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In this paper, it is shown that there is a gap in the paper [Chidume, C. E., Shahzad, N.: Weak convergence theorems for a finite family of strict pseudo-contractions. Nonlinear Anal., 72, 1257-1265 (2010)], consequently, the main results of the paper do not hold in uniformly smooth Banach spaces. Meanwhile, it is also shown that the main results (Lemma 3.4, Theorems 3.5-3.6, 3.8-3.9) in the paper [Cholamjiak, P., Suantai, S.: Weak convergence theorems for a countable family of strict pseudo-contractions in Banach spaces. Fixed Point Theory Appl., 2010, Article ID 632137, 16 pages (2010)] do not hold in L for p > 3. Finally, some modified results are presented in the setting of uniformly smooth and uniformly convex Banach spaces which include L for p ≥ 2 as special cases. Furthermore, our arguments are also different from the ones given by the authors above. [ABSTRACT FROM AUTHOR]

Let p be a prime, m ≥ 2, and ( m, p( p – 1)) = 1. In this paper, we will calculate explicitly the Gauss sum $$ G{\left( \chi \right)} = {\sum {_{{x \in \mathbb{F}^{ * }_{q} }} \chi {\left( x \right)}\zeta ^{{T{\left( x \right)}}}_{p} } } $$ in the case of [ (ℤ/ mℤ)* : 〈 p〉] = 4, and −1 ∉ 〈 p〉, where q = p f , $$ f = \frac{{\varphi {\left( m \right)}}} {4} $$ , χ is a multiplicative character of $${\Bbb F}$$ q with order m, and T is the trace map for $${\Bbb F}$$ q / $${\Bbb F}$$ p . Under the assumptions [ (ℤ/ mℤ)* : 〈 p〉] = 4 and −1 ∉ 〈 p〉, the decomposition field of p in the cyclotomic field ℚ(ζ m ) is an imaginary quartic (abelian) field. And G (χ) is an integer in K. We deal with the case where K is cyclic in this paper and leave the non–cyclic case to the next paper. [ABSTRACT FROM AUTHOR]

Let X* be a free monoid over an alphabet X and W be a finite language over X. Let S( W) be the Rees quotient X*/ I( W), where I( W) is the ideal of X* consisting of all elements of X* that are not subwords of W. Then S( W) is a finite monoid with zero and is called the discrete syntactic monoid of W. W is called finitely based if the monoid S( W) is finitely based. In this paper, we give some sufficient conditions for a monoid to be non-finitely based. Using these conditions and other results, we describe all finitely based 2-limited words over a three-element alphabet. Furthermore, an explicit algorithm is given to decide that whether or not a 2-limited word in which there are exactly two non-linear letters is finitely based. [ABSTRACT FROM AUTHOR]

Kernel functions play an important role in defining new search directions for primal-dual interior-point algorithm for solving linear optimization problems. In this paper we present a new kernel function which yields an algorithm with the best known complexity bound for both large- and small-update methods. [ABSTRACT FROM AUTHOR]

An algebra A is said to be congruence permutable if any two congruences on it are permutable. This property has been investigated in several varieties of algebras, for example, de Morgan algebras, p-algebras, Kn,o-algebras. In this paper, we study the class of symmetric extended de Morgan algebras that are congruence permutable. In particular we consider the case where A is finite, and show that A is congruence permutable if and only if it is isomorphic to a direct product of finitely many simple algebras. [ABSTRACT FROM AUTHOR]

In this paper, we offer a graded equivalence between the quotient categories defined by any graded Morita–Takeuchi context via certain modifications of the graded cotensor functors. As a consequence, we show a commutative diagram that establish the relation between the closed objects of the categories gr C and M C , where C is a graded coalgebra. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the expected number of steps in solving multi–discrete logarithm problems over a group of elliptic curves with prime order by using Pollard's rho method and parallel collision search algorithm. We prove that when using these algorithms to compute discrete logarithms, the knowledge gained through computing many logarithms does not make it easier for finding other logarithms. Hence in an elliptic cryptosystem, it is safe for many users to share the same curve, with different private keys. [ABSTRACT FROM AUTHOR]