14,048 research outputs found
Polynomial systems supported on circuits and dessins d'enfants
We study polynomial systems whose equations have as common support a set C of
n+2 points in Z^n called a circuit. We find a bound on the number of real
solutions to such systems which depends on n, the dimension of the affine span
of the minimal affinely dependent subset of C, and the "rank modulo 2" of C. We
prove that this bound is sharp by drawing so-called dessins d'enfant on the
Riemann sphere. We also obtain that the maximal number of solutions with
positive coordinates to systems supported on circuits in Z^n is n+1, which is
very small comparatively to the bound given by the Khovanskii fewnomial
theorem.Comment: 19 pages, 5 figures, Section 3.1 revised, minor changes in other
section
String Theory, Loop Quantum Gravity and Eternalism
Eternalism, the view that what we regard locally as being located in the past, the present and the future equally exists, is the best ontological account of temporal existence in line with special and general relativity. However, special and general relativity are not fundamental theories and several research programs aim at finding a more fundamental theory of quantum gravity weaving together all we know from relativistic physics and quantum physics. Interestingly, some of these approaches assert that time is not fundamental. If time is not fundamental, what does it entail for eternalism and the standard debate over existence in time? First, I will argue that the non-fundamentality of time to be found in string theory entails standard eternalism. Second, I will argue that the non-fundamentality of time to be found in loop quantum gravity entails atemporal eternalism, namely a novel position in the spirit of standard eternalism
Priority Monism Beyond Spacetime
I will defend two claims. First, Schaffer's priority monism is in tension with many research programs in quantum gravity. Second, priority monism can be modified into a view more amenable to this physics. The first claim is grounded in the fact that promising approaches to quantum gravity such as loop quantum gravity or string theory deny the fundamental reality of spacetime. Since fundamental spacetime plays an important role in Schaffer's priority monism by being identified with the fundamental structure, namely the cosmos, the disappearance of spacetime in these views might undermine classical priority monism. My second claim is that priority monism can avoid this issue with two moves: first, in dropping one of its core assumptions, namely that the fundamental structure is spatio-temporal, second, by identifying the connection between the non-spatio-temporal structure and the derivative spatio-temporal structure with mereological composition
The No Self View and the Meaning of Life
Several philosophers, both in Buddhist and Western philosophy, claim that the self does not exist. The no-self view may, at first glance, appear to be a reason to believe that life is meaningless. In the present article, I argue indirectly in favor of the no-self view by showing that it does not entail that life is meaningless. I then examine Buddhism and argue, further, that the no-self view may even be construed as partially grounding an account of the meaning of life
The geometry of proper quaternion random variables
Second order circularity, also called properness, for complex random
variables is a well known and studied concept. In the case of quaternion random
variables, some extensions have been proposed, leading to applications in
quaternion signal processing (detection, filtering, estimation). Just like in
the complex case, circularity for a quaternion-valued random variable is
related to the symmetries of its probability density function. As a
consequence, properness of quaternion random variables should be defined with
respect to the most general isometries in , i.e. rotations from .
Based on this idea, we propose a new definition of properness, namely the
-properness, for quaternion random variables using invariance
property under the action of the rotation group . This new definition
generalizes previously introduced properness concepts for quaternion random
variables. A second order study is conducted and symmetry properties of the
covariance matrix of -proper quaternion random variables are
presented. Comparisons with previous definitions are given and simulations
illustrate in a geometric manner the newly introduced concept.Comment: 14 pages, 3 figure
Spacetime Emergence in Quantum Gravity: Functionalism and the Hard Problem
Spacetime functionalism is the view that spacetime is a functional structure implemented by a more fundamental ontology. Lam and Wüthrich have recently argued that spacetime functionalism helps to solve the epistemological problem of empirical coherence in quantum gravity and suggested that it also (dis)solves the hard problem of spacetime, namely the problem of offering a picture consistent with the emergence of spacetime from a non-spatio-temporal structure. First, I will deny that spacetime functionalism solves the hard problem by showing that it comes in various species, each entailing a different attitude towards, or answer to, the hard problem. Second, I will argue that the existence of an explanatory gap, which grounds the hard problem, has not been correctly taken into account in the literature
Euler Characteristic of real nondegenerate tropical complete intersections
We define nondegenerate tropical complete intersections imitating the
corresponding definition in complex algebraic geometry. As in the complex
situation, all nonzero intersection multiplicity numbers between tropical
hypersurfaces defining a nondegenerate tropical complete intersection are equal
to 1. The intersection multiplicity numbers we use are sums of mixed volumes of
polytopes which are dual to cells of the tropical hypersurfaces. We show that
the Euler characteristic of a real nondegenerate tropical complete intersection
depends only on the Newton polytopes of the tropical polynomials which define
the intersection. Basically, it is equal to the usual signature of a complex
complete intersection with same Newton polytopes, when this signature is
defined. The proof reduces to the toric hypersurface case, and uses the notion
of -polynomials of complex varieties.Comment: Version 1: slight revision of a preprint which appeared on our
webpages on April 2007, version 2: abstract expande
Descartes' Rule of Signs for Polynomial Systems supported on Circuits
We give a multivariate version of Descartes' rule of signs to bound the
number of positive real roots of a system of polynomial equations in n
variables with n+2 monomials, in terms of the sign variation of a sequence
associated both to the exponent vectors and the given coefficients. We show
that our bound is sharp and is related to the signature of the circuit.Comment: 25 pages, 3 figure
Topological types of real regular jacobian elliptic surfaces
We present the topological classification of real parts of real regular
elliptic surfaces with a real section.Comment: 17 pages, 7 figures, to appear in Geometriae Dedicat
Higher Order Statistsics of Stokes Parameters in a Random Birefringent Medium
We present a new model for the propagation of polarized light in a random
birefringent medium. This model is based on a decomposition of the higher order
statistics of the reduced Stokes parameters along the irreducible
representations of the rotation group. We show how this model allows a detailed
description of the propagation, giving analytical expressions for the
probability densities of the Mueller matrix and the Stokes vector throughout
the propagation. It also allows an exact description of the evolution of
averaged quantities, such as the degree of polarization. We will also discuss
how this model allows a generalization of the concepts of reduced Stokes
parameters and degree of polarization to higher order statistics. We give some
notes on how it can be extended to more general random media
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