arXiv subjects
Publications and source records attributed to Steen Pedersen.
Consider a real valued function defined, but not differentiable at some point. We use sequences approaching the point of interest to define and study sequential concepts of secant and cord derivatives of the function at the point of interest. If the function is the celebrated Weierstrass function, it follows from some of our results that the set cord derivatives at any point coincides with the extended real line.
We study two classes of extension problems, and their interconnections: (i) Extension of positive definite (p.d.) continuous functions defined on subsets in locally compact groups $G$; (ii) In case of Lie groups, representations of the associated Lie algebras $La\left(G\right)$ by unbounded skew-Hermitian operators acting in a reproducing kernel Hilbert space (RKHS) $\mathscr{H}_{F}$. Why extensions? In science, experimentalists frequently gather spectral data in cases when the observed data is limited, for example limited by the precision of instruments; or on account of a variety of other limiting external factors. Given this fact of life, it is both an art and a science to still produce solid conclusions from restricted or limited data. In a general sense, our monograph deals with the mathematics of extending some such given partial data-sets obtained from experiments. More specifically, we are concerned with the problems of extending available partial information, obtained, for example, from sampling. In our case, the limited information is a restriction, and the extension in turn is the full positive definite function (in a dual variable); so an extension if available will be an everywhere defined generating function for the exact probability distribution which reflects the data; if it were fully available. Such extensions of local information (in the form of positive definite functions) will in turn furnish us with spectral information. In this form, the problem becomes an operator extension problem, referring to operators in a suitable reproducing kernel Hilbert spaces (RKHS). In our presentation we have stressed hands-on-examples. Extensions are almost never unique, and so we deal with both the question of existence, and if there are extensions, how they relate back to the initial completion problem.
We study two extension problems, and their interconnections: (i) extension of positive definite (p.d.) continuous functions defined on subsets in locally compact groups $G$; and (ii) (in case of Lie groups $G$) representations of the associated Lie algebras $La\left(G\right)$, i.e., representations of $La\left(G\right)$ by unbounded skew-Hermitian operators acting in a reproducing kernel Hilbert space $\mathscr{H}_{F}$ (RKHS). Our analysis is non-trivial even if $G=\mathbb{R}^{n}$, and even if $n=1$. If $G=\mathbb{R}^{n}$, (ii), we are concerned with finding systems of strongly commuting selfadjoint operators $\left\{ T_{i}\right\} $ extending a system of commuting Hermitian operators with common dense domain in $\mathscr{H}_{F}$. Specifically, we consider partially defined positive definite (p.d.) continuous functions $F$ on a fixed group. From $F$ we then build a reproducing kernel Hilbert space $\mathscr{H}_{F}$, and the operator extension problem is concerned with operators acting in $\mathscr{H}_{F}$, and with unitary representations of $G$ acting on $\mathscr{H}_{F}$. Our emphasis is on the interplay between the two problems, and on the harmonic analysis of our RKHSs $\mathscr{H}_{F}$.
We establish a formula yielding the Hausdorff measure for a class of non-self-similar Cantor sets in terms of the canonical covers of the Cantor set.
Messenger RNA encodes a sequence of amino acids by using codons. For most amino acids there are multiple synonymous codons that can encode the amino acid. The translation speed can vary from one codon to another, thus there is room for changing the ribosome speed while keeping the amino acid sequence and hence the resulting protein. Recently, it has been noticed that the choice of the synonymous codon, via the resulting distribution of slow- and fast-translated codons, affects not only on the average speed of one ribosome translating the messenger RNA (mRNA) but also might have an effect on nearby ribosomes by affecting the appearance of "traffic jams" where multiple ribosomes collide and form queues. To test this "context effect" further, we here investigate the effect of the sequence of synonymous codons on the ribosome traffic by using a ribosome traffic model with codon-dependent rates, estimated from experiments. We compare the ribosome traffic on wild type sequences and sequences where the synonymous codons were swapped randomly. By simulating translation of 87 genes, we demonstrate that the wild type sequences, especially those with a high bias in codon usage, tend to have the ability to reduce ribosome collisions, hence optimizing the cellular investment in the translation apparatus. The magnitude of such reduction of the translation time might have a significant impact on the cellular growth rate and thereby have importance for the survival of the species.
We investigate the skew-adjoint extensions of a partial derivative operator acting in the direction of one of the sides a unit square. We investigate the unitary equivalence of such extensions and the spectra of such extensions. It follows from our results, that such extensions need not have discete spectrum. We apply our techniques to the problem of finding commuting skew-adjoint extensions of the partial derivative operators acting in the directions of the sides of the unit square. While our results are most easily stated for the unit square, they are established for a larger class of domains, including certain fractal domains.
Let C be a Cantor set. For a real number t let C+t be the translate of C by t, We say two real numbers s,t are equivalent if the intersection of C and C+s is a translate of the intersection of C and C+t. We consider a class of Cantor sets determined by similarities with one fixed positive contraction ratio. For this class of Cantor set, we show that an "initial segment" of the intersection of C and C+t is a self-similar set with contraction ratios that are powers of the contraction ratio used to describe C as a self- similar set if and only if t is equivalent to a rational number. Our results are new even for the middle thirds Cantor set.
We establish formulas for bounds on the Haudorff measure of the intersection of certain Cantor sets with their translates. As a consequence we obtain a formula for the Hausdorff dimensions of these intersections.
We study restriction and extension theory for semibounded Hermitian operators in the Hardy space of analytic functions on the disk D. Starting with the operator zd/dz, we show that, for every choice of a closed subset F in T=bd(D) of measure zero, there is a densely defined Hermitian restriction of zd/dz corresponding to boundary functions vanishing on F. For every such restriction operator, we classify all its selfadjoint extension, and for each we present a complete spectral picture. We prove that different sets F with the same cardinality can lead to quite different boundary-value problems, inequivalent selfadjoint extension operators, and quite different spectral configurations. As a tool in our analysis, we prove that the von Neumann deficiency spaces, for a fixed set F, have a natural presentation as reproducing kernel Hilbert spaces, with a Hurwitz zeta-function, restricted to FxF, as reproducing kernel.
We present a model for spectral theory of families of selfadjoint operators, and their corresponding unitary one-parameter groups (acting in Hilbert space.) The models allow for a scale of complexity, indexed by the natural numbers $\mathbb{N}$. For each $n\in\mathbb{N}$, we get families of selfadjoint operators indexed by: (i) the unitary matrix group U(n), and by (ii) a prescribed set of $n$ non-overlapping intervals. Take $Ω$ to be the complement in $\mathbb{R}$ of $n$ fixed closed finite and disjoint intervals, and let $L^{2}(Ω)$ be the corresponding Hilbert space. Moreover, given $B\in U(n)$, then both the lengths of the respective intervals, and the gaps between them, show up as spectral parameters in our corresponding spectral resolutions within $L^{2}(Ω)$. Our models have two advantages: One, they encompass realistic features from quantum theory, from acoustic wave equations and their obstacle scattering; as well as from harmonic analysis. Secondly, each choice of the parameters in our models, $n\in\mathbb{N}$, $B\in U(n)$, and interval configuration, allows for explicit computations, and even for closed-form formulas: Computation of spectral resolutions, of generalized eigenfunctions in $L^{2}(Ω)$ for the continuous part of spectrum, and for scattering coefficients. Our models further allow us to identify embedded point-spectrum (in the continuum), corresponding, for example, to bound-states in scattering, to trapped states, and to barriers in quantum scattering. The possibilities for the discrete atomic part of spectrum includes both periodic and non-periodic distributions.
Studying unitary one-parameter groups in Hilbert space (U(t),H), we show that a model for obstacle scattering can be built, up to unitary equivalence, with the use of translation representations for L2-functions in the complement of two finite and disjoint intervals. The model encompasses a family of systems (U (t), H). For each, we obtain a detailed spectral representation, and we compute the scattering operator, and scattering matrix. We illustrate our results in the Lax-Phillips model where (U (t), H) represents an acoustic wave equation in an exterior domain; and in quantum tunneling for dynamics of quantum states.
We study the momentum operator defined on the disjoint union of two intervals. Even in one dimension, the question of two non-empty open and non-overlapping intervals has not been worked out in a way that extends the cases of a single interval and gives a list of the selfadjoint extensions. Starting with zero boundary conditions at the four endpoints, we characterize the selfadjoint extensions and undertake a systematic and complete study of the spectral theory of the selfadjoint extensions. In an application of our extension theory to harmonic analysis, we offer a new family of spectral pairs. Compared to earlier studies, it yields a more direct link between spectrum and geometry.
We consider some properties of the intersection of deleted digits Cantor sets with their translates. We investigate conditions on the set of digits such that, for any t between zero and the dimension of the deleted digits Cantor set itself, the set of translations such that the intersection has Hausdorff dimension equal to t is dense in the set F of translations such that the intersection is non-empty. We make some simple observations regarding properties of the set F, in particular, we characterize when F is an interval, in terms of conditions on the digit set.
Individual mRNAs are translated by multiple ribosomes that initiate translation with a few seconds interval. The ribosome speed is codon dependant, and ribosome queuing has been suggested to explain specific data for translation of some mRNAs in vivo. By modelling the stochastic translation process as a traffic problem, we here analyze conditions and consequences of collisions and queuing. The model allowed us to determine the on-rate (0.8 to 1.1 initiations per sec) and the time (1 sec) the preceding ribosome occludes initiation for Escherichia coli lacZ mRNA in vivo. We find that ribosome collisions and queues are inevitable consequences of a stochastic translation mechanism that reduce the translation efficiency substantially on natural mRNAs. The cells minimize collisions by having its mRNAs being unstable and by a highly selected codon usage in the start of the mRNA. The cost of mRNA breakdown is offset by the concomitant increase in translational efficiency.
This paper introduces Fourier duality for a class of affine iterated function systems (IFS) T_i. These systems are determined by a finite family of contractive affine maps in R^d. Our Fourier duality applies to the resulting probability measure mu which is fixed by (T_i). When the IFS is given, the support of the associated mu is a compact set X in R^d, typically a fractal. Our Fourier duality refers to the Hilbert space L^2(X, mu): We show that under a certain unitarity condition involving a pair of affine iterated function systems (T_i) and (S_j) it is possible to recursively construct a Fourier bases in the Hilbert space L^2(X, mu) with the Fourier basis for one depending on the other.
In the first section we review recent results on the harmonic analysis of fractals generated by iterated function systems with emphasis on spectral duality. Classical harmonic analysis is typically based on groups whereas the fractals are most often not groups. We show that nonetheless those fractals that come from iteration of affine mappings in R^d have a spectral duality which is instead based on approximation and a certain dual affine system on the Fourier transform side. The present work is based on iteration of frame estimates (which have been studied earlier for regions in R^d). Our emphasis is on new results regarding the interplay between the limit-fractal on the one hand, and on the other the corresponding regions in R^d which generate iterated function systems of contractive affine mappings. As an application of our frame results, we obtain a classification of a certain type of spectral pairs.
Let $ Ω\subset R^d $ have finite positive Lebesgue measure, and let $ \mathcal{L}^{2}(Ω) $ be the corresponding Hilbert space of $ \mathcal{L}^{2} $-functions on $ Ω$. We shall consider the exponential functions $ e_λ $ on $ Ω$ given by $ e_λ(x)=e^{i2πλx} $. If these functions form an orthogonal basis for $ \mathcal{L}^{2}(Ω) $, when $ λ$ ranges over some subset $ Λ$ in $ R^d $, then we say that $ (Ω,Λ) $ is a spectral pair, and that $ Λ$ is a spectrum. We conjecture that $ (Ω,Λ) $ is a spectral pair if and only if the translates of some set $ Ω' $ by the vectors of $ Λ$ tile $ R^d $. In the special case of $ Ω=I^d $, the $ d $-dimensional unit cube, we prove this conjecture, with $ Ω'=I^d $, for $ d \leq 3 $, describing all the tilings by $ I^d $, and for all $ d $ when $ Λ$ is a discrete periodic set. In an appendix we generalize the notion of spectral pair to measures on a locally compact abelian group and its dual.
We prove that no ball admits a non-harmonic orthogonal basis of exponentials. We use a combinatorial result, originally studied by Erd\H os, which says that the number of distances determined by $n$ points in ${\Bbb R}^d$ is at least $C_d n^{\frac{1}{d}+ε_d}$, $ε_d>0$.