Displaying similar documents to “On continuous functions with no unilateral derivatives”

Elementary moves for higher dimensional knots

Dennis Roseman (2004)

Fundamenta Mathematicae

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For smooth knottings of compact (not necessarily orientable) n-dimensional manifolds in n + 2 (or n + 2 ), we generalize the notion of knot moves to higher dimensions. This reproves and generalizes the Reidemeister moves of classical knot theory. We show that for any dimension there is a finite set of elementary isotopies, called moves, so that any isotopy is equivalent to a finite sequence of these moves.

On the Signatures of Torus Knots

Maciej Borodzik, Krzysztof Oleszkiewicz (2010)

Bulletin of the Polish Academy of Sciences. Mathematics

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We study properties of the signature function of the torus knot T p , q . First we provide a very elementary proof of the formula for the integral of the signature over the circle. We also obtain a closed formula for the Tristram-Levine signature of a torus knot in terms of Dedekind sums.

On a functional-differential equation related to Golomb's self-described sequence

Y.-F. S. Pétermann, J.-L. Rémy, I. Vardi (1999)

Journal de théorie des nombres de Bordeaux

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The functional-differential equation f ' ( t ) = 1 / f ( f ( t ) ) is closely related to Golomb’s self-described sequence F , 1 , 1 , 2 , 2 , 2 , 3 , 3 , 2 , 4 , 4 , 4 3 , 5 , 5 , 5 , 3 , 6 , 6 , 6 , 6 , 4 , . We describe the increasing solutions of this equation. We show that such a solution must have a nonnegative fixed point, and that for every number p 0 there is exactly one increasing solution with p as a fixed point. We also show that in general an initial condition doesn’t determine a unique solution: indeed the graphs of two distinct increasing solutions cross each...

Legendrian and transverse twist knots

John B. Etnyre, Lenhard L. Ng, Vera Vértesi (2013)

Journal of the European Mathematical Society

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In 1997, Chekanov gave the first example of a Legendrian nonsimple knot type: the m ( 5 2 ) knot. Epstein, Fuchs, and Meyer extended his result by showing that there are at least n different Legendrian representatives with maximal Thurston-Bennequin number of the twist knot K - 2 n with crossing number 2 n + 1 . In this paper we give a complete classification of Legendrian and transverse representatives of twist knots. In particular, we show that K - 2 n has exactly n 2 2 Legendrian representatives with maximal Thurston–Bennequin...

Some non-trivial PL knots whose complements are homotopy circles

Greg Friedman (2007)

Fundamenta Mathematicae

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We show that there exist non-trivial piecewise linear (PL) knots with isolated singularities S n - 2 S , n ≥ 5, whose complements have the homotopy type of a circle. This is in contrast to the case of smooth, PL locally flat, and topological locally flat knots, for which it is known that if the complement has the homotopy type of a circle, then the knot is trivial.

Quandles and symmetric quandles for higher dimensional knots

Seiichi Kamada (2014)

Banach Center Publications

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A symmetric quandle is a quandle with a good involution. For a knot in ℝ³, a knotted surface in ℝ⁴ or an n-manifold knot in n + 2 , the knot symmetric quandle is defined. We introduce the notion of a symmetric quandle presentation, and show how to get a presentation of a knot symmetric quandle from a diagram.

Representations of (1,1)-knots

Alessia Cattabriga, Michele Mulazzani (2005)

Fundamenta Mathematicae

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We present two different representations of (1,1)-knots and study some connections between them. The first representation is algebraic: every (1,1)-knot is represented by an element of the pure mapping class group of the twice punctured torus PMCG₂(T). Moreover, there is a surjective map from the kernel of the natural homomorphism Ω:PMCG₂(T) → MCG(T) ≅ SL(2,ℤ), which is a free group of rank two, to the class of all (1,1)-knots in a fixed lens space. The second representation is parametric:...

On the closure of spaces of sums of ridge functions and the range of the X -ray transform

Jan Boman (1984)

Annales de l'institut Fourier

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For a R n { 0 } and Ω an open bounded subset of R n definie L p ( Ω , a ) as the closed subset of L p ( Ω ) consisting of all functions that are constant almost everywhere on almost all lines parallel to a . For a given set of directions a ν R n { 0 } , ν = 1 , ... , m , we study for which Ω it is true that the vector space ( * ) L p ( Ω , a 1 ) + + L p ( Ω , a m ) is a closed subspace of L p ( Ω ) . This problem arizes naturally in the study of image reconstruction from projections (tomography). An essentially equivalent problem is to decide whether a certain matrix-valued differential operator...

Brunnian local moves of knots and Vassiliev invariants

Akira Yasuhara (2006)

Fundamenta Mathematicae

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K. Habiro gave a neccesary and sufficient condition for knots to have the same Vassiliev invariants in terms of C k -moves. In this paper we give another geometric condition in terms of Brunnian local moves. The proof is simple and self-contained.

Extending Peano derivatives

Hajrudin Fejzić, Jan Mařík, Clifford E. Weil (1994)

Mathematica Bohemica

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Let H [ 0 , 1 ] be a closed set, k a positive integer and f a function defined on H so that the k -th Peano derivative relative to H exists. The major result of this paper is that if H has finite Denjoy index, then f has an extension, F , to [ 0 , 1 ] which is k times Peano differentiable on [ 0 , 1 ] with f i = F i on H for i = 1 , 2 , ... , k .

On blocks of arithmetic progressions with equal products

N. Saradha (1997)

Journal de théorie des nombres de Bordeaux

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Let f ( X ) [ X ] be a monic polynomial which is a power of a polynomial g ( X ) [ X ] of degree μ 2 and having simple real roots. For given positive integers d 1 , d 2 , , m with < m and gcd ( , m ) = 1 with μ m + 1 whenever m < 2 , we show that the equation f ( x ) f ( x + d 1 ) f ( x + ( k - 1 ) d 1 ) = f ( y ) f ( y + d 2 ) f ( y + ( m k - 1 ) d 2 ) with f ( x + j d 1 ) 0 for 0 j < k has only finitely many solutions in integers x , y and k 1 except in the case m = μ = 2 , = k = d 2 = 1 , f ( X ) = g ( X ) , x = f ( y ) + y .

A Whitney extension theorem in L p and Besov spaces

Alf Jonsson, Hans Wallin (1978)

Annales de l'institut Fourier

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The classical Whitney extension theorem states that every function in Lip ( β , F ) , F R n , F closed, k < β k + 1 , k a non-negative integer, can be extended to a function in Lip ( β , R n ) . Her Lip ( β , F ) stands for the class of functions which on F have continuous partial derivatives up to order k satisfying certain Lipschitz conditions in the supremum norm. We formulate and prove a similar theorem in the L p -norm. The restrictions to R d , d < n , of the Bessel potential spaces in R n and the Besov or generalized Lipschitz...