Orthonormal System Research Articles

Parameterization of rotations in reference frames with redundant crystallographic axes

Parameterizations of rotations in three-dimensional space are fundamental for description of crystallographic textures. Such parameterizations (e.g., Rodrigues parameters) are usually specified using orthonormal coordinate systems, whereas bases of crystal lattices are generally non-orthogonal. In the case of crystals with hexagonal or rhombohedral lattices, the reference frames involve redundant crystallographic axes, and hence a question arises about feasibility of the generalization of the rotation parameterizations to such frames. The definition of Rodrigues parameters can be extended so they are directly linked to non-Cartesian bases of crystal lattices. The new Rodrigues parameters are contra- or covariant components of vectors specified with respect to exactly the same lattice basis as atomic positions in a unit cell. The generalized formalism allows for using redundant crystallographic axes. Also the orientation matrices can be represented in such frames. The Rodrigues parameterization in non-Cartesian coordinate frames is convenient for crystallographic applications because the generalized parameters are directly related to indices of rotation-invariant lattice directions and to Miller indices of rotation-invariant lattice planes. In the case of the hexagonal and rhombohedral lattices, the frames with redundant axes are used to account for lattice symmetry, but one may apply such frames for other reasons. They can be convenient for handling arbitrary symmetries, in particular symmetries arising in description of some phenomena or symmetries of physical processes. The practicality of frames is illustrated by an alternative description of body-centered lattices, a formula for lattice rotation in deformation by slip, and a new interpretation of indexing of single crystal diffraction patterns.

Completely Sidon sets in $$C^*$$ C ∗ -algebras

A sequence in a \(C^*\)-algebra A is called completely Sidon if its span in A is completely isomorphic to the operator space version of the space \(\ell _1\) (i.e. \(\ell _1\) equipped with its maximal operator space structure). The latter can also be described as the span of the free unitary generators in the (full) \(C^*\)-algebra of the free group \(\mathbb {F}_\infty \) with countably infinitely many generators. Our main result is a generalization to this context of Drury’s classical theorem stating that Sidon sets are stable under finite unions. In the particular case when \(A=C^*(G)\) the (maximal) \(C^*\)-algebra of a discrete group G, we recover the non-commutative (operator space) version of Drury’s theorem that we recently proved. We also give several non-commutative generalizations of our recent work on uniformly bounded orthonormal systems to the case of von Neumann algebras equipped with normal faithful tracial states.

Modified Spherical Harmonics in Four Dimensions

The classical theory of spherical harmonics on the unit sphere S is well-known. In an earlier paper, entitled “Modified Spherical Harmonics”, we dealt with a modification of this theory, adapted to the half-sphere $$S_{+}$$ , in case of three dimensions. In the present paper we extend these results to the four-dimensional case. Although the results look quite similar, their proofs are not. In $$\mathbb {R}^4 =\left\{ (x,y,t,s) \right\} $$ the Laplace equation $$\Delta h=0$$ will be replaced by the equation $$s\Delta u+2\,\frac{\partial u}{\partial s}=0$$ . Homogeneous polynomial solutions of this equation, if restricted to the half-sphere $$S_{+}=\left\{ (x,y,t,s): x^2 + y^2 + t^2 + s^2=1, s > 0 \right\} $$ are called modified spherical harmonics. Endowed with a non-Euclidean scalar product on $$S_{+}$$ , these functions behave like the classical spherical harmonics on the full sphere in $$\mathbb {R}^4$$ . We shall give an explicit expression for the corresponding zonal harmonics and dwell on their connection with the Poisson-type kernel, adapted to the above differential equation. Finally we shall give an explicit orthonormal system of modified spherical harmonics.

Reliability of Color Doppler Ultrasound Imaging for the Assessment of Anterolateral Thigh Flap Perforators: A Prospective Study of 30 Perforators.

Diagnostic, II.

Slice Regular Malmquist–Takenaka System in the Quaternionic Hardy Spaces

In this paper the slice regular analogue of the Malmquist–Takenaka system is introduced. It is proved that, under certain restrictions regarding to the parameters of the system, they form a complete orthonormal system in the quaternionic Hardy spaces of the unit ball. The properties of associated projection operator are also studied.

A Note on Sidon Sets in Bounded Orthonormal Systems

We give a simple example of an n-tuple of orthonormal elements in $$L_2$$ (actually martingale differences) bounded by a fixed constant, and hence subgaussian with a fixed constant but that are Sidon only with constant $$\approx \sqrt{n}$$ . This is optimal. The first example of this kind was given by Bourgain and Lewko, but with constant $$\approx \sqrt{\log n}$$ . We also include the analogous $$n\times n$$ -matrix valued example, for which the optimal constant is $$\approx n$$ . We deduce from our example that there are two n-tuples each Sidon with constant 1, lying in orthogonal linear subspaces and such that their union is Sidon only with constant $$\approx \sqrt{n}$$ . This is again asymptotically optimal. We show that any martingale difference sequence with values in $$[-1,1]$$ is “dominated” in a natural sense (related to our results) by any sequence of independent, identically distributed, symmetric $$\{-1,1\}$$ -valued variables (e.g. the Rademacher functions). We include a self-contained proof that any sequence $$(\varphi _n)$$ that is the union of two Sidon sequences lying in orthogonal subspaces is such that $$(\varphi _n\otimes \varphi _n \otimes \varphi _n\otimes \varphi _n)$$ is Sidon.

Unconditional Convergence of Fourier Series for Functions of Bounded Variation

This article concerns the unconditional convergence a.e. of Fourier series with respect to general orthonormal systems. We find certain conditions to be satisfied by the functions in the orthonormal system so that the Fourier series of each function of finite variation unconditionally converge a.e. The results are best possible.

Vector-valued extensions for fractional integrals of Laguerre expansions

We prove some vector-valued inequalities for fractional integrals defined for several orthonormal systems of Laguerre functions. On the one hand, we obtain weighted Lp-Lq vector-valued extensions, in a multidimensional setting, for negative powers of the operator related to so-called Laguerre expansions of Hermite type. On the other hand, we give necessary and sufficient conditions for vector-valued Lp-Lq estimates related to negative powers of the Laguerre operator associated to expansions of convolution type, in a one-dimensional setting. Both types of vector-valued inequalities are based on estimates of the kernel with precise control of the parameters involved. As an application, mixed norm estimates for fractional integrals related to the harmonic oscillator are deduced. © Instytut Matematyczny PAN, 2018.

Investigation of eigenvibrations of a loaded bar

The differential eigenvalue problem describing eigenvibrations of a bar with fixed ends and attached load at an interior point is investigated. This problem has an increasing sequence of positive simple eigenvalues with limit point at infinity. To the sequence of eigenvalues, there corresponds a complete orthonormal system of eigenfunctions. We formulate limit differential eigenvalue problems and prove the convergence of the eigenvalues and eigenfunctions of the initial problem to the corresponding eigenvalues and eigenfunctions of the limit problems as load mass tending to infinity. The original differential eigenvalue problem is approximated by the finite difference method on a uniform grid. Error estimates for approximate eigenvalues and eigenfunctions are established. Theoretical results are illustrated by numerical experiments for a model problem. Investigations of this paper can be generalized for the cases of more complicated and important problems on eigenvibrations of beams, plates and shells with attached loads.

Orthonormal spline systems on ℝ with zero means as bases in H 1(ℝ)

We present a simple geometric characterization of knot sequences for which the corresponding orthonormal spline system of arbitrary order r with zero means is a basis in the Hardy space H 1(ℝ).

Orthogonal polynomials in the cumulative Ord family and its application to variance bounds

ABSTRACTThis article presents and reviews several basic properties of the Cumulative Ord family of distributions; this family contains all the commonly used discrete distributions. A complete classification of the Ord family of probability mass functions is related to the orthogonality of the corresponding Rodrigues polynomials. Also, for any random variable X of this family and for any suitable function g in , the article provides useful relationships between the Fourier coefficients of g (with respect to the orthonormal polynomial system associated to X) and the Fourier coefficients of the forward difference of g (with respect to another system of polynomials, orthonormal with respect to another distribution of the system). Finally, using these properties, a class of bounds for the variance of is obtained, in terms of the forward differences of g. These bounds unify and improve several existing results.

Functional series representable as a sum of two universal series

Series with respect to systems Φ{φn(x)}n=1∞ of measurable and almost everywhere finite functions are discussed. A necessary and sufficient condition for representing any series with respect to a system Φ as a sum of two universal series is formulated. A consequence of the condition is that any series with respect to an arbitrary complete and orthonormal system Φ is a sum of two universal series.

Functions of the Lipschitz class and summability of Fourier series

In this paper we investigate summability of the general orthonormal systems (GOS) of the functions f(x) ∈ Lip 1.

Convergence of the Fourier Series of Lip 1 Functions with Respect to General Orthonormal Systems

We establish sufficient conditions that should be satisfied by functions of a general orthonormal system (ONS) {φ n (x)} in order that the Fourier series in this system be convergent almost everywhere on [0, 1] for any function from the class Lip 1. It is shown that the obtained conditions are best possible in a certain sense.

Quadrature finite element method for elliptic eigenvalue problems

A positive semi-definite eigenvalue problem for second-order self-adjoint elliptic differential operator definedon a bounded domain in the planewith smooth boundary and Dirichlet boundary condition is considered. This problem has a nondecreasing sequence of positive eigenvalues of finite multiplicity with a limit point at infinity. To the sequence of eigenvalues, there corresponds an orthonormal system of eigenfunctions. The original differential eigenvalue problem is approximated by the finite element method with numerical integration and Lagrange curved triangular finite elements of arbitrary order. Error estimates for approximate eigenvalues and eigenfunctions are established.

Orthogonal series estimates on strong spatial mixing data

We study a nonparametric regression model for sample data which is defined on an N-dimensional lattice structure and which is assumed to be strong spatial mixing: we use design adapted multidimensional Haar wavelets which form an orthonormal system w.r.t. the empirical measure of the sample data. For such orthonormal systems, we consider a nonparametric hard thresholding estimator. We give sufficient criteria for the consistency of this estimator and we derive rates of convergence. The theorems reveal that our estimator is able to adapt to the local smoothness of the underlying regression function and the design distribution. We illustrate our results with simulated examples.

Spectral cluster bounds for orthonormal systems and oscillatory integral operators in Schatten spaces

We generalize the Lp spectral cluster bounds of Sogge for the Laplace–Beltrami operator on compact Riemannian manifolds to systems of orthonormal functions. The optimality of these new bounds is also discussed. These spectral cluster bounds follow from Schatten-type bounds on oscillatory integral operators.

On the absolute convergence of Fourier series with respect to general orthonormal systems

Abstract The problems of absolute convergence of multiple Fourier series with respect to both general orthonormal systems and multiple Haar series are studied.

Mass spectrometry based metabolomics for in vitro systems pharmacology: pitfalls, challenges, and computational solutions

IntroductionMass spectrometry based metabolomics has become a promising complement and alternative to transcriptomics and proteomics in many fields including in vitro systems pharmacology. Despite several merits, metabolomics based on liquid chromatography mass spectrometry (LC–MS) is a developing area that is yet attached to several pitfalls and challenges. To reach a level of high reliability and robustness, these issues need to be tackled by implementation of refined experimental and computational protocols.ObjectivesThis study illustrates some key pitfalls in LC–MS based metabolomics and introduces an automated computational procedure to compensate for them.MethodNon-cancerous mammary gland derived cells were exposed to 27 chemicals from four pharmacological classes plus a set of six pesticides. Changes in the metabolome of cell lysates were assessed after 24 h using LC–MS. A data processing pipeline was established and evaluated to handle issues including contaminants, carry over effects, intensity decay and inherent methodology variability and biases. A key component in this pipeline is a latent variable method called OOS-DA (optimal orthonormal system for discriminant analysis), being theoretically more easily motivated than PLS-DA in this context, as it is rooted in pattern classification rather than regression modeling.ResultThe pipeline is shown to reduce experimental variability/biases and is used to confirm that LC–MS spectra hold drug class specific information.ConclusionLC–MS based metabolomics is a promising methodology, but comes with pitfalls and challenges. Key difficulties can be largely overcome by means of a computational procedure of the kind introduced and demonstrated here. The pipeline is freely available on www.github.com/stephanieherman/MS-data-processing.

On a Sidon-type inequality for discrete orthonormal systems

A Sidon-type inequality is proved for discrete orthonormal systems of special formwhose particular case is theWalsh system.