Distributed Lag Estimation When the Parameter Space is Explicitly Infinite- Dimensional

AbstractThe main purpose of this paper is to show that ideas of deformation theory can be applied to ‘infinite-dimensional geometry’. We develop the deformation theory of Brody curves. A Brody curve is a kind of holomorphic map from the complex plane to the projective space. Since the complex plane is not compact, the parameter space of the deformation can be infinite-dimensional. As an application we prove a lower bound on the mean dimension of the space of Brody curves.

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Robust Hyperbolic Chaos in Froude Pendulum with Delayed Feedback and Periodic Braking

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127419300350 ◽

2019 ◽

Vol 29 (12) ◽

pp. 1930035

Author(s):

Sergey P. Kuznetsov ◽

Yuliya V. Sedova

Keyword(s):

Parameter Space ◽

Invariant Subspaces ◽

Constant Angular Velocity ◽

Diagnostic Tools ◽

Small Perturbations ◽

Infinite Dimensional ◽

Dimensional State Space ◽

Froude Pendulum ◽

Numerical Research ◽

Hyperbolic Chaos

We indicate a possibility of implementing hyperbolic chaos using a Froude pendulum that is able to produce self-oscillations due to the suspension on a shaft rotating at constant angular velocity, in the presence of time-delay feedback and of periodic braking by the application of additional frictional force. We formulate a mathematical model and carry out its numerical research. In the parameter space we reveal areas of chaotic and regular dynamics using the analysis of Lyapunov exponents and some other diagnostic tools. It is shown that there are regions in the parameter space where the Poincaré stroboscopic map has an attractor, which is a kind of Smale–Williams solenoid embedded in the infinite-dimensional state space. We confirm the hyperbolicity of the attractor by numerical calculations including the analysis of angles of intersections of stable and unstable invariant subspaces of vectors of small perturbations for trajectories on the attractor and verify the absence of tangencies between these subspaces.

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Gluing an Infinite Number of Instantons

Nagoya Mathematical Journal ◽

10.1017/s0027763000009466 ◽

2007 ◽

Vol 188 ◽

pp. 107-131 ◽

Cited By ~ 4

Author(s):

Masaki Tsukamoto

Keyword(s):

Gauge Theory ◽

Parameter Space ◽

Infinite Number ◽

Moduli Spaces ◽

Dimensional Parameter ◽

Yang Mills ◽

Infinite Dimensional ◽

Alternating Method ◽

One Step ◽

Infinite Energy

AbstractThis paper is one step toward infinite energy gauge theory and the geometry of infinite dimensional moduli spaces. We generalize a gluing construction in the usual Yang-Mills gauge theory to an “infinite energy” situation. We show that we can glue an infinite number of instantons, and that the resulting ASD connections have infinite energy in general. Moreover they have an infinite dimensional parameter space. Our construction is a generalization of Donaldson’s “alternating method”.

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Conformal invariance of white noise

Nagoya Mathematical Journal ◽

10.1017/s0027763000021383 ◽

1985 ◽

Vol 98 ◽

pp. 87-98 ◽

Cited By ~ 11

Author(s):

Takeyuki Hida ◽

Ke-Seung Lee ◽

Sheu-San Lee

Keyword(s):

Probability Theory ◽

White Noise ◽

Parameter Space ◽

Conformal Invariance ◽

Rotation Group ◽

Gaussian Random Fields ◽

Conformal Group ◽

Time Parameter ◽

Infinite Dimensional ◽

Finite Dimensional

The remarkable link between the structure of the white noise and that of the infinite dimensional rotation group has been exemplified by various approaches in probability theory and harmonic analysis. Such a link naturally becomes more intricate as the dimension of the time-parameter space of the white noise increases. One of the powerful method to illustrate this situation is to observe the structure of certain subgroups of the infinite dimensional rotation group that come from the diffeomorphisms of the time-parameter space, that is the time change. Indeed, those subgroups would shed light on the probabilistic meanings hidden behind the usual formal observations. Moreover, the subgroups often describe the way of dependency for Gaussian random fields formed from the white noise as the time-parameter runs over the basic parameter space.The main purpose of this note is to introduce finite dimensional subgroups of the infinite dimensional rotation group that have important probabilistic meanings and to discuss their roles in probability theory. In particular, we shall see that the conformal invariance of white noise can be described in terms of the conformal group which is a finite dimensional Lie subgroup of the infinite dimensional rotation group.

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Making a Quantum Universe: Symmetry and Gravity

Universe ◽

10.3390/universe6110194 ◽

2020 ◽

Vol 6 (11) ◽

pp. 194

Author(s):

Houri Ziaeepour

Keyword(s):

Hilbert Space ◽

Parameter Space ◽

Quantum Dynamics ◽

Degrees Of Freedom ◽

Elementary Particles ◽

Infinite Dimensional ◽

Quantum Universe ◽

Theoretical Evidence ◽

Fundamental Forces ◽

The Universe

So far, none of attempts to quantize gravity has led to a satisfactory model that not only describe gravity in the realm of a quantum world, but also its relation to elementary particles and other fundamental forces. Here, we outline the preliminary results for a model of quantum universe, in which gravity is fundamentally and by construction quantic. The model is based on three well motivated assumptions with compelling observational and theoretical evidence: quantum mechanics is valid at all scales; quantum systems are described by their symmetries; universe has infinite independent degrees of freedom. The last assumption means that the Hilbert space of the Universe has SU(N→∞)≅areapreservingDiff.(S2) symmetry, which is parameterized by two angular variables. We show that, in the absence of a background spacetime, this Universe is trivial and static. Nonetheless, quantum fluctuations break the symmetry and divide the Universe to subsystems. When a subsystem is singled out as reference—observer—and another as clock, two more continuous parameters arise, which can be interpreted as distance and time. We identify the classical spacetime with parameter space of the Hilbert space of the Universe. Therefore, its quantization is meaningless. In this view, the Einstein equation presents the projection of quantum dynamics in the Hilbert space into its parameter space. Finite dimensional symmetries of elementary particles emerge as a consequence of symmetry breaking when the Universe is divided to subsystems/particles, without having any implication for the infinite dimensional symmetry and its associated interaction-percived as gravity. This explains why gravity is a universal force.

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