International Journal of Inclusive Education

Abstract We investigate the failure of the Stone-Weierstrass theorem focusing on the existence of large dimensional vector spaces within the set $$\mathcal {C}(L, \mathbb {K}) {\setminus } \overline{\mathcal {A}}$$ C ( L , K ) \ A ¯ , where L is a compact Hausdorff space and $$\mathcal {A}$$ A is a self-adjoint subalgebra of $$\mathcal {C}(L, \mathbb {K})$$ C ( L , K ) that vanishes nowhere on L but does not necessarily separate the points of L. We address the problem of finding the precise codimension of $$\overline{\mathcal {A}}$$ A ¯ in a broad setting, which allows us to describe the lineability of $$\mathcal {C}(L, \mathbb {K}) {\setminus } \overline{\mathcal {A}}$$ C ( L , K ) \ A ¯ in detail. Our analysis yields both affirmative and negative results regarding the lineability of this set. Furthermore, we also study the set $$(\mathcal {C}(\partial {D}, \mathbb {C}) {\setminus } \overline{\text {Pol}(\partial {D})}) \cup \{0\}$$ ( C ( ∂ D , C ) \ Pol ( ∂ D ) ¯ ) ∪ { 0 } , where $$\text {Pol}(\partial {D})$$ Pol ( ∂ D ) is the set of all complex polynomials in one variable restricted to the boundary of the unit disk. Recent lineability properties are also taken into account.

We investigate connections between upper/lower estimates for Banach lattices and the notion of relative s-decomposability, which has roots in interpolation theory. To get a characterization of relatively s-decomposable Banach lattices in terms of the above estimates, we assign to each Banach lattice X two sequence spaces $$X_{U}$$ and $$X_{L}$$ that are largely determined by the set of p, for which $$l_{p}$$ is finitely lattice representable in X. As an application, we obtain an orbital factorization of relative K-functional estimates for Banach couples $$\vec {X} =(X_{0},X_{1})$$ and $$\vec {Y}=(Y_{0},Y_{1})$$ through some suitable couples of weighted $$L_{{p}}$$ -spaces provided if $$X_{i},Y_{i}$$ are relatively s -decomposable for $$i=0,1$$ . Also, we undertake a detailed study of the properties of optimal upper and lower sequence spaces $$X_{U}$$ and $$X_{L}$$ , and, in particular, prove that these spaces are rearrangement invariant. In the Appendix, a description of the optimal upper sequence space for a separable Orlicz space as a certain intersection of some special Musielak-Orlicz sequence spaces is given.

A well-known boundary observability inequality for the elasticity system establishes that the energy of the system can be estimated from the solution on a sufficiently large part of the boundary for sufficiently large time. This inequality is relevant in different contexts as the exact boundary controllability, the boundary stabilization or some inverse source problems. Here we show that a corresponding boundary observability inequality for the spectral collocation approximation of the linear elasticity system in a d-dimensional cube also holds, uniformly with respect to the discretization parameter. This property is essential to prove that natural numerical approaches of the previous problems based on replacing the elasticity system by the collocation discretization will give successful approximations of the continuous counterparts. As an application we obtain the boundary controllability of the discrete system resulting when approximating the elasticity system with this numerical method, uniformly with respect to the discretization parameter. We also give numerical evidences of the convergence of these discrete controls to a boundary control of the limit 2d-elasticity system in a square domain.

A Christ–Kiselev maximal theorem is proved for linear operators between quasi-Banach function lattices satisfying certain lattice geometrical conditions. The result is further explored for weighted Lorentz spaces, classical Lorentz spaces, and Wiener amalgams of Lebesgue function and sequence spaces. Extensions are made to Köthe dual operators and to operators on interpolation spaces of quasi-Banach function lattices. Several applications to maximal Fourier operators are presented.

This paper focuses on the properties of quasisymmetric self–mappings of $$\mathbb {R}^n$$ with $$n\ge 2$$ . We obtain several geometric characterizations of quasisymmetric homeomorphisms in terms of ring domains and all pairs of disjoint continua by using conformal moduli and geometric moduli recently introduced by Tukia and Väisälä. As an application, we establish six necessary and sufficient conditions for a homeomorphism to be a quasimöbius transformation of the Riemann sphere $$\mathbb {\overline{R}}^n$$ onto itself.

We give applications of known and new Liouville type theorems to universal singularity and decay estimates for non scale invariant elliptic problems, including Lane–Emden and Schrödinger type systems. This applies to various classes of nonlinearities with regular variation and possibly different behaviors at 0 and $$\infty $$ . To this end, we adapt the method from Souplet (Discrete Contin Dyn Syst 43:1702–1734, 2023) to elliptic systems, which relies on a generalized rescaling technique and on doubling arguments from Poláčik et al (Duke Math J 139:555–579, 2007). This is in particular facilitated by new Liouville type theorems in the whole space and in a half-space, for elliptic problems without scale invariance, that we obtain. Our results apply to some non-cooperative systems, for which maximum principle based techniques such as moving planes do not apply. To prove these Liouville type theorems, we employ two methods, respectively based on Pohozaev-type identities combined with functional inequalities on the unit sphere, and on reduction to a scalar equation by proportionality of components. In turn we survey the known methods for proving Liouville-type theorems for superlinear elliptic equations and systems, and list some of the typical known results for (Sobolev subcritical) systems. In the case of scalar equations, we also revisit the classical Gidas–Spruck integral Bernstein method, providing some improvements which turn out to be efficient for certain nonlinearities, and we next compare the performances of various methods on a benchmark example.

AbstractBuilding on previous work by Caicedo and the second author, we develop a method that decides the existence of units of finite order in blocks of $$\mathbb {Z}_p G$$ Z p G of defect 1. This allows us to prove that if p is a prime and G is a finite group whose Sylow p-subgroup has order p, then any unit u of $$\mathbb {Z}G$$ Z G of order p is conjugate to an element of $$\pm G$$ ± G within $$\mathbb {Q}G$$ Q G . This is a special case of the Zassenhaus conjecture. We also prove some new results on units of finite order in $$\mathbb {Z}{{\text {PSL}}}(2,q)$$ Z PSL ( 2 , q ) for certain q, and construct a unit of order 15 in $$V(\mathbb {Z}_{(3,5)}{{\text {PSL}}}(2,16))$$ V ( Z ( 3 , 5 ) PSL ( 2 , 16 ) ) which is a 3- and 5-local counterexample to the Zassenhaus conjecture, raising the hope that our methods may lead to a global counterexample among simple groups.

We introduce two constructions to obtain left-invariant Ricci-flat pseudo-Riemannian metrics on nilpotent Lie groups, one based on gradings, the other on filtrations, both depending on the combinatorics of the set of weights. As an application, we show that every nilpotent Lie algebra of dimension up to 7 and every nice nilpotent Lie algebra of dimension up to 9 admit an indefinite Ricci-flat metric.

Abstract We study germs of hypersurfaces $$(Y,0)\subset (\mathbb {C}^{n+1},0)$$ ( Y , 0 ) ⊂ ( C n + 1 , 0 ) that can be described as the image of $${\mathscr {A}}$$ A -finite mappings $$f:(X,S)\rightarrow (\mathbb {C}^{n+1},0)$$ f : ( X , S ) → ( C n + 1 , 0 ) defined on an icis (X, S) of dimension n. We extend the definition of the Jacobian module given by Fernández de Bobadilla, Nuño-Ballesteros and Peñafort-Sanchis when $$X=\mathbb {C}^n$$ X = C n , which controls the image Milnor number $$\mu _I(X,f)$$ μ I ( X , f ) . We apply these results to prove the case $$n=2$$ n = 2 of the generalised Mond conjecture, which states that $${\mu _I(X,f)\ge \text{codim}_{\mathscr {A}_e}(X,f)}$$ μ I ( X , f ) ≥ codim A e ( X , f ) , with equality if (Y, 0) is weighted homogeneous.

The aim of this paper is to obtain the automorphism groups of compact non-orientable Klein surfaces of topological genus 4 with $$k>0$$ boundary components. These groups clearly depend on the number of boundary components, that is, the value of k. The starting point of the paper is the analogous result for surfaces without boundary components, that is $$k=0$$ , obtained by Bujalance, Etayo and Martínez (2014), since it can be proven that a group that acts on a surface with boundary components also acts on surfaces of the same topological type without boundary components. The proof is parallel to the approach followed by Bujalance, Etayo and Martínez (2023) for surfaces of topological genus 3. It is worth noting that as the topological genus of the surface increases, so do the order of the groups of automorphisms and the complexity of their algebraic structure, and hence the difficulty of the computational problem.

AbstractIn 2012  G. Bartolini and M. Izquierdo showed that all equisymmetric strata of Broughton’s stratification, corresponding to the actions of groups of order 2 and 3 on Riemann surfaces of genus g are contained in the same connected component $$\mathcal {C}_g^{2,3}$$ C g 2 , 3 of the singular locus $$\mathcal {S}_g$$ S g . The other components that were known to the literature were those composed of points of certain single strata. This motivated the question of whether the singular locus can contain connected components different from $$\mathcal {C}_g^{2,3}$$ C g 2 , 3 and being the union of at least two strata. The existence of such components was proved for infinitely many genera by G. Gromadzki and the author. In this paper, we will describe and classify all types of connected components of the singular locus. In particular, we prove that every connected component different than $$\mathcal {C}_g^{2,3}$$ C g 2 , 3 coincides with some Cornalba component, and we give necessary and sufficient conditions for a cyclic action to define such a component.

AbstractIn this paper, we will study Neumann (p, q)-eigenvalue problem for the weighted p-Laplace operator in outward Hölder cuspidal domains. The suggested method is based on the composition operators on weighted Sobolev spaces.

We extend the scope of the shadowing lemma, initially proposed for generalized hyperbolic invertible operators on Banach spaces [3], to encompass both noninvertible operators and semigroups of operators. Moreover, we demonstrate that certain operators with the positive bounded shadowing property do not fall under the category of generalized hyperbolic operators (and similarly for semigroups). As a result, this provides a partial negative answer to a conjecture in [9].

This article is devoted to studying complex algebraic sets under (global) blow-spherical equivalence. This equivalence lives strictly between semialgebraic bi-Lipschitz equivalence and topological equivalence. The main results of this article are complete classifications of complex algebraic curves. Firstly, we present a complete classification of complex algebraic curves under blow-spherical homeomorphisms at infinity and, then, we present a complete classification of complex algebraic curves under (global) blow-spherical homeomorphisms. For the classification at infinity we also present a classification with normal forms. We also present several properties of the blow-spherical equivalence. For instance, we prove that the degree of curves is preserved under blow-spherical homeomorphisms at infinity. Another property presented here is a Bernstein-type result which says that a pure dimensional complex algebraic set which is blow-spherical homeomorphic at infinity to a Euclidean space must be an affine linear subspace. We also prove that the LNE property of complex algebraic curves is invariant under blow-spherical homeomorphisms.