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We prove that if u is a locally Lipschitz continuous function on an open set chi subset of Rn + 1 satisfying the nonlinear heat equation partial derivative(t)u = Delta(vertical bar u vertical bar(p-1) u), p > 1, weakly away from the zero set u(-1) (0) in chi, then u is a weak solution to this equation in all of chi.
For n∈N , let Xn={a1,a2,…,an} be an n-element set and let F=(Xn;<f) be a fence, also called a zigzag poset. As usual, we denote by In the symmetric inverse semigroup on Xn. We say that a transformation α∈In is fence-preserving if x<fy implies that xα<fyα, for all x,y in the domain of α. In this paper, we study the semigroup PFIn of all partial fence-preserving injections of Xn and its subsemigroup IFn={α∈PFIn:α−1∈PFIn}. Clearly, IFn is an inverse semigroup and contains all regular elements of PFIn. We characterize the Green’s relations for the semigroup IFn. Further, we prove that the semigroup IFn is generated by its elements with rank ≥n−2. Moreover, for n∈2N, we find the least generating set and calculate the rank of IFn.
Abelian duality is realized naturally by combining differential cohomology and locally covariant quantum field theory. This leads to a -algebra of observables, which encompasses the simultaneous discretization of both magnetic and electric fluxes. We discuss the assignment of physically well-behaved states on this algebra and the properties of the associated GNS triple. We show that the algebra of observables factorizes as a suitable tensor product of three -algebras: the first factor encodes dynamical information, while the other two capture topological data corresponding to electric and magnetic fluxes. On the former factor and in the case of ultra-static globally hyperbolic spacetimes with compact Cauchy surfaces, we exhibit a state whose two-point correlation function has the same singular structure of a Hadamard state. Specifying suitable counterparts also on the topological factors, we obtain a state for the full theory, ultimately implementing Abelian duality transformations as Hilbert space isomorphisms.
From monthly mean observatory data spanning 1957-2014, geomagnetic field secular variation values were calculated by annual differences. Estimates of the spherical harmonic Gauss coefficients of the core field secular variation were then derived by applying a correlation based modelling. Finally, a Fourier transform was applied to the time series of the Gauss coefficients. This process led to reliable temporal spectra of the Gauss coefficients up to spherical harmonic degree 5 or 6, and down to periods as short as 1 or 2 years depending on the coefficient. We observed that a k(-2) slope, where k is the frequency, is an acceptable approximation for these spectra, with possibly an exception for the dipole field. The monthly estimates of the core field secular variation at the observatory sites also show that large and rapid variations of the latter happen. This is an indication that geomagnetic jerks are frequent phenomena and that significant secular variation signals at short time scales - i.e. less than 2 years, could still be extracted from data to reveal an unexplored part of the core dynamics.
We give a new and very short proof of a theorem of Greiner asserting that a positive and contractive -semigroup on an -space is strongly convergent in case it has a strictly positive fixed point and contains an integral operator. Our proof is a streamlined version of a much more general approach to the asymptotic theory of positive semigroups developed recently by the authors. Under the assumptions of Greiner's theorem, this approach becomes particularly elegant and simple. We also give an outlook on several generalisations of this result.
For an arbitrary euclidean field F we introduce a central extension (G(F), Phi) of SL(2, F) admitting a left-ordering and study its algebraic properties. The elements of G(F) are order preserving bijections of the convex hull of Q in F. If F = R then G(F) is isomorphic to the classical universal covering group of the Lie group SL(2, R). Among other results we show that G(F) is a perfect group which possesses a rank 1 cone of exceptional type. We also prove that its centre is an infinite cyclic group and investigate its normal subgroups.
The knowledge of the largest expected earthquake magnitude in a region is one of the key issues in probabilistic seismic hazard calculations and the estimation of worst-case scenarios. Earthquake catalogues are the most informative source of information for the inference of earthquake magnitudes. We analysed the earthquake catalogue for Central Asia with respect to the largest expected magnitudes m(T) in a pre-defined time horizon T-f using a recently developed statistical methodology, extended by the explicit probabilistic consideration of magnitude errors. For this aim, we assumed broad error distributions for historical events, whereas the magnitudes of recently recorded instrumental earthquakes had smaller errors. The results indicate high probabilities for the occurrence of large events (M >= 8), even in short time intervals of a few decades. The expected magnitudes relative to the assumed maximum possible magnitude are generally higher for intermediate-depth earthquakes (51-300 km) than for shallow events (0-50 km). For long future time horizons, for example, a few hundred years, earthquakes with M >= 8.5 have to be taken into account, although, apart from the 1889 Chilik earthquake, it is probable that no such event occurred during the observation period of the catalogue.
This paper concerns integral varifolds of arbitrary dimension in an open subset of Euclidean space satisfying integrability conditions on their first variation. Firstly, the study of pointwise power decay rates almost everywhere of the quadratic tilt-excess is completed by establishing the precise decay rate for two-dimensional integral varifolds of locally bounded first variation. In order to obtain the exact decay rate, a coercive estimate involving a height-excess quantity measured in Orlicz spaces is established. Moreover, counter-examples to pointwise power decay rates almost everywhere of the super-quadratic tilt-excess are obtained. These examples are optimal in terms of the dimension of the varifold and the exponent of the integrability condition in most cases, for example if the varifold is not two-dimensional. These examples also demonstrate that within the scale of Lebesgue spaces no local higher integrability of the second fundamental form, of an at least two-dimensional curvature varifold, may be deduced from boundedness of its generalised mean curvature vector. Amongst the tools are Cartesian products of curvature varifolds.
This paper is concerned with the filtering problem in continuous time. Three algorithmic solution approaches for this problem are reviewed: (i) the classical Kalman-Bucy filter, which provides an exact solution for the linear Gaussian problem; (ii) the ensemble Kalman-Bucy filter (EnKBF), which is an approximate filter and represents an extension of the Kalman-Bucy filter to nonlinear problems; and (iii) the feedback particle filter (FPF), which represents an extension of the EnKBF and furthermore provides for a consistent solution in the general nonlinear, non-Gaussian case. The common feature of the three algorithms is the gain times error formula to implement the update step (to account for conditioning due to the observations) in the filter. In contrast to the commonly used sequential Monte Carlo methods, the EnKBF and FPF avoid the resampling of the particles in the importance sampling update step. Moreover, the feedback control structure provides for error correction potentially leading to smaller simulation variance and improved stability properties. The paper also discusses the issue of nonuniqueness of the filter update formula and formulates a novel approximation algorithm based on ideas from optimal transport and coupling of measures. Performance of this and other algorithms is illustrated for a numerical example.
Particle filters (also called sequential Monte Carlo methods) are widely used for state and parameter estimation problems in the context of nonlinear evolution equations. The recently proposed ensemble transform particle filter (ETPF) [S. Reich, SIAM T. Sci. Comput., 35, (2013), pp. A2013-A2014[ replaces the resampling step of a standard particle filter by a linear transformation which allows for a hybridization of particle filters with ensemble Kalman filters and renders the resulting hybrid filters applicable to spatially extended systems. However, the linear transformation step is computationally expensive and leads to an underestimation of the ensemble spread for small and moderate ensemble sizes. Here we address both of these shortcomings by developing second order accurate extensions of the ETPF. These extensions allow one in particular to replace the exact solution of a linear transport problem by its Sinkhorn approximation. It is also demonstrated that the nonlinear ensemble transform filter arises as a special case of our general framework. We illustrate the performance of the second-order accurate filters for the chaotic Lorenz-63 and Lorenz-96 models and a dynamic scene-viewing model. The numerical results for the Lorenz-63 and Lorenz-96 models demonstrate that significant accuracy improvements can be achieved in comparison to a standard ensemble Kalman filter and the ETPF for small to moderate ensemble sizes. The numerical results for the scene-viewing model reveal, on the other hand, that second-order corrections can lead to statistically inconsistent samples from the posterior parameter distribution.
We prove that the Atiyah–Singer Dirac operator in L2 depends Riesz continuously on L∞ perturbations of complete metrics g on a smooth manifold. The Lipschitz bound for the map depends on bounds on Ricci curvature and its first derivatives as well as a lower bound on injectivity radius. Our proof uses harmonic analysis techniques related to Calderón’s first commutator and the Kato square root problem. We also show perturbation results for more general functions of general Dirac-type operators on vector bundles.
We consider the Cauchy problem for the heat equation in a cylinder C (T) = X x (0, T) over a domain X in R (n) , with data on a strip lying on the lateral surface. The strip is of the form S x (0, T), where S is an open subset of the boundary of X. The problem is ill-posed. Under natural restrictions on the configuration of S, we derive an explicit formula for solutions of this problem.
We study biased Maker-Breaker positional games between two players, one of whom is playing randomly against an opponent with an optimal strategy. In this work we focus on the case of Breaker playing randomly and Maker being "clever". The reverse scenario is treated in a separate paper. We determine the sharp threshold bias of classical games played on the edge set of the complete graph , such as connectivity, perfect matching, Hamiltonicity, and minimum degree-1 and -2. In all of these games, the threshold is equal to the trivial upper bound implied by the number of edges needed for Maker to occupy a winning set. Moreover, we show that CleverMaker can not only win against asymptotically optimal bias, but can do so very fast, wasting only logarithmically many moves (while the winning set sizes are linear in n).
We equip the space of lattice cones with a coproduct which makes it a cograded, coaugmented, connnected coalgebra. The exponential generating sum and exponential generating integral on lattice cones can be viewed as linear maps on this space with values in the space of meromorphic germs with linear poles at zero. We investigate the subdivision properties-reminiscent of the inclusion-exclusion principle for the cardinal on finite sets-of such linear maps and show that these properties are compatible with the convolution quotient of maps on the coalgebra. Implementing the algebraic Birkhoff factorization procedure on the linear maps under consideration, we factorize the exponential generating sum as a convolution quotient of two maps, with each of the maps in the factorization satisfying a subdivision property. A direct computation shows that the polar decomposition of the exponential generating sum on a smooth lattice cone yields an Euler-Maclaurin formula. The compatibility with subdivisions of the convolution quotient arising in the algebraic Birkhoff factorization then yields the Euler-Maclaurin formula for any lattice cone. This provides a simple formula for the interpolating factor by means of a projection formula.
In this paper we present a Bayesian framework for interpolating data in a reproducing kernel Hilbert space associated with a random subdivision scheme, where not only approximations of the values of a function at some missing points can be obtained, but also uncertainty estimates for such predicted values. This random scheme generalizes the usual subdivision by taking into account, at each level, some uncertainty given in terms of suitably scaled noise sequences of i.i.d. Gaussian random variables with zero mean and given variance, and generating, in the limit, a Gaussian process whose correlation structure is characterized and used for computing realizations of the conditional posterior distribution. The hierarchical nature of the procedure may be exploited to reduce the computational cost compared to standard techniques in the case where many prediction points need to be considered.
Ancient genomes have revolutionized our understanding of Holocene prehistory and, particularly, the Neolithic transition in western Eurasia. In contrast, East Asia has so far received little attention, despite representing a core region at which the Neolithic transition took place independently ~3 millennia after its onset in the Near East. We report genome-wide data from two hunter-gatherers from Devil’s Gate, an early Neolithic cave site (dated to ~7.7 thousand years ago) located in East Asia, on the border between Russia and Korea. Both of these individuals are genetically most similar to geographically close modern populations from the Amur Basin, all speaking Tungusic languages, and, in particular, to the Ulchi. The similarity to nearby modern populations and the low levels of additional genetic material in the Ulchi imply a high level of genetic continuity in this region during the Holocene, a pattern that markedly contrasts with that reported for Europe.
We show a connection between the CDE′ inequality introduced in Horn et al. (Volume doubling, Poincaré inequality and Gaussian heat kernel estimate for nonnegative curvature graphs. arXiv:1411.5087v2, 2014) and the CDψ inequality established in Münch (Li–Yau inequality on finite graphs via non-linear curvature dimension conditions. arXiv:1412.3340v1, 2014). In particular, we introduce a CDφψ inequality as a slight generalization of CDψ which turns out to be equivalent to CDE′ with appropriate choices of φ and ψ. We use this to prove that the CDE′ inequality implies the classical CD inequality on graphs, and that the CDE′ inequality with curvature bound zero holds on Ricci-flat graphs.
In this note, we consider the semigroup O(X) of all order endomorphisms of an infinite chain X and the subset J of O(X) of all transformations alpha such that vertical bar Im(alpha)vertical bar = vertical bar X vertical bar. For an infinite countable chain X, we give a necessary and sufficient condition on X for O(X) = < J > to hold. We also present a sufficient condition on X for O(X) = < J > to hold, for an arbitrary infinite chain X.
This article assesses the distance between the laws of stochastic differential equations with multiplicative Levy noise on path space in terms of their characteristics. The notion of transportation distance on the set of Levy kernels introduced by Kosenkova and Kulik yields a natural and statistically tractable upper bound on the noise sensitivity. This extends recent results for the additive case in terms of coupling distances to the multiplicative case. The strength of this notion is shown in a statistical implementation for simulations and the example of a benchmark time series in paleoclimate.