Valery A. Kotov and Serge V. Kotov

"According to General Relativity, a stellar binary generates gravitational waves at a primary frequency twice the orbital one; these waves however have not yet been detected. If the Universe contains gravitational radiation at discrete frequency(ies) - particularly with the period of 160 minutes discovered in the 1970s in the Sun, corresponding resonances might be found in the distribution of orbital frequencies of binaries. With this in mind, we analyse all available data on orbital frequencies of close binaries of the Galaxy. In the frequency range 5 to 160 µHz, we find one significant frequency 104.2 µHz - at the 4 sigma confidence level -which modulates the distribution of about 5000 binaries with periods P < 5.5 d. The corresponding "resonant" period, 160.0 +/- 0.5 min, coincides with that of solar pulsation 160.0 min. The question on its origin and also the hypothesis of a cosmological nature of the oscillation are briefly discussed."

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A waveform is the shape of a signal as it changes with time, position, phase, or another independent variable. It may represent voltage, sound pressure, displacement, light intensity, or another measurable quantity.

Common periodic waveforms include sine, square, triangle, sawtooth, and pulse waves. Each has a distinct shape and harmonic spectrum.

Waveforms are described using amplitude, frequency, period, phase, duty cycle, offset, rise time, and other parameters. Fourier analysis can represent complex waveforms as combinations of sinusoidal components.

Waveform analysis is used in mathematics, electronics, acoustics, communications, control systems, seismology, music, imaging, and computer graphics.

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A wave is a disturbance or variation that transfers energy and information through space or through a medium. The material carrying a mechanical wave may oscillate locally without travelling with the wave over long distances.

Waves can be transverse, longitudinal, surface, standing, travelling, linear, or nonlinear. Electromagnetic and gravitational waves can propagate through vacuum, while sound and water waves require a material medium.

Important properties include amplitude, frequency, period, wavelength, phase, speed, and polarisation. Waves can reflect, refract, diffract, interfere, disperse, and form resonances.

Wave behaviour is central to acoustics, optics, communications, oceanography, seismology, quantum mechanics, astronomy, and many other areas of science and engineering.

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The wave equation is a partial differential equation that describes the propagation of waves through space and time. It relates the second time derivative of a field to its spatial curvature.

Its solutions include travelling waves, standing waves, pulses, and superpositions of these forms. The propagation speed depends on the physical medium or field being modelled.

Boundary and initial conditions determine the permitted solutions. On strings, membranes, pipes, and cavities, these conditions produce normal modes and characteristic resonance frequencies.

The wave equation is used in acoustics, electromagnetism, elasticity, water waves, seismology, quantum theory, and general relativity. Numerical methods are often required for complex geometries and varying media.

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Translational symmetry is the property of an object, pattern, or system remaining unchanged after being shifted by a particular distance and direction. Repeating the translation produces a regular sequence of equivalent positions.

A one-dimensional repeating pattern has translational symmetry along a line. Crystals and tilings may have two- or three-dimensional translation lattices.

In physics, continuous translational symmetry means that the laws do not depend on absolute position. Through Noether's theorem, this symmetry is associated with conservation of linear momentum.

Translational symmetry is important in geometry, crystallography, group theory, solid-state physics, periodic functions, wave propagation, and the classification of repeating structures.

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The spectrum of an operator is the set of scalar values for which the operator minus that scalar times the identity does not have a bounded inverse. It generalises the collection of eigenvalues familiar from finite-dimensional matrices.

In finite-dimensional linear algebra, the spectrum consists exactly of the eigenvalues. In infinite-dimensional spaces, it may also include values that are not associated with ordinary eigenvectors.

The spectrum can be divided into point, continuous, and residual parts, depending on the behaviour of the operator and its range. For self-adjoint operators, the spectrum is real.

Operator spectra are fundamental in functional analysis, differential equations, quantum mechanics, vibration theory, stability analysis, and the study of waves and resonances.

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A sine wave is a smooth periodic oscillation described by the sine function. It is characterised by amplitude, frequency, period, wavelength, and phase.

The waveform rises and falls symmetrically about its mean value. Its instantaneous rate of change and accumulated area are represented by related sine and cosine functions.

Sine waves are mathematically fundamental because linear systems preserve their frequency: the output may change in amplitude and phase but remains sinusoidal. More complicated periodic signals can be decomposed into sums of sine waves.

They model simple harmonic motion, alternating current, sound, light, radio signals, water waves, vibrations, and many other oscillatory processes.

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A power spectrum describes how the power or variance of a signal is distributed across frequency. It reveals which frequencies contribute most strongly to an observed time series or spatial pattern.

For deterministic signals, the power spectrum is related to the squared magnitude of the Fourier transform. For random processes, it is commonly described by the power spectral density.

Sharp peaks may indicate periodic components, while broad distributions can represent noise or irregular fluctuations. The resolution and reliability of estimates depend on record length, sampling, windowing, and averaging.

Power spectra are used in acoustics, astronomy, geophysics, neuroscience, communications, climate research, vibration analysis, and many other forms of time-series analysis.

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Phase describes the position of a point within one complete cycle of a periodic wave or oscillation. It is usually expressed as an angle in degrees or radians, with one cycle corresponding to 360 degrees or 2π radians.

Two waves with the same frequency may have a phase difference. Waves in phase reach corresponding peaks and troughs together, while waves half a cycle apart are in antiphase.

Phase differences determine how waves combine. Constructive interference occurs when phases align, while destructive interference occurs when peaks and troughs oppose one another.

Phase is important in acoustics, optics, electronics, communications, rotating machinery, Fourier analysis, quantum mechanics, and any system involving coupled oscillations.

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A permutation is an arrangement of objects in a particular order or, more generally, a one-to-one mapping of a set onto itself. Changing the order produces a different permutation.

A set with n distinct elements has n factorial possible permutations. Restrictions or repeated elements change the counting formulas used.

Permutations can be written in row notation or decomposed into cycles. Their composition forms the symmetric group, one of the central structures in group theory.

Permutations are used in combinatorics, probability, algebra, algorithms, cryptography, scheduling, ranking, genetics, and the analysis of symmetries.

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