Maximum Genus of Strong Embeddings

The strong embedding conjecture states that any 2-connected graph has a strong embedding on some surface. It implies the circuit double cover conjecture: Any 2-connected graph has a circuit double cover.Conversely, it is not true. But for a 3-regular graph, the two conjectures are equivalent. In this paper, a characterization of graphs having a strong embedding with exactly 3 faces, which is the strong embedding of maximum genus, is given. In addition, some graphs with the property are provided. More generally, an upper bound of the maximum genus of strong embeddings of a graph is presented too. Lastly, it is shown that the interpolation theorem is true to planar Halin graph.     

Uniqueness of PL Minimal Surfaces

Using a standard fact in hyperbolic geometry, we give a simple proof of the uniqueness of PL minimal surfaces, thus filling in a gap in the original proof of Jaco and Rubinstein. Moreover, in order to clarify some ambiguity, we sharpen the definition of PL minimal surfaces, and prove a technical lemma on the Plateau problem in the hyperbolic space.     

The Upper Envelope of Positive Self-Similar Markov Processes

We establish integral tests in connection with laws of the iterated logarithm at 0 and at +∞, for the upper envelope of positive self-similar Markov processes. Our arguments are based on the Lamperti representation and on the study of the upper envelope of the future infimum due to the author (see Pardo in Stoch. Stoch. Rep. 78:123–155, [2006]). These results extend laws of the iterated logarithm for Bessel processes due to Dvoretsky and Erdős (Proceedings of the Second Berkeley Symposium, [1951]) and stable Lévy processes with no positive jumps conditioned to stay positive due to Bertoin (Stoch. Process. Appl. 55:91–100, [1995]). Research supported by a grant from CONACYT (Mexico).     

Spectral Characterizations of Cliford Minimal Hypersurfaces

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Tube Formulas and Complex Dimensions of Self-Similar Tilings

We use the self-similar tilings constructed in (Pearse in Indiana Univ. Math J. 56(6):3151–3169, 2007) to define a generating function for the geometry of a self-similar set in Euclidean space. This tubularzeta function encodes scaling and curvature properties related to the complement of the fractal set, and the associated system of mappings. This allows one to obtain the complex dimensions of the self-similar tiling as the poles of the tubularzeta function and hence develop a tube formula for self-similar tilings in ℝ^d. The resulting power series in εis a fractal extension of Steiner’s classical tube formula for convex bodies K⊆ℝ ^d . Our sum has coefficients related to the curvatures of the tiling, and contains terms for each integer i=0,1,…,d−1, just as Steiner’s does. However, our formula also contains a term for each complex dimension. This provides further justification for the term “complex dimension”. It also extends several aspects of the theory of fractal strings to higher dimensions and sheds new light on the tube formula for fractals strings obtained in (Lapidus and van Frankenhuijsen in Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings, 2006).     

Limiting Cases of a Superstring in AdS5×S5

The superstring action in AdS ₅×S ⁵ depends on two parameters: the inverse string tension and the radius R. The standard AdS/CFT correspondence requires that the string action depend on only the combination . After reviewing previous work on the light-cone superstring, we derive the explicit form of the action at =0. Its zero-mode part coincides with the superparticle action in AdS ₅×S ⁵, and the =0 string spectrum, as expected, must therefore include the protected type-IIB supergravity states. Following recent suggestions, we conjecture that such a tensionless string spectrum must also contain higher-spin massless states in AdS ₅. We also discuss the case of another parameterization of the string action that admits the straightforward flat-space limit R, but the limits R0 and are not equivalent in this case. The limit R0 then corresponds to shrinking S ⁵ to the size zero, simultaneously freezing the fluctuations of the radial coordinate of AdS ₅. This case results in a nonstandard AdS/CFT correspondence picture.     

Bouligand Dimension of Fractal Curve of Almost Periodic Series

in this paper, we obtain the calculating formulas for Bouligand dimension offractal curves represented by some almost periodic function series.     

Existence of the Self-Similar Solutions in the Heat Flow of Harmonic Maps

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On the Lusternik–Schnirelmann category of Peano continua

We define the LS-category _catg${\mathrm{cat}}_g$ by means of covers of a space by general subsets, and show that this definition coincides with the classical Lusternik–Schnirelmann category for compact metric ANR spaces. We apply this result to give short dimension theoretic proofs of the Grossman–Whitehead theorem and Dranishnikov?s theorem. We compute _catg${\mathrm{cat}}_g$ for some fractal Peano continua such as Menger spaces and Pontryagin surfaces.     

The Groups of Symmetric Genus σ ≤ 8

Let G be a finite group. The symmetric genus σ (G) is the minimum genus of any compact Riemann surface on which G acts faithfully as a group of automorphisms. Here we classify the groups of symmetric genus σ, for all values of σ such that 4 ≤ σ ≤ 8. In addition, we obtain some general results about the partial presentations that groups acting on surfaces must have. We show that a group with even genus and no “large order” elements in its Sylow 2-subgroup has restrictions on its Sylow 2-subgroup. As a consequence, we show that if G is a 2-group with positive symmetric genus, then σ(G) is odd. The software package MAGMA was employed to help with the calculations, and the MAGMA library of small groups was essential to the classification.     

The Spectral Characterizations of Some Minimal Submanifolds of a Sphere

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Towards a Characterization of Self-Similar Tilings in Terms of Derived Voronoï Tessellations

In this paper, a technique for analyzing levels of hierarchy in a tiling of Euclidean space is presented. Fixing a central configuration P of tiles in , a `derived Voronoï' tessellation _P is constructed based on the locations of copies of P in . A family of derived Voronoï tilings is formed by allowing the central configurations to vary through an infinite number of possibilities. The family will normally be an infinite one, but we show that for a self-similar tiling it is finite up to similarity. In addition, we show that if the family is finite up to similarity, then is pseudo-self-similar. The relationship between self-similarity and pseudo-self-similarity is not well understood, and this is the obstruction to a complete characterization of self-similarity via our method. A discussion and conjecture on the connection between the two forms of hierarchy for tilings is provided.     

A Class of Dust-Like Self-Similar Solutions of the Massless Einstein�CVlasov System

In this paper the existence of a class of self-similar solutions of the Einstein–Vlasov system is proved. The initial data for these solutions are not smooth, with their particle density being supported in a submanifold of codimension one. They can be thought of as intermediate between smooth solutions of the Einstein–Vlasov system and dust. The motivation for studying them is to obtain insights into possible violation of weak cosmic censorship by solutions of the Einstein–Vlasov system. By assuming a suitable form of the unknowns it is shown that the existence question can be reduced to that of the existence of a certain type of solution of a four-dimensional system of ordinary differential equations depending on two parameters. This solution starts at a particular point P ₀ and converges to a stationary solution P ₁ as the independent variable tends to infinity. The existence proof is based on a shooting argument and involves relating the dynamics of solutions of the four-dimensional system to that of solutions of certain two- and three-dimensional systems obtained from it by limiting processes. The spacetimes constructed do not constitute a counterexample to cosmic censorship since they are not asymptotically flat. They should be seen as the first step on a possible path towards constructing solutions of importance for understanding the issue of the formation of naked singularities in general relativity.     

Intersections of curve systems and the crossing number ofC 5 ×C 5

It and are two families of pairwise disjoint simple closed curves in the plane such that each curve in intersects each curve in , then the total number of points of intersection in is at least 2(m−1)n, where , and this bound is best possible. We use this to show that the cartesian product of two 5-cycles has crossing number 15.     

Equivalent Separating Curves on Closed Surface of Genus 2

Let {A,B} be a complete system of the closed orientable surface F of genus 2. A simple closed curve C on F is separating with respect to (A, B) if it is disjoint from A∪B and it cuts F into two once-punctured tori X, Y with A(?)X, B(?)Y. Letγbe a simple closed curve on F which is disjoint from A∪B and intersects C essentially in two points. In this paper, we show that up to isotopy, {hnγ(C):n∈Z} is the set containing all the simple closed curves on F which is separating with respect to (A,B), where hγis the Dehn twist alongγon F. This also shows how two simple closed curves on F which are separating with respect to (A,B) are related. The result can be applied to yield all Haken spheres of a Heegaard splitting V∪F W which are weakly equivalent to a given Heken sphere of the splitting.