We construct an explicit example of an asymptotically conformal chord-arc curve that fails to be asymptotically smooth. This implies that a function belonging to both the little Bloch space and BMOA does not necessarily lie in VMOA , and that a strongly quasisymmetric homeomorphism which is symmetric is not necessarily strongly symmetric. We also provide a complete characterization of asymptotically smooth curves in terms of asymptotic conformality and uniform approximability.
In this paper, we study the normalized solutions and regularity of solutions for a class of the Biharmonic equations with p-Laplacian as follows {[ Δ ^2u+Δ _p u=μ |u|^q-2u+λ u, in ℝ^N,; ∫ _ℝ^N|u|^2dx=c, ]. where μ >0 , λ <0 is a Lagrange multiplier, 2
In this paper, we prove that the Artinian Gorenstein 𝕂-algebra A_F_s of codimension n, socle degree d and Macaulay dual generator F_s := ℓ_1^d + … + ℓ_s^d ∈𝕂[X_1, …, X_n] where ℓ _1, ⋯ , ℓ_s are general linear forms satisfies the strong Lefschetz property (SLP). This result allows us to study whether the Waring rank of F_s is exactly s. Furthermore, we show that A_F_s is the doubling of a suitable 0-dimensional scheme Z_F_s in ℙ^n-1, the so-called tight annihilating scheme of A_F_s, and we compute the minimal free resolution of A_F_s in terms of the minimal free R-resolution of I(Z_F_s). Finally, we determine the linear general Jordan type of A_F_s.
Every finite group G acts as an automorphism group of several bordered Klein surfaces. The minimal genus of these surfaces is called the real genus ρ (G) of the group G. It is known that all odd positive integers are the real genus of some group. On the contrary, not all even integers are. C. L. May compiled a series of families of groups, from which he obtained arithmetic sequences of even numbers which are real genus of some group, covering a large part of the even numbers. In particular, it results that 2, 12 and 24 are not the real genus of any group. May asked on whether this is a question of small numbers, or else there are other gaps in the spectrum of the real genus, that is to say, numbers N such that there are no groups of real genus N. Recently, it has been proved that 72 is not the real genus of a group. The next two numbers on which the question remained unsolved are 84 and 108. In the present work we prove that there is no group of each real genus 84 and 108.
The classical Gregory coefficients are also known as the (reciprocal) logarithmic numbers, the Cauchy numbers of the first kind or the Bernoulli numbers of the second kind. In this paper, we define Gregory coefficients of arbitrary order via the reciprocal of high powers of the natural logarithm and examine their many elegant properties analogous to those of the classical Gregory coefficients. In particular, we obtain several identities involving infinite series with higher-order Gregory coefficients and Euler’s (also known as Euler–Mascheroni’s) constant.
In the first part of the paper we show that every closed subspace of JT or JT^* contains ℓ _2 complemented in JT or JT^* respectively, and JT contains uncomplemented copies of ℓ _2 . As a result, the predual ℬ of JT, as well as the spaces JT and JT^* , are subprojective and superprojective. In the second part, we prove that every weakly Cauchy sequence that is not weakly convergent in JT has a subsequence equivalent to the basis of J. Hence, every non-reflexive subspace of JT contains an isomorphic copy of J, and every Schauder basic sequence in JT has a subsequence which is equivalent either to the basis of ℓ _2 or to the basis of J. Moreover these subspaces may be selected to be complemented in JT.
In this paper we study the Cauchy problem for semilinear parabolic system with nonconstant coefficient singular initial data {[ U_t-Δ U=μ _1|U|^p-1U+β |U|^r-1U|V|^r+1, x∈ℝ^N,t>0,; V_t-Δ V=μ _2|V|^p-1V+β |U|^r+1|V|^r-1V, x∈ℝ^N,t>0,; U(x,0)=λ _1 a(x/|x|)|x|^-2/(p-1), x∈ℝ^N∖{0},; V(x,0)=λ _2 b(x/|x|)|x|^-2/(p-1), x∈ℝ^N∖{0},; ]. where N≥ 2 , p=2r+1 , μ _1,μ _2,β >0 , λ _1,λ _2>0 are constant parameters, a≥ 0≢0 , b≥ 0≢0 . We demonstrate that when 2
We consider a degenerate p-Laplacian equation, with a reaction that exhibits the competing effects of a parametric concave term and of a sign-changing convex perturbation. Using variational tools and critical groups, we prove an existence and multiplicity result which is global in the parameter. In the process, we also prove some general results of independent interest.
Let f:M→ M be a homeomorphism on a closed manifold M. Let M be a universal covering of M and f be a lifting of f to M . We prove that topological stability for f on M induces relative topological stability for f on M . Conversely, we also prove that the relative topological stability for f with a leafwise condition induces topological stability for f. Finally, we construct a homeomorphism on M which is relatively N-expansive, but is not relatively (N-1) -expansive.
In this paper, we consider the semilinear elliptic equation ε ^2Δ u+V(x)(|u-1|^p-1)=0, u>0, u∈ H^1(ℝ^2), where ε >0 is a small parameter, the power p>1 , V is a smooth positive function. Under the appropriate gap condition, the problem admits a solution u_ε concentrating along a closed curve Γ , which is stationary and nondegenerate with respect to the weighted length functional ∫ _Γ V^1/2 .
A lambda-translator in H-2 & times;R is a surface whose mean curvature H satisfies H = < N, partial derivative(z)>+lambda, where N is the unit normal of the surface, partial derivative(z) is the vertical Killing vector field and lambda is an element of R. In this paper, we study how the geometry of the boundary of a compact lambda translator affects the shape of the surface, asking under what conditions the symmetries of the boundary are inherited by the whole surface. Due to the product structure of H-2 & times; R and the geometry of H-2, we distinguish between different notions of graphs and reflections. We provide conditions on the boundary curve of the surface to ensure that an embedded compact lambda-translator is a graph. Finally, we present estimates for the area of a vertical graph lambda-translator in terms of its height and volume.
Quasi-isometries are a versatile type of maps that preserve the large-scale geometry of spaces, while introducing significant local distortions. Following Kanai’s work, which established the invariance of various analytic and geometric properties under quasi-isometries, this paper generalizes isoperimetric and Sobolev inequalities for exponents less than the manifold’s dimension, proving both that they are equivalent and preserved by quasi-isometries.
Our main goal in this paper is to extend mathematical ideas of Gromov concerning the phenomenon of symplectic squeezing from real to p-adic geometry. Let n⩾ 2 be an integer and let p be a prime number. We prove that the analog of Gromov’s non-squeezing theorem does not hold for p-adic embeddings: for any p-adic absolute value R, the entire p-adic space (ℚ_p)^2n is symplectomorphic to the p-adic cylinder Z_p^2n(R) of radius R, showing a degree of flexibility which stands in contrast with the real case. However, some rigidity remains: we prove that the p-adic affine analog of Gromov’s result still holds. We will also show that in the nonlinear situation, if the p-adic embeddings are equivariant with respect to a torus action, then non-squeezing holds, which generalizes a recent result by Figalli, Palmer and the second author. This allows us to introduce equivariant p-adic analytic symplectic capacities, of which the p-adic equivariant Gromov width is an example.
A λ -translator in ℍ^2×ℝ is a surface whose mean curvature H satisfies H= ⟨ N,∂ _z⟩ +λ , where N is the unit normal of the surface, ∂ _z is the vertical Killing vector field and λ∈ℝ . In this paper, we study how the geometry of the boundary of a compact λ -translator affects the shape of the surface, asking under what conditions the symmetries of the boundary are inherited by the whole surface. Due to the product structure of ℍ^2×ℝ and the geometry of ℍ^2 , we distinguish between different notions of graphs and reflections. We provide conditions on the boundary curve of the surface to ensure that an embedded compact λ -translator is a graph. Finally, we present estimates for the area of a vertical graph λ -translator in terms of its height and volume.
Approximation properties of shift–invariant subspaces and their dilations in Sobolev spaces are studied. Indeed, we characterize those shift–invariant subspaces that provide a fixed simultaneous approximation order and/or simultaneous density order. Here, we work in a multidimensional context and consider dilations of shift–invariant subspaces by powers of a fix expansive linear map. We put emphasis on the finitely generated shift–invariant subspaces. To give our results on simultaneous density order we need the notion of approximate continuity associated to the considered expansive linear map. Since the linear maps can be aniisotropic, our conditions depend of such dilations.
We introduce and study a strict monotonicity property of the norm in solid Banach lattices of real functions that prevents such spaces from having the local diameter two property. Then we show that any strictly convex 1-symmetric norm on c_0(Γ ) possesses this property. In the opposite direction, we show that any Banach space which is strictly convex renormable and contains a complemented copy of c_0(ℕ), admits an equivalent strictly convex norm for which the space has the local diameter two property. In particular, this enables us to construct a strictly convex norm on c_0(Γ ), where Γ is uncountable, for which the space has a 1-unconditional basis and the local diameter two property.
We consider evolution equation with fractional Schrödinger operators in Morrey spaces. We prove order preserving properties of the associated semigroup in Morrey scale. We prove monotonicity of the semigroup with respect to Morrey’s potentials and give some precise estimates of its exponential growth. We show that the Arendt-Batty condition on the potential is necessary for exponential decay of Morrey’s norms of the semigroup and find a large class of dissipative potentials for which it is also sufficient.
In this paper I study properties of the generators _γ of non-local Dirichlet forms ℰ^μ_γ on ultrametric spaces which are the path space of simple stationary Bratteli diagrams. The measures used to define the Dirichlet forms are taken to be the Gibbs measures μ_ψ associated to Hölder continuous potentials ψ for one-sided shifts. I also define a cohomology H_lc(X_B) for X_B which can be seen as dual to the homology of Bowen and Franks. Besides studying spectral properties of _γ, I show that for γ large enough (with sharp bounds depending on the diagram and the measure theoretic entropy h_μ_ψ of μ_ψ) there is a unique ℰ^μ_γ-minimizing representative of any class c∈ H_lc(X_B).
By a coprime commutator in a profinite group G we mean any element of the form [x, y], where x,y∈ G and (|x|,|y|)=1 . It is well-known that the subgroup generated by the coprime commutators of G is precisely the pronilpotent residual γ _∞ (G) . There are several recent works showing that finiteness conditions on the set of coprime commutators have strong impact on the properties of γ _∞ (G) and, more generally, on the structure of G. In this paper we show that if the set of coprime commutators of a profinite group G is covered by countably many procyclic subgroups, then γ _∞ (G) is finite-by-procyclic. In particular, it follows that G is finite-by-pronilpotent-by-abelian.
