Fault Detection of Planetary Gears Based on Signal Space Constellations

A new method to process the vibration signal acquired by an accelerometer placed in a planetary gearbox housing is proposed, which is useful to detect potential faults. The method is based on the phenomenological model and consists of the projection of the healthy vibration signals onto an orthonormal basis. Low pass components representation and Gram–Schmidt’s method are conveniently used to obtain such a basis. Thus, the measured signals can be represented by a set of scalars that provide information on the gear state. If these scalars are within a predefined range, then the gear can be diagnosed as correct; in the opposite case, it will require further evaluation. The method is validated using measured vibration signals obtained from a laboratory test bench.

Banach Actions Preserving Unconditional Convergence

Let A,X,Y be Banach spaces and A×X→Y, (a,x)↦ax be a continuous bilinear function, called a Banach action. We say that this action preserves unconditional convergence if for every bounded sequence (an)n∈ω in A and unconditionally convergent series ∑n∈ωxn in X, the series ∑n∈ωanxn is unconditionally convergent in Y. We prove that a Banach action A×X→Y preserves unconditional convergence if and only if for any linear functional y*∈Y* the operator Dy*:X→A*, Dy*(x)(a)=y*(ax) is absolutely summing. Combining this characterization with the famous Grothendieck theorem on the absolute summability of operators from ℓ1 to ℓ2, we prove that a Banach action A×X→Y preserves unconditional convergence if A is a Hilbert space possessing an orthonormal basis (en)n∈ω such that for every x∈X, the series ∑n∈ωenx is weakly absolutely convergent. Applying known results of Garling on the absolute summability of diagonal operators between sequence spaces, we prove that for (finite or infinite) numbers p,q,r∈[1,∞] with 1r≤1p+1q, the coordinatewise multiplication ℓp×ℓq→ℓr preserves unconditional convergence if and only if one of the following conditions holds: (i) p≤2 and q≤r, (ii) 2<p<q≤r, (iii) 2<p=q<r, (iv) r=∞, (v) 2≤q<p≤r, (vi) q<2<p and 1p+1q≥1r+12.

Coherent states for a system of an electron moving in a plane: case of discrete spectrum

In this work, we construct different classes of coherent states related to a quantum system, recently studied in [1], of an electron moving in a plane in uniform external magnetic and electric fields which possesses both discrete and continuous spectra. The eigenfunctions are realized as an orthonormal basis of a suitable Hilbert space appropriate for building the related coherent states. These latter are achieved in the context where we consider both spectra purely discrete obeying the criteria that a family of coherent states must satisfies.

A few remarks on Pimsner–Popa bases and regular subfactors of depth 2

We prove that a finite index regular inclusion of $II_1$ -factors with commutative first relative commutant is always a crossed product subfactor with respect to a minimal action of a biconnected weak Kac algebra. Prior to this, we prove that every finite index inclusion of $II_1$ -factors which is of depth 2 and has simple first relative commutant (respectively, is regular and has commutative or simple first relative commutant) admits a two-sided Pimsner–Popa basis (respectively, a unitary orthonormal basis).

Anisotropic Metamaterials Filled Rectangular Metallic Waveguides Propagation Study and Analysis

In this paper, we propose a novel generalized S-matrix characterization approach. The goal is to keep track of all observed discontinuities as efficiently as possible. In terms of reflection value, the proposed control strategy is based on transmission coefficients and one-axis rectangular guides. We successfully manipulate metal rectangular waveguide filters with both geometrical and physical discontinuity. Lossless discontinuity is depicted as a periodic structure that contains Metamaterials. The modal development of transverse fields provides the basis for the generalized S-matrix approach. The approach works by breaking down electromagnetic fields for each of the guides that make up the discontinuity on an orthonormal basis. When the Galerkin method is used, the matrix of diffraction of the junction is obtained directly.

The Similarities and Distances of Growth Rates Related to COVID-19 Between Different Countries Based on Spectral Analysis

The COVID-19 pandemic has taken more than 1.78 million of lives across the globe. Identifying the underlying evolutive patterns between different countries would help us single out the mutated paths and behavior of this virus. I devise an orthonormal basis which would serve as the features to relate the evolution of one country's cases and deaths to others another's via coefficients from the inner product. Then I rank the coefficients measured by the inner product via the featured frequencies. The distances between these ranked vectors are evaluated by Manhattan metric. Afterwards, I associate each country with its nearest neighbor which shares the evolutive pattern via the distance matrix. Our research shows such patterns is are not random at all, i.e., the underlying pattern could be contributed to by some factors. In the end, I perform the typical cosine similarity on the time-series data. The comparison shows our mechanism differs from the typical one, but is also related to each it in some way. These findings reveal the underlying interaction between countries with respect to cases and deaths of COVID-19.

Stable Calculation of Krawtchouk Functions from Triplet Relations

Deployment of the recurrence relation or difference equation to generate discrete classical orthogonal polynomials is vulnerable to error propagation. This issue is addressed for the case of Krawtchouk functions, i.e., the orthonormal basis derived from the Krawtchouk polynomials. An algorithm is proposed for stable determination of these functions. This is achieved by defining proper initial points for the start of the recursions, balancing the order of the direction in which recursions are executed and adaptively restricting the range over which equations are applied. The adaptation is controlled by a user-specified deviation from unit norm. The theoretical background is given, the algorithmic concept is explained and the effect of controlled accuracy is demonstrated by examples.

Digital Bandpass Modulation Methods

This chapter presents mathematical models of baseband and bandpass digital communication systems based on binary and quaternary phase-shift keying, frequency-shift keying, and quadrature amplitude modulation. The systems are deduced as special cases from the general generic system structure and the related theory of orthonormal basis functions. The systems are uniquely presented using mathematical operators and detailed derivatives for signals in time and frequency domains at the system’s vital points, that is, the transmitter, the receiver, and the noise generator, using the concepts of both stochastic (continuous and discrete) and deterministic (continuous and discrete) signal processing. The vital characteristics of the system and its blocks are expressed in terms of amplitude spectral density, autocorrelation functions, power and energy spectral densities, and bit error probability.