## Recursos de colección

1. #### Asymptotics for the size of the largest component scaled to “logn” in inhomogeneous random graphs

Turova, Tatyana S.
We study inhomogeneous random graphs in the subcritical case. Among other results, we derive an exact formula for the size of the largest connected component scaled by log n, with n being the size of the graph. This generalizes a result for the “rank-1 case”. We also investigate branching processes associated with these graphs. In particular, we discover that the same well-known equation for the survival probability, whose positive solution determines the asymptotics of the size of the largest component in the supercritical case, also plays a crucial role in the subcritical case. However, now it is the negative solutions...

2. #### Encomplexed Brown invariant of real algebraic surfaces in ℝP3

Björklund, Johan
We construct an invariant of parametrized generic real algebraic surfaces in ℝP3 which generalizes the Brown invariant of immersed surfaces from smooth topology. The invariant is constructed using self-intersections, which are real algebraic curves with points of three local characters: the intersection of two real sheets, the intersection of two complex conjugate sheets or a Whitney umbrella. In Kirby and Melvin (Local surgery formulas for quantum invariants and the Arf invariant, in Proceedings of the Casson Fest, Geom. Topol. Monogr. 7, pp. 213–233, Geom. Topol. Publ., Coventry, 2004) the Brown invariant was expressed through a self-linking number of the self-intersection. We...

3. #### Duality and distance formulas in spaces defined by means of oscillation

Perfekt, Karl-Mikael
For the classical space of functions with bounded mean oscillation, it is well known that $\operatorname{VMO}^{**} = \operatorname{BMO}$ and there are many characterizations of the distance from a function f in $\operatorname{BMO}$ to $\operatorname{VMO}$. When considering the Bloch space, results in the same vein are available with respect to the little Bloch space. In this paper such duality results and distance formulas are obtained by pure functional analysis. Applications include general Möbius invariant spaces such as QK-spaces, weighted spaces, Lipschitz–Hölder spaces and rectangular $\operatorname{BMO}$ of several variables.

4. #### An essay on Bergman completeness

Chen, Bo-Yong
We give first of all a new criterion for Bergman completeness in terms of the pluricomplex Green function. Among several applications, we prove in particular that every Stein subvariety in a complex manifold admits a Bergman complete Stein neighborhood basis, which improves a theorem of Siu. Secondly, we give for hyperbolic Riemann surfaces a sufficient condition for when the Bergman and Poincaré metrics are quasi-isometric. A consequence is an equivalent characterization of uniformly perfect planar domains in terms of growth rates of the Bergman kernel and metric. Finally, we provide a noncompact Bergman complete pseudoconvex manifold without nonconstant negative plurisubharmonic...

5. #### Proper holomorphic embeddings of finitely connected planar domains into ℂ n

Majcen, Irena
In this paper we consider proper holomorphic embeddings of finitely connected planar domains into ℂn that approximate given proper embeddings on smooth curves. As a side result we obtain a tool for approximating a $\mathcal{C}^{\infty}$ diffeomorphism on a polynomially convex set in ℂn by an automorphism of ℂn that satisfies some additional properties along a real embedded curve.

6. #### Biharmonic PNMC submanifolds in spheres

Balmuş, Adina; Montaldo, Stefano; Oniciuc, Cezar
We obtain several rigidity results for biharmonic submanifolds in $\mathbb{S}^{n}$ with parallel normalized mean curvature vector fields. We classify biharmonic submanifolds in $\mathbb{S}^{n}$ with parallel normalized mean curvature vector fields and with at most two distinct principal curvatures. In particular, we determine all biharmonic surfaces with parallel normalized mean curvature vector fields in $\mathbb{S}^{n}$. ¶ Then we investigate, for (not necessarily compact) proper-biharmonic submanifolds in $\mathbb{S}^{n}$, their type in the sense of B.-Y. Chen. We prove that (i) a proper-biharmonic submanifold in $\mathbb{S}^{n}$ is of 1-type or 2-type if and only if it has constant mean curvature f=1 or f∈(0,1), respectively; and (ii) there...

7. #### An improved Riemann mapping theorem and complexity in potential theory

Bell, Steven R.
We discuss applications of an improvement on the Riemann mapping theorem which replaces the unit disc by another “double quadrature domain,” i.e., a domain that is a quadrature domain with respect to both area and boundary arc length measure. Unlike the classic Riemann mapping theorem, the improved theorem allows the original domain to be finitely connected, and if the original domain has nice boundary, the biholomorphic map can be taken to be close to the identity, and consequently, the double quadrature domain is close to the original domain. We explore some of the parallels between this new theorem and the...

8. #### On the Carleson duality

Hytönen, Tuomas; Rosén, Andreas
As a tool for solving the Neumann problem for divergence-form equations, Kenig and Pipher introduced the space ${\mathcal{X}}$ of functions on the half-space, such that the non-tangential maximal function of their L2 Whitney averages belongs to L2 on the boundary. In this paper, answering questions which arose from recent studies of boundary value problems by Auscher and the second author, we find the pre-dual of ${\mathcal{X}}$, and characterize the pointwise multipliers from ${\mathcal{X}}$ to L2 on the half-space as the well-known Carleson-type space of functions introduced by Dahlberg. We also extend these results to Lp generalizations of the space ${\mathcal{X}}$....

9. #### An extension property of the Bourgain–Pisier construction

Suárez de la Fuente, Jesús
It is proved that the natural embedding of a separable Banach space X into the corresponding Bourgain–Pisier space extends $\mathcal{L}_{\infty}$-valued operators.

10. #### Möbius homogeneous hypersurfaces with two distinct principal curvatures in Sn+1

Li, Tongzhu; Ma, Xiang; Wang, Changping
The purpose of this paper is to classify the Möbius homogeneous hypersurfaces with two distinct principal curvatures in Sn+1 under the Möbius transformation group. Additionally, we give a classification of the Möbius homogeneous hypersurfaces in S4.

11. #### Area-preserving isotopies of self-transverse immersions of S1 in ℝ2

Karlsson, Cecilia
Let C and C′ be two smooth self-transverse immersions of S1 into ℝ2. Both C and C′ subdivide the plane into a number of disks and one unbounded component. An isotopy of the plane which takes C to C′ induces a one-to-one correspondence between the disks of C and C′. An obvious necessary condition for there to exist an area-preserving isotopy of the plane taking C to C′ is that there exists an isotopy for which the area of every disk of C equals that of the corresponding disk of C′. In this paper we show that this is also...

12. #### Verdier specialization via weak factorization

Aluffi, Paolo
Let X⊂V be a closed embedding, with V∖X nonsingular. We define a constructible function ψX, V on X, agreeing with Verdier’s specialization of the constant function 1V when X is the zero-locus of a function on V. Our definition is given in terms of an embedded resolution of X; the independence of the choice of resolution is obtained as a consequence of the weak factorization theorem of Abramovich–Karu–Matsuki–Włodarczyk. The main property of ψX, V is a compatibility with the specialization of the Chern class of the complement V∖X. With the definition adopted here, this is an easy consequence of standard intersection theory....

13. #### On the h-triangles of sequentially (S r ) simplicial complexes via algebraic shifting

Pournaki, Mohammad Reza; Seyed Fakhari, Seyed Amin; Yassemi, Siamak
Recently, Haghighi, Terai, Yassemi, and Zaare-Nahandi introduced the notion of a sequentially (Sr) simplicial complex. This notion gives a generalization of two properties for simplicial complexes: being sequentially Cohen–Macaulay and satisfying Serre’s condition (Sr). Let Δ be a (d−1)-dimensional simplicial complex with Γ(Δ) as its algebraic shifting. Also let (hi, j(Δ))0≤j≤i≤d be the h-triangle of Δ and (hi, j(Γ(Δ)))0≤j≤i≤d be the h-triangle of Γ(Δ). In this paper, it is shown that for a Δ being sequentially (Sr) and for every i and j with 0≤j≤i≤r−1, the equality hi, j(Δ)=hi, j(Γ(Δ)) holds true.

14. #### A new generalization of the Lelong number

Lagerberg, Aron
We will introduce a quantity which measures the singularity of a plurisubharmonic function φ relative to another plurisubharmonic function ψ, at a point a. We denote this quantity by νa, ψ(φ). It can be seen as a generalization of the classical Lelong number in a natural way: if ψ=(n−1)log| ⋅ −a|, where n is the dimension of the set where φ is defined, then νa, ψ(φ) coincides with the classical Lelong number of φ at the point a. The main theorem of this article says that the upper level sets of our generalized Lelong number, i.e. the sets of the form {z: νz,...

15. #### Calculation of local formal Fourier transforms

We calculate the local Fourier transforms for connections on the formal punctured disk, reproducing the results of J. Fang and C. Sabbah using a different method. Our method is similar to Fang’s, but more direct.

16. #### Simplicity of eigenvalues in Anderson-type models

Naboko, Sergey; Nichols, Roger; Stolz, Günter
We show almost sure simplicity of eigenvalues for several models of Anderson-type random Schrödinger operators, extending methods introduced by Simon for the discrete Anderson model. These methods work throughout the spectrum and are not restricted to the localization regime. We establish general criteria for the simplicity of eigenvalues which can be interpreted as separately excluding the absence of local and global symmetries, respectively. The criteria are applied to Anderson models with matrix-valued potential as well as with single-site potentials supported on a finite box.

17. #### Asymptotic porosity of planar harmonic measure

Graczyk, Jacek; Świa̧tek, Grzegorz
We study the distribution of harmonic measure on connected Julia sets of unicritical polynomials. Harmonic measure on a full compact set in ℂ is always concentrated on a set which is porous for a positive density of scales. We prove that there is a topologically generic set $\mathcal{A}$ in the boundary of the Mandelbrot set such that for every $c\in \mathcal{A}$, β>0, and λ∈(0,1), the corresponding Julia set is a full compact set with harmonic measure concentrated on a set which is not β-porous in scale λn for n from a set with positive density amongst natural numbers.

18. #### Volume formula for a ℤ2-symmetric spherical tetrahedron through its edge lengths

Kolpakov, Alexander; Mednykh, Alexander; Pashkevich, Marina
The present paper considers volume formulæ, as well as trigonometric identities, that hold for a tetrahedron in 3-dimensional spherical space of constant sectional curvature +1. The tetrahedron possesses a certain symmetry: namely rotation of angle π in the middle points of a certain pair of its skew edges.

19. #### Noise correlation bounds for uniform low degree functions

Austrin, Per; Mossel, Elchanan
We study correlation bounds under pairwise independent distributions for functions with no large Fourier coefficients. Functions in which all Fourier coefficients are bounded by δ are called δ-uniform. The search for such bounds is motivated by their potential applicability to hardness of approximation, derandomization, and additive combinatorics. ¶ In our main result we show that $\operatorname{\mathbb {E}}[f_{1}(X_{1}^{1},\ldots,X_{1}^{n}) \ldots f_{k}(X_{k}^{1},\ldots,X_{k}^{n})]$ is close to 0 under the following assumptions: [start-list]*the vectors $\{ (X_{1}^{j},\ldots,X_{k}^{j}) : 1 \leq j \leq n\}$ are independent identically distributed, and for each j the vector $(X_{1}^{j},\ldots,X_{k}^{j})$ has a pairwise independent distribution;*the functions fi are uniform;*the functions fi are of...

20. #### Fonction maximale centrée de Hardy–Littlewood sur les espaces hyperboliques

Li, Hong-Quan; Lohoué, Noël
On montre que la fonction maximale centrée de Hardy–Littlewood, M, sur les espaces hyperboliques réels $\mathbb{H}^{n} = \mathbb{R}^{{\mathchoice {\raise .17ex\hbox {$\scriptstyle +$}}{\raise .17ex\hbox {$\scriptstyle +$}}{\raise .1ex\hbox {$\scriptscriptstyle +$}}{\scriptscriptstyle +}}} \times \mathbb{R}^{n - 1}$, satisfait l’inégalité de type faible $\| M f \|_{L^{1, \infty}} \leq A (n \log {n}) \| f\|_{1}$ pour toute f∈L1(ℍn), où A>0 est une constante indépendante de la dimension n.

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