Mathematical Institute of the Slovak Academy of Sciences

Abstract For rectangular confidence regions for the mean values of multivariate normal distributions the following conjecture of 0. J. Dunn [3], [4] is proved: Such a confidence region constructed for the case of independent coordinates is, at the same time, a conservative confidence region for any case of dependent coordinates. This result is based on an inequality for the probabilities of rectangles in normal distributions, which permits one to factor out the probability for any single coordinate.

Abstract The Yang-Mills functional over a Riemann surface is studied from the point of view of Morse theory. The main result is that this is a ‘perfect' functional provided due account is taken of its gauge symmetry. This enables topological conclusions to be drawn about the critical sets and leads eventually to information about the moduli space of algebraic bundles over the Riemann surface. This in turn depends on the interplay between the holomorphic and unitary structures, which is analysed in detail.

Organometal trihalide perovskite based solar cells have exhibited the highest efficiencies to‐date when incorporated into mesostructured composites. However, thin solid films of a perovskite absorber should be capable of operating at the highest efficiency in a simple planar heterojunction configuration. Here, it is shown that film morphology is a critical issue in planar heterojunction CH 3 NH 3 PbI 3‐ x Cl x solar cells. The morphology is carefully controlled by varying processing conditions, and it is demonstrated that the highest photocurrents are attainable only with the highest perovskite surface coverages. With optimized solution based film formation, power conversion efficiencies of up to 11.4% are achieved, the first report of efficiencies above 10% in fully thin‐film solution processed perovskite solar cells with no mesoporous layer.

Abstract We present a self-contained account of the ideas of R. Penrose connecting four-dimensional Riemannian geometry with three-dimensional complex analysis. In particular we apply this to the self-dual Yang-Mills equations in Euclidean 4-space and compute the number of moduli for any compact gauge group. Results previously announced are treated with full detail and extended in a number of directions.

Abstract This paper is mainly concerned with a re-examination of the basic postulates and the consequent procedure for the construction of the constitutive equations of material behaviour in thermomechanics. However, the implication of the basic postulates and the significance of the related procedure for the development of the constitutive equations is also illustrated in some detail in the context of flow of heat in a rigid solid with particular reference to the propagation of thermal waves at finite speed. More specifically, after briefly examining the nature of the basic equations of motion for a system of particles within the scope of the classical newtonian mechanics, the basic postulates of the purely mechanical theory for a continuum (including its specialization for a rigid body) is re-examined. This includes some differences from the usual procedure on the subject. Next, thermal variables are introduced and after observing a useful analogy between the thermal and mechanical variables, a discussion of a theory of heat (or a purely thermal theory) is provided which differs from the usual development in the classical thermodynamics. A detailed application of the latter development is then made to the problem of heat flow in a stationary rigid solid using several different and well-motivated constitutive equations. Special cases of these include linearized theories of the classical heat flow by conduction and of heat flow transmitted as thermal waves. The remainder of the paper is concerned with thermal mechanical theory of deformable media along with discussions of a number of related issues on the subject.

Abstract The theory of quantum computational networks is the quantum generalization of the theory of logic circuits used in classical computing machines. Quantum gates are the generalization of classical logic gates. A single type of gate, the universal quantum gate, together with quantum ‘unit wires’, is adequate for constructing networks with any possible quantum computational property.

Csiszár and Körner's book is widely regarded as a classic in the field of information theory, providing deep insights and expert treatment of the key theoretical issues. It includes in-depth coverage of the mathematics of reliable information transmission, both in two-terminal and multi-terminal network scenarios. Updated and considerably expanded, this new edition presents unique discussions of information theoretic secrecy and of zero-error information theory, including the deep connections of the latter with extremal combinatorics. The presentations of all core subjects are self contained, even the advanced topics, which helps readers to understand the important connections between seemingly different problems. Finally, 320 end-of-chapter problems, together with helpful solving hints, allow readers to develop a full command of the mathematical techniques. It is an ideal resource for graduate students and researchers in electrical and electronic engineering, computer science and applied mathematics.

Abstract The last ten years have seen rapid advances in the understanding of differentiable four-manifolds not least of which has been the discovery of new 'exotic' manifolds. These results have had far-reaching consequences in geometry, topology, and mathematical physics and have proved to be a mainspring of current mathematical research. This book provides a lucid and accessible account to the modern study of the geometry of four-manifolds. Consequently, it will form required reading for all those mathematicians and theoretical physicists whose research touches on this topic. Prerequisites are a firm grounding in differential geometry and topology as might be gained from the first year of a graduate course. The authors present both a thorough treatment of the main lines of these developments in four-manifold topology - notably the definition of new invariants of four-manifolds - and also a wide-ranging treatment of relevant topics from geometry and global analysis. All of the main theorems about Yang-Mills instantons on four-manifolds are proved in detail. On the geometric side, the book contains a new proof of the classification of instantons on the four-sphere, together with an extensive discussion of the differential geometry of holomorphic vector bundles. At the end of the book the different strands of the theory are brought together in the proofs of results which settle long-standing problems in four-manifolds topology and which are close to the frontiers of current research.

On decrit une construction qui attribue une solution de l'equation de Korteweg-de Vries a chaque point d'un certain grassmannien de dimension infinie. On determine quelle classe on obtient par cette methode

Organic-inorganic perovskites such as CH3NH3PbI3 are promising materials for a variety of optoelectronic applications, with certified power conversion efficiencies in solar cells already exceeding 21%. Nevertheless, state-of-the-art films still contain performance-limiting non-radiative recombination sites and exhibit a range of complex dynamic phenomena under illumination that remain poorly understood. Here we use a unique combination of confocal photoluminescence (PL) microscopy and chemical imaging to correlate the local changes in photophysics with composition in CH3NH3PbI3 films under illumination. We demonstrate that the photo-induced 'brightening' of the perovskite PL can be attributed to an order-of-magnitude reduction in trap state density. By imaging the same regions with time-of-flight secondary-ion-mass spectrometry, we correlate this photobrightening with a net migration of iodine. Our work provides visual evidence for photo-induced halide migration in triiodide perovskites and reveals the complex interplay between charge carrier populations, electronic traps and mobile halides that collectively impact optoelectronic performance.

A well-known result of Schur [9] asserts that the diagonal elements (al,..., an) of annxn Hermitian matrix A satisfy a system of linear inequalities involving the eigenvalues (Xi,..., Xn). In geometric terms, regarding a and k as points in R " and allowing the symmetric group £ „ to act by permutation of coordinates, this result

In a series of papers W. F. Sheppard (1912, 1914) has considered the approximate representation of equidistant, equally weighted, and uncorrelated observations under the following assumptions:– (i) The data being u 1 , u 2 , …, u n , the representation is to be given by linear combinations (ii) The linear combinations are to be such as would reproduce any set of values that were already values of a polynomial of degree not higher than the k th. (iii) The sum of squared coefficients which measures the mean square error of y i , is to be a minimum for each value of i .

Abstract The book is written for students of mathematics and physics who have a basic knowledge of analysis and linear algebra. It can be used as a textbook for courses and/or seminars in functional analysis. Starting from metric spaces it proceeds quickly to the central results of the field, including the theorem of HahnBanach. The spaces (p Lp (X,(), C(X)' and Sobolov spaces are introduced. A chapter on spectral theory contains the Riesz theory of compact operators, basic facts on Banach and C*-algebras and the spectral representation for bounded normal and unbounded self-adjoint operators in Hilbert spaces. An introduction to locally convex spaces and their duality theory provides the basis for a comprehensive treatment of Fréchet spaces and their duals. In particular recent results on sequences spaces, linear topological invariants and short exact sequences of Fréchet spaces and the splitting of such sequences are presented. These results are not contained in any other book in this field.

Abstract The current state of the art in relativistic calculation of atomic structures is surveyed. The theory is modelled on the practice in non-relativistic calculations, using many-particle wave functions built from Dirac central field spinors. The Hamiltonian includes quantum electrodynamic effects in the form of the Breit approximation for the interaction energy of two electrons. Within the limits for which this is valid, it is possible to construct matrices for one- and two-particle operators and hence to perform atomic structure calculations which automatically include the major relativistic effects. The theory can be greatly simplified by using Racah's tensor operators. Major applications have utilized the Hartree or Hartree-Fock methods, and the relevant equations are formulated in detail. Numerical Hartree-Fock solutions for the average of the ground configuration have now been obtained for most elements with atomic number less than 103, and some solutions have also been obtained for ions. The same methods have been used to investigate the likely configurations of superheavy elements with atomic numbers as high as 168. A few calculations have also been done using a relativistic analogue of Roothaan's method. Methods of solution and numerical results are described for both types of calculation. No attempt has been made to describe the perturbation techniques due to Layzer and Bahcall, since an adequate review of recent work in this area has been given by Doyle. A brief description is also given of some problems in which it has been necessary to take relativistic effects into account including, in particular, elastic scattering of low and medium energy electrons from heavy elements, the calculation of x-ray scattering factors, and the probability of electron shake-off following sudden perturbations.

Huntington's disease (HD) is a neurodegenerative disorder caused by an unstable CAG repeat. For patients at risk, participating in predictive testing and learning of having CAG expansion, a major unanswered question shifts from "Will I get HD?" to "When will it manifest?" Using the largest cohort of HD patients analyzed to date (2913 individuals from 40 centers worldwide), we developed a parametric survival model based on CAG repeat length to predict the probability of neurological disease onset (based on motor neurological symptoms rather than psychiatric onset) at different ages for individual patients. We provide estimated probabilities of onset associated with CAG repeats between 36 and 56 for individuals of any age with narrow confidence intervals. For example, our model predicts a 91% chance that a 40-year-old individual with 42 repeats will have onset by the age of 65, with a 95% confidence interval from 90 to 93%. This model also defines the variability in HD onset that is not attributable to CAG length and provides information concerning CAG-related penetrance rates.

Recent Letters and Comments discuss the quantization of self-dual two-dimensional Lagrangeans which describe systems that are not explicitly canonical. We make some remarks on the most efficient method for exhibiting the canonical structure.

Monte Carlo methods are a very general and useful approach for the estimation of expectations arising from stochastic simulation. However, they can be computationally expensive, particularly when the cost of generating individual stochastic samples is very high, as in the case of stochastic PDEs. Multilevel Monte Carlo is a recently developed approach which greatly reduces the computational cost by performing most simulations with low accuracy at a correspondingly low cost, with relatively few simulations being performed at high accuracy and a high cost. In this article, we review the ideas behind the multilevel Monte Carlo method, and various recent generalizations and extensions, and discuss a number of applications which illustrate the flexibility and generality of the approach and the challenges in developing more efficient implementations with a faster rate of convergence of the multilevel correction variance.

The notion of multiple Wiener $i\prime ltegral$ was introduced first by N.

Abstract The geometric approach to quantization was introduced by Kostant and Souriau more than twenty years ago. It has given valuable and lasting insights into the relationship between classical and quantum systems. It continues to be a popular research topic. The ideas have proved useful in pure mathematics, notably in representation theory, as well as in theoretical physics. The most recent applications have been in conformal field theory and in the Jones-Witten theory of knots. The original edition of this book was published in 1980. For this edition it has been completely revised and extensively rewritten. The presentation has been simplified and a large number of examples have been added. The material on field theory has been expanded.

We explore the use of deep learning hierarchical models for problems in financial prediction and classification. Financial prediction problems – such as those presented in designing and pricing securities, constructing portfolios, and risk management – often involve large data sets with complex data interactions that currently are difficult or impossible to specify in a full economic model. Applying deep learning methods to these problems can produce more useful results than standard methods in finance. In particular, deep learning can detect and exploit interactions in the data that are, at least currently, invisible to any existing financial economic theory. Copyright © 2016 John Wiley & Sons, Ltd.