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We will show in this paper that if λ is very close to 1, then can be attained, where M is a compact–manifold with boundary. This result gives a counter–example to the conjecture of de Figueiredo and Ruf in their paper titled "On an inequality by Trudinger and Moser and related elliptic equations" ( Comm. Pure. Appl. Math., 55, 135–152, 2002). [ABSTRACT FROM AUTHOR]

Let E be a cookie-cutter set with dim H E = s. It is known that the Hausdorff s-measure and the packing s-measure of the set E are positive and finite. In this paper, we prove that for a gauge function g the set E has positive and finite Hausdorff g-measure if and only if 0 < lim inf t→0 $$ \tfrac{{g(t)}} {{t^s }} $$ < ∞. Also, we prove that for a doubling gauge function g the set E has positive and finite packing g-measure if and only if 0 < lim sup t→0 $$ \tfrac{{g(t)}} {{t^s }} $$ < ∞. [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]

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]

Since the Leibniz–Newton formula for derivatives cannot be used in local fields, it is important to investigate the new concept of derivatives in Walsh–analysis, or harmonic analysis on local fields. On the basis of idea of derivatives introduced by Butzer, Schipp and Wade, Weisz has proved that the maximal operators of the one–dimensional dyadic derivative and integral are bounded from the dyadic Hardy space H p,q to L p,q , of weak type ( L 1 , L 1 ), and the corresponding maximal operators of the two–dimensional case are of weak type $$ {\left( {H^{\# }_{1} ,L_{1} } \right)} $$ . In this paper, we show that these maximal operators are bounded both on the dyadic Hardy spaces H p and the hybrid Hardy spaces $$ \begin{array}{*{20}c} {{H^{\# }_{p} }} & {{0 < p \leqslant 1}} \\ \end{array} $$ . [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]

A right R–module E over a ring R is said to be maximally injective in case for any maximal right ideal m of R, every R–homomorphism f : m → E can be extended to an R–homomorphism f' : R → E. In this paper, we first construct an example to show that maximal injectivity is a proper generalization of injectivity. Then we prove that any right R–module over a left perfect ring R is maximally injective if and only if it is injective. We also give a partial affirmative answer to Faith's conjecture by further investigating the property of maximally injective rings. Finally, we get an approximation to Faith's conjecture, which asserts that every injective right R–module over any left perfect right self–injective ring R is the injective hull of a projective submodule. [ABSTRACT FROM AUTHOR]

This work focuses on the second type of generalized Feigenbaum's equation where φ( x) is C-increasing function on [0, 1] and satisfies that φ(0) = 0,0 < φ′( x) < 1 ( x ∈ [0, 1]). Using constructive method, we discuss the existence of C-single-valley solutions whose derivatives are not equal to 0 on origin of the above equation. [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]