## Recursos de colección

#### Project Euclid (Hosted at Cornell University Library) (192.066 recursos)

Publicacions Matemàtique

1. #### Right Engel elements of stability groups of general series in vector spaces

Wehrfritz, B. A. F.
Let $V$ be an arbitrary vector space over some division ring $D$, $\mathbf{L}$ a general series of subspaces of $V$ covering all of $V\backslash \{0\}$ and $S$ the full stability subgroup of $\mathbf{L}$ in $\operatorname{GL}(V)$. We prove that always the set of bounded right Engel elements of $S$ is equal to the $\omega$-th term of the upper central series of $S$ and that the set of right Engel elements of $S$ is frequently equal to the hypercentre of $S$.

2. #### Localization genus

Møller, Jesper M.; Scherer, Jérôme
Which spaces look like an $n$-sphere through the eyes of the $n$-th Postnikov section functor and the $n$-connected cover functor? The answer is what we call the Postnikov genus of the $n$-sphere. We define in fact the notion of localization genus for any homotopical localization functor in the sense of Bousfield and Dror Farjoun. This includes exotic genus notions related for example to Neisendorfer localization, or the classical Mislin genus, which corresponds to rationalization.

3. #### Continuity of solutions to space-varying pointwise linear elliptic equations

Bandara, Lashi
We consider pointwise linear elliptic equations of the form $\mathrm{L}_x u_x = \eta_x$ on a smooth compact manifold where the operators $\mathrm{L}_x$ are in divergence form with real, bounded, measurable coefficients that vary in the space variable $x$. We establish $\mathrm{L}^{2}$-continuity of the solutions at $x$ whenever the coefficients of $\mathrm{L}_x$ are $\mathrm{L}^{\infty}$-continuous at $x$ and the initial datum is $\mathrm{L}^{2}$-continuous at $x$. This is obtained by reducing the continuity of solutions to a homogeneous Kato square root problem. As an application, we consider a time evolving family of metrics $\mathrm{g}_t$ that is tangential to the Ricci flow almost-everywhere along geodesics when starting with a smooth initial metric. Under...

4. #### Blow-up solutions for some nonlinear elliptic equations involving a Finsler-Laplacian

Della Pietra, Francesco; di Blasio, Giuseppina
In this paper we prove existence results and asymptotic behavior for strong solutions $u\in W^{2,2}_{\operatorname{loc}}(\Omega)$ of the nonlinear elliptic problem $$\label{abstr} \begin{cases} -\Delta_{H}u+H(\nabla u)^{q}+\lambda u=f&\text{in }\Omega,\\ u\rightarrow +\infty &\text{on }\partial\Omega, \end{cases}\tag{P}$$ where $H$ is a suitable norm of $\mathbb R^{n}$, $\Omega\subset\mathbb R^{n}$ is a bounded domain, $\Delta_{H}$ is the Finsler Laplacian, $1\lt q\le 2$, $\lambda>0$, and $f$ is a suitable function in $L^{\infty}_{\operatorname{loc}}$. Furthermore, we are interested in the behavior of the solutions when $\lambda\rightarrow 0^{+}$, studying the so-called ergodic problem associated to (P). A key role in order to study the ergodic problem will be played by local gradient estimates for (P).

5. #### Equigeneric and equisingular families of curves on surfaces

Dedieu, T.; Sernesi, E.
We investigate the following question: let $C$ be an integral curve contained in a smooth complex algebraic surface $X$; is it possible to deform $C$ in $X$ into a nodal curve while preserving its geometric genus? ¶ We affirmatively answer it in most cases when $X$ is a Del Pezzo or Hirzebruch surface (this is due to Arbarello and Cornalba, Zariski, and Harris), and in some cases when $X$ is a $K3$ surface. Partial results are given for all surfaces with numerically trivial canonical class. We also give various examples for which the answer is negative.

6. #### An example pertaining to the failure of the Besicovitch-Federer structure Theorem in Hilbert space

De Pauw, Thierry
We give an example, in the infinite dimensional separable Hilbert space, of a purely unrectifiable Borel set with finite nonzero one dimensional Hausdorff measure, whose projection is nonnegligible in a set of directions which is not Aronszajn null.

7. #### Low energy canonical immersions into hyperbolic manifolds and standard spheres

del Rio, Heberto; Santos, Walcy; Simanca, Santiago R.
We consider critical points of the global $L^2$-norm of the second fundamental form, and of the mean curvature vector of isometric immersions of compact Riemannian manifolds into a fixed background Riemannian manifold, as functionals over the space of deformations of the immersion. We prove new gap theorems for these functionals into hyperbolic manifolds, and show that the celebrated gap theorem for minimal immersions into the standard sphere can be cast as a theorem about their critical points having constant mean curvature function, and whose second fundamental form is suitably small in relation to it. In this case, the various minimal submanifolds that occur at the pointwise upper bound...

8. #### A formula for the Dubrovnik polynomial of rational knots

Caprau, Carmen and; Urabe, Katherine
We provide a formula for the Dubrovnik polynomial of a rational knot in terms of the entries of the tuple associated with a braid-form diagram of the knot. Our calculations can be easily carried out using a computer algebra system.

9. #### A Marcinkiewicz integral type characterization of the Sobolev space

Hajłasz, Piotr; Liu, Zhuomin
In this paper we present a new characterization of the Sobolev space $W^{1,p}$, $1\lt p\lt \infty$ which is a higher dimensional version of a result of Waterman [32]. We also provide a new and simplified proof of a recent result of Alabern, Mateu, and Verdera [2]. Finally, we generalize the results to the case of weighted Sobolev spaces with respect to a Muckenhoupt weight.

10. #### Automorphism groups of simplicial complexes of infinite-type surfaces

Hernández, Jesús Hernández; Valdez, Ferrán
Let $S$ be an orientable surface of infinite genus with a finite number of boundary components. In this work we consider the curve complex $\mathcal{C}(S)$, the nonseparating curve complex $\mathcal{N}(S)$, and the Schmutz graph $\mathcal{G}(S)$ of $S$. When all topological ends of $S$ carry genus, we show that all elements in the automorphism groups $\operatorname{Aut}(\mathcal{C}(S))$, $\operatorname{Aut}(\mathcal{N}(S))$, and $\operatorname{Aut}(\mathcal{G}(S))$ are geometric, i.e., these groups are naturally isomorphic to the extended mapping class group $\operatorname{MCG}^{*}(S)$ of the infinite surface $S$. Finally, we study rigidity phenomena within $\operatorname{Aut}(\mathcal{C}(S))$ and $\operatorname{Aut}(\mathcal{N}(S))$.

11. #### Bilinear weighted Hardy inequality for nonincreasing functions

Křepela, Martin
We characterize the validity of the bilinear Hardy inequality for nonincreasing functions $\|f^{**} g^{**}\|_{L^q(w)} \le C \|f\|_{\Lambda^{p_1}(v_1)}\|g\|_{\Lambda^{p_2}(v_2)},$ in terms of the weights $v_1$, $v_2$, $w$, covering the complete range of exponents $p_1,p_2,q\in (0,\infty]$. The problem is solved by reducing it into the iterated Hardy-type inequalities \begin{align*} \left( \int\limits_0^\infty \biggl( \int\limits_0^x (g^{**}(t))^\alpha \varphi(t)\,\mathrm{d}t \biggr)^\frac{\beta}{\alpha} \psi(x)\,\mathrm{d}x \right)^\frac{1}{\beta} & \le C \biggl( \int\limits_0^\infty (g^*(x))^\gamma \omega(x) \,\mathrm{d}x \biggr)^\frac{1}{\gamma}, \\ \left( \int\limits_0^\infty \biggl( \int\limits_x^\infty (g^{**}(t))^\alpha \varphi(t)\,\mathrm{d}t \biggr)^\frac{\beta}{\alpha} \psi(x)\,\mathrm{d}x \right)^\frac{1}{\beta} & \le C \biggl( \int\limits_0^\infty (g^*(x))^\gamma \omega(x) \,\mathrm{d}x \biggr)^\frac{1}{\gamma}. \end{align*} Validity of these inequalities is characterized here for $0\lt\alpha\le\beta\lt\infty$ and $0\lt\gamma\lt\infty$.

12. #### Common zeros preserving maps on vector-valued function spaces and Banach modules

Let $X$, $Y$ be Hausdorff topological spaces, and let $E$ and $F$ be Hausdorff topological vector spaces. For certain subspaces $A(X, E)$ and $A(Y,F)$ of $C(X,E)$ and $C(Y,F)$ respectively (including the spaces of Lipschitz functions), we characterize surjections $S,T\colon A(X,E) \rightarrow A(Y,F)$, not assumed to be linear, which jointly preserve common zeros in the sense that $Z(f-f') \cap Z(g-g') \neq \emptyset$ if and only if $Z(Sf-Sf') \cap Z(Tg-Tg') \neq \emptyset$ for all $f,f',g,g'\in A(X,E)$. Here $Z(\cdot)$ denotes the zero set of a function. Using the notion of point multipliers we extend the notion of zero set for the elements of a Banach module and give a representation for surjective...

13. #### Convergence of functions of self-adjoint operators and applications

Brown, Lawrence G.
The main result (roughly) is that if $(H_i)$ converges weakly to $H$ and if also $f(H_i)$ converges weakly to $f(H)$, for a single strictly convex continuous function $f$, then $(H_i)$ must converge strongly to $H$. One application is that if $f({\operatorname{pr}}(H))={\operatorname{pr}} (f(H))$, where ${\operatorname{pr}}$ denotes compression to a closed subspace $M$, then $M$ must be invariant for $H$. A consequence of this is the verification of a conjecture of Arveson, that Theorem 9.4 of [Arv] remains true in the infinite dimensional case. And there are two applications to operator algebras. If $h$ and $f(h)$ are both quasimultipliers, then $h$ must be a multiplier. Also (still roughly stated), if $h$...

14. #### Global well-posedness and symmetries for dissipative active scalar equations with positive-order couplings

Ferreira, Lucas C. F.; Lima, Lidiane S. M.
We consider a family of dissipative active scalar equations outside the $L^{2}$-space. This was introduced in [7] and its velocity fields are coupled with the active scalar via a class of multiplier operators which morally behave as derivatives of positive order. We prove global well-posedness and time-decay of solutions, without smallness assumptions, for initial data belonging to the critical Lebesgue space $L^{\frac{n}{2\gamma-\beta}}(\mathbb{R}^{n})$ which is a class larger than that of the above reference. Symmetry properties of solutions are investigated depending on the symmetry of initial data and coupling operators.

15. #### Röver's Simple Group Is of Type $F_\infty$

Belk, James; Matucci, Francesco
We prove that Claas Röver's Thompson-Grigorchuk simple group $V\mathcal{G}$ has type $F_\infty$. The proof involves constructing two complexes on which $V\mathcal{G}$ acts: a simplicial complex analogous to the Stein complex for $V$, and a polysimplicial complex analogous to the Farley complex for $V$. We then analyze the descending links of the polysimplicial complex, using a theorem of Belk and Forrest to prove increasing connectivity.

16. #### Integral Restriction for Bilinear Operators

Zhao, Weiren; Wang, Meng; Zhao, Guoping
We consider the integral domain restriction operator $T_{\Omega}$ for certain bilinear operator $T$. We obtain that if $(s,p_1,p_2)$ satisfies $\frac{1}{p_1}+\frac{1}{p_2}\geq \frac{2}{\min\{1,s\}}$ and $\|T\|_{L^{p_1}\times L^{p_2}\rightarrow L^s}\lt\infty$, then $\|T_{\Omega}\|_{L^{p_1}\times L^{p_2}\rightarrow L^s}\lt\infty$. For some special domain $\Omega$, this property holds for triplets $(s,p_1,p_2)$ satisfying $\frac{1}{p_1}+\frac{1}{p_2}\gt\frac{1}{\min\{1,s\}}$.

17. #### The Kato square root problem follows from an extrapolation property of the Laplacian

Egert, Moritz; Haller-Dintelmann, Robert; Tolksdorf, Patrick
On a domain $\Omega \subseteq \mathbb{R}^d$ we consider second-order elliptic systems in divergence-form with bounded complex coefficients, realized via a sesquilinear form with domain $\mathrm{H}_0^1(\Omega) \subseteq \mathcal{V} \subseteq \mathrm{H}^1(\Omega)$. Under very mild assumptions on~$\Omega$ and $\mathcal{V}$ we show that the solution to the Kato Square Root Problem for such systems can be deduced from a regularity result for the fractional powers of the negative Laplacian in the same geometric setting. This extends earlier results of McIntosh [25] and Axelsson-Keith-McIntosh [6] to non-smooth coefficients and domains.

18. #### A monotonicity formula for minimal sets with a sliding boundary condition

David, G.
We prove a monotonicity formula for minimal or almost minimal sets for the Hausdorff measure $\mathcal H^d$, subject to a sliding boundary constraint where competitors for $E$ are obtained by deforming $E$ by a one-parameter family of functions $\varphi_t$ such that $\varphi_t(x) \in L$ when $x\in E$ lies on the boundary $L$. In the simple case when $L$ is an affine subspace of dimension $d-1$, the monotone or almost monotone functional is given by $F(r) = r^{-d} \mathcal H^d(E \cap B(x,r)) + r^{-d} \mathcal H^d(S \cap B(x,r))$, where $x$ is any point of $E$ (not necessarily on $L$) and $S$ is the shade of $L$ with a light...

19. #### Vitali's theorem without uniform boundedness

Dieu, Nguyen Quang; Manh, Phung Van; Bang, Pham Hien; Hung, Le Thanh
Let $\{f_m\}_{m \ge 1}$ be a sequence of holomorphic functions defined on a bounded domain $D \subset \mathbb C^n$ or a sequence of rational functions $(1 \le \deg r_m \le m)$ defined on $\mathbb C^n$. We are interested in finding sufficient conditions to ensure the convergence of $\{f_m\}_{m \ge 1}$ on a large set provided the convergence holds pointwise on a not too small set. This type of result is inspired from a theorem of Vitali which gives a positive answer for uniformly bounded sequence.

20. #### Summation of Coefficients of Polynomials on $\ell_{p}$ Spaces

Dimant, Verónica; Sevilla-Peris, Pablo
We investigate the summability of the coefficients of $m$-homogeneous polynomials and $m$-linear mappings defined on $\ell_{p}$ spaces. In our research we obtain results on the summability of the coefficients of $m$-linear mappings defined on $\ell_{p_{1}} \times \dotsb \times \ell_{p_{m}}$. The first results in this respect go back to Littlewood [17] and Bohnenblust and [6] for bilinear and $m$-linear forms on $c_{0}$, and Hardy and Littlewood [15] and Praciano-Pereira [20] for bilinear and $m$-linear forms on arbitrary $\ell_{p}$ spaces. Our results recover and in some case complete these old results through a general approach on vector valued $m$-linear mappings.

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