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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]

In this paper, we investigate a decidable case of the term semiunification problem. In our case, the arity of any functional symbol in terms equals one (with a possible exception for outer symbols of terms, for which no restrictions are imposed). We propose an algorithm constructing the most general solution of the semiunification problem. In addition, we prove an upper bound on the height of this solution; this upper bound is linear with respect to heights of terms in the initial problem. Our method reduces the problem for terms to a special system of equations in free semigroups and solves the latter system. Bibliography: 10 titles. [ABSTRACT FROM AUTHOR]

Let ( Ω, μ) be a σ-finite measure space and Φ : Ω × [0,∞) → [0,∞] be a Musielak–Orlicz function. Denote by L Φ( Ω) the Musielak–Orlicz space generated by Φ. We prove that the Amemiya norm equals the Orlicz norm in L Φ( Ω). [ABSTRACT FROM AUTHOR]

The article analyzes enough conditions on the lower order coefficients of the parabolic equations in the second-order. It discusses the solution of the Cauchy problem in classes of rising initial functions stables x-uniformly to zero on each compact set K ∈ RN. It presents the reality of coefficients αik = αki (i, k = 1, 2, ..., N) and the satisfaction of the uniform parabolicity conditions; Ǝ there exist constants ɻ0 0 and ɻ1.