Harmonic calculus on fractals---A measure geometric approach II
M.
Zähle
3407-3423
Abstract: Riesz potentials of fractal measures $\mu$ in metric spaces and their inverses are introduced. They define self-adjoint operators in the Hilbert space $L_2(\mu)$ and the former are shown to be compact. In the Euclidean case the corresponding spectral asymptotics are derived with Besov space methods. The inverses of the Riesz potentials are fractal pseudodifferential operators. For the order two operator the spectral dimension coincides with the Hausdorff dimension of the underlying fractal.
Notes on limits of Sobolev spaces and the continuity of interpolation scales
Mario
Milman
3425-3442
Abstract: We extend lemmas by Bourgain-Brezis-Mironescu (2001), and Maz'ya-Shaposhnikova (2002), on limits of Sobolev spaces, to the setting of interpolation scales. This is achieved by means of establishing the continuity of real and complex interpolation scales at the end points. A connection to extrapolation theory is developed, and a new application to limits of Sobolev scales is obtained. We also give a new approach to the problem of how to recognize constant functions via Sobolev conditions.
Rhombic embeddings of planar quad-graphs
Richard
Kenyon;
Jean-Marc
Schlenker
3443-3458
Abstract: Given a finite or infinite planar graph all of whose faces have degree $4$, we study embeddings in the plane in which all edges have length $1$, that is, in which every face is a rhombus. We give a necessary and sufficient condition for the existence of such an embedding, as well as a description of the set of all such embeddings. RÉSUMÉ. Etant donné un graphe planaire, fini ou infini, dont toutes les faces sont de degré $4$, on étudie ses plongements dans le plan dont toutes les arêtes sont de longueur $1$, c'est à dire dont toutes les faces sont des losanges. On donne une condition nécessaire et suffisante pour l'existence d'un tel plongement, et on décrit l'ensemble de ces plongements.
How to obtain transience from bounded radial mean curvature
Steen
Markvorsen;
Vicente
Palmer
3459-3479
Abstract: We show that Brownian motion on any unbounded submanifold $P$ in an ambient manifold $N$ with a pole $p$ is transient if the following conditions are satisfied: The $p$-radial mean curvatures of $P$ are sufficiently small outside a compact set and the $p$-radial sectional curvatures of $N$ are sufficiently negative. The `sufficiency' conditions are obtained via comparison with explicit transience criteria for radially drifted Brownian motion in warped product model spaces.
Cremer fixed points and small cycles
Lia
Petracovici
3481-3491
Abstract: Let $\lambda= e^{2\pi i \alpha}$, $\alpha \in \mathbb{R}\setminus \mathbb{Q}$, and let $(p_n/q_n)$ denote the sequence of convergents to the regular continued fraction of $\alpha$. Let $f$ be a function holomorphic at the origin, with a power series of the form $f(z)= \lambda z+\sum _{n=2}^{\infty}a_nz^n$. We assume that for infinitely many $n$ we simultaneously have (i) $\log \log q_{n+1} \geq 3\log q_n$, (ii) the coefficients $a_{1+q_n}$ stay outside two small disks, and (iii) the series $f(z)$ is lacunary, with $a_j=0$ for $2+q_n\leq j \leq q_n^{1+q_n}-1$. We then prove that $f(z)$ has infinitely many periodic orbits in every neighborhood of the origin.
Fixed point index in symmetric products
José
M.
Salazar
3493-3508
Abstract: Let $U$ be an open subset of a locally compact metric ANR $X$ and let $f:U \rightarrow X$ be a continuous map. In this paper we study the fixed point index of the map that $f$ induces in the $n$-symmetric product of $X$, $F_{n}(X)$. This index can detect the existence of periodic orbits of period $\leq n$ of $f$, and it can be used to obtain the Euler characteristic of the $n$-symmetric product of a manifold $X$, $\chi(F_{n}(X))$. We compute $\chi(F_{n}(X))$ for all orientable compact surfaces without boundary.
Tangent algebraic subvarieties of vector fields
Juan
B.
Sancho de Salas
3509-3523
Abstract: An algebraic commutative group $G$ is associated to any vector field $D$ on a complete algebraic variety $X$. The group $G$ acts on $X$ and its orbits are the minimal subvarieties of $X$ which are tangent to $D$. This group is computed in the case of a vector field on $\mathbb{P}_n$.
Clones from creatures
Martin
Goldstern;
Saharon
Shelah
3525-3551
Abstract: We show that (consistently) there is a clone $\mathcal{C}$ on a countable set such that the interval of clones above $\mathcal{C}$ is linearly ordered and has no coatoms.
Smooth projective varieties with extremal or next to extremal curvilinear secant subspaces
Sijong
Kwak
3553-3566
Abstract: We intend to give a classification of smooth nondegenerate projective varieties admitting extremal or next to extremal curvilinear secant subspaces. Gruson, Lazarsfeld and Peskine classified all projective integral curves with extremal secant lines. On the other hand, if a locally Cohen-Macaulay variety $X^{n}\subset \mathbb{P}^{n+e}$ of degree $d$meets with a linear subspace $L$ of dimension $\beta$ at finite points, then $\operatorname{length} {(X\cap L)}\le d-e+\beta$ as a finite scheme. A linear subspace $L$ for which the above length attains maximal possible value is called an extremal secant subspace and such $L$ for which $\operatorname{length}{(X\cap L)}= d-e+\beta -1$ is called a next to extremal secant subspace. In this paper, we show that if a smooth variety $X$ of degree $d\ge 6$ has extremal or next to extremal curvilinear secant subspaces, then it is either Del Pezzo or a scroll over a curve of genus $g\le 1$. This generalizes the results of Gruson, Lazarsfeld and Peskine (1983) for curves and the work of M-A. Bertin (2002) who classified smooth higher dimensional varieties with extremal secant lines. This is also motivated and closely related to establishing an upper bound for the Castelnuovo-Mumford regularity and giving a classification of the varieties on the boundary.
Homological and finiteness properties of picture groups
Daniel
S.
Farley
3567-3584
Abstract: Picture groups are a class of groups introduced by Guba and Sapir. Known examples include Thompson's groups $F$, $T$, and $V$. In this paper, a large class of picture groups is proved to be of type $F_{\infty}$. A Morse-theoretic argument shows that, for a given picture group, the rational homology vanishes in almost all dimensions.
Comparing Castelnuovo-Mumford regularity and extended degree: The borderline cases
Uwe
Nagel
3585-3603
Abstract: Castelnuovo-Mumford regularity and any extended degree function can be thought of as complexity measures for the structure of finitely generated graded modules. A recent result of Doering, Gunston, and Vasconcelos shows that both can be compared in the case of a graded algebra. We extend this result to modules and analyze when the estimate is in fact an equality. A complete classification is obtained if we choose as extended degree the homological or the smallest extended degree. The corresponding algebras are characterized in three ways: by relations among the algebra generators, by using generic initial ideals, and by their Hilbert series.
Depth and cohomological connectivity in modular invariant theory
Peter
Fleischmann;
Gregor
Kemper;
R.
James
Shank
3605-3621
Abstract: Let $G$ be a finite group acting linearly on a finite-dimensional vector space $V$ over a field $K$ of characteristic $p$. Assume that $p$ divides the order of $G$ so that $V$ is a modular representation and let $P$ be a Sylow $p$-subgroup for $G$. Define the cohomological connectivity of the symmetric algebra $S(V^*)$ to be the smallest positive integer $m$ such that $H^m(G,S(V^*))\not=0$. We show that $\min\left\{\dim_K(V^P) + m + 1,\dim_K(V)\right\}$is a lower bound for the depth of $S(V^*)^G$. We characterize those representations for which the lower bound is sharp and give several examples of representations satisfying the criterion. In particular, we show that if $G$ is $p$-nilpotent and $P$ is cyclic, then, for any modular representation, the depth of $S(V^*)^G$is $\min\left\{\dim_K(V^P) + 2,\dim_K(V)\right\}$.
Suspensions of crossed and quadratic complexes, co-H-structures and applications
Fernando
Muro
3623-3653
Abstract: Crossed and quadratic modules are algebraic models of the 2-type and the 3-type of a space, respectively. In this paper we compute a purely algebraic suspension functor from crossed to quadratic modules which sends a 2-type to the 3-type of its suspension. We also give some applications in homotopy theory and group theory.
Asymptotic behaviour of arithmetically Cohen-Macaulay blow-ups
Huy
Tài
Hà;
Ngô
Viêt
Trung
3655-3672
Abstract: This paper addresses problems on arithmetic Macaulayfications of projective schemes. We give a surprising complete answer to a question poised by Cutkosky and Herzog. Let $Y$ be the blow-up of a projective scheme $X = \operatorname{Proj} R$ along the ideal sheaf of $I \subset R$. It is known that there are embeddings $Y \cong \operatorname{Proj} k[(I^e)_c]$for $c \ge d(I)e + 1$, where $d(I)$ denotes the maximal generating degree of $I$, and that there exists a Cohen-Macaulay ring of the form $k[(I^e)_c]$(which gives an arithmetic Macaulayfication of $X$) if and only if $H^0(Y,\mathcal{O}_Y) = k$, $H^i(Y,\mathcal{O}_Y) = 0$ for $i = 1,..., \dim Y-1$, and $Y$ is equidimensional and Cohen-Macaulay. We show that under these conditions, there are well-determined invariants $\varepsilon$ and $e_0$ such that $k[(I^e)_c]$ is Cohen-Macaulay for all $c > d(I)e + \varepsilon$ and $e > e_0$, and that these bounds are the best possible. We also investigate the existence of a Cohen-Macaulay Rees algebra of the form $R[(I^e)_ct]$. If $R$ has negative $a^*$-invariant, we prove that such a Cohen-Macaulay Rees algebra exists if and only if $\pi_*\mathcal{O}_Y = \mathcal{O}_X$, $R^i\pi_*\mathcal{O}_Y = 0$ for $i > 0$, and $Y$ is equidimensional and Cohen-Macaulay. Moreover, these conditions imply the Cohen-Macaulayness of $R[(I^e)_ct]$ for all $c > d(I)e + \varepsilon$ and $e> e_0$.
Degeneration of linear systems through fat points on $K3$ surfaces
Cindy
De Volder;
Antonio
Laface
3673-3682
Abstract: In this paper we introduce a technique to degenerate $K3$surfaces and linear systems through fat points in general position on $K3$surfaces. Using this degeneration we show that on generic $K3$ surfaces it is enough to prove that linear systems with one fat point are non-special in order to obtain the non-speciality of homogeneous linear systems through $n = 4^u9^w$fat points in general position. Moreover, we use this degeneration to obtain a result for homogeneous linear systems through $n = 4^u9^w$ fat points in general position on a general quartic surface in $\mathbb{P}^3$.
Asymptotic properties of convolution operators and limits of triangular arrays on locally compact groups
Yves
Guivarc'h;
Riddhi
Shah
3683-3723
Abstract: We consider a locally compact group $G$ and a limiting measure $\mu$of a commutative infinitesimal triangular system (c.i.t.s.) $\Delta$of probability measures on $G$. We show, under some restrictions on $G$, $\mu$ or $\Delta$, that $\mu$ belongs to a continuous one-parameter convolution semigroup. In particular, this result is valid for symmetric c.i.t.s. $\Delta$ on any locally compact group $G$. It is also valid for a limiting measure $\mu$ which has `full' support on a Zariski connected $\mathbb{F}$-algebraic group $G$, where $\mathbb{F}$ is a local field, and any one of the following conditions is satisfied: (1) $G$ is a compact extension of a closed solvable normal subgroup, in particular, $G$ is amenable, (2) $\mu$ has finite one-moment or (3) $\mu$ has density and in case the characteristic of $\mathbb{F}$ is positive, the radical of $G$ is $\mathbb{F}$-defined. We also discuss the spectral radius of the convolution operator of a probability measure on a locally compact group $G$, we show that it is always positive for any probability measure on $G$, and it is also multiplicative in case of symmetric commuting measures.
Complex immersions in Kähler manifolds of positive holomorphic $k$-Ricci curvature
Fuquan
Fang;
Sérgio
Mendonça
3725-3738
Abstract: The main purpose of this paper is to prove several connectedness theorems for complex immersions of closed manifolds in Kähler manifolds with positive holomorphic $k$-Ricci curvature. In particular this generalizes the classical Lefschetz hyperplane section theorem for projective varieties. As an immediate geometric application we prove that a complex immersion of an $n$-dimensional closed manifold in a simply connected closed Kähler $m$-manifold $M$ with positive holomorphic $k$-Ricci curvature is an embedding, provided that $2n\ge m+k$. This assertion for $k=1$ follows from the Fulton-Hansen theorem (1979).
Applications of the Wold decomposition to the study of row contractions associated with directed graphs
Elias
Katsoulis;
David
W.
Kribs
3739-3755
Abstract: Based on a Wold decomposition for families of partial isometries and projections of Cuntz-Krieger-Toeplitz-type, we extend several fundamental theorems from the case of single vertex graphs to the general case of countable directed graphs with no sinks. We prove a Szego-type factorization theorem for CKT families, which leads to information on the structure of the unit ball in free semigroupoid algebras, and show that joint similarity implies joint unitary equivalence for such families. For each graph we prove a generalization of von Neumann's inequality which applies to row contractions of operators on Hilbert space which are related to the graph in a natural way. This yields a functional calculus determined by quiver algebras and free semigroupoid algebras. We establish a generalization of Coburn's theorem for the $\mathrm{C}^*$-algebra of a CKT family, and prove a universality theorem for $\mathrm{C}^*$-algebras generated by these families. In both cases, the $\mathrm{C}^*$-algebras generated by quiver algebras play the universal role.
Lagrangian tori in homotopy elliptic surfaces
Tolga
Etgü;
David
McKinnon;
B.
Doug
Park
3757-3774
Abstract: Let $E(1)_K$ denote the symplectic four-manifold, homotopy equivalent to the rational elliptic surface, corresponding to a fibred knot $K$ in $S^3$ constructed by R. Fintushel and R. J. Stern in 1998. We construct a family of nullhomologous Lagrangian tori in $E(1)_K$ and prove that infinitely many of these tori have complements with mutually non-isomorphic fundamental groups if the Alexander polynomial of $K$ has some irreducible factor which does not divide $t^n-1$ for any positive integer $n$. We also show how these tori can be non-isotopically embedded as nullhomologous Lagrangian submanifolds in other symplectic $4$-manifolds.
Renorming James tree space
Petr
Hájek;
Jan
Rychtár
3775-3788
Abstract: We show that the James tree space $JT$ can be renormed to be Lipschitz separated. This negatively answers the question of J. Borwein, J. Giles and J. Vanderwerff as to whether every Lipschitz separated Banach space is an Asplund space.
Differentiation evens out zero spacings
David
W.
Farmer;
Robert
C.
Rhoades
3789-3811
Abstract: If $f$ is a polynomial with all of its roots on the real line, then the roots of the derivative $f'$ are more evenly spaced than the roots of $f$. The same holds for a real entire function of order 1 with all its zeros on a line. In particular, we show that if $f$ is entire of order 1 and has sufficient regularity in its zero spacing, then under repeated differentiation the function approaches, after normalization, the cosine function. We also study polynomials with all their zeros on a circle, and we find a close analogy between the two situations. This sheds light on the spacing between zeros of the Riemann zeta-function and its connection to random matrix polynomials.
On Ore's conjecture and its developments
Ilaria
Del Corso;
Roberto
Dvornicich
3813-3829
Abstract: The $p$-component of the index of a number field $K$, ${ \rm ind}_p(K)$, depends only on the completions of $K$ at the primes over $p$. More precisely, ${\rm ind}_p(K)$ equals the index of the $\mathbb{Q} _p$-algebra $K\otimes\mathbb{Q} _p$. If $K$ is normal, then $K\otimes\mathbb{Q} _p\cong L^n$ for some $L$ normal over $\mathbb{Q} _p$ and some $n$, and we write $I_p(nL)$ for its index. In this paper we describe an effective procedure to compute $I_p(nL)$ for all $n$ and all normal and tamely ramified extensions $L$ of $\mathbb{Q} _p$, hence to determine ${\rm ind}_p(K)$ for all Galois number fields that are tamely ramified at $p$. Using our procedure, we are able to exhibit a counterexample to a conjecture of Nart (1985) on the behaviour of $I_p(nL)$.