obtained for d a (θ, ?)-derivation.     

A Characterization of Generalized Derivations on Prime Rings

Let R be a noncommutative prime ring, U be the left Utumi quotient ring of R, and k, m, n, r be fixed positive integers. If there exist a generalized derivation G and a derivation g (which is independent of G) of R such that [G(x^m)xⁿ + xⁿg(x^m), x^r]_k = 0, for all x ∈ R, then there exists a ∈ U such that G(x) = ax, for all x ∈ R. As a consequence of the result in the present article, one may obtain Theorem 1 in Demir and Argaç [10 Demir, Ç., Argaç, N. (2010). A result on generalized derivations with Engel conditions on one-sided ideals. J. Korean Math. Soc. 47(3):483–494.[Crossref], [Web of Science ®] , [Google Scholar]].     

Note on roots location of a symmetric polynomial with respect to the imaginary axis

In the theory of the separation of roots of algebraic equations, the well-known Routh–Hurwitz–Fujiwara theorem enables us to separate the complex roots of a polynomial with complex coefficients in terms of the inertia of a related Hermitian matrix. Unfortunately, it fails if the polynomial has a nontrivial factor which is symmetric with respect to the imaginary axis. In this article, we present a method to overcome the fault and formulate the inertia of a scalar polynomial with complex coefficients in terms of the inertia of several Hermitian matrices based on a factorization of a monic symmetric polynomial into products of monic symmetric polynomials with only simple roots in the complex plane and on computing the inertia of each factor by means of a subtle perturbation.     

Discrete unbounded sets in a finite dimensional space and beyond

Discrete infinite sets in a finite dimensional space, i.e., infinite sets without finite limit points appear in various branches of analysis (zero and pole sets of meromorphic functions, various models in the mathematical theory of quasicrystals, and so on). Here we introduce some notions and present some new theorems connected with such sets.