Real coordinate space

In mathematics, the real coordinate space of dimension n, denoted Rn or \R^n, is the set of the n-tuples of real numbers, that is the set of all sequences of n real numbers. Special cases are called the real line R1 and the real coordinate plane R2. With component-wise addition and scalar multiplication, it is a real vector space, and its elements are called coordinate vectors. The coordinates over any basis of the elements of a real vector space form a real coordinate space of the same dimension as that of the vector space. Similarly, the Cartesian coordinates of the points of a Euclidean space of dimension n form a real coordinate space of dimension n. These one to one correspondences between vectors, points and coordinate vectors explain the names of coordinate space and coordinate vector. It allows using geometric terms and methods for studying real coordinate

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Conditional probabilities in the excursion set theory: generic barriers and non-Gaussian initial conditions

Andrea De Simone, Michele Maggiore

The excursion set theory, where density perturbations evolve stochastically with the smoothing scale, provides a method for computing the dark matter halo mass function. The computation of the mass function is mapped into the so-called first-passage time problem in the presence of a moving barrier. The excursion set theory is also a powerful formalism to study other properties of dark matter haloes such as halo bias, accretion rate, formation time, merging rate and the formation history of haloes. This is achieved by computing conditional probabilities with non-trivial initial conditions, and the conditional two-barrier first-crossing rate. In this paper we use the path integral formulation of the excursion set theory to calculate analytically these conditional probabilities in the presence of a generic moving barrier, including the one describing the ellipsoidal collapse, and for both Gaussian and non-Gaussian initial conditions. While most of our analysis associated with Gaussian initial conditions assumes Markovianity (top-hat in momentum space smoothing, rather than generic filters), the non-Markovianity of the random walks induced by non-Gaussianity is consistently accounted for. We compute, for a generic barrier, the first two scale-independent halo bias parameters, the conditional mass function and the halo formation time probability, including the effects of non-Gaussianities. We also provide the expression for the two-constant-barrier first-crossing rate when non-Markovian effects are induced by a top-hat filter function in real space.

Blackwell Publishing Ltd, Oxford2011

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Spin-1/2 kagome Heisenberg antiferromagnet with strong breathing anisotropy

Frédéric Mila, Seyed Saeed Soyouf Jahromi

We study the zero-temperature phase diagram of the spin-1/2 Heisenberg model with breathing anisotropy (i.e., with different coupling strength on the upward and downward triangles) on the kagome lattice. Our study relies on large scale tensor network simulations based on infinite projected entangled-pair state and infinite projected entangled-simplex state methods adapted to the kagome lattice. Our energy analysis suggests that the U(1) algebraic quantum spin-liquid (QSL) ground-state of the isotropic Heisenberg model is stable up to very large breathing anisotropy until it breaks down to a critical lattice-nematic phase that breaks rotational symmetry in real space through a first-order quantum phase transition. Our results also provide further insight into the recent experiment on vanadium oxyfluoride compounds which has been shown to be relevant platforms for realizing QSL in the presence of breathing anisotropy.

SCIPOST FOUNDATION2020

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The perception of potential: interference, dimensionality and knowledge

Leonardo Laurence Impett

This paper presents a system that investigates the sonification of wave interaction in a performance space and its interaction with a live performer – the illumination of sonic activity within a real space, in contrast to conventional ALife algorithmic, event- or material-based approaches. The model maintains three parallel representations of the entire live/virtual system: wavespace, symbol space and performance space. The cross-modal analysis and representation of behavior is important to the evolution of the system, which displays emergence on multiple levels of structure. Micro-evolution takes place within the population of wave-emitting and –listening agents. A higher level of structure emerges from their aggregate in interaction with the live performer, and a formal level as symbol space learns from the performer. Cross-modal representation is seen as a significant factor in the evolution of Western art music, in the development of multi-leveled structure and of work that affords many dimensions of engagement. We discuss the nature of knowledge produced through working with such systems and the role of the subject in ALife-generated knowledge. New models of simulation-derived knowledge are seen as important to cultural understanding.

2013

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Espace euclidien

En mathématiques, un espace euclidien est un objet algébrique permettant de généraliser de façon naturelle la géométrie traditionnelle développée par Euclide, dans ses Éléments. Une géométrie de cett

Espace vectoriel

vignette|Dans un espace vectoriel, on peut additionner deux vecteurs. Par exemple, la somme du vecteur v (en bleu) et w (en rouge) est v + w. On peut aussi multiplier un vecteur, comme le vecteur w qu

Variété topologique

En topologie, une variété topologique est un espace topologique, éventuellement séparé, assimilable localement à un espace euclidien. Les variétés topologiques constituent une classe importante des

MATH-123(b): Geometry

Ce cours donne une introduction à la géométrie des courbes et des surfaces.

PHYS-403: Computer simulation of physical systems I

The two main topics covered by this course are classical molecular dynamics and the Monte Carlo method.

PHYS-310: Solid state physics II

This course gives an introduction into Solid State Physics (crystal structure of materials, electronic and magnetic properties, thermal and electronic transport). The course material is at the level of Ashcroft & Mermin and is addressed to the 3rd year students in Physics.