Commutativity for a certain class of rings

We discuss the commutativity of certain rings with unity 1 and one-sideds-unital rings under each of the following conditions:x ^r [x ^s ,y]=±[x,y ^t ]x ⁿ x ^r [x ^s ,y]=±x ⁿ [x,y ^t ]x ^r [x ^s ,y]=±[x,y ^t ]y ^m , andx ^r [x ^s ,y]=±y ^m [x,y ^t ], wherer, n, andm are non-negative integers andt>1,s are positive integers such that eithers, t are relatively prime ors[x,y]=0 implies [x,y]=0. Further, we improve the result of [6, Theorem 3] and reprove several recent results.     

On structure and commutativity of certain periodic near rings

In the present paper we first establish decomposition theorems for near rings satisfying either of the properties xy = x^my^pxⁿ or xy = y^mx^pyⁿ, where m≥1, n≥1, p≥1 are positive integers depending on the pair of near ring elements x,y; and further, we investigate commutativity of such near rings. Moreover, it is also shown that under some additional hypotheses, such nearrings turn out to be commutative rings.     

Commutativity of One Sided S- Unital Rings

In the present paper we extend some commutativity theorems for rings as follows: Let m > 1, n and k be fixed non- negative integers, and let R be a left or right s- unital ring satisfying the polynomial identity [xⁿ]y ? y^mx^k,x] = 0. Then R is commutative. Under appropriate conditions the commutativity of R has also been proved for the case m = 1.     

Commutativity of rings with polynomial constraints

Let p, q and r be fixed non-negative integers. In this note, it is shown that if R is left (right) s-unital ring satisfying , respectively) where , then R is commutative. Moreover, commutativity of R is also obtained under different sets of constraints on integral exponents. Also, we provide some counterexamples which show that the hypotheses are not altogether superfluous. Thus, many well-known commutativity theorems become corollaries of our results.     

ON PSEUDO-COMMUTATMTY AND COMMUTATIVITY IN RINGS

Abstract

A ring R is called pseudo-commutative if for each x,y ε R there exists an integer n = n(x, y) for which xy = nyx. We first show that a generalization of a commutativity condition of Chacron and Thierrin implies pseudo-commutativity in rings; we then study pseudo-commutativity and commutativity in rings with constraints of the form xy = σk_iyⁱxⁱ, where the k_i are integers.     

Commutativity of Rings with Constraints Involving a Subset

Suppose that R is an associative ring with identity 1, J(R) the Jacobson radical of R, and N(R) the set of nilpotent elements of R. Let m 1 be a fixed positive integer and R an m-torsion-free ring with identity 1. The main result of the present paper asserts that R is commutative if R satisfies both the conditions(i) [x ^m, y ^m] = 0 for all and(ii) [(xy) ^m + y ^m x ^m, x] = 0 = [(yx) ^m + x ^m y ^m, x], for all This result is also valid if (i) and (ii) are replaced by (i) [x ^m, y ^m] = 0 for all and (ii) [(xy) ^m + y ^m x ^m, x] = 0 = [(yx) ^m + x ^m y ^m, x] for all Other similar commutativity theorems are also discussed.     

Commutativity theorems in rings with involution

In this paper, we investigate commutativity of ring R with involution ? which admits a derivation satisfying certain algebraic identities. Some well-known results characterizing commutativity of prime rings have been generalized. Finally, we provide examples to show that various restrictions imposed in the hypotheses of our theorems are not superfluous.     

Filtering on manifolds

We coasider a partially observable diffusion process (x _t,y_t)t0 whose unobservable componentx _t lives on a submanifold M ofR ⁿ . We present some general conditions under which the conditional law ofx _t, given the observationsy _s ,s [0,t], admits a density w.r.t. a given measure on M. We characterize the analytical properties of this density by using appropriate Sobolev spaces.Research supported by the Hungarian National Foundation of Scientific Research No. 2290.     

Commutativity of rings through a Streb's result

In this paper we investigate commutativity of rings with unity satisfying any one of the properties:

A wavelike functional equation of Pexider type

Summary. Let (G, +) and (H, +) be abelian groups such that the equation 2u = v 2u = v is solvable in both G and H. It is shown that if f₁, f₂, f₃, f₄, : G ×G ? H f_1, f_2, f_3, f_4, : G \times G \longrightarrow H satisfy the functional equation f₁(x + t, y + s) + f₂(x - t, y - s) = f₃(x + s, y - t) + f₄(x - s, y + t) for all x, y, s, t ? G x, y, s, t \in G , then f₁, f₂, f₃, and f₄ are given by f₁ = w + h, f₂ = w - h, f₃ = w + k, f₄ = w - k where w : G ×G ? H w : G \times G \longrightarrow H is an arbitrary solution of f (x + t, y + s) + f (x - t, y - s) = f (x + s, y - t) + f (x - s, y + t) for all x, y, s, t ? G x, y, s, t \in G , and h, k : G ×G ? H h, k : G \times G \longrightarrow H are arbitrary solutions of D_y,t³g(x,y) = 0 \Delta_{y,t}^{3}g(x,y) = 0 and D_x,t³g(x,y) = 0 \Delta_{x,t}^{3}g(x,y) = 0 for all x, y, s, t ? G x, y, s, t \in G .     

On the Invariant Fields and Rings of Some Groups of Polynomial Automorphisms

Abstract

We consider the group G of C-automorphisms of C(x, y) (resp. C[x, y]) generated by s, t such that t(x) = y, t(y) = x and s(x) = x, s(y) = ? y + u(x) where u ∈ C[x] is of degree k ≥ 2. Using Galois's theory, we show that the invariant field and the invariant algebra of G are equal to C.     

On Radicals with Amitsur Property

Abstract

A radical γ has the Amitsur property, if γ(A[x]) = (γ(A[x]) ∩ A)[x] for every ring A. To any radical γ with Amitsur property we construct the smallest radical γ ^x which coincides with γ on polynomial rings. Distinct special radicals with Amitsur property are given which coincide on simple rings and on polynomial rings, answering thus a stronger version of M. Ferrero's problem. Radicals γ with Amitsur property are characterized which satisfy A[x, y] ∈ γ whenever A[x] ∈ γ.     

Characterizations of generalized quadrangles by generalized homologies

Let S = (P, B, I) be a generalized quadrangle of order (s, t). For x, y P, x y, let (x, y) be the group of all collineations of S fixing x and y linewise. If z {x, y}, then the set of all points incident with the line xz (resp. yz) is denoted by (resp. ). The generalized quadrangle S = (P, B, I) is said to be (x, y)-transitive, x y, if (x, y) is transitive on each set and . If S = (P, B, I) is a generalized quadrangle of order (s, t), s > 1 and t > 1, which is (x, y)-transitive for all x, y P with x y, then it is proved that we have one of the following: (i) S W(s), (ii) S Q(4, s), (iii) S H(4, s), (iv) S Q(5, s), (v) s = t² and all points are regular.     

Polar factorization of maps on Riemannian manifolds

Let (M,g) be a connected compact manifold, C³ smooth and without boundary, equipped with a Riemannian distance d(x,y). If s : M ? M s : M \to M is merely Borel and never maps positive volume into zero volume, we show s = t °u s = t \circ u factors uniquely a.e. into the composition of a map t(x) = exp_x[-?y(x)] t(x) = {\rm exp}_x[-\nabla\psi(x)] and a volume-preserving map u : M ? M u : M \to M , where y: M ? \bold R \psi : M \to {\bold R} satisfies the additional property that (y^c)^c = y (\psi^c)^c = \psi with y^c(y) :=inf{c(x,y) - y(x) | x ? M} \psi^c(y) :={\rm inf}\{c(x,y) - \psi(x)\,\vert\,x \in M\} and c(x,y) = d²(x,y)/2. Like the factorization it generalizes from Euclidean space, this non-linear decomposition can be linearized around the identity to yield the Hodge decomposition of vector fields.¶The results are obtained by solving a Riemannian version of the Monge--Kantorovich problem, which means minimizing the expected value of the cost c(x,y) for transporting one distribution f 3 0 f \ge 0 of mass in L¹(M) onto another. Parallel results for other strictly convex cost functions c(x,y) 3 0 c(x,y) \ge 0 of the Riemannian distance on non-compact manifolds are briefly discussed.     

On the law S ( S ( x , y ), T ( x , y )) = S ( x , y ) of fuzzy logic

Motivated by some functional models arising in fuzzy logic, when classical boolean relations between sets are generalized, we study the functional equation S(S(x, y), T(x, y)) = S(x, y), where S is a continuous t-conorm and T is a continuous t-norm. Some interesting methods for solving this type of equations are introduced.     

New Logarithmic Sobolev Inequalities and an ɛ‐Regularity Theorem for the Ricci Flow

In this note, we prove an ?‐regularity theorem for the Ricci flow. Let (Mⁿ,g(t)) with t ? [?T,0] be a Ricci flow, and let H_x0(y,s) be the conjugate heat kernel centered at some point (x₀,0) in the final time slice. By substituting H_x0(?,s) into Perelman's W‐functional, we obtain a monotone quantity W_x0(s) that we refer to as the pointed entropy. This satisfies W_x0(s) ≤ 0, and W_x0(s) = 0 if and only if (Mⁿ,g(t)) is isometric to the trivial flow on Rⁿ. Then our main theorem asserts the following: There exists ? > 0, depending only on T and on lower scalar curvature and μ‐entropy bounds for the initial slice (Mⁿ,g(?T)) such that W_x0(s) ≥ ?? implies |Rm| ≤ r^?2 on P_{? r}(x₀,0), where r² ≡ |s| and P_ρ(x,t) ≡ B_ρ(x,t) × (t?ρ²,t] is our notation for parabolic balls. The main technical challenge of the theorem is to prove an effective Lipschitz bound in x for the s‐average of W_x(s). To accomplish this, we require a new log‐Sobolev inequality. Perelman's work implies that the metric measure spaces (Mⁿ,g(t),dvol_g(t)) satisfy a log‐Sobolev; we show that this is also true for the heat kernel weighted spaces (Mⁿ,g(t),H_x0(?,t)dvol_g(t)). Our log‐Sobolev constants for these weighted spaces are in fact universal and sharp. The weighted log‐Sobolev has other consequences as well, including certain average Gaussian upper bounds on the conjugate heat kernel. © 2014 Wiley Periodicals, Inc.     

On almost-quasi-commutative rings

A ring R is called almost-quasi-commutative if for each x, y ∈ R there exist nonzero relatively prime integers j = j(x, y) and k = k(x, y) and a non-negative integer n = n(x, y) such that jxy = k(yx) ⁿ . We establish some general properties of such rings, study commutativity of almost-quasi-commutative R, and consider several examples.     

Some commutativity criteria for prime rings with differential identities on Jordan ideals

In this article, we present some commutativity theorems for a ring R equipped with a generalized derivation satisfying certain differential identities on Jordan ideals of R. Some related results for prime rings are also discussed. Finally; we provide examples to show that the assumed restrictions are not superfluous.     

A Symmetric Presentation for J 1

Abstract

We give a computer-free proof that the sporadic simple group J ₁ is a isomorphic to the progenitor 2*⁵ : A ₅ factorized over a single relation. Precisely, we prove that J ₁ is defined by the presentation ?x, y, t ∣ x ⁵ = y ³ = (xy)² = 1 = t ² = [y, t] = [y, t ^{x 3} ] = (xt)⁷?.     

On the Oscillation of Second Order Neutral Differential Equations of Emden-Fowler Type

Some oscillation criteria are established by the averaging technique for the second order neutral delay differential equation of Emden-Fowler type (a(t)x￠(t))￠+q₁(t)| y(t-s₁)|^a sgn y(t-s₁) +q₂(t)| y(t-s₂)|^b sgn y(t-s₂)=0,    t 3 t₀,(a(t)x'(t))'+q_1(t)| y(t-\sigma_1)|^{\alpha}\,{\rm sgn}\,y(t-\sigma_1) +q_2(t)| y(t-\sigma_2)|^{\beta}\,{\rm sgn}\,y(t-\sigma_2)=0,\quad t \ge t_0, where x(t) = y(t) + p(t)y(t − τ), τ, σ₁ and σ₂ are nonnegative constants, α > 0, β > 0, and a, p, q ₁, q₂ ? C([t₀, ￥), \Bbb R)q_2\in C([t_0, \infty), {\Bbb R}) . The results of this paper extend and improve some known results. In particular, two interesting examples that point out the importance of our theorems are also included.