No full-text available
To read the full-text of this research,
you can request a copy directly from the author.
On the efficient computation of multidimensional singular sums
Abstract
This thesis is concerned with the development of the singular Euler–Maclaurin expansion, a novel method that allows for the efficient evaluation of large sums over values of functions with singularities. The method offers an approximation to the sum whose runtime is independent of the number of summands and whose error falls of exponentially with the expansion order. Hereby, a powerful numerical tool is provided whose applications range from fast multidimensional summation methods in numerical analysis over the analysis of long-range interactions in condensed matter systems to the evaluation of partition functions in quantum physics. The numerical performance of the new method is demonstrated by precisely computing the forces in a topological defect in a one-dimensional chain of long-range interacting particles. Furthermore, prototypical sums in an infinite two-dimensional lattice are efficiently evaluated. In the derivation of the multidimensional expansion, a deep connection between our new method to analytical number theory is revealed. This connection provides tools for the efficient computation of the operator coefficients that appear in the expansion. On the other hand, the expansion yields new globally convergent representations of the Riemann zeta function and its generalisation to higher dimensions.
... We show that the discrete lattice problem can be separated into a term that describes its continuous analog, the continuum contribution, and a term that includes all information about the microstructure, the lattice contribution, hence demonstrating equivalence between lattice and continuum. To this end, we apply the recently developed singular Euler-Maclaurin (SEM) expansion [13][14][15], which generalizes the 300-year old Euler-Maclaurin summation formula, and extend it to nonlinear and multiatomic systems. The singular lattice sum is expressed in terms of an integral and a lattice contribution described by a differential operator, both of which are efficiently computable. ...
... The representation of the lattice contribution in Eq. (4) is called the SEM expansion, a full derivation of which is provided in Refs. [13][14][15]. For = R d the SEM operator D ( ) takes the particularly simple form ...
... The Hadamard finite-part integral is the natural extension of the standard integral to functions that exhibit nonintegrable power-law singularities (see the original publication of Hadamard [135] or Ref. [26]). Here, we follow the notation of Ref. [15]. For ...
Continuum limits are a powerful tool in the study of many-body systems, yet their validity is often unclear when long-range interactions are present. In this paper, we rigorously address this issue and put forth an exact representation of long-range interacting lattices that separates the model into a term describing its continuous analog, the integral contribution, and a term that fully resolves the microstructure, the lattice contribution. For any system dimension, for any lattice, for any power-law interaction, and for linear, nonlinear, and multiatomic lattices, we show that the lattice contribution can be described by a differential operator based on the multidimensional generalization of the Riemann zeta function, namely, the Epstein zeta function. We employ our representation in Fourier space to solve the important problem of long-range interacting unconventional superconductors. We derive a generalized Bardeen-Cooper-Schrieffer gap equation and find emerging exotic phases in two-dimensional superconductors with topological phase transitions. Finally, we utilize nonequilibrium Higgs spectroscopy to analyze the impact of long-range interactions on the collective excitations of the condensate. We show that the interactions can be used to fine tune the Higgs mode's stability, ranging from exponential decay of the oscillation amplitude up to complete stabilization. By providing a unifying framework for long-range interactions on a lattice, both classical and quantum, our research can guide the search for exotic phases of matter across different fields.
... We show that the discrete lattice problem can be separated into a term that describes its continuous analog, the continuum contribution, and a term that includes all information about the microstructure, the lattice contribution, hence demonstrating equivalence between lattice and continuum. To this end, we apply the recently developed Singular Euler-Maclaurin (SEM) expansion [13][14][15], which generalizes the 300-year old Euler-Maclaurin summation formula, and extend it to nonlinear and multiatomic systems. The singular lattice sum is expressed in terms of an integral and a lattice contribution described by a differential operator, both of which are efficiently computable. ...
... with D a differential operator of infinite order, D ( ) its truncation up to order 2 + 1, and ∆ the Laplacian. The representation of the lattice contribution in Eq. (4) is called the Singular Euler-Maclaurin (SEM) expansion, a full derivation of which is provided in [13][14][15]. For Ω = R d the SEM operator D ( ) takes the particularly simple form ...
... The Hadamard finite-part integral is the natural extension of the standard integral to functions that exhibit non-integrable power-law singularities, see the original publication of Hadamard [134] or Ref. [20]. Here, we follow the notation of Ref. [15]. For ...
Continuum limits are a powerful tool in the study of many-body systems, yet their validity is often unclear when long-range interactions are present. In this work, we rigorously address this issue and put forth an exact representation of long-range interacting lattices that separates the model into a term describing its continuous analog, the integral contribution, and a term that fully resolves the microstructure, the lattice contribution. For any system dimension, any lattice, any power-law interaction, and for linear, nonlinear, and multi-atomic lattices, we show that the lattice contribution can be described by a differential operator based on the multidimensional generalization of the Riemann zeta function, namely the Epstein zeta function. We employ our representation in Fourier space to solve the important problem of long-range interacting unconventional superconductors. We derive a generalized Bardeen-Cooper-Schrieffer gap equation and find emerging exotic phases in two-dimensional superconductors with topological phase transitions. Finally, we utilize non-equilibrium Higgs spectroscopy to analyze the impact of long-range interactions on the collective excitations of the condensate. We show that the interactions can be used to fine-tune the Higgs mode's stability, ranging from exponential decay of the oscillation amplitude up to complete stabilization.
... We show that the discrete lattice problem can be separated into a term that describes its continuous analogue, the continuum contribution, and a term that includes all information about the microstructure, the lattice contribution, hence demonstrating an equivalence between lattice and continuum. To this end, we apply the recently developed Singular Euler-Maclaurin (SEM) expansion [13][14][15], which generalizes the 300-year old Euler-Maclaurin summation formula, and extend it to nonlinear and multiatomic systems. The singular lattice sum is expressed in terms of an integral and a lattice contribution described by a differential operator, both of which are efficiently computable. ...
... with D a differential operator of infinite order, D ( ) its truncation up to order 2 + 1, and ∆ the Laplacian. The representation of the lattice contribution in Eq. (4) is called the Singular Euler-Maclaurin (SEM) expansion, a full derivation of which is provided in [13][14][15]. For Ω = R d the SEM operator D ( ) takes the particularly simple form ...
... The Hadamard finite-part integral is the natural extension of the standard integral to functions that exhibit non-integrable power-law singularities, see the original publication of Hadamard [94] or Ref. [28]. Here, we follow the notation of Ref. [15]. For f x (y) = g(y) |y − x| ν , with g sufficiently differentiable, the Hadamard integral of f x over a domain Ω is defined by subtracting the Taylor series of g up to order Ω\Bε(x) f x (y) dy − H ν,ε g (x) , with the differential operator H ν,ε = kmax k=0 1 k! R d \Bε y, ∇ k |y| ν dy, ν ∈ C \ (N + d). ...
Continuum limits are a powerful tool in the study of many-body systems, yet their validity is often unclear when long-range interactions are present. In this work, we rigorously address this issue and put forth an exact representation of long-range interacting lattices that separates the model into a term describing its continuous analogue, the integral contribution, and a term that fully resolves the microstructure, the lattice contribution. For any system dimension, any lattice, any power-law interaction and for linear, nonlinear, and multi-atomic lattices, we show that the lattice contribution can be described by a differential operator based on the multidimensional generalization of the Riemann zeta function, namely the Epstein zeta function. We determine the conditions under which this contribution becomes particularly relevant, demonstrating the existence of quasi scale-invariant lattice contributions in a wide range of fundamental physical phenomena. Our representation provides a broad set of tools for studying the analytical properties of the system and it yields an efficient numerical method for the evaluation of the arising lattice sums. We benchmark its performance by computing classical forces and energies in three important physical examples, in which the standard continuum approximation fails: Skyrmions in a two-dimensional long-range interacting spin lattice, defects in ion chains, and spin waves in a three-dimensional pyrochlore lattice with dipolar interactions. We demonstrate that our method exhibits the accuracy of exact summation at the numerical cost of an integral approximation, allowing for precise simulations of long-range interacting systems even at macroscopic scales. Finally, we apply our analytical tool set to the study of quantum spin lattices and derive anomalous quantum spin wave dispersion relations due to long-range interactions in arbitrary dimensions.
This work discusses two periods of transformation of the Proto-Kartvelian population: the first (L–XXV centuries BC) when the entire population spoke one Proto-Kartvelian language and lived in a relatively large area; the second period – (XXV–X centuries BC), when the population divided into three parts: Proto-Svan; speaking the Colchian-Georgian language and the third part was
scattered on the European continent.
The second period is described by two different mathematical models: a part of the Proto-Kartvelian speaking population went to Europe and slowly began the process of their assimilation on the European continent. The unknown function that determines the number of Proto-Kartvelian speaking people in Europe at the time is described by a Pearl - Verhulst-type mathematical model with variable coefficients that also take the assimilation process into account. The analytical
solution of the Cauchy problem is found in quadratures.
The population that remained primarily in former Asia and the Caucasus region was gradually divided into two groups: those who spoke the Proto-Svan and those who spoke Colchian-Georgian languages. To describe their interference and development, a mathematical model is used, which is described by a nonlinear dynamic system with nonlinear terms of self-limitation and takes into
account the unnatural reduction of the Colchian-Georgian population as a result of hostilities with neighboring peoples. For a dynamic system without nonlinear terms of self-constraint, in the case of certain relationships between variable coefficients, the first integral was found, by means of which the Bernoulli equation with variable coefficients was obtained for one of the unknown functions. In the case of constant coefficients of the dynamic system, for certain dependences
between the coefficients, the dynamic system follows the system of Lotka–Volterra equations, with corresponding periodic solutions.
For the general mathematical model (nonlinear terms of self-limitation and unnatural reduction of the Colchian-Georgian population due to hostilities with neighboring peoples) in two cases of certain interdependencies between constant coefficients, it is shown that the divergence of an unknown vector-function in the physically meaningful first quarter of the phase plane changes the sign when passing through some half-direct one. Taking into account the principle (theorem) of Bendixson, theorems have been proved, on the variability of the divergence of the vector field and the existence of closed trajectories in some singly connected domain of the point located on this half-direct (starting point of the trajectory).
Thus, for the general dynamic system, with some dependencies between constant coefficients, it is shown that there is no assimilation of the Proto-Svan population by the Colchian-Georgian population and these two populations (Colchian-Georgian and Proto-Svan) coexist peacefully in virtually the same region due to the transformation of the Proto-Kartvelian population.
We extend the classical Euler–Maclaurin (EM) expansion to sums over multidimensional lattices that involve functions with algebraic singularities. This offers a tool for a precise and fast evaluation of singular sums that appear in multidimensional long-range interacting systems. We find that the approximation error decays exponentially with the expansion order for band-limited functions and that the runtime is independent of the number of particles. First, the EM summation formula is generalised to lattices in higher dimensions, assuming a sufficiently regular summand function. We then develop this new expansion further and construct the singular Euler–Maclaurin expansion in higher dimensions, an extension of our previous work in one dimension, which remains applicable and useful even if the summand function includes a singular function factor. We connect our method to analytical number theory and show that all operator coefficients can be efficiently computed from derivatives of the Epstein zeta function. Finally we demonstrate the numerical performance of the expansion and efficiently compute singular lattice sums in infinite two-dimensional lattices, which are of relevance in condensed matter, statistical, and quantum physics. An implementation in mathematica is provided online along with this article.
Complex topological configurations are fertile ground for exploring emergent phenomena and exotic phases in condensed-matter physics. For example, the recent discovery of polarization vortices and their associated complex-phase coexistence and response under applied electric fields in superlattices of (PbTiO3)n/(SrTiO3)n suggests the presence of a complex, multi-dimensional system capable of interesting physical responses, such as chirality, negative capacitance and large piezo-electric responses1–3. Here, by varying epitaxial constraints, we discover room-temperature polar-skyrmion bubbles in a lead titanate layer confined by strontium titanate layers, which are imaged by atomic-resolution scanning transmission electron microscopy. Phase-field modelling and second-principles calculations reveal that the polar-skyrmion bubbles have a skyrmion number of +1, and resonant soft-X-ray diffraction experiments show circular dichroism, confirming chirality. Such nanometre-scale polar-skyrmion bubbles are the electric analogues of magnetic skyrmions, and could contribute to the advancement of ferroelectrics towards functionalities incorporating emergent chirality and electrically controllable negative capacitance.
New q-series in the spirit of Jacobi have been found in a publication first published in 1884 written in Russian and translated into English in 1928. This work was found by chance and appears to be almost totally unknown. From these entirely new q-series, fresh lattice sums have been discovered and are presented here.
The Euler–Maclaurin (EM) summation formula is used in many theoretical studies and numerical calculations. It approximates the sum (Formula presented.) of values of a function f by a linear combination of a corresponding integral of f and values of its higher-order derivatives (Formula presented.). An alternative (Alt) summation formula is proposed, which approximates the sum by a linear combination of integrals only, without using high-order derivatives of f. Explicit and rather easy to use bounds on the remainder are given. Extensions to multi-index summation and to sums over lattice polytopes are indicated. Applications to summing possibly divergent series are presented. The Alt formula will in most cases outperform, or greatly outperform, the EM summation formula in terms of the execution time and memory use. One of the advantages of the Alt calculations is that, in contrast with the EM ones, they can be almost completely parallelized. Illustrative examples are given. In one of the examples, where an array of values of the Hurwitz generalized zeta function is computed with high accuracy, it is shown that both our implementation of the EM summation formula and, especially, the Alt formula perform much faster than the built-in Mathematica command HurwitzZeta[].
We give analytical results pertaining to the distributions of zeros of a class of sums which involve complex powers of the distance to points in a two-dimensional square lattice and trigonometric functions of their angle. Let C(0, 1; s) denote the product of the Riemann zeta function and the Catalan beta function, and let C(1, 4m; s) denote a particular set of angular sums. We then introduce a function that is the quotient of the angular lattice sums C(1, 4m; s) with C(0, 1; s), and use its properties to prove that C(1, 4m; s) obeys the Riemann hypothesis for any m if and only if C(0, 1; s) obeys the Riemann hypothesis. We furthermore prove that if the Riemann hypothesis holds, then C(1, 4m; s) and C(0, 1; s) have the same distribution of zeros on the critical line (in a sense made precise in the proof). We also show that if C(0, 1; s) obeys the Riemann hypothesis and all its zeros on the critical line have multiplicity one, then all the zeros of every C(1, 4m; s) have multiplicity one. We give numerical results illustrating these and other results.
Our understanding of the “long range” electrodynamic, electrostatic, and polar interactions that dominate the organization of small objects at separations beyond an interatomic bond length is reviewed. From this basic-forces perspective, a large number of systems are described from which one can learn about these organizing forces and how to modulate them. The many practical systems that harness these nanoscale forces are then surveyed. The survey reveals not only the promise of new devices and materials, but also the possibility of designing them more effectively.
We review the relation between Loop Quantum Gravity on a fixed graph and
discrete models of gravity. We compare Regge and twisted geometries, and
discuss discrete actions based on twisted geometries and on the discretization
of the Plebanski action. We discuss the role of discrete geometries in the spin
foam formalism, with particular attention to the definition of the simplicity
constraints.
We consider a general class of discrete nonlinear Schrödinger equations (DNLS) on the lattice hZ with mesh size h > 0. In the continuum limit when h → 0, we prove that the limiting dynamics are given by a nonlinear Schrödinger equation (NLS) on R with the fractional Laplacian (−Δ)[superscript α] as dispersive symbol. In particular, we obtain that fractional powers 1/2 < α < 1 arise from long-range lattice interactions when passing to the continuum limit, whereas the NLS with the usual Laplacian −Δ describes the dispersion in the continuum limit for short-range or quick-decaying interactions (e. g., nearest-neighbor interactions).
Our results rigorously justify certain NLS model equations with fractional Laplacians proposed in the physics literature. Moreover, the arguments given in our paper can be also applied to discuss the continuum limit for other lattice systems with long-range interactions.
We study a nonlocal Frenkel-Kontorova model that describes a one-dimensional chain of atoms moving in a periodic external potential and repulsing one another according to a long-range law, e.g., the power law ~x-n. The investigation is carried out both numerically and analytically in approximations of a weak or strong bond between atoms. Static characteristics of kinks (topological solitons) such as the effective mass, shape, and amplitude of the Peierls potential, the interaction energy of kinks, and the creation energy of kink-antikink pairs are calculated for different (exponential and power with n=1 and 3) laws of the interparticle interaction and various concentrations of atoms, i.e., ratio between the external potential period and the average spacing of atoms in the chain. The anharmonicity of the interaction potential between atoms is shown to result in differences between kink and antikink parameters, which are proportional to the value of the anharmonicity that rises with increasing exponent n of the interaction potential as well as at a changeover from a complex to a simpler unit cell. It is noted that at a power law of the interparticle repulsion this law describes also the asymptotics of the kink shape as well as the interaction energy of the kinks. Because of this, the dependence of, e.g., the amplitude of the Peierls potential versus the atom concentration, is similar to the ``devil's staircase.'' The applicability of the extended Frenkel-Kontorova model for describing diffusion characteristics of a quasi-one-dimensional layer adsorbed on a crystal surface is discussed.
Hamiltonians with inverse square interaction potential occur in the study of a variety of physical systems and exhibit a rich mathematical structure. In this talk we briefly mention some of the applications of such Hamiltonians and then analyze the case of the N-body rational Calogero model as an example. This model has recently been shown to admit novel solutions, whose properties are discussed.
The classical ground state of the Frenkel–Kontorova model is investigated for the case of a globally deformable substrate potential, whose amplitude is a function of the position of all particles. This system serves as an idealised description of a one-dimensional crystal of trapped ions in an optical cavity, where the deformable potential originates from the cavity back action due to the dynamical Stark shift. In the limit of a low relative substrate depth, a sine-Gordon field theory is obtained with coefficients that are functionals of the chain configuration. It is shown that the back action modifies the phase diagram, where the order and scaling of the phase-transition can be controlled by the cavity parameters. Due to the back action, the kinks, which constitute topological soliton solutions in the sine-Gordon model, are shown to exhibit two stable profiles with qualitatively different properties. The choice between the stable configurations is made collectively, all kinks taking either one form or the other.
The family of two-dimensional (2D) materials grows day by day, hugely expanding the scope of possible phenomena to be explored in two dimensions, as well as the possible van der Waals (vdW) heterostructures that one can create. Such 2D materials currently cover a vast range of properties. Until recently, this family has been missing one crucial member: 2D magnets. The situation has changed over the past 2 years with the introduction of a variety of atomically thin magnetic crystals. Here we will discuss the difference between magnetic states in 2D materials and in bulk crystals and present an overview of the 2D magnets that have been explored recently. We will focus on the case of the two most studied systems—semiconducting CrI3 and metallic Fe3GeTe2—and illustrate the physical phenomena that have been observed. Special attention will be given to the range of new van der Waals heterostructures that became possible with the appearance of 2D magnets, offering new perspectives in this rapidly expanding field. Until recently, the family of 2D materials was missing one crucial member — 2D magnets. This Review presents an overview of the 2D magnets studied so far and of the heterostructures that can be realized by combining them with other 2D materials.
Hierarchical matrices are an efficient framework for large-scale fully populated matrices arising, e.g., from the finite element discretization of solution operators of elliptic boundary value problems. In addition to storing such matrices, approximations of the usual matrix operations can be computed with logarithmic-linear complexity, which can be exploited to setup approximate preconditioners in an efficient and convenient way. Besides the algorithmic aspects of hierarchical matrices, the main aim of this book is to present their theoretical background. The book contains the existing approximation theory for elliptic problems including partial differential operators with nonsmooth coefficients. Furthermore, it presents in full detail the adaptive cross approximation method for the efficient treatment of integral operators with non-local kernel functions. The theory is supported by many numerical experiments from real applications.
This book provides a concrete introduction to quantum fields on a lattice: a precise and non-perturbative definition of quantum field theory obtained by replacing continuous space-time by a discrete set of points on a lattice. The path integral on the lattice is explained in concrete examples using weak and strong coupling expansions. Fundamental concepts such as 'triviality' of Higgs fields and confinement of quarks and gluons into hadrons are described and illustrated with the results of numerical simulations. The book also provides an introduction to chiral symmetry and chiral gauge theory, as well as quantized non-abelian gauge fields, scaling and universality. Based on the lecture notes of a course given by the author, this book contains many explanatory examples and exercises, and is suitable as a textbook for advanced undergraduate and graduate courses.
One-dimensional chains of laser-cooled ions have recently been confined in the fields of electromagnetic traps. This paper considers the minimum energy states of this one-dimensional (1D) form of condensed matter. Molecular dynamics simulations of the minimum energy states are compared to a density functional theory of the inhomogeneous crystal. Unlike 2D and 3D inhomogeneous Coulombic systems, where mean-field theory works well in describing the overall density variation on scales large compared to an interparticle spacing, we show that correlations are essential in determining the density of the 1D Coulomb chain. On the other hand, the long-range interactions that contribute to the mean field must also be kept.
Following the successful launch of the book on 'Photonic Crystals' in 2004 [1], this special issue of physica status solidi again comprises the work of research groups from the corresponding Priority Programme of the German Research Foundation and some of their close collaborators. Three years on, the understanding of photonic crystal properties has improved across numerous materials systems, applications were devised and developed approaching the industrial stage, and attention extended to the new areas of plasmonic crystals and metamaterials. The current issue presents a collection of Feature Articles reporting on the progress in this active field.
In the tradition of the first publication, the Cover Picture shows an overlay of a 3D photonic crystal made from macroporous silicon, a simulation of a waveguide splitter, and the lasing modes of a planar photonic crystals laser made from III–V-compound semiconductors (Picture courtesy of conImago, http://www.conimago.de). (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
In this paper we compare G(p), the Mellin transform (together with its analytic continuation), and , the related Hadamard finite-part integral of a function g(x), which decays exponentially at infinity and has specified singular behavior at the origin. Except when p is a nonpositive integer, these coincide. When p is a nonpositive integer, is well defined, but G(p) has a pole. We show that the terms in the Laurent expansion about this pole can be simply expressed in terms of the Hadamard
finite-part integral of a related function. This circumstance is exploited to provide a conceptually uniform proof of the
various generalizations of the Euler-Maclaurin expansion for the quadrature error functional.
Euler–Maclaurin formulas for a polytope express the sum of the values of a function over the lattice points in the polytope in terms of integrals of the function and its derivatives over faces of the polytope or its expansions. Exact Euler–Maclaurin formulas [A.G. Khovanskii, A.V. Pukhlikov, Algebra and Analysis 4 (1992) 188–216; S.E. Cappell, J.L. Shaneson, Bull. Amer. Math. Soc. 30 (1994) 62–69; C. R. Acad. Sci. Paris Sér. I Math. 321 (1995) 885–890; V. Guillemin, J. Differential Geom. 45 (1997) 53–73; M. Brion, M. Vergne, J. Amer. Math. Soc. 10 (2) (1997) 371–392] apply to exponential or polynomial functions; Euler–Maclaurin formulas with remainder [Y. Karshon, S. Sternberg, J. Weitsman, Proc. Natl. Acad. Sci. 100 (2) (2003) 426–433; Duke Math. J. 130 (3) (2005) 401–434] apply to more general smooth functions.In this paper we review these results and present proofs of the exact formulas obtained by these authors, using elementary methods. We then use an algebraic formalism due to Cappell and Shaneson to relate the different formulas.
The existence conditions of the zeta function of a pseudodifferential operator and the definition of determinant thereby obtained are reviewed, as well as the concept of multiplicative anomaly associated with the determinant and its calculation by means of the Wodzicki residue. Exponentially fast convergent formulas – valid in the whole of the complex plane and yielding the pole positions and residua – that extend the ones by Chowla and Selberg for the Epstein zeta function (quadratic form) and by Barnes (affine form) are then given. After briefly recalling the zeta function regularization procedure in quantum field theory, some applications of these expressions in physics are described.
The two-body potential of systems with long-range interactions decays at large distances as , with , where d is the space dimension. Examples are: gravitational systems, two-dimensional hydrodynamics, two-dimensional elasticity, charged and dipolar systems. Although such systems can be made extensive, they are intrinsically non additive. Moreover, the space of accessible macroscopic thermodynamic parameters might be non convex. The violation of these two basic properties is at the origin of ensemble inequivalence, which implies that specific heat can be negative in the microcanonical ensemble and temperature jumps can appear at microcanonical first order phase transitions. The lack of convexity implies that ergodicity may be generically broken. We present here a comprehensive review of the recent advances on the statistical mechanics and out-of-equilibrium dynamics of systems with long-range interactions. The core of the review consists in the detailed presentation of the concept of ensemble inequivalence, as exemplified by the exact solution, in the microcanonical and canonical ensembles, of mean-field type models. Relaxation towards thermodynamic equilibrium can be extremely slow and quasi-stationary states may be present. The understanding of such unusual relaxation process is obtained by the introduction of an appropriate kinetic theory based on the Vlasov equation. Comment: 118 pages, review paper, added references, slight change of content
Topological solitons occur in many nonlinear classical field theories. They are stable, particle-like objects, with finite mass and a smooth structure. Examples are monopoles and Skyrmions, Ginzburg-Landau vortices and sigma-model lumps, and Yang-Mills instantons. This book is a comprehensive survey of static topological solitons and their dynamical interactions. Particular emphasis is placed on the solitons which satisfy first-order Bogomolny equations. For these, the soliton dynamics can be investigated by finding the geodesics on the moduli space of static multi-soliton solutions. Remarkable scattering processes can be understood this way. The book starts with an introduction to classical field theory, and a survey of several mathematical techniques useful for understanding many types of topological soliton. Subsequent chapters explore key examples of solitons in one, two, three and four dimensions. The final chapter discusses the unstable sphaleron solutions which exist in several field theories.
A Planck-scale minimal observable length appears in many approaches to quantum gravity. It is sometimes argued that this minimal length might conflict with Lorentz invariance, because a boosted observer could see the minimal length further Lorentz contracted. We show that this is not the case within loop quantum gravity. In loop quantum gravity the minimal length (more precisely, minimal area) does not appear as a fixed property of geometry, but rather as the minimal (nonzero) eigenvalue of a quantum observable. The boosted observer can see the same observable spectrum, with the same minimal area. What changes continuously in the boost transformation is not the value of the minimal length: it is the probability distribution of seeing one or the other of the discrete eigenvalues of the area. We discuss several difficulties associated with boosts and area measurement in quantum gravity. We compute the transformation of the area operator under a local boost, propose an explicit expression for the generator of local boosts and give the conditions under which its action is unitary. Comment: 12 pages, 3 figures
An Elementary View of Euler's Summation Formula
T. M. Apostol. An Elementary View of Euler's Summation Formula.
The American Mathematical Monthly, 106(5):409-418, 1999.
Polyharmonic Functions. Oxford mathematical monographs
- N Aronszajn
- T M Creese
- L J Lipkin
N. Aronszajn, T. M. Creese, and L. J. Lipkin. Polyharmonic
Functions. Oxford mathematical monographs. Clarendon Press,
1983.
Physics of longrange interacting systems
- A Campa
- T Dauxois
- D Fanelli
- S Ruffo
A. Campa, T. Dauxois, D. Fanelli, and S. Ruffo. Physics of longrange interacting systems. OUP Oxford, 2014.
Zetafunktionen und elektrostatische Gitterpotentiale
- O Emersleben
O. Emersleben. Zetafunktionen und elektrostatische Gitterpotentiale. I. Phys. Z, 24:73-80, 1923.
- P Epstein
P. Epstein. Zur Theorie allgemeiner Zetafunktionen. II. Math.
Ann., 56:615-644, 1903.
- P Epstein
P. Epstein. Zur Theorie allgemeiner Zetafunktionen. II. Math.
Ann., 63:205-216, 1906.
- L Hörmander
L. Hörmander. The Analysis of Linear Partial Differential Operators II: Differential Operators with Constant Coefficients. Classics
in Mathematics. Springer, 2005.
- L Hörmander
L. Hörmander. The Analysis of Linear Partial Differential Operators III: Pseudo-Differential Operators. Classics in Mathematics.
Springer, 2007.