On the divergence of Fourier series in the spaces ϕ(<Emphasis Type="Italic">L</Emphasis>) containing <Emphasis Type="Italic">L</Emphasis>

The paper deals with the question of the divergence of Fourier series in function spaces wider than L = L[?π, π], but narrower than L^p = L^p[?π, π] for all p ∈ (0, 1). It is proved that the recent results of Filippov on the generalization to the space ?(L) of Kolmogorov’s theorem on the convergence of Fourier series in L^p, p ∈ (0, 1), cannot be improved.     

Holomorphic factorization of polynomials

We give necessary and sufficient conditions for a holomorphic factorization of an irreducible polynomial P(s, λ), s ∈ Cⁿ, λ ∈ C, in a domain Ω ? Cⁿ which is connected with the ordering of the real part of the roots of the equation P(s, λ) = 0, s ∈ Ω.     

Interpolation and duality of generalized grand Morrey spaces on quasi-metric measure spaces

Let θ ∈ (0, 1), λ ∈ [0, 1) and p, p ₀, p ₁ ∈ (1,∞] be such that (1 ? θ)/p ₀ + θ/p ₁ = 1/p, and let φ, φ₀, φ₁ be some admissible functions such that φ, φ₀ ^p/p0 and φ₁ ^p/p1 are equivalent. We first prove that, via the ± interpolation method, the interpolation L _φ0 ^p0),λ (X), L _φ1 ^{p1), λ} (X), θ> of two generalized grand Morrey spaces on a quasi-metric measure space X is the generalized grand Morrey space L _φ ^p),λ (X). Then, by using block functions, we also find a predual space of the generalized grand Morrey space. These results are new even for generalized grand Lebesgue spaces.     

Positive solutions of <Emphasis Type="Italic">p</Emphasis>-th Yamabe type equations on graphs

Let G = (V,E) be a finite connected weighted graph, and assume 1 ? α ? p ? q. In this paper, we consider the p-th Yamabe type equation ―?_pu+hu^q―1 = λfu^α―1 on G, where ?_p is the p-th discrete graph Laplacian, h < 0 and f > 0 are real functions defined on all vertices of G. Instead of H. Ge’s approach [Proc. Amer. Math. Soc., 2018, 146(5): 2219–2224], we adopt a new approach, and prove that the above equation always has a positive solution u > 0 for some constant λ ∈ ?. In particular, when q = p, our result generalizes Ge’s main theorem from the case of α ? p > 1 to the case of 1 ? α ? p, It is interesting that our new approach can also work in the case of α ? p > 1.     

Behavior of the Fourier–Walsh coefficients of a corrected function

We prove that, given a sequence {ak}_k=1^∞ with a_k ↓ 0 and {ak}_k=1^∞ ? l₂, reals 0 < ε < 1 and p ∈ [1, 2], and f ∈ L^p(0, 1), we can find f ∈ L^p(0, 1) with mes{f ≠ f < ε whose nonzero Fourier–Walsh coefficients c_k(f) are such that |c_k(f)| = a_k for k ∈ spec(f).     

On the periodicity of continued fractions in elliptic fields

Article [1] raised the question of the finiteness of the number of square-free polynomials f ∈ ?[h] of fixed degree for which \(\sqrt f \) has periodic continued fraction expansion in the field ?((h)) and the fields ?(h)(\(\sqrt f \)) are not isomorphic to one another and to fields of the form ?(h)\(\left( {\sqrt {c{h^n} + 1} } \right)\), where c ∈ ?* and n ∈ ?. In this paper, we give a positive answer to this question for an elliptic field ?(h)(\(\sqrt f \)) in the case deg f = 3.     

Characterization of convex domains in ℂ<Superscript>2</Superscript> with non-compact automorphism group

In the field of several complex variables, the Greene-Krantz Conjecture, whose consequences would be far reaching, has yet to be proven. The conjecture is as follows: Let D be a smoothly bounded domain in ?ⁿ. Suppose there exists {g _j} ? Aut(D) such that {g _j(z)} accumulates at a boundary point p ∈ ?D for some z ∈ D. Then ?D is of finite type at p. In this paper, we prove the following result, yielding further evidence to the probable veracity of this important conjecture: Let D be a bounded convex domain in ?² with C ² boundary. Suppose that there is a sequence {g _j} ? Aut(D) such that {g _j(z)} accumulates at a boundary point for some point z ∈ D. Then if p ∈ ?D is such an orbit accumulation point, ?D contains no non-trivial analytic variety passing through p.     

Degree sum of a pair of independent edges and <Emphasis Type="Italic">Z</Emphasis><Subscript>3</Subscript>-connectivity

Let G be a 2-edge-connected simple graph on n vertices. For an edge e = uv ∈ E(G), define d(e) = d(u) + d(v). Let F denote the set of all simple 2-edge-connected graphs on n ≥ 4 vertices such that G ∈ F if and only if d(e) + d(e’) ≥ 2n for every pair of independent edges e, e’ of G. We prove in this paper that for each G ∈ F, G is not Z ₃-connected if and only if G is one of K _2,n?2, K _3,n?3, K _2,n?2 ⁺ , K _3,n?3 ⁺ or one of the 16 specified graphs, which generalizes the results of X. Zhang et al. [Discrete Math., 2010, 310: 3390–3397] and G. Fan and X. Zhou [Discrete Math., 2008, 308: 6233–6240].     

An implicit degree condition for relative length of long paths and cycles in graphs

For a graph G, we denote by p(G) and c(G) the number of vertices of a longest path and a longest cycle in G, respectively. For a vertex v in G, id(v) denotes the implicit degree of v. In this paper, we obtain that if G is a 2-connected graph on n vertices such that the implicit degree sum of any three independent vertices is at least n + 1, then either G contains a hamiltonian path, or c(G) ≥ p(G) ? 1.     

Results of Diophantine approximation by unlike powers of primes

Let k be an integer with k ≥ 6: Suppose that λ₁, λ₂,..., λ₅ be nonzero real numbers not all of the same sign, satisfying that λ₁/λ₂ is irrational, and suppose that η is a real number. In this paper, for any ε > 0; we consider the inequality |λ₁p₁ + λ₂p ₂ ² + λ₃p ₃ ³ + λ₄p ₄ ⁴ + λ₅p ₅ ^k + η | < (max p_j)^-σ(k)+ε has infinitely many solutions in prime variables p₁, p₂,...,p₅, where σ(k) depends on k. Our result gives an improvement of the recent result. Furthermore, using the similar method in this paper, we can refine some results on Diophantine approximation by unlike powers of primes, and get the related problem.     

Free groups and ends of graphs

Let X be a connected graph. An automorphism of X is said to be parabolic if it leaves no finite subset of vertices in X invariant and fixes precisely one end of X and hyperbolic if it leaves no finite subset of vertices in X invariant and fixes precisely two ends of X. Various questions concerning dynamics of parabolic and hyperbolic automorphisms are discussed.The set of ends which are fixed by some hyperbolic element of a group G acting on X is denoted by ?(G). If G contains a hyperbolic automorphism of X and G fixes no end of X, then G contains a free subgroup F such that ?(F) is dense in ?(G) with respect to the natural topology on the ends of X.As an application we obtain the following: A group which acts transitively on a connected graph and fixes no end has a free subgroup whose directions are dense in the end boundary.     

Сильная суммируeмость интегралов Фурье и сопяжённых им в точках Лебега порядка <Emphasis Type="Italic">p</Emphasis> ∈ (1,2]

In the paper, the behavior of (φ, λ)-strong means of Fourier integrals and their conjugates is investigated at Lebesgue’s points of order p ∈ (1, 2]. Based on the results obtained, consequences are deduced for the φ-strong summability of Fourier integrals and their conjugates; in particular, by the logarithmic and (C, α)-methods of summability at Lebesgue’s points of order p ∈ (1, 2].     

Equiconvergence of expansions in multiple Fourier series and in fourier integrals with “lacunary sequences of partial sums”

We investigate the equiconvergence on T^N = [?π, π)^N of expansions in multiple trigonometric Fourier series and in the Fourier integrals of functions f ∈ L_p(T^N) and g ∈ L_p(R^N), p > 1, N ≥ 3, g(x) = f(x) on T^N, in the case where the “partial sums” of these expansions, i.e., S_n(x; f) and J_α(x; g), respectively, have “numbers” n ∈ Z^N and α ∈ R^N (n_j = [α_j], j = 1,..., N, [t] is the integral part of t ∈ R¹) containing N ? 1 components which are elements of “lacunary sequences.”     

On the completeness of the system of root vectors of the Sturm-Liouville operator with general boundary conditions

We study general boundary value problems with nondegenerate characteristic determinant Δ(λ) for the Sturm-Liouville equation on the interval [0, 1]. Necessary and sufficient conditions for the completeness of root vectors are obtained in terms of the potential. In particular, it is shown that if Δ(λ) ≠ const, q(·) ∈ C ^k [0, 1] for some k ? 0, and q ^(k)(0) ≠ (?1) ^k q ^(k)(1), then the system of root vectors is complete and minimal in L ^p [0, 1] for p ∈ [1,∞).     

Hereditary normality of a space of the form ℱ(<Emphasis Type="Italic">X</Emphasis>)

Assuming the continuum hypothesis we construct an example of a nonmetrizable compact set X with the following properties(1) X ⁿ is hereditarily separable for all n ∈ ?(2) X ⁿ \ Δ _n is perfectly normal for every n ∈ ?, where Δ _n is the generalized diagonal of X ⁿ , i.e., the set of points with at least two equal coordinates(3) for every seminormal functor ? that preserves weights and the points of bijectivity the space ? _k (X) is hereditarily normal, where k is the second smallest element of the power spectrum of the functor ?; in particular, X ² and λ ₃ X are hereditarily normal.Our example of a space of this type strengthens the well-known example by Gruenhage of a nonmetrizable compact set whose square is hereditarily normal and hereditarily separable.     

Convergence analysis of the rectangular Morley element scheme for second order problem in arbitrary dimensions

We present the convergence analysis of the rectangular Morley element scheme utilised on the second order problem in arbitrary dimensions. Specifically, we prove that the convergence of the scheme is of O(h) order in energy norm and of O(h²) order in L² norm on general d-rectangular triangulations. Moreover, when the triangulation is uniform, the convergence rate can be of O(h²) order in energy norm, and the convergence rate in L² norm is still of O(h²) order, which cannot be improved. Numerical examples are presented to demonstrate our theoretical results.     

Blow-up of critical norms for the 3-D Navier-Stokes equations

Let u =(uh, u3) be a smooth solution of the 3-D Navier-Stokes equations in R3× [0, T). It was proved that if u3 ∈ L∞(0, T;˙B-1+3/p p,q(R3)) for 3 p, q ∞ and uh∈ L∞(0, T; BMO-1(R3)) with uh(T) ∈ VMO-1(R3), then u can be extended beyond T. This result generalizes the recent result proved by Gallagher et al.(2016), which requires u ∈ L∞(0, T;˙B-1+3/pp,q(R3)). Our proof is based on a new interior regularity criterion in terms of one velocity component, which is independent of interest.     

On the smoothness of the conjugacy between circle maps with a break

For any α ∈ (0, 1), c ∈ ?₊ \ {1} and γ > 0 and for Lebesgue almost all irrational ρ ∈ (0, 1), any two C ^2+α -smooth circle diffeomorphisms with a break, with the same rotation number ρ and the same size of the breaks c, are conjugate to each other via a C ¹-smooth conjugacy whose derivative is uniformly continuous with modulus of continuity ω(x) = A|log x|^?γ for some A > 0.     

Implicit differential equations with impasse singularities and singular perturbation problems

This paper concerns families of constrained differential systems having the form {fx1-1} where h: ? ⁿ → ? is a C ^r -function, (α ₁, …, α _n ) is a C ^r vector field on ? ⁿ , r ≥ 1, x ∈ ? ⁿ and λ is a real parameter. Using singular perturbation techniques we present some results on the dynamics of the system around a point (x, λ) of the impasse set λ = h(x). In addition, planar constrained systems with non-regular impasse surfaces are considered.     

Tikhonov–Phillips regularizations in linear models with blurred design

The paper deals with recovering an unknown vector β ∈ ? ^p based on the observations Y = Xβ + ? ξ and Z = X + σζ, where X is an unknown n × p matrix with n ≥ p, ξ ∈ ? ^p is a standard white Gaussian noise, ζ is an n × p matrix with i.i.d. standard Gaussian entries, and ?, σ ∈ ?⁺ are known noise levels. It is assumed that X has a large condition number and p is large. Therefore, in order to estimate β, the simple Tikhonov–Phillips regularization (ridge regression) with a data-driven regularization parameter is used. For this estimation method, we study the effect of noise σζ on the quality of recovering Xβ using concentration inequalities for the prediction error.