Xiao-Xiong Gan

• Morgan State University

In this paper we provide necessary and sufficient conditions for a formal power series g over R or C to have formal power series g1/k such that (g1/k)k=g for all positive integers k or for some positive integers. The algorithms of computing such formal root series are also provided. Moreover, we investigate the uniqueness of formal root series if t...

In this article we consider the topology on the set of formal Laurent series induced by the ultrametric defined via the order. In particular, we establish that the product of formal Laurent series, considered in [GAN, X. X. - BUGAJEWSKI, D.:On formal Laurent series, Bull. Braz. Math. Soc. 42 (2011), 415-437], is not continuous. We also show some ap...

In the first part of the paper we examine mappings of higher order from a general point of view, that is, in normed spaces of bounded real-valued functions defined on RR. Particular attention is paid to the relation of such mappings with the so-called autonomous superposition operators. Next we investigate mappings of higher order in Banach spaces...

This article introduces the (just-in-time) JIT-transportation problem, which requires that all demanded goods be shipped to their destinations on schedule, at a zero or minimal destination-storage cost. The JIT-transportation problem is a special goal programming problem with discrete constraints. This article provides a mathematical model for such...

Several kinds of formal Laurent series have been introduced with some restrictions so far. This paper systematically sets up a natural definition and structure of formal Laurent series without those restrictions, including introducing a multiplication between formal Laurent series. This paper also provides some results on the algebraic structure of...

This paper establishes a new assignment problem called the time limit assignment problem (TLAP) which pursues the best overall efficiency only if all tasks have been completed in time or very close to the deadline. The time limit assignment problem contains one discrete goal constraint, and therefore it is a special goal programming problem. This p...

Given a formal power series g(x) = P 1 n=0 bn x n and a nonunitf(x) = P 1 n=1 an x n , it is well-known that the derivative of the composition of formal power series satisfies the Chain Rule, that is, (g - f) 0 = g 0 (f) ¢ f 0 . It is also proved that the right distributive law for formal power series exists if the composed series, such as f above,...

Given a formal power series g(x) = b0 +b1x +b2x2 +··· and a nonunit f(x)= a1x + a2x2 +··· , it is well known that the composition of g with f , g(f(x)), is a formal power series. If the formal power series f above is not a nonunit, that is, the constant term of f is not zero, the existence of the composition g(f(x)) has been an open problem for man...

This paper generalizes Marcinkiewicz’s universal primitive on pointwise a.e. convergence directly to higher-dimensional spaces. It is also proved that the set of all universal primitive functions with respect to some given nonzero null sequence is residual and, hence, dense in the Banach space C(In, Rm) n, m ε N.

This paper generalizes Marcinkiewicz’s universal primitive on pointwise a.e. convergence directly to higher-dimensional spaces. It is also proved that the set of all universal primitive functions with respect to some given nonzero null sequence is residual and, hence, dense in the Banach space C ( I n , R m ) ∀ n , m ∈ N C({I^n},{\mathbb {R}^m})\fo...

For n ∈ N n \in \mathbb {N} and I = [ 0 , 1 ] I = [0,1] , let I n {I^n} be the unit cube and λ n {\lambda ^n} the Lebesgue measure in R n {\mathbb {R}^n} . It is proved that if f : I n → R n f:{I^n} \to {\mathbb {R}^n} and F 0 : I n → R {F_0}:{I^n} \to \mathbb {R} are continuous and ε > 0 \varepsilon > 0 , then there exist a continuous F : I n → R...

For n ∈ N and I = [ 0, 1 ], let In be the unit cube and λn the Lebesgue measure in Rn. It is proved that if f: In → Rn and F0: In → R are continuous and $\varepsilon > 0$, then there exist a continuous F: In → R and an open set $W \subset (I^n)^\circ$ with λn(W) = 1 such that $\text{(i)}\, \nabla F \text{exists and is continuous on} W,$ $\text{(ii)...

Given a formal power series g(x )= b0 + b1x + b2x 2 + ··· and a nonunit f(x )= a1x + a2x 2 + ···, it is well known that the composition of g with f, g(f(x)) , is a formal power series. Under the definition of the derivative of formal power series, the Chain rule of the composition is also true if the composed formal power series is a nonunit. If th...

Let ε>0 and let f:I n →ℝ n be a continuous mapping for any n∈ℕ. Then there exists an approximate antigradient F [see the author and K. R. Stromberg, Proc. Am. Math. Soc. 119, No. 4, 1201-1209 (1993; preceding review)] of f with respect to ε. In this paper, we are going to show that we may also recapture some of those antigradients from its gradient...