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It has been widely discussed whether the existence of other universes than the known universe is a purely metaphysical consideration. This paper argues to the contrary that the existence of other universes is a meaningful hypothesis, which can be tested based on observable data. The paper applies the perspective of observer bias as an approach for assigning probabilities to the mutually excluding hypotheses of universe vs. multiverse, i.e., whether the known universe is the only universe, or just one universe in an ensemble of universes, a so-called multiverse. The basic idea of the paper relies on the following consideration. In a multiverse, the vast majority of observers would live in universes that are more life friendly. Thus, conditional on there being a multiverse, we should expect to find ourselves in a universe with values of the fundamental parameters that provide particularly fertile grounds for life. In contrast, if there is only a single universe, it may well be the case that there is still a few observers even if the parameters are far from ideal for life, and in that case, those observers will find values of the parameters that are not ideal for intelligent life. It may well be the case that, among all parameter configurations that allow life at least somewhere, those that are not ideal for life far outnumber those that are ideal for life. Based on this elementary consideration, the paper proceeds to propose a quantitative framework to determine probabilities for either hypothesis. In particular, it is described how a future ab initio determination of some of the factors in the Drake equation may be used to infer which one of the two hypotheses is the more likely. A quantitative approach to this end is proposed. Expressing these factors of the Drake equation in the context of the two hypotheses, a general likelihood approach is first described. Then, to support intuition, example computations are provided, illustrating how an actual hypothesis test would work in practice.

Cerebral oedema develops after anoxic brain injury. In two models of asphyxial and asystolic cardiac arrest without resuscitation, we found that oedema develops shortly after anoxia secondary to terminal depolarizations and the abnormal entry of CSF. Oedema severity correlated with the availability of CSF with the age-dependent increase in CSF volume worsening the severity of oedema. Oedema was identified primarily in brain regions bordering CSF compartments in mice and humans. The degree of ex vivo tissue swelling was predicted by an osmotic model suggesting that anoxic brain tissue possesses a high intrinsic osmotic potential. This osmotic process was temperature-dependent, proposing an additional mechanism for the beneficial effect of therapeutic hypothermia. These observations show that CSF is a primary source of oedema fluid in anoxic brain. This novel insight offers a mechanistic basis for the future development of alternative strategies to prevent cerebral oedema formation after cardiac arrest.

Supported by chl a satellite data in the North Atlantic (and phytoplankton division rates computed from that data), the disturbance-recovery hypothesis for the initiation of phytoplankton blooms posits that the change in chl a concentration is proportional to the relative change in the phytoplankton division rate. We used this hypothesis, introduced by Behrenfeld, as a principal model assumption and constructed a non-autonomous ordinary differential equation model for seasonally varying chl a concentrations. Our quantitative comparison between model simulations and in situ measurements of chl a and primary production collected from a Swedish fjord was 2-fold: first, using approximate Bayesian computations, we found distributions of values for the 3 model parameters that best described the chl a data. Then, we validated our model by comparing the simulated (not fitted) division rate to the division rate determined from the data. Our minimalistic model was able to capture (1) the yearly trend in the chl a concentration, (2) the pattern of growth and decline in the phytoplankton division rate, and (3) the decreasing trend in the relative change of the division rate exhibited in the data for several individual years. Moreover, the modeling efficiency was positive (between 0.3 and 0.9 with an average of 0.63) for all 11 yr included in this study. We conclude that the change in chl a concentration being proportional to the relative change in the division rate is a possible explanation for the bloom dynamics in the Gullmar fjord. In addition, our work provides a simple and empirically based differential equation for representing yearly dynamics of primary production, e.g. for generating ecological hypotheses using models of other trophic levels.

This report studies the biomechanics, in particular thin-film lubrication, of an artificial knee joint. We study whether using a porous (but still stress resilient) material, or simply adding to the current bearing design a large number of small drilled holes, makes it possible to entrap a lubricating fluid, in a way similar to what the living tissue does. The report concludes that such a design is indeed possible and may lead to less friction in low-load situation; but that the high-load situation is less clear. Also, the report considers only sliding movement of the bearing, not a combination of sliding and rolling.

During the last decade, there has been an increasing interest in the coupling between the acute inflammatory response and the Hypothalamic–Pituitary–Adrenal (HPA) axis. The inflammatory response is activated acutely by pathogen- or damage-related molecular patterns, whereas the HPA axis maintains a long-term level of the stress hormone cortisol which is also anti-inflammatory. A new integrated model of the interaction between these two subsystems of the inflammatory system is proposed and coined the integrated inflammatory stress (ITIS) model. The coupling mechanisms describing the interactions between the subsystems in the ITIS model are formulated based on biological reasoning and its ability to describe clinical data. The ITIS model is calibrated and validated by simulating various scenarios related to endotoxin (LPS) exposure. The model is capable of reproducing human data of tumor necrosis factor alpha, adrenocorticotropic hormone (ACTH) and cortisol and suggests that repeated LPS injections lead to a deficient response. The ITIS model predicts that the most extensive response to an LPS injection in ACTH and cortisol concentrations is observed in the early hours of the day. A constant activation results in elevated levels of the variables in the model while a prolonged change of the oscillations in ACTH and cortisol concentrations is the most pronounced result of different LPS doses predicted by the model.

Helicity plays a unique role as an integral invariant of a dynamical system. In this paper, the concept of helicity in the general setting of Hamiltonian dynamics is discussed. It is shown, through examples, how the conservation of overall helicity can imply a bound on other quantities of the motion in a nontrivial way.

Spreading edema after strokeThe brain is enveloped in a cushion of cerebrospinal fluid (CSF), which normally provides protection and helps to remove metabolic waste. CSF transport has also recently been shown to play unexpected roles in neurodegeneration and sleep. Mestreet al.used multimodal in vivo imaging in rodents and found that, after a stroke, an abnormally large volume of CSF rushes into the brain, causing swelling (see the Perspective by Moss and Williams). This influx of CSF is caused by constrictions of arteries triggered by a well-known propagating chemical reaction-diffusion wave called spreading depolarization. CSF transport can thus play a role in brain swelling after stroke.Science, this issue p.eaax7171; see also p.1195

Sestinas are poems of 39 lines comprising six verses of six lines each, and a three line final verse or ‘envoi’. The structure of the sestina is built around word repetition rather than strict rhyme. Each verse uses the same set line ending words, but in a permuted order. The form of the permutation is highly specific, and is equivalent to iteration of the tent map. This paper considers for which number N of verses, other than 6, can a sestina-like poem be formed. That is, which N will the prescribed permutation lead to a poem of N verses where no two verses have the same order of their end words. In so doing, a link is found between permutation groups, chaotic dynamics, and Cunningham numbers.

This paper presents a brief review of how industrial mathematics, inspired by the Oxford Study Group activity, organized itself in Europe, gave rise to the European Consortium for Mathematics in Industry, the series of European Study Groups with Industry, and to new modes of productive contacts between industry and applied mathematicians in academia.

Modeling Clinic for Industrial Mathematics (MODCLIM) is a Strategic Partnership for the Development of Training Workshops and Modeling Clinic for Industrial Mathematics, funded through the European Commission under the Erasmus Plus Program, Key Action 2: Cooperation for innovation and the exchange of good practices.

Modern society is increasingly faced with problems arising from overcrowded and congested motorways, and control of large crowds (sometimes failing, resulting in casualties, even loss of life). Over-engineering or oversizing available space is not always an option, and there is a need for mathematically sophisticated solutions arising from dynamical systems modelling.

We study the stability and dynamics of melting icebergs. Specifically, we address the ‘toppling’ or ‘rollover’ observed for floating icebergs. The rollover is thought to occur because the ocean melts the iceberg from below, causing its overall mass and mass distribution to change with time. We model the evolution of equilibrium positions for a general homogeneous body afloat in an ideal fluid, as this homogeneous body is subjected to ‘melting’, i.e. a slow removal of material from the submerged part. If this process is the dominating melting mechanism, can the likelihood of a toppling be inferred from observing only the above-surface part? We show here that some information about the evolution of stability due to melting can be inferred from the surface geometry of the iceberg.

We study finite metric spaces with elements picked from, and distances consistent with, ambient Riemannian manifolds. The concepts of negative type and strictly negative type are reviewed, and the conjecture that hyperbolic spaces are of strictly negative type is settled, in the affirmative. The technique of the proof is subsequently applied to show that every compact manifold of negative type must have trivial fundamental group, and to obtain a necessary criterion for product manifolds to be of negative type.

Around 1900, J. H. Jeans suggested that the “abnormal” specific heats observed in diatomic gases, specifically the lack of contribution to the heat capacity from the internal vibrational degrees of freedom, in apparent violation of the equipartition theorem, might be caused by the large separation between the time scale for the vibration and the time scale associated with a typical binary collision in the gas. We consider here a simple 1D model and show how, when these time scales are well separated, the collisional dynamics is constrained by a many- particle adiabatic invariant. The effect is that the collisional energy exchanges betgween the translational and the vibrational degrees of freedom are slowed down by an exponential factor (as Jeans conjectured). A metastable situation thus occurs, in which the fast vibrational degrees of freedom effectively do not contribute to the specific heat. Hence, the observed “freezing out” of the vibrational degrees of freedom could in principle be explained in terms of classical mechanics. We discuss the phenomenon analytically, on the basis of an approximation introduced by Landau and Teller (1936) for a related phenomenon, and estimate the time scale for the evolution to statistical equilibrium. The theoretical analysis is supported by numerical examples.

We prove that, if a finite metric space is of strictly negative type, then its transfinite diameter is uniquely realized by the infinite extender (load vector). Finite metric spaces that have this property include all spaces on two, three, or four points, all trees, and all finite subspaces of Euclidean spaces. We prove that, if the distance matrix is both hypermetric and regular, then it is of strictly negative type. We show that the strictly negative type finite subspaces of spheres are precisely those which do not contain two pairs of antipodal points. In connection with an open problem raised by Kelly, we conjecture that all finite subspaces of hyperbolic spaces are hypermetric and regular, and hence of strictly negative type.

Equation-free methods make possible an analysis of the evolution of a few coarse-grained or macroscopic quantities for a detailed and realistic model with a large number of fine-grained or microscopic variables, even though no equations are explicitly given on the macroscopic level. This will facilitate a study of how the model behavior depends on parameter values including an understanding of transitions between different types of qualitative behavior. These methods are introduced and explained for traffic jam formation and emergence of oscillatory pedestrian counter flow in a corridor with a narrow door.

Small oscillations of a rotating elasticum with a mass at the free end are investigated with Poincare-Lindstedt series. It is shown that the mass moves on a figure-eight shaped curve in a direction determined by the sign of the angular velocity and hence that the Coriolis force influences the motion.

Content: Classical mechanics: Calculus of variations, Lagrange’s equations, Symmetries and Noether’s theorem, Hamilton’s equations, cannonical transformations, integrable systems, pertubation theory. Symplectic integration: Numerical integrators, symplectic integrators, main theorem on symplectic integrators.

A recurrent problem in organic chemistry is the generation of new molecular structures that conform to some predetermined set of structural constraints that are imposed in an endeavor to build certain required properties into the newly generated structure. An example of this is the pharmacophore model, used in medicinal chemistry to guide de novo design or selection of suitable structures from compound databases. We propose here a method that efficiently links up a selected number of required atom positions while at the same time directing the emergent molecular skeleton to avoid forbidden positions. The linkage process takes place on a lattice whose unit step length and overall geometry is designed to match typical architectures of organic molecules. We use an optimization method to select from the many different graphs possible. The approach is demonstrated in an example where crystal structures of the same (in this case rigid) ligand complexed with different proteins are available.

The previous chapters have dealt with problems of translational motion. We now begin an investigation of motions where rotation is an important ingredient. We start by associating with one particle the so-called angular momentum, defined with respect to some reference point often taken to be at rest in an inertial frame. These considerations will then be extended to a system of particles. Forming the angular momentum for a system of particles with respect to the center of mass (CM), we shall see that the equation of motion for this angular momentum has a simple form, even if CM is accelerating.

In the previous chapters we have considered rotations of rigid bodies possessing a high degree of symmetry. We have seen that the rotational motion of such a body around an axis fixed in inertial space and fixed relative to the body is as simple to describe as the motion of a single particle moving along a straight line. The same equations govern the motion in these two cases, when p is substituted for L, and v is substituted for ω, etc.

As mentioned, we will here assume some knowledge of elementary mechanics and the concepts within that theory. As the reader progresses through the book, he or she will find these concepts treated in substantial detail.

In this chapter we investigate the motion of the center of mass for a system of particles, acted upon by external forces. As an important special case we shall consider the motion of the center of mass of the system of particles when no external forces act on the system.

This section contains substantial mathematical derivations. The results - not the derivations - are crucial to an insight into fundamental aspects of the Newtonian theory of motion. The main result can be expressed in a simple and elegant manner, and the physical content of this important result is quite transparent (see Equation (6.36)).

For a system of N charged particles moving in a homogeneous, sufficiently strong magnetic field, a many-particle adiabatic invariant constrains the collisional exchange of energy between the degrees of freedom perpendicular to and parallel to the magnetic field. A description of this phenomenon in terms of Hamilton dynamics is given. The relation to the Equipartition Theorem of statistical mechanics is briefly discussed.

Since 1998 European Study Groups have been held in Denmark, and Danish companies from LEGO and NOVO to very small high-tech firms have participated. Here, I briefly describe the history, the organization and the format of the Danish Study Groups, and highlight a few problem solutions. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Our own galaxy is called the Milky Way. The galaxy closest to us is the Andromeda Galaxy. Light from the Andromeda reaching us today initiated out there some 2 000 000 years ago. About this time man was at the beginning of his long journey through evolution.

In Newtonian mechanics, dynamical systems are described by differential equations which, if supplied with initial conditions, can be integrated to a unique solution at any future time t. A unique solution means that for each set of initial conditions there exists only one solution. A set of initial conditions determines exactly one solution. Dynamical systems having this property are called deterministic. See Example 12.8.

It was an important event in the history of physics when Sir Isaac Newton in 1687 published his book Philosophiae Naturalis Principia Mathematica.

Since the time of Galileo’s famous experiments with freely falling bodies it has been known that all material bodies, at the same place on Earth, fall with the same acceleration when air resistance is disregarded. Through many experiments this has become a well documented experimental fact.

This chapter contains nothing new. Through a series of examples we wish to illustrate the application of the basic theorems of mechanics. We start by reviewing classical mechanics in the form of ten “laws” or theorems.

Preface. Dynamical Zeta Functions V. Baladi. The Global Dynamics of Impact Oscillators C. Budd. Grazing in Impact Oscillators C. Budd. Topological Entropy of Unimodal Maps A. Douady. Henon Mappings in the Complex Domain J. H. Hubbard, R. W. Oberste-Vorth. Symbolic Dynamics, Group Automorphisms and Markov Partitions B. Kitchens. A Monotonicity Conjecture for Real Cubic Maps S. P. Dawson, R. Galeeva, J. Milnor, C. Tresser. Dynamics of Ordinary Differential Equations C. Sparrow. Real Bounds in Complex Dynamics S. van Strien. Homoclinic Bifurcations and Strange Attractors M. Viana. Introduction to Hyperbolic Dynamics J.-C. Yoccoz. Ergodic Theory of Differentiable Dynamical Systems Lai-Sang Young. Index.

The general treatment of the motion of a rigid body is rather involved and we shall in this book consider only certain special cases. The insights to be gained can be applied in virtually all of theoretical physics, and have important technological ramifications. Applications of the theorems derived stretch from studies of the spin of the electron and rotating atomic nuclei, to investigations of the motion of planets and galaxies.

As a preparation for later chapters we shall now review the description of the motion of the Earth relative to the fixed stars.

In this chapter we will try to summarize some of the ideas and opinions on which Newtonian physics rests. These views have been substantially revised in the 20th century, largely due to the works of Einstein in his special and general theory of relativity.

Publisher Summary This chapter discusses the influence of the invariant on the collisional relaxation of the electron velocity distribution in a strongly magnetized pure electron plasma. The results of molecular dynamics simulations are presented that corroborate the existence of the adiabatic invariant and the value of the calculated rate. The results of recent experiments (with cryogenic pure electron plasmas) that are in agreement with the theory are also described. In a study described in the chapter, the isolated collision of two electrons in a strong magnetic field was considered and the exponentially small exchange of energy between the parallel and the perpendicular degrees of freedom was calculated. The energy exchange was used to calculate the equipartition rate for a plasma where parallel temperature and perpendicular temperature were not equal. Boltzmann-like collision operator was employed, which treated collisions and separated binary interactions.