Minimal Peano curve

A Peano curve p(x) with maximum square-to-linear ratio |p(x)?p(y)|²/|x?y| equal to 5 2/3 is constructed; this ratio is smaller than that of the classical Peano-Hilbert curve, whose maximum square-to-linear ratio is 6. The curve constructed is of fractal genus 9 (i.e., it is decomposed into nine fragments that are similar to the whole curve) and of diagonal type (i.e., it intersects a square starting from one corner and ending at the opposite corner). It is proved that this curve is a unique (up to isometry) regular diagonal Peano curve of fractal genus 9 whose maximum square-to-linear ratio is less than 6. A theory is developed that allows one to find the maximum square-to-linear ratio of a regular Peano curve on the basis of computer calculations.     

Minimal Self-Similar Peano Curve of Genus 5 × 5

Doklady Mathematics - The paper presents a plane regular fractal Peano curve with a Euclidean square-to-line ratio (L2-locality) of $$5\frac{{43}}{{73}}$$ , which is minimal among all known curves...     

Attainment of maximum cube-to-linear ratio for three-dimensional Peano curves

The class of so-called q-adic Peano curves is defined, which is large enough to include the polyfractal curves. The cube-to-linear ratio for this class attains its maximum value, which can be effectively determined by an exhaustive search implementable on modern computers.     

<Emphasis Type="Italic">L</Emphasis><Subscript>∞</Subscript>-Locality of Three-Dimensional Peano Curves

A theory and corresponding algorithms are developed for fast and exact calculation of the L_∞-locality (i.e., the greatest cube-to-linear ratio in the maximum metric) for polyfractal three-dimensional Peano curves.     

Fractal peano curves

We show that the theory of iterated function systems (i.f.s.s) can be used to construct and geometrically describe Peano curves. We present this point of view by exhibiting i.f.s.s whose attractors are the graphs of some well-known Peano curves.     

Peano kernel associated with the polyharmonic mean value property in the annulus

The well known mean value property for polyharmonic functions in the ball (see Picone [9], Bramble-Payne [2]) is generalized to the case of an annular domain. We prove that the Peano kernel arising from this new mean value property has definite sign. It is a polyspline in the sense of [8]. The present results generalize former investigations concerning the Peano kernel associated with the mean value property of polyharmonic functions in the ball.     

Chaotic spread spectrum watermark of optimal space-filling curves

Spread spectrum watermarking scheme is becoming an important research subject. In this paper, we present a method based on Peano–Hilbert space-filling curves for enhancing the robustness. Peano–Hilbert curve is a continuous mapping from one-dimensional space onto two-dimensional space. It is useful in many applications including quantum mechanics even, and preserves optimal locality. At the same time, we utilize a specified chaotic dynamic system–ICMIC map, which shows lowpass properties when the controlling parameter is devised. In this case, the watermarking detection resorts to the Neyman–Pearson criterion based on some statistical assumptions. Experimental results show that the proposed scheme can work well under JPEG compression and resist line-removal test.     

On the form of witness terms

We investigate the development of terms during cut-elimination in first-order logic and Peano arithmetic for proofs of existential formulas. The form of witness terms in cut-free proofs is characterized in terms of structured combinations of basic substitutions. Based on this result, a regular tree grammar computing witness terms is given and a class of proofs is shown to have only elementary cut-elimination.     

On Affine Surface That can be Completed by A Smooth Curve

In C⁶, we consider a non linear system of differential equations with four invariants: two quadrics, a cubic and a quartic. Using Enriques-Kodaira classification of algebraic surfaces, we show that the affine surface obtained by setting these invariants equal to constants is the affine part of an abelian surface. This affine surface is completed by gluing to it a one genus 9 curve consisting of two isomorphic genus 3 curves intersecting transversely in 4 points.     

Geometric realizations for Dyck's regular map on a surface of genus 3

Klein's and Dyck's regular maps on Riemann surfaces of genus 3 were one impetus for the theory of regular maps, automorphic functions, and algebraic curves. Recently a polyhedral realization inE ³ of Klein's map was discovered [18], thereby underlining the strong analogy to the icosahedron. In this paper we show that Dyck's map can be realized inE ³ as a polyhedron of Kepler-Poinsot-type, i.e., with maximal symmetry and minimal self-intersections. Furthermore some closely related polyhedra and a Kepler-Poinsot-type realization of Sherk's regular map of genus 5 are discussed.     

Universal space in the Cartesian product of Peano curves without free arcs

We prove that there is the universal space for the class of n-dimensional separable metric spaces in the Cartesian product K₁×?×Kn+1 of Peano curves without free arcs. It is also shown that the set of embeddings of any n-dimensional separable metric space X into this universal space is a residual set in C(X,K₁×?×Kn+1). Other properties of product of Peano curves without free arcs are also proved.     

Copulae,Self-Affine Functions,and Fractal Dimensions

We use the fact that the functions defined on the unit interval whose graphs support a copula are those that are Lebesgue-measure-preserving in order to characterize self-affine functions whose graphs are the support of a copula. This result allows computation of the Hausdorff, packing, and box-counting dimensions. The discussion is applied to classic examples such as the Peano and Hilbert curves, and the results are extended to discontinuous self-affine functions.     

Dimensions of the coordinate functions of space-filling curves

The graphs of coordinate functions of space-filling curves such as those described by Peano, Hilbert, Pólya and others, are typical examples of self-affine sets, and their Hausdorff dimensions have been the subject of several articles in the mathematical literature. In the first half of this paper, we describe how the study of dimensions of self-affine sets was motivated, at least in part, by these coordinate functions and their natural generalizations, and review the relevant literature. In the second part, we present new results on the coordinate functions of Pólya's one-parameter family of space-filling curves. We give a lower bound for the Hausdorff dimension of their graphs which is fairly close to the box-counting dimension. Our techniques are largely probabilistic. The fact that the exact dimension remains elusive seems to indicate the need for further work in the area of self-affine sets.     

Teichmüller Curves Through Fermat Curves

The billiard in a regular n-gon is known to give rise to a Teichmüller curve. For odd n, we compute the genus of this curve, a number field over which the curve may be defined and branched covering relations between certain pairs of these curves. If n is a power of a prime congruent to 3 or 5 modulo 8, the Teichmüller curve may be defined over the rationals. Received: June 2006, Revision: October 2006, Accepted: November 2006     

Exhibiting SHA[2] on hyperelliptic Jacobians

We discuss approaches to computing in the Shafarevich-Tate group of Jacobians of higher genus curves, with an emphasis on the theory and practice of visualisation. Especially for hyperelliptic curves, this often enables the computation of ranks of Jacobians, even when the 2-Selmer bound does not bound the rank sharply. This was previously only possible for a few special cases. For curves of genus 2, we also demonstrate a connection with degree 4 del Pezzo surfaces, and show how the Brauer-Manin obstruction on these surfaces can be used to compute members of the Shafarevich-Tate group of Jacobians. We derive an explicit parametrised infinite family of genus 2 curves whose Jacobians have nontrivial members of the Shafarevich-Tate group. Finally, we prove that under certain conditions, the visualisation dimension for order 2 cocycles of Jacobians of certain genus 2 curves is 4 rather than the general bound of 32.     

Infinite approximate Peano derivatives

In this paper we introduce approximate Peano derivatives with infinite values allowed, and we show that these derivatives are Baire one, and possess the Darboux and Denjoy-Clarkson properties. Also we show that if they are bounded from above or below on an interval, then the corresponding ordinary derivatives exist and equal the approximate Peano derivatives.

     

Interstitial and pseudo gaps in models of Peano Arithmetic

In this paper we study the automorphism groups of models of Peano Arithmetic. Kossak, Kotlarski, and Schmerl [9] shows that the stabilizer of an unbounded element a of a countable recursively saturated model of Peano Arithmetic M is a maximal subgroup of Aut(M) if and only if the type of a is selective. We extend this result by showing that if M is a countable arithmetically saturated model of Peano Arithmetic, Ω ? M is a very good interstice, and a ∈ Ω, then the stabilizer of a is a maximal subgroup of Aut(M) if and only if the type of a is selective and rational (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computing genus 2 curves from invariants on the Hilbert moduli space

We give a new method for generating genus 2 curves over a finite field with a given number of points on the Jacobian of the curve. We define two new invariants for genus 2 curves as values of modular functions on the Hilbert moduli space and show how to compute them. We relate them to the usual three Igusa invariants on the Siegel moduli space and give an algorithm to construct curves using these new invariants. Our approach simplifies the complex analytic method for computing genus 2 curves for cryptography and reduces the amount of computation required.