A Quadruple Integral Involving Chebyshev Polynomials Tn(x): Derivation and Evaluation

The aim of the current document is to evaluate a quadruple integral involving the Chebyshev polynomial of the first kind Tn(x) and derive in terms of the Hurwitz-Lerch zeta function. Special cases are evaluated in terms of fundamental constants. The zero distribution of almost all Hurwitz-Lerch zeta functions is asymmetrical. All the results in this work are new.

Physical Origins of Universal Mobility in MOSFET Inversion Layers: Conservation Laws

The salient properties of charge flow (or current) along the MOSFET’s inversion layer are shown to be consilient with a river’s flow in a gravitational potential field, insofar as both are fundamentally governed by energy conservation principles, and their laminar and turbulent conditions determined by friction losses at shallow depths. We establish analytically that the low-field, "universal" effective mobility, μ_eff<b>, </b>long reported to vary as ~(E*_T)^-1/3 for transversal fields below 0.5 MV/cm, is manifestation and consequence of both energy and momentum conservation under laminar flow conditions and quantum mechanical effects, in which case the inversion layer’s mean thickness also varies as ~(E*_T)^-1/3 up to a maximum value E*_T ≈ 0.35 MV/cm at 300K, determined only by interface "terrain" amplitude and fundamental constants.

Physical Origins of Universal Mobility in MOSFET Inversion Layers: Conservation Laws

The salient properties of charge flow (or current) along the MOSFET’s inversion layer are shown to be consilient with a river’s flow in a gravitational potential field, insofar as both are fundamentally governed by energy conservation principles, and their laminar and turbulent conditions determined by friction losses at shallow depths. We establish analytically that the low-field, "universal" effective mobility, μ_eff<b>, </b>long reported to vary as ~(E*_T)^-1/3 for transversal fields below 0.5 MV/cm, is manifestation and consequence of both energy and momentum conservation under laminar flow conditions and quantum mechanical effects, in which case the inversion layer’s mean thickness also varies as ~(E*_T)^-1/3 up to a maximum value E*_T ≈ 0.35 MV/cm at 300K, determined only by interface "terrain" amplitude and fundamental constants.

A Quadruple Integral Involving Product of the Struve Hv(βt) and Parabolic Cylinder Du(αx) Functions

The objective of the present paper is to obtain a quadruple infinite integral. This integral involves the product of the Struve and parabolic cylinder functions and expresses it in terms of the Hurwitz–Lerch Zeta function. Almost all Hurwitz-Lerch Zeta functions have an asymmetrical zero distributionSpecial cases in terms fundamental constants and other special functions are produced. All the results in the work are new.

Quadruple Integral Involving the Logarithm and Product of Bessel Functions Expressed in Terms of the Lerch Function

In this paper, we have derived and evaluated a quadruple integral whose kernel involves the logarithm and product of Bessel functions of the first kind. A new quadruple integral representation of Catalan’s G and Apéry’s ζ(3) constants are produced. Some special cases of the result in terms of fundamental constants are evaluated. All the results in this work are new.

A Note on a Triple Integral

A closed form expression for a triple integral not previously considered is derived, in terms of the Lerch function. Almost all Lerch functions have an asymmetrical zero-distribution. The kernel of the integral involves the product of the logarithmic, exponential, quotient radical, and polynomial functions. Special cases are derived in terms of fundamental constants; results are summarized in a table. All results in this work are new.

The fundamental constants of physics and the International System of Units

Air Canada managed to have a passenger aircraft run out of fuel in mid-air due to confusion about metric units (Stephenson in Mars climate orbiter mishap investigation board phase I report, NASA, 1999), and NASA lost an entire spacecraft due to a misunderstanding amongst engineers about the units used in the propulsion system design (Witkin in Jet’s fuel ran out after metric conversion errors, The New York Times, 1983). Measurements only make sense if the units are correct and well-defined. A unit of measurement is a definite magnitude of a quantity, defined by convention or law. Any other quantity of that kind can then be expressed as a multiple or submultiple of the unit of measurement. The Egyptians used the Farao as definite magnitude, while many years later, the french revolutionists introduced the earth as a reference and laid the foundations for the modern decimal system. Since recently, we have a truly universal and stable system that uses physics’s natural constants and laws to define the base units of measurement. This paper explains how this new concept works and how it is implemented in practice.

Mellin Transform of Logarithm and Quotient Function with Reducible Quartic Polynomial in Terms of the Lerch Function

A class of definite integrals involving a quotient function with a reducible polynomial, logarithm and nested logarithm functions are derived with a possible connection to contact problems for a wedge. The derivations are expressed in terms of the Lerch function. Special cases are also derived in terms fundamental constants. The majority of the results in this work are new.

On the scarcity of redshifted OH and millimetre-band molecular absorption

ABSTRACT Despite much searching, redshifted decimetre- and millimetre-band absorption by molecular gas remains very rare, limited to just six systems at zabs ≳ 0.05. Detection of these transitions can yield precise diagnostics of the conditions of the star-forming gas in the earlier Universe, the hydroxyl (OH) radical being of particular interest as in the λ = 18 cm ground state there are four different transitions located close to neutral hydrogen 21- cm and thus detectable with the Square Kilometre Array and its pathfinders. The four transitions of OH have very different dependences on the fundamental constants, thus having much potential in testing for any evolution in these over large look-back times. By collating the photometry in a uniform manner, we confirm our previous hypothesis that the normalized OH absorption strength is correlated with the optical–near-infrared red colour of the sightline. Applying this to the published searches, we find that all, but one (J0414+054), have simply not been searched sufficiently deeply. We suggest that this is due to the standard selection of sources with reliable optical redshifts introducing a bias against those with enough dust with which to shield the molecular gas. For the single source searched to sufficient depth, we have reason to suspect that the high degree of reddening arises from another system along the sightline, thus not being inconsistent with our hypothesis. We also show that the same optical redshift bias can account for the scarcity of millimetre-band absorption.