Angles Between Infinite Dimensional Subspaces with Applications to the Rayleigh-Ritz and Alternating Projectors Methods

We define angles from-to and between infinite dimensional subspaces of a Hilbert space, inspired by the work of E. J. Hannan, 1961/1962 for general canonical correlations of stochastic processes. The spectral theory of selfadjoint operators is used to investigate the properties of the angles, e.g., to establish connections between the angles corresponding to orthogonal complements. The classical gaps and angles of Dixmier and Friedrichs are characterized in terms of the angles. We introduce principal invariant subspaces and prove that they are connected by an isometry that appears in the polar decomposition of the product of corresponding orthogonal projectors. Point angles are defined by analogy with the point operator spectrum. We bound the Hausdorff distance between the sets of the squared cosines of the angles corresponding to the original subspaces and their perturbations. We show that the squared cosines of the angles from one subspace to another can be interpreted as Ritz values in the Rayleigh-Ritz method, where the former subspace serves as a trial subspace and the orthogonal projector of the latter subspace serves as an operator in the Rayleigh-Ritz method. The Hausdorff distance between the Ritz values, corresponding to different trial subspaces, is shown to be bounded by a constant times the gap between the trial subspaces. We prove a similar eigenvalue perturbation bound that involves the gap squared. Finally, we consider the classical alternating projectors method and propose its ultimate acceleration, using the conjugate gradient approach. The corresponding convergence rate estimate is obtained in terms of the angles. We illustrate a possible acceleration for the domain decomposition method with a small overlap for the 1D diffusion equation.Comment: 22 pages. Accepted to Journal of Functional Analysi

A Time-Dependent Dirichlet-Neumann Method for the Heat Equation

We present a waveform relaxation version of the Dirichlet-Neumann method for parabolic problem. Like the Dirichlet-Neumann method for steady problems, the method is based on a non-overlapping spatial domain decomposition, and the iteration involves subdomain solves with Dirichlet boundary conditions followed by subdomain solves with Neumann boundary conditions. However, each subdomain problem is now in space and time, and the interface conditions are also time-dependent. Using a Laplace transform argument, we show for the heat equation that when we consider finite time intervals, the Dirichlet-Neumann method converges, similar to the case of Schwarz waveform relaxation algorithms. The convergence rate depends on the length of the subdomains as well as the size of the time window. In this discussion, we only stick to the linear bound. We illustrate our results with numerical experiments.Comment: 9 pages, 5 figures, Lecture Notes in Computational Science and Engineering, Vol. 98, Springer-Verlag 201

Hvilke krav stiller likestillings- og diskrimineringsloven til likelønn?

Avhandlingen søker å undersøke hvilke plikter og rettigheter som ligger i likelønnsvernet innenfor norsk rett. Det tas utgangspunkt i det norske lovverket og de internasjonale regler som har betydning for rettstilstanden. Det undersøkes også hvordan lovverket håndheves i praksis. Hoveddelen av avhandlingen har en rettsdogmatisk tilnærming til likelønnsspørsmålet. Avslutningsvis kommer også en rettspolitisk drøftelse av temaet

Practical Attacks on NESHA-256

Abstract. NESHA-256 is a cryptographic hash function designed by Esmaeili et al. and presented at WCC \u2709. We show that NESHA-256 is highly insecure

Den hybride arbeidshverdagen - Betydningen av kontorløsning for jobbengasjement og organisasjonstilhørighet sett i lys av ansattes personlighetskarakteristika

Som en konsekvens av covid-19-pandemien ble store deler av norsk arbeidsliv satt til å arbeide fra hjemmekontor i mars 2020. Denne erfaringen har ført til at flere ønsker å benytte seg av en mer fleksibel kontorløsning, også etter pandemien. Den foreliggende studien undersøker hvordan ulike kontorløsninger, herunder hjemmekontor, hybridkontor og kontorplass hos arbeidsgiver er forbundet med jobbengasjement og organisasjonstilhørighet hos ansatte. Hybridkontor defineres som en løsning hvor ansatte arbeider noe på hjemmekontor og noe på kontorplass hos arbeidsgiver. I tillegg undersøker studien hvorvidt personlighetstrekkene ekstroversjon og nevrotisisme modererer sammenhengene mellom kontorløsning og engasjement, samt tilhørighet. Studien bruker jobbkarakteristikamodellen, konservering av ressursteori og person-job fit teorien for å forklare hvordan faktorene er relaterer til hverandre. Data ble samlet inn i to større, statlige, norske bedrifter ved bruk av spørreskjema og ble analysert videre ved bruk av SPSS. For de direkte sammenhengene ble det benyttet ANCOVA-analyser mens interaksjonsanalysene ble gjennomført i PROCESS Macro modell 2 for å undersøke hvorvidt personlighetstrekkene fungerer som moderatorer. Resultatene viser at ansatte som skårer høyt på nevrotisisme opplever mindre organisasjonstilhørighet på hjemmekontor sammenlignet med bruk av hybridkontor. Det ble ikke funnet sammenhenger mellom de andre undersøkte faktorene. Studien gir imidlertid et unikt innblikk i hvordan kontorløsninger påvirker de ansatte og gir verdifull informasjon som kan brukes ved utforming av «Den nye arbeidshverdagen».As a result of the COVID-19 pandemic, large parts of the Norwegian working force were required to work at home in March 2020. This experience seems to have led to a preference for a more flexible work solution also after the pandemic. This study investigates how different office solutions, including home office, hybrid office and employer's office, are associated with workers job engagement and organizational commitment. Hybrid office is defined as an office solution where the employee is working part of the time from home and part of the time from the employer's office. Furthermore, it examines how extraversion and neuroticism moderate the relationships between office solution and job engagement, as well as organizational commitment. The Job Characteristics Model, Conservation of Resources theory and person-job fit theory were used to explain the relationships in question. The data was collected with a questionnaire survey, in two larger Norwegian governmental organizations and furthermore analyzed using SPSS. The direct relationships were investigated using ANCOVA-analysis whereas interaction effects, where analyzed with PROCESS Macro model 2 to investigate whether the analyzed personality types acted as moderators. The results show that workers who score high on neuroticism experience less organizational commitment when working from home compared to a hybrid way of working. No other relationships were found regarding the other study variables. Nevertheless, this study provides a unique insight into how working solutions affect workers. This is valuable information that can be used when designing “The New Way of Working”.Masteroppgave i arbeids- og organisasjonspsykologiMAPSYK345MAPS-AOPMAPS-PSY

Finite Size Scaling of Domain Chaos

Numerical studies of the domain chaos state in a model of rotating Rayleigh-Benard convection suggest that finite size effects may account for the discrepancy between experimentally measured values of the correlation length and the predicted divergence near onset

Convergence Acceleration via Combined Nonlinear-Condensation Transformations

A method of numerically evaluating slowly convergent monotone series is described. First, we apply a condensation transformation due to Van Wijngaarden to the original series. This transforms the original monotone series into an alternating series. In the second step, the convergence of the transformed series is accelerated with the help of suitable nonlinear sequence transformations that are known to be particularly powerful for alternating series. Some theoretical aspects of our approach are discussed. The efficiency, numerical stability, and wide applicability of the combined nonlinear-condensation transformation is illustrated by a number of examples. We discuss the evaluation of special functions close to or on the boundary of the circle of convergence, even in the vicinity of singularities. We also consider a series of products of spherical Bessel functions, which serves as a model for partial wave expansions occurring in quantum electrodynamic bound state calculations.Comment: 24 pages, LaTeX, 12 tables (accepted for publication in Comput. Phys. Comm.

Prediction Properties of Aitken's Iterated Delta^2 Process, of Wynn's Epsilon Algorithm, and of Brezinski's Iterated Theta Algorithm

The prediction properties of Aitken's iterated Delta^2 process, Wynn's epsilon algorithm, and Brezinski's iterated theta algorithm for (formal) power series are analyzed. As a first step, the defining recursive schemes of these transformations are suitably rearranged in order to permit the derivation of accuracy-through-order relationships. On the basis of these relationships, the rational approximants can be rewritten as a partial sum plus an appropriate transformation term. A Taylor expansion of such a transformation term, which is a rational function and which can be computed recursively, produces the predictions for those coefficients of the (formal) power series which were not used for the computation of the corresponding rational approximant.Comment: 34 pages, LaTe

Spectral Theory for Perturbed Krein Laplacians in Nonsmooth Domains

We study spectral properties for $H_{K,\Omega}$, the Krein--von Neumann extension of the perturbed Laplacian $-\Delta+V$ defined on $C^\infty_0(\Omega)$, where $V$ is measurable, bounded and nonnegative, in a bounded open set $\Omega\subset\mathbb{R}^n$ belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class $C^{1,r}$, $r>1/2$. In particular, in the aforementioned context we establish the Weyl asymptotic formula #\{j\in\mathbb{N} | \lambda_{K,\Omega,j}\leq\lambda\} = (2\pi)^{-n} v_n |\Omega| \lambda^{n/2}+O\big(\lambda^{(n-(1/2))/2}\big) {as} \lambda\to\infty, where $v_n=\pi^{n/2}/ \Gamma((n/2)+1)$ denotes the volume of the unit ball in $\mathbb{R}^n$, and $\lambda_{K,\Omega,j}$, $j\in\mathbb{N}$, are the non-zero eigenvalues of $H_{K,\Omega}$, listed in increasing order according to their multiplicities. We prove this formula by showing that the perturbed Krein Laplacian (i.e., the Krein--von Neumann extension of $-\Delta+V$ defined on $C^\infty_0(\Omega)$) is spectrally equivalent to the buckling of a clamped plate problem, and using an abstract result of Kozlov from the mid 1980's. Our work builds on that of Grubb in the early 1980's, who has considered similar issues for elliptic operators in smooth domains, and shows that the question posed by Alonso and Simon in 1980 pertaining to the validity of the above Weyl asymptotic formula continues to have an affirmative answer in this nonsmooth setting.Comment: 60 page

A high-performance matrix-matrix multiplication methodology for CPU and GPU architectures

Current compilers cannot generate code that can compete with hand-tuned code in efficiency, even for a simple kernel like matrix–matrix multiplication (MMM). A key step in program optimization is the estimation of optimal values for parameters such as tile sizes and number of levels of tiling. The scheduling parameter values selection is a very difficult and time-consuming task, since parameter values depend on each other; this is why they are found by using searching methods and empirical techniques. To overcome this problem, the scheduling sub-problems must be optimized together, as one problem and not separately. In this paper, an MMM methodology is presented where the optimum scheduling parameters are found by decreasing the search space theoretically, while the major scheduling sub-problems are addressed together as one problem and not separately according to the hardware architecture parameters and input size; for different hardware architecture parameters and/or input sizes, a different implementation is produced. This is achieved by fully exploiting the software characteristics (e.g., data reuse) and hardware architecture parameters (e.g., data caches sizes and associativities), giving high-quality solutions and a smaller search space. This methodology refers to a wide range of CPU and GPU architectures