Spatiotemporal dynamics of continuum neural fields

We survey recent analytical approaches to studying the spatiotemporal dynamics of continuum neural fields. Neural fields model the large-scale dynamics of spatially structured biological neural networks in terms of nonlinear integrodifferential equations whose associated integral kernels represent the spatial distribution of neuronal synaptic connections. They provide an important example of spatially extended excitable systems with nonlocal interactions and exhibit a wide range of spatially coherent dynamics including traveling waves oscillations and Turing-like patterns

The Zero Bound on Interest Rates and Optimal Monetary Policy

This paper considers the consequences for monetary policy of the zero floor for nominal interest rates. The zero bound can be a significant constraint on the ability of a central bank to combat deflation. The paper shows, in the context of an intertemporal equilibrium model, that open-market operations, even of "unconventional" types, are ineffective if future policy is expected to be purely forward looking. However, a credible commitment to the right sort of history-dependent policy can largely mitigate the distortions created by the zero bound. In the model, optimal policy involves a commitment to adjust interest rates so as to achieve a time-varying price-level target, when this is consistent with the zero bound. The paper also discusses ways in which other central bank actions, although irrelevant apart from their effects on expectations, may help to make a central bank's commitment to its target more credible.macroeconomics, Zero Bound, Interest Rates, Optimal Monetary Policy

Higher uniformity of bounded multiplicative functions in short intervals on average

Let $\lambda$ denote the Liouville function. We show that, as $X \rightarrow \infty$, $$\int_{X}^{2X} \sup_{\substack{P(Y)\in \mathbb{R}[Y]\\ deg(P)\leq k}} \Big | \sum_{x \leq n \leq x + H} \lambda(n) e(-P(n)) \Big |\ dx = o ( X H)$$ for all fixed $k$ and $X^{\theta} \leq H \leq X$ with $0 < \theta < 1$ fixed but arbitrarily small. Previously this was only established for $k \leq 1$. We obtain this result as a special case of the corresponding statement for (non-pretentious) $1$-bounded multiplicative functions that we prove. In fact, we are able to replace the polynomial phases $e(-P(n))$ by degree $k$ nilsequences $\overline{F}(g(n) \Gamma)$. By the inverse theory for the Gowers norms this implies the higher order asymptotic uniformity result $$\int_{X}^{2X} \| \lambda \|_{U^{k+1}([x,x+H])}\ dx = o ( X )$$ in the same range of $H$. We present applications of this result to patterns of various types in the Liouville sequence. Firstly, we show that the number of sign patterns of the Liouville function is superpolynomial, making progress on a conjecture of Sarnak about the Liouville sequence having positive entropy. Secondly, we obtain cancellation in averages of $\lambda$ over short polynomial progressions $(n+P_1(m),\ldots, n+P_k(m))$, which in the case of linear polynomials yields a new averaged version of Chowla's conjecture. We are in fact able to prove our results on polynomial phases in the wider range $H\geq \exp((\log X)^{5/8+\varepsilon})$, thus strengthening also previous work on the Fourier uniformity of the Liouville function.Comment: 104 page

Wave Propagation in Materials for Modern Applications

In the recent decades, there has been a growing interest in micro- and nanotechnology. The advances in nanotechnology give rise to new applications and new types of materials with unique electromagnetic and mechanical properties. This book is devoted to the modern methods in electrodynamics and acoustics, which have been developed to describe wave propagation in these modern materials and nanodevices. The book consists of original works of leading scientists in the field of wave propagation who produced new theoretical and experimental methods in the research field and obtained new and important results. The first part of the book consists of chapters with general mathematical methods and approaches to the problem of wave propagation. A special attention is attracted to the advanced numerical methods fruitfully applied in the field of wave propagation. The second part of the book is devoted to the problems of wave propagation in newly developed metamaterials, micro- and nanostructures and porous media. In this part the interested reader will find important and fundamental results on electromagnetic wave propagation in media with negative refraction index and electromagnetic imaging in devices based on the materials. The third part of the book is devoted to the problems of wave propagation in elastic and piezoelectric media. In the fourth part, the works on the problems of wave propagation in plasma are collected. The fifth, sixth and seventh parts are devoted to the problems of wave propagation in media with chemical reactions, in nonlinear and disperse media, respectively. And finally, in the eighth part of the book some experimental methods in wave propagations are considered. It is necessary to emphasize that this book is not a textbook. It is important that the results combined in it are taken “from the desks of researchers“. Therefore, I am sure that in this book the interested and actively working readers (scientists, engineers and students) will find many interesting results and new ideas

Introduction to Vassiliev Knot Invariants

This book is a detailed introduction to the theory of finite type (Vassiliev) knot invariants, with a stress on its combinatorial aspects. It is intended to serve both as a textbook for readers with no or little background in this area, and as a guide to some of the more advanced material. Our aim is to lead the reader to understanding by means of pictures and calculations, and for this reason we often prefer to convey the idea of the proof on an instructive example rather than give a complete argument. While we have made an effort to make the text reasonably self-contained, an advanced reader is sometimes referred to the original papers for the technical details of the proofs. Version 3: some typos and inaccuracies are corrected.Comment: 512 pages, thousands picture

Computational photochemistry of heteroaromatic biomolecules : photodynamic therapy and ultrafast relaxation

This thesis focuses on the photochemistry of heteroaromatic biomolecules. These molecular systems have a rich photochemistry and take part in photochemical reactions that have many very topical applications. Small heteroaromatics constitute important biological building blocks and are therefore a fundamental components of living organisms. Even though these compounds absorb light very efficiently, they also have ultrafast relaxation processes available to them. This means that they can remove the absorbed energy very fast and avoid harmfull photoproducts forming, which can lead to cell damage. Larger heteroaromatics have a similarly efficient absorption of electromagnetic light, and are present in compounds that are responsible for the harvesting of energy in nature, for example the chlorophyll molecule in green plants and bacteria. If large heteroaromatics are artificially presented to living cells however, the excess energy absorbed by these systems may also cause cell damage. This destructive force can however be utilised in therapy forms where there is a need to get rid of unwanted cells, such as in anti-cancer therapy. A form of therapy based on this principle is photodynamic therapy. The use of computational chemistry in the investigations of photochemical phenomena has increased following the improvements in the efficiency of computers and algorithms. Modern techniques have now reached a stage where ultrafast relaxation processes can be calculated for small heteroaromatics. As the experimental community has also reached a stage where these compounds can be probed using ultrafast laser experiments, there is a need for computational input to aid in the interpretation of the data of these phenomena. This thesis will present computational results concerning the relaxation dynamics of important small heteroaromatic biomolecules, and discuss them in terms of experimental data collected by collaborative groups. For the development of molecules to be used in photodynamic therapy, a lot of work is needed to ensure safety for use in human beings. With the computational chemistry community now being able to carry out absorption studies for large heteroaromatics, computational structure-absorption relationships can aid the development of this form of therapy. At the limits of modern photochemistry, methods are also appearing that can be used for studies of ultrafast relaxation in larger systems. These computations could contribute hugely to the understanding of the behaviour of these types of systems and aid their development. In a large component of this thesis, new structure-absorption relationships are presented for interesting heteroaromatics with potential for use in photodynamic therapy. One section is also devoted to exploratory work using methods that have not before been used in systems that are larger in size, and presents some promising results as well as current challenges in the field

Precision Lattice Calculation of Kaon Decays with Möbius Domain Wall Fermions

We report our recent development in algorithms and progress in measurements in lattice QCD. The algorithmic development includes the forecasted force gradient integrator, and further theoretical development and implementation of the Möbius domain wall fermions. These new technologies make it practical to simulate large 48^3*96 and 64^3*128 lattice ensembles with (5.5fm)^3 boxes and 140MeV pion. The calculation was performed using the Möbius domain wall fermions and the Iwasaki gauge action. Simulated directly at physical quark masses, these ensembles are of great value for our ongoing and future lattice measurement projects.With the help of measurement techniques such as the eigCG algorithm and the all mode averaging method, we perform a direct, precise lattice calculation of the semileptonic kaon decay K→πlν using these newly generated high quality lattice ensembles. Our main result is the form factor f^+_{Kπ}(q^2) evaluated directly at zero momentum transfer q^2=0. Free of various systematic errors, this new result can be used to determine the CKM matrix element Vus to a very high precision when combined with experimental input. The calculation also provides results for various low energy strong interaction constants such as the pseudoscalar decay constants f_K and f_π, and the neutral kaon mixing matrix element B_K. These calculations are naturally performed by reusing the propagators calculated for the kaon semileptonic decay mentioned above. So they come with no or very low additional cost. The results allow us to also determine these important low energy constants on the lattice to unprecedented accuracy

Precision Lattice Calculation of Kaon Decays with Möbius Domain Wall Fermions

We report our recent development in algorithms and progress in measurements in lattice QCD. The algorithmic development includes the forecasted force gradient integrator, and further theoretical development and implementation of the Möbius domain wall fermions. These new technologies make it practical to simulate large 48^3*96 and 64^3*128 lattice ensembles with (5.5fm)^3 boxes and 140MeV pion. The calculation was performed using the Möbius domain wall fermions and the Iwasaki gauge action. Simulated directly at physical quark masses, these ensembles are of great value for our ongoing and future lattice measurement projects.With the help of measurement techniques such as the eigCG algorithm and the all mode averaging method, we perform a direct, precise lattice calculation of the semileptonic kaon decay K→πlν using these newly generated high quality lattice ensembles. Our main result is the form factor f^+_{Kπ}(q^2) evaluated directly at zero momentum transfer q^2=0. Free of various systematic errors, this new result can be used to determine the CKM matrix element Vus to a very high precision when combined with experimental input. The calculation also provides results for various low energy strong interaction constants such as the pseudoscalar decay constants f_K and f_π, and the neutral kaon mixing matrix element B_K. These calculations are naturally performed by reusing the propagators calculated for the kaon semileptonic decay mentioned above. So they come with no or very low additional cost. The results allow us to also determine these important low energy constants on the lattice to unprecedented accuracy