Spectral Manifolds Research Articles

Volcanic Micro-Earthquake Classification With Spectral Manifolds in Low-Dimensional Latent Spaces

Volcanic Micro-Earthquake Classification With Spectral Manifolds in Low-Dimensional Latent Spaces

Atrial Fibrillation Detection With Spectral Manifolds in Low-Dimensional Latent Spaces

Atrial Fibrillation Detection With Spectral Manifolds in Low-Dimensional Latent Spaces

The spectrum of delay differential equations with multiple hierarchical large delays

<p style='text-indent:20px;'>We prove that the spectrum of the linear delay differential equation <inline-formula><tex-math id="M1">\begin{document}$ x'(t) = A_{0}x(t)+A_{1}x(t-\tau_{1})+\ldots+A_{n}x(t-\tau_{n}) $\end{document}</tex-math></inline-formula> with multiple hierarchical large delays <inline-formula><tex-math id="M2">\begin{document}$ 1\ll\tau_{1}\ll\tau_{2}\ll\ldots\ll\tau_{n} $\end{document}</tex-math></inline-formula> splits into two distinct parts: the strong spectrum and the pseudo-continuous spectrum. As the delays tend to infinity, the strong spectrum converges to specific eigenvalues of <inline-formula><tex-math id="M3">\begin{document}$ A_{0} $\end{document}</tex-math></inline-formula>, the so-called asymptotic strong spectrum. Eigenvalues in the pseudo-continuous spectrum however, converge to the imaginary axis. We show that after rescaling, the pseudo-continuous spectrum exhibits a hierarchical structure corresponding to the time-scales <inline-formula><tex-math id="M4">\begin{document}$ \tau_{1}, \tau_{2}, \ldots, \tau_{n}. $\end{document}</tex-math></inline-formula> Each level of this hierarchy is approximated by spectral manifolds that can be easily computed. The set of spectral manifolds comprises the so-called asymptotic continuous spectrum. It is shown that the position of the asymptotic strong spectrum and asymptotic continuous spectrum with respect to the imaginary axis completely determines stability. In particular, a generic destabilization is mediated by the crossing of an <inline-formula><tex-math id="M5">\begin{document}$ n $\end{document}</tex-math></inline-formula>-dimensional spectral manifold corresponding to the timescale <inline-formula><tex-math id="M6">\begin{document}$ \tau_{n} $\end{document}</tex-math></inline-formula>.

Spectral (isotropic) manifolds and their dimension

A set of symmetric matrices whose ordered vector of eigenvalues belongs to a fixed set in Rn is called spectral or isotropic. In this paper, we establish that every locally symmetric submanifold M of Rn gives rise to a spectral manifold, for k ∈ {2, 3, . . . , ∞, ω}. An explicit formula for the dimension of the spectral manifold in terms of the dimension and the intrinsic properties of M is derived.

Lie Group Methods for Eigenvalue Function

By considering a C∞ structure on the ordered non-increasing of elements of Rn, we show that it is a differentiable manifold. By using of Lie groups, we show that eigenvalue function is a submersion. This fact is used to prove some results. These results is applied to prove a few facts about spectral manifolds and spectral functions. Orthogonal matrices act on the real symmetric matrices as a Lie transformation group. This fact, also, is used to prove the results.

Quanta of geometry: noncommutative aspects.

In the construction of spectral manifolds in noncommutative geometry, a higher degree Heisenberg commutation relation involving the Dirac operator and the Feynman slash of real scalar fields naturally appears and implies, by equality with the index formula, the quantization of the volume. We first show that this condition implies that the manifold decomposes into disconnected spheres, which will represent quanta of geometry. We then refine the condition by involving the real structure and two types of geometric quanta, and show that connected spin manifolds with large quantized volume are then obtained as solutions. The two algebras M_{2}(H) and M_{4}(C) are obtained, which are the exact constituents of the standard model. Using the two maps from M_{4} to S^{4} the four-manifold is built out of a very large number of the two kinds of spheres of Planckian volume. We give several physical applications of this scheme such as quantization of the cosmological constant, mimetic dark matter, and area quantization of black holes.

Geometry and the quantum: basics

Motivated by the construction of spectral manifolds in noncommutative geometry, we introduce a higher degree Heisenberg commutation relation involving the Dirac operator and the Feynman slash of scalar fields. This commutation relation appears in two versions, one sided and two sided. It implies the quantization of the volume. In the one-sided case it implies that the manifold decomposes into a disconnected sum of spheres which will represent quanta of geometry. The two sided version in dimension 4 predicts the two algebras M_2(H) and M_4(C) which are the algebraic constituents of the Standard Model of particle physics. This taken together with the non-commutative algebra of functions allows one to reconstruct, using the spectral action, the Lagrangian of gravity coupled with the Standard Model. We show that any connected Riemannian Spin 4-manifold with quantized volume >4 (in suitable units) appears as an irreducible representation of the two-sided commutation relations in dimension 4 and that these representations give a seductive model of the "particle picture" for a theory of quantum gravity in which both the Einstein geometric standpoint and the Standard Model emerge from Quantum Mechanics. Physical applications of this quantization scheme will follow in a separate publication.

High pressure luminescence spectra of CaMoO4:Pr3+

Steady state and time resolved luminescence measurements of CaMoO4 doped with Pr3+ as a function of hydrostatic pressure in the 1–175 kbar range are presented. It has been observed that with increasing pressure the spectral features shift towards lower energies, the decay times of both 3P0 and 1D2 emitting levels become shorter and the intensity of the 3P0 emission decreases to complete quenching at about 110 kbar, whereas that of the 1D2 emission increases in the 0–100 kbar range and then rapidly decreases when the pressure exceeds 127 kbar. A variation of the structure of the spectral manifolds indicates that a pressure induced phase transition of the host lattice occurs in the 80–100 kbar range. The quenching of the luminescence and the shortening of the decay times have been accounted for by means of a model that takes into account the role played by a praseodymium trapped exciton in the excited state dynamics of the investigated material.

Dichotomy spectra and Morse decompositions of linear nonautonomous differential equations

Recently, the existence of Morse decompositions for nonautonomous dynamical systems was shown for three different time domains: the past, the future and—in the linear case—the entire time. In this article, notions of exponential dichotomy are discussed with respect to the three time domains. It is shown that an exponential dichotomy gives rise to an attractor–repeller pair in the projective space, which is a building block of a Morse decomposition. Moreover, based on the notions of exponential dichotomy, dichotomy spectra are introduced, and it is proved that the corresponding spectral manifolds lead to Morse decompositions in the projective space.

A definition of spectrum for differential equations on finite time

Hyperbolicity of an autonomous rest point is characterised by its linearization not having eigenvalues on the imaginary axis. More generally, hyperbolicity of any solution which exists for all times can be defined by means of Lyapunov exponents or exponential dichotomies. We go one step further and introduce a meaningful notion of hyperbolicity for linear systems which are defined for finite time only, i.e. on a compact time interval. Hyperbolicity now describes the transient dynamics on that interval. In this framework, we provide a definition of finite-time spectrum, study its relations with classical concepts, and prove an analogue of the Sacker–Sell spectral theorem: For a d-dimensional system the spectrum is non-empty and consists of at most d disjoint (and often compact) intervals. An example illustrates that the corresponding spectral manifolds may not be unique, which in turn leads to several challenging questions.

Spectral gaps, inertial manifolds and kinematic dynamos

Inertial manifolds are desirable objects when ones wishes a dynamical process to behave asymptotically as a finite-dimensional ones. Recently [Physica D 194 (2004) 297] these manifolds are constructed for the kinematic dynamo problem with time-periodic velocity. It turns out, however, that the conditions imposed on the fluid velocity to guarantee the existence of inertial manifolds are too demanding, in the sense that they imply that all the solutions tend exponentially to zero. The inertial manifolds are meaningful because they represent different decay rates, but the classical dynamos where the magnetic field is maintained or grows are not covered by this approach, at least until more refined estimates are found.

Spectral barriers and inertial manifolds for time- discretized dissipative equations

The existence of inertial manifolds for dissipative nonlinear evolutionary equationsunder time discretizations of the semi-implicit Euler method is proved by using a generalized concept of spectral barrier for time-discretized equations.

DICHOTOMY SPECTRUM OF NONAUTONOMOUS LINEAR STOCHASTIC DIFFERENTIAL EQUATIONS

We investigate a concept of dichotomy spectrum for nonautonomous linear stochastic differential equations, which is defined with sample-wise exponential dichotomy of the two-parameter flow generated by the equation. We use random norm and cohomology to capture the nature of the stochastic nonuniformity. The main result is our spectral theorem stating that the dichotomy spectrum consists of compact random intervals with corresponding spectral manifolds, which are Oseledets spaces if the equation generates a random dynamical system. The dichotomy spectrum is nonrandom and equals the Lyapunov spectrum if the stochastic differential equation is Lyapunov regular.

Dichotomy Spectrum for Nonautonomous Differential Equations

For nonautonomous linear differential equations ⋅x=A(t) x with locally integrable A: R→R N×N the so-called dichotomy spectrum is investigated in this paper. As the closely related dichotomy spectrum for skew product flows with compact base (Sacker–Sell spectrum) our dichotomy spectrum for nonautonomous differential equations consists of at most N closed intervals, which in contrast to the Sacker–Sell spectrum may be unbounded. In the constant coefficients case these intervals reduce to the real parts of the eigenvalues of A. In any case the spectral intervals are associated with spectral manifolds comprising solutions with a common exponential growth rate. The main result of this paper is a spectral theorem which describes all possible forms of the dichotomy spectrum.

Spectroscopy of interchromophoric interactions in self-organized porphyrin and chlorin complexes

Spectral-luminescent properties of multimolecular complexes (triads and pentads, complexation constants range from 5·106 to 5·107 M−1) formed by two-fold coordination of dipyridyl and tetra-pyridyl substituted porphyrin or related molecules with Zn-porphyrin and Zn-chlorin dimers having various spacers between macrocycles (-CH2-CH2- or phenyl ring) have been studied in methylcyclohexane solutions in a temperature range from 140 to 360 K. The red shift of Zn-dimer electronic Q- and B-bands (Δv≤550 cm−1) upon complexation with pyridyl containing ligands is explained in terms of extra-ligation which influences the relative position for HOMO's a1u and a2u according to the four-orbital model. The splitting of Zn-dimer B-bands (ΔE≥600 cm−1) as well as the significant transformation of these bands in trimeric and pentameric complexes (redistribution of the absorption band intensities in Soret region and appearance of complicated splitted spectral manifolds Δv≤1900 cm−1) is connected with excitonic interactions of strong B-transitions of π-conjugated macrocycles included in the complex. Observed experimental splittings are compared with theoretical values calculated using the point dipole approximation and a computer-simulated geometry of the complexes under investigation. It has been observed that mutual influence of π-electronic macrocycles in the complexes leads to the red shift of porphyrin extra-ligand free base Q-bands (Δv≈120 cm−1) and is accompanied by quenching a fluorescence of certain components.

The Single-Valued Extension Property and Spectral Manifolds

We discuss the relation between the single-valued extension property (that is, Dunford’s property (A)) and spectral manifolds ${X_T}(F)$ of a bounded linear operator. In particular, we prove that Dunford’s property (C) implies the property (A). We also prove that if $T \in B(X)$ has the property $({\beta ^{\ast }})$ introduced by Fong, then $X_{{T^{\ast }}}^{\ast }(F) = {X_T}{(\mathbb {C}\backslash F)^ \bot }$ for every closed set $F$ in the complex plane $\mathbb {C}$.

The single-valued extension property and spectral manifolds

We discuss the relation between the single-valued extension property (that is, Dunford’s property (A)) and spectral manifolds X T ( F ) {X_T}(F) of a bounded linear operator. In particular, we prove that Dunford’s property (C) implies the property (A). We also prove that if T ∈ B ( X ) T \in B(X) has the property ( β ∗ ) ({\beta ^{\ast }}) introduced by Fong, then X T ∗ ∗ ( F ) = X T ( C ∖ F ) ⊥ X_{{T^{\ast }}}^{\ast }(F) = {X_T}{(\mathbb {C}\backslash F)^ \bot } for every closed set F F in the complex plane C \mathbb {C} .

Spectral Manifolds of Bounded S-Decomposable Operators

Spectral Manifolds of Bounded S-Decomposable Operators

Spectral manifolds of bounded 𝑆-decomposable operators

We prove some properties of spectral manifolds of a bounded S S -decomposable operator on a complex Banach space. Also, we prove a new characterization of S S -decomposability.

On the spectral manifolds of the simple unilateral shift and its adjoint

On the spectral manifolds of the simple unilateral shift and its adjoint