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We are concerned with the following class of equations with exponential nonlinearities: which is related to the Tzitzéica equation. Here $$h_1, h_2$$ are two smooth positive functions. The purpose of the paper is to initiate the analytical study of the above equation and to give a quite complete picture both for what concerns the blow-up phenomena and the existence issue. In the first part of the paper we provide a quantization of local blow-up masses associated to a blowing-up sequence of solutions. Next we exclude the presence of blow-up points on the boundary under the Dirichlet boundary conditions. In the second part of the paper we consider the Tzitzéica equation on compact surfaces: we start by proving a sharp Moser-Trudinger inequality related to this problem. Finally, we give a general existence result. [ABSTRACT FROM AUTHOR]

In this paper, it is proved that with at most $$ O{\left( {N^{{\frac{{65}} {{66}}}} } \right)} $$ exceptions, all even positive integers up to N are expressible in the form $$ p^{2}_{2} + p^{3}_{3} + p^{4}_{4} + p^{5}_{5} $$ . This improves a recent result $$ O{\left( {N^{{\frac{{19193}} {{19200}} + \varepsilon }} } \right)} $$ due to C. Bauer. [ABSTRACT FROM AUTHOR]

The main result of the paper is a theorem, using which a new proof of Roth’s theorem is obtained, a new solvability criterion for the matrix equation AX-YB = C is proved, a formula for a particular solution of the latter is derived, and the least of the orders of square nonsingular matrices containing a given rectangular matrix as a submatrix is determined. Bibliography: 5 titles. [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]

In this paper, we shall prove that for any orientable 3–manifold M, there is a link L = K ∪ K 1 ∪ K 2 ∪ K 3 with four components in M, such that the complement of L, say M L , contains separating essential closed surfaces of all positive genera. [ABSTRACT FROM AUTHOR]

H. P. Gumm and T. Schröder stated a conjecture that the preservation of preimages by a functor T for which | T1| = 1 is equivalent to the satisfaction of the class equality $${{\mathcal {HS}}({\sf K}) = {\mathcal {SH}}({\sf K})}$$ for any class K of T-coalgebras. Although T. Brengos and V. Trnková gave a positive answer to this problem for a wide class of Set-endofunctors, they were unable to find the full solution. Using a construction of a rigid unary algebra we prove $${{\mathcal {HS}} \neq {\mathcal {SH}}}$$ for a class of Set-endofunctors not preserving non-empty preimages; these functors have not been considered previously. [ABSTRACT FROM AUTHOR]

We consider the problem of testing whether a given system of equations over a fixed finite semigroup S has a solution. For the case where S is a monoid, we prove that the problem is computable in polynomial time when S is commutative and is the union of its subgroups but is NP-complete otherwise. When S is a monoid or a regular semigroup, we obtain similar dichotomies for the restricted version of the problem where no variable occurs on the right-hand side of each equation. We stress connections between these problems and constraint satisfaction problems. In particular, for any finite domain D and any finite set of relations Γ over D, we construct a finite semigroup SΓ such that CSP(Γ) is polynomial-time equivalent to the satifiability problem for systems of equations over SΓ. [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]