Search results for "eigenvalues"

Musical pitch quantization as an eigenvalue problem

2020

How can discrete pitches and chords emerge from the continuum of sound? Using a quantum cognition model of tonal music, we prove that the associated Schrödinger equation in Fourier space is invariant under continuous pitch transpositions. However, this symmetry is broken in the case of transpositions of chords, entailing a discrete cyclic group as transposition symmetry. Our research relates quantum mechanics with music and is consistent with music theory and seminal insights by Hermann von Helmholtz.

PT Symmetry and Weyl Asymptotics

2012

For a class of PT-symmetric operators with small random perturbations, the eigenvalues obey Weyl asymptotics with probability close to 1. Consequently, when the principal symbol is nonreal, there are many nonreal eigenvalues.

Complex powers and non-compact manifolds

2002

We study the complex powers $A^{z}$ of an elliptic, strictly positive pseudodifferential operator $A$ using an axiomatic method that combines the approaches of Guillemin and Seeley. In particular, we introduce a class of algebras, ``extended Weyl algebras,'' whose definition was inspired by Guillemin's paper on the subject. An extended Weyl algebra can be thought of as an algebra of ``abstract pseudodifferential operators.'' Many algebras of pseudodifferential operators are extended Weyl algebras. Several results typical for algebras of pseudodifferential operators (asymptotic completeness, construction of Sobolev spaces, boundedness between apropriate Sobolev spaces, >...) generalize to…

Corrigendum to “Imaginary time propagation code for large-scale two-dimensional eigenvalue problems in magnetic fields” [Comput. Phys. Comm. 184(3) (…

2016

Zur Existenz von Lösungen gewisser Randwertaufgaben

1971

With the aid of some known results about integral equations of the Hammerstein type there is proofed an existence theorem for the following class of boundary value problems−y″−l 2 y′=f(x,y),y(a)=y(b)=0,l 2>0 mit|f(x, y)|=0,l 3 (x)>0. The existence range is determined by the greatest eigenvalue of some linear problem.

Weak commutation relations of unbounded operators and applications

2011

Four possible definitions of the commutation relation $[S,T]=\Id$ of two closable unbounded operators $S,T$ are compared. The {\em weak} sense of this commutator is given in terms of the inner product of the Hilbert space $\H$ where the operators act. Some consequences on the existence of eigenvectors of two number-like operators are derived and the partial O*-algebra generated by $S,T$ is studied. Some applications are also considered.

Kernel manifold alignment for domain adaptation

2016

The wealth of sensory data coming from different modalities has opened numerous opportu- nities for data analysis. The data are of increasing volume, complexity and dimensionality, thus calling for new methodological innovations towards multimodal data processing. How- ever, multimodal architectures must rely on models able to adapt to changes in the data dis- tribution. Differences in the density functions can be due to changes in acquisition conditions (pose, illumination), sensors characteristics (number of channels, resolution) or different views (e.g. street level vs. aerial views of a same building). We call these different acquisition modes domains, and refer to the adaptation proble…

Topology of synaptic connectivity constrains neuronal stimulus representation, predicting two complementary coding strategies

2022

In motor-related brain regions, movement intention has been successfully decoded from in-vivo spike train by isolating a lower-dimension manifold that the high-dimensional spiking activity is constrained to. The mechanism enforcing this constraint remains unclear, although it has been hypothesized to be implemented by the connectivity of the sampled neurons. We test this idea and explore the interactions between local synaptic connectivity and its ability to encode information in a lower dimensional manifold through simulations of a detailed microcircuit model with realistic sources of noise. We confirm that even in isolation such a model can encode the identity of different stimuli in a lo…

A Novel Solution to Find the Dynamic Response of an Euler–Bernoulli Beam Fitted with Intraspan TMDs under Poisson Type Loading

2020

This contribution considers a virtual experiment on the vibrational response of rail and road bridges equipped with smart devices in the form of damping elements to mitigate vibrations. The internal damping of the bridge is considered a discontinuity that contain a dashpot. Exact complex eigenvalues and eigenfunctions are derived from a characteristic equation built as the determinant of a 4 x 4 matrix

The impact of sample reduction on PCA-based feature extraction for supervised learning

2006

"The curse of dimensionality" is pertinent to many learning algorithms, and it denotes the drastic raise of computational complexity and classification error in high dimensions. In this paper, different feature extraction (FE) techniques are analyzed as means of dimensionality reduction, and constructive induction with respect to the performance of Naive Bayes classifier. When a data set contains a large number of instances, some sampling approach is applied to address the computational complexity of FE and classification processes. The main goal of this paper is to show the impact of sample reduction on the process of FE for supervised learning. In our study we analyzed the conventional PC…