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1.
Using the fixed point method, we prove the Hyers–Ulam stability of homomorphisms in complex Banach algebras and complex Banach Lie algebras and also of derivations on complex Banach algebras and complex Banach Lie algebras for the general Jensen-type functional equation f(α xβ y) + f(α x ? β y) = 2α f(x) for any \({\alpha, \beta \in \mathbb{R}}\) with \({\alpha, \beta \neq 0}\) . Furthermore, we prove the hyperstability of homomorphisms in complex Banach algebras for the above functional equation with αβ = 1.  相似文献   

2.
We consider quadratic functions f that satisfy the additional equation y2 f(x) =  x2 f(y) for the pairs \({ (x,y) \in \mathbb{R}^2}\) that fulfill the condition P(x, y) =  0 for some fixed polynomial P of two variables. If P(x, y) =  axbyc with \({ a , b , c \in \mathbb{R}}\) and \({(a^2 + b^2)c \neq 0}\) or P(x,y) =  x n ? y with a natural number \({n \geq 2}\), we prove that f(x) =  f(1) x2 for all \({x \in \mathbb{R}}\). Some related problems, admitting quadratic functions generated by derivations, are considered as well.  相似文献   

3.
It is shown that, when α < ?1, the Laguerre operator ly = xy′' + (1 + α ? x)y′ requires more than one boundary condition at x = 0 in order to be self-adjoint. The more negative α, the more boundary conditions required.  相似文献   

4.
We study the existence and uniqueness of the following kind of backward stochastic differential equation, $$x(t) + \int_t^T {f(x(s),y(s),s)ds + \int_t^T {y(s)dW(s) = X,} }$$ under local Lipschitz condition, where (Ω, ?,P, W(·), ?t) is a standard Wiener process, for any given (x, y),f(x, y, ·) is an ?t-adapted process, andX is ?t-measurable. The problem is to look for an adapted pair (x(·),y(·)) that solves the above equation. A generalized matrix Riccati equation of that type is also investigated. A new form of stochastic maximum principle is obtained.  相似文献   

5.
Jack Maney 《代数通讯》2013,41(9):3496-3513
Let R be an integral domain, and let x ∈ R be a nonzero nonunit that can be written as a product of irreducibles. Coykendall and Maney (to appear), defined the irreducible divisor graph of x, denoted G(x), as follows. The vertices of G(x) are the nonassociate irreducible divisors of x (each from a pre-chosen coset of the form π U(R) for π ∈ R irreducible). Given distinct vertices y and z, we put an edge between y and z if and only if yz|x. Finally, if y n |x but y n+1 ? x, then we put n ? 1 loops on the vertex y.

In this article, inspired by the approach of the authors from Akhtar and Lee (to appear Akhtar , R. , Lee , L. Homology of zero divisors . To appear in Rocky Mountain J. Math.  [Google Scholar]), we study G(x) using homology. A connection is found between H 1 and the cycle space of G(x). We also characterize UFDs via these homology groups.  相似文献   

6.
An algebra with two binary operations · and +  that are commutative, associative, and idempotent is called a bisemilattice. A bisemilattice that satisfies Birkhoff’s equation x · (x + y) =  x + (x · y) is a Birkhoff system. Each bisemilattice determines, and is determined by, two semilattices, one for the operation +  and one for the operation ·. A bisemilattice for which each of these semilattices is a chain is called a bichain. In this note, we characterize the finite bichains that are weakly projective in the variety of Birkhoff systems as those that do not contain a certain three-element bichain. As subdirectly irreducible weak projectives are splitting, this provides some insight into the fine structure of the lattice of subvarieties of Birkhoff systems.  相似文献   

7.
In a classic Markov decision problem of Derman et al. (Oper. Res. 23(6):1120–1130, 1975) an investor has an initial capital x from which to make investments, the opportunities for which occur randomly over time. An investment of size y results in profit P(y), and the aim is maximize the sum of the profits obtained within a given time t. The problem is similar to a groundwater management problem of Burt (Manag. Sci. 11(1):80–93, 1964), the notorious bomber problem of Klinger and Brown (Stochastic Optimization and Control, pp. 173–209, 1968), and types of fighter problems addressed by Weber (Stochastic Dynamic Optimization and Applications in Scheduling and Related Fields, p. 148, 1985), Shepp et al. (Adv. Appl. Probab. 23:624–641, 1991) and Bartroff et al. (Adv. Appl. Probab. 42(3):795–815, 2010a). In all these problems, one is allocating successive portions of a limited resource, optimally allocating y(x,t), as a function of remaining resource x and remaining time t. For their investment problem, Derman et al. (Oper. Res. 23(6):1120–1130, 1975) proved that an optimal policy has three monotonicity properties: (A) y(x,t) is nonincreasing in t, (B) y(x,t) is nondecreasing in x, and (C) x?y(x,t) is nondecreasing in x. Theirs is the only problem of its type for which all three properties are known to be true. In the bomber problem the status of (B) is unresolved. For the general fighter problem the status of (A) is unresolved. We survey what is known about these exceedingly difficult problems. We show that (A) and (C) remain true in the bomber problem, but that (B) is false if we very slightly relax the assumptions of the usual model. We give other new results, counterexamples and conjectures for these problems.  相似文献   

8.
In 2001, Kawarabayashi proved that for any odd integer k ≥ 3, if a k-connected graph G is \({K^{-}_{4}}\) -free, then G has a k-contractible edge. He pointed out, by a counterexample, that this result does not hold when k is even. In this paper, we have proved the following two results on the subject: (1) For any even integer k ≥ 4, if a k-connected graph G is \({K_{4}^{-}}\) -free and d G (x) + d G (y) ≥ 2k + 1 hold for every two adjacent vertices x and y of V(G), then G has a k-contractible edge. (2) Let t ≥ 3, k ≥ 2t – 1 be integers. If a k-connected graph G is \({(K_{1}+(K_{2} \cup K_{1, t}))}\) -free and d G (x) + d G (y) ≥ 2k + 1 hold for every two adjacent vertices x and y of V(G), then G has a k-contractible edge.  相似文献   

9.
Jianhua Zhou 《代数通讯》2013,41(9):3724-3730
Let L, L′ be Lie algebras over a commutative ring R. A R-linear mapping f: L → L′ is called a triple homomorphism from L to L′ if f([x, [y, z]]) = [f(x), [f(y), f(z)]] for all x, y, z ∈ L. It is clear that homomorphisms, anti-homomorphisms, and sums of homomorphisms and anti-homomorphisms are all triple homomorphisms. We proved that, under certain assumptions, these are all triple homomorphisms.  相似文献   

10.
We characterize solutions ${f, g : \mathbb{R} \to \mathbb{R}}$ of the functional equation f(x + g(x)y) = f(x)f(y) under the assumption that f is locally bounded above at each point ${x \in \mathbb{R}}$ . Our result refers to Go?a?b and Schinzel (Publ Math Debr 6:113–125, 1959) and Wo?od?ko (Aequationes Math 2:12–29, 1968).  相似文献   

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