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Applications in physics of the multiplicative anomaly formula involving some basic differential operators

http://www.arxiv.org/abs/hep-th/9804118
...ng that such situation may be quite widespread in mathematical physics. However, the consequences of the existence of the determinant anomaly have often been overlooked....
Emilio Elizalde, Guido Cognola, Sergio Zerbini


Summation formulae for the bilateral basic hypergeometric series ${}_1ψ_1 ( a; b; q, z )$)$

http://www.arxiv.org/abs/1603.06657
...We give summation formulae for the bilateral basic hypergeometric series ${}_1\psi_1( a; b; q, z )$ through Ramanujan s summation formula, which are generalizations of nontrivial identities found in the physics of three-dimensional Abelian mirror sym...
Hironori Mori, Takeshi Morita


What is the sign of $\hbar$

http://www.arxiv.org/abs/1007.0323
...We present an elementary argument showing that the sign of $\hbar$ in the basic formulation of Quantum Mechanics can be changed without incurring in any physical consequences....
Massimo Testa


Selberg s trace formula: an introduction

http://www.arxiv.org/abs/math/0407288
...These lecture notes provide a basic introduction to Selberg s trace formula. We discuss the simplest possible case: the spectrum of the Laplacian on a compact Riemannian surface of constant negative curvature. (To appear in Springer LN...
Jens Marklof


Reflection of light from a moving mirror: derivation of the relativistic Doppler formula without Lorentz transformations

http://www.arxiv.org/abs/physics/0605100
...vation does not involve Lorentz transformations, length contractions and time dilations, and therefore is conceptually simpler than the standard derivations in physics textbooks. This discussion can be useful for teaching introductory physics and als...
Malik Rakhmanov


Sakharov s induced gravity: a modern perspective

http://www.arxiv.org/abs/gr-qc/0204062
...Sakharov s 1967 notion of induced gravity is currently enjoying a significant resurgence. The basic idea, originally presented in a very brief 3-page paper with a total of 4 formulas, is that gravity is not fundamental in the sense of particle physic...
Matt Visser


ADM analysis and massive gravity

http://www.arxiv.org/abs/1302.0687
...on of the curvature components. After that we review the basic problems associated with attempts of constructing a viable massive gravity theory. And finally, we present the metric formulations of ghost-free massive gravity models, and comment on exi...
Alexey Golovnev


Teraelectronvolt Astronomy

http://www.arxiv.org/abs/1006.5210
...Ground-based gamma-ray astronomy, which provides access to the TeV energyrange, is a young and rapidly developing discipline. Recent discoveries in thiswaveband have important consequences for a wide range of topics in astrophysicsand astroparticle p...


Localization and Conjectures from String Duality

http://www.arxiv.org/abs/math-ph/0701057
...simple moduli spaces. It is a key technique in the proof of the general mirror formulas, the proof of the Hori-Vafa formulas for explicit expressions of basic hypergeometric series of homogeneous manifolds, the proof of the Mari\ no-Vafa formula, its...
Kefeng Liu


Cosmology as Condensed Matter Physics

http://www.arxiv.org/abs/gr-qc/9511076
...We note that in general there exist two basic aspects in any branch of physics, including cosmology - one dealing with the attributes of basic constituents and forces of nature, the other dealing with how structures arise from them and how they evolv...
B. L. Hu



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Central or centripetal acceleration


orbital velocity

$\displaystyle{a}_{{c}}=\frac{{v}^{{2}}}{{r}};\ \ \ {a}_{{c}}=-\omega^{{2}}{r}$

omega radial velocity

Particle motion (classical mechanics)


Newtonian force

$\displaystyle{F}={m}\ddot{{r}}=\dot{{p}}$

F force , m mass of particle , r particle position vector

momentum

$\displaystyle{p}={m}\dot{{r}}={m}{v}$

p momentum

Kinetic energy

$\displaystyle{T}=\frac{{{1}}}{{{2}}}{m}{v}^{{2}}$

T kinetic energy , v particle velocity

Static fields (Electrostatics)


Electrostatic potential

$\displaystyle{E}=-\nabla\phi$

E electric field , electrostatic potential

Static fields (electrostatics)


Electrostatic force between two static charges: Coulombs law

$\displaystyle{F}_{{2}}=\frac{{{q}_{{1}}{q}_{{2}}}}{{{4}\pi\in_{{0}}{{r}_{{12}}^{{2}}}}}\hat{{r}}_{{12}}$

F2 force on q2 , q1,2 charges, r12 vector from 1 to 2, r^ is the unit vector

Gravitation (Newtonian gravitation)


Newtons law of gravitation

$\displaystyle{F}_{{1}}=\frac{{{G}{m}_{{1}}{m}_{{2}}}}{{{{r}_{{12}}^{{2}}}}}\hat{{r}}_{{12}}$

m1,2 masses , F1 force on m1 (=-F2) , r12 vector from m1 to m2 , ^ unit vector

Newtons field equations

$\displaystyle{g}=-\nabla\phi$

G constant of gravitation , g gravitational field strength

Propagation of light


Doppler effect

$\displaystyle{v}'{/}{v}=\gamma{\left({1}+{v}{/}{c}{\cos{\propto}}\right)}$

v frequency received in S , v' frequency emitted in S' , . Arrival angle in S

Special relativity


mass energy relation

$\displaystyle{E}={m}\cdot{c}^{{2}}$



Equations of movement


movement with constant acceleration

$\displaystyle{x}={x}_{{0}}+{v}{0}{t}+\frac{{1}}{{2}}{a}{t}^{{2}};\ \ \ \ {v}={v}_{{0}}+{a}\cdot{t}\ \ \ \ {v}^{{2}}={\left({v}_{{0}}\right)}^{{2}}+{2}{a}{\left({x}−{x}{0}\right)};\ \ \ \ {a}=\frac{{{d}{v}}}{{\left.{d}{t}\right.}}$

v velocity

uniform movement

$\displaystyle{x}=\vec{{v}}_{{0}}\cdot{t}$

v constant velocity

Newtons first law


An object that is at rest will stay at rest unless an external force acts upon it. An object that is in motion will not change its velocity unless an external force acts upon it.

$\displaystyle\sum{\mathbf{{{F}}}}={0};\frac{{{d}\vec{{v}}}}{{{\left.{d}{t}\right.}}}={0}$

F: force, p momentum

Newtons second law




$\displaystyle{F}=\frac{{{d}{p}}}{{\left.{d}{t}\right.}}$

F: force, p momentum

Charges in the Magnetic Field (Magnetism)


Magnitude of the Magnetic Force on a Moving Charge

$\displaystyle{F}={q}{v}{B}{\sin{\theta}}$

theta is the angle between direction of movement and magnetic field. If B and F are perpendicular the expression simplifies to `F=q*v*B`

Radius of the Circle Described by a Charged Particle Moving Perpendicular to a Magnetic Field

$\displaystyle{r}=\frac{{{m}{v}}}{{{q}{B}}}\theta$



Magnetic Force on a Current

$\displaystyle{F}={I}{l}{B}{\sin{\theta}}$



Magnetic Field Created by a Current

$\displaystyle{B}=\frac{{\mu_{{o}}}}{{{2}\pi}}\frac{{{I}}}{{{r}}}$



Magnetic Force on a Moving Charge, also called lorentz force

$\displaystyle{F}={q}{\left({v}\times{B}\right)}$



Optics


Frequency of an Electromagnetic Wave

$\displaystyle{f}=\frac{{{c}}}{{\lambda}}$



Law of Reflection

$\displaystyle\theta_{{{I}{n}{c}{i}{d}{e}{n}{c}{e}}}=\theta_{{{r}{e}{f}{l}{e}{c}{t}{i}{o}{n}}}$



Index of Refraction

$\displaystyle{n}=\frac{{{c}}}{{\upsilon}}$



Snell’s Law

$\displaystyle{n}_{{1}}{{\sin{\theta}}_{{1}}=}{n}_{{2}}{{\sin{\theta}}_{{2}}}$



Critical Angle for total reflection

$\displaystyle\theta_{{c}}={\arcsin{{\left(\frac{{{n}_{{2}}}}{{{n}_{{1}}}}\right)}}}$



Focal Length for a Spherical Concave Mirror

$\displaystyle{f}=\frac{{{R}}}{{{2}}}$



Mirror and Lens Equation

$\displaystyle\frac{{{1}}}{{{d}}}+\frac{{{1}}}{{{d}^{{i}}}}=\frac{{{1}}}{{{f}}}$



Magnification

$\displaystyle{m}=\frac{{{h}^{{i}}}}{{{h}}}=\frac{{-{d}^{{i}}}}{{{d}}}$



Maxima for Single Slit Diffraction

$\displaystyle{d}{\sin{\theta}}={\left({n}+\frac{{{1}}}{{{2}}}\right)}\lambda,\text{where n is an integer}$



Minima for Single Slit Diffraction

$\displaystyle{d}{\sin{\theta}}={n}\lambda,\text{where n is an integer}$



Kirchhoffs Laws


Kirchhoffs current law (KCL): The junction between several circuit elements is called a node

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The sum of the currents values at a node is zero.

$\displaystyle\Sigma{I}_{{i}}={0}$



Kirchhoffs voltage law (KVL): The sum of voltages at a current loop is zero.

$\displaystyle\Sigma{v}_{{1}}={0}$



DC Circuits with Capacitor and Resistance


two Resistors in Parallel

$\displaystyle\frac{{{1}}}{{{R}_{{1}}}}=\frac{{{1}}}{{{R}_{{1}}}}=\frac{{{1}}}{{{R}_{{2}}}}$



Power Dissipated in a Resistor

$\displaystyle{P}={I}{V}={I}^{{2}}{R}=\frac{{{V}^{{2}}}}{{{R}}}$



Heat Dissipated in a Resistor

$\displaystyle{H}={I}^{{2}}{R}{t}$



two Resistors in Series

$\displaystyle{R}_{{1}}={R}_{{1}}+{R}_{{2}}$



Stored Energy of a Capacitor

$\displaystyle{U}=\frac{{{1}}}{{{2}}}{Q}\Delta{V}=\frac{{{1}}}{{{2}}}{Q}{V}=\frac{{{Q}^{{2}}}}{{{2}{C}}}=\frac{{{1}}}{{{2}}}{Q}{V}$



two Capacitors in Series

$\displaystyle\frac{{{1}}}{{{C}_{{t}}}}=\frac{{{1}}}{{{C}_{{1}}}}+\frac{{{1}}}{{{C}_{{2}}}}$



two Capacitors in Parallel

$\displaystyle{C}_{{t}}={C}_{{1}}+{C}_{{2}}$



Capacitance


Capacitance

$\displaystyle{C}=\frac{{Q}}{{V}}$



Parallel plate capacitor; ε is the permittivity in farad per meter (F/m).

$\displaystyle{C}=\epsilon\cdot\frac{{A}}{{l}}$



Parallel circuit rules




$\displaystyle\frac{{1}}{{L}_{{T}}}=\frac{{1}}{{L}_{{1}}}+\frac{{1}}{{L}_{{2}}}+\frac{{1}}{{L}_{{3}}}+…$





$\displaystyle{V}_{{T}}={V}_{{1}}+{V}_{{2}}+{V}_{{3}}+\ldots$





$\displaystyle{I}_{{T}}={I}_{{1}}={I}_{{2}}={I}_{{3}}=\ldots$





$\displaystyle\frac{{1}}{{R}_{{T}}}=\frac{{1}}{{R}_{{1}}}+\frac{{1}}{{R}_{{2}}}+\frac{{1}}{{R}_{{3}}}+\ldots$





$\displaystyle{C}_{{T}}={C}_{{1}}+{C}_{{2}}+{C}_{{3}}+…$



Voltage division


Voltage division

$\displaystyle{V}_{{1}}={V}_{{T}}\cdot\frac{{R}_{{1}}}{{{R}_{{1}}+{R}_{{2}}+{R}_{{3}}+\ldots}}$



Current division


Current division

$\displaystyle{I}_{{1}}={I}_{{T}}\cdot\frac{{{R}_{{2}}+{R}_{{3}}\ldots}}{{{R}_{{1}}+{R}_{{2}}+{R}_{{3}}+\ldots}}$



Ohms law


Ohms law

$\displaystyle{I}=\frac{{{V}}}{{{R}}}$



Joules law


Joules law

$\displaystyle{P}={V}\cdot{I}={I}^{{2}}\cdot{R}=\frac{{V}^{{2}}}{{R}}$



Series circuit rules




$\displaystyle{V}_{{T}}={V}_{{1}}+{V}_{{2}}+{V}_{{3}}+\ldots$





$\displaystyle{I}_{{T}}={I}_{{1}}={I}_{{2}}={I}_{{3}}=\ldots$





$\displaystyle{R}_{{T}}={R}_{{1}}+{R}_{{2}}+{R}_{{3}}+\ldots$





$\displaystyle\frac{{1}}{{C}_{{T}}}=\frac{{1}}{{C}_{{1}}}+\frac{{1}}{{C}_{{2}}}+\frac{{1}}{{C}_{{3}}}+…$





$\displaystyle{L}_{{T}}={L}_{{1}}+{L}_{{2}}+{L}_{{3}}+…$



Permittivity


Permittivity; ε0 is the permittivity in vacuum

$$



epsilon=epsilon_0*epsilon_

$$

εr is the relative permittivity or dielectric constant

Capacitor


Current of capacitor

$\displaystyle{I}_{{C}}{\left({t}\right)}={C}{d}\frac{{{V}_{{C}}{\left({t}\right)}}}{{\left.{d}{t}\right.}}$



Voltage of capacitor

$\displaystyle{V}_{{C}}{\left({t}\right)}={V}_{{C}}{\left({0}\right)}+\frac{{1}}{{C}}\int{I}_{{c}}{\left({t}\right)}\cdot{\left.{d}{t}\right.}$



Voltage of capacitor

$\displaystyle{V}_{{L}}{\left({t}\right)}={L}{d}\frac{{{I}_{{L}}{\left({t}\right)}}}{{\left.{d}{t}\right.}}$



Energy of capacitor

$\displaystyle{W}_{{C}}={C}\cdot\frac{{V}^{{2}}}{{2}}$



Inductor


Current of inductor

$\displaystyle{I}_{{l}}{\left({t}\right)}={I}_{{{L}}}{\left({0}\right)}+\frac{{1}}{{L}}\int{V}_{{L}}{\left({t}\right)}\cdot{\left.{d}{t}\right.}$



Energy of inductor

$\displaystyle{W}_{{L}}={L}\cdot\frac{{I}^{{2}}}{{2}}$



Definition of Work




$\displaystyle{W}=\vec{{F}}\cdot\vec{{s}}$

F is the force along the way s

Compton Scattering




$\displaystyle\lambda'-\lambda={\frac{{{h}}}{{{m}_{{e}}{c}}}}{\left({1}-{\cos{\theta}}\right)}$

lambda initial wavelength, lambda' wavelength after scattering, me electron mass, c speed of light, theta scattering angle

Strength of the homogenous E-Field




$\displaystyle{E}=\frac{{F}}{{Q}}$

F is the force on a charge Q

Definition of Power




$\displaystyle{P}=\frac{{W}}{{t}}$

power is defined as the work performed in time time t divided by t

Photo Effect


Kmax is the maximum kinetic energy of an ejected photon

$\displaystyle{T}_{{\max}}={h}{f}-\phi;\ \ \ \ \phi={h}{{f}_{{0}}\ }\ \ \ {K}_{\max}={h}{\left({f}-{f}_{{0}}\right)}$

f0 is the threshold frequency of the metal

Angular (radial) Velocity




$\displaystyle\omega=\frac{{{d}\phi}}{{{\left.{d}{t}\right.}}}=\frac{{{2}\pi}}{{T}}\ \ \ \ \text{(For circular movement) }\ $



Definition of Energy


Energy is the ability of a system to perform work.

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Due to energy conservation the total energy of a system is a constant, although energy may transform from one form into another.

Fluids and Solids


Density

$\displaystyle\rho=\frac{{m}}{{V}}$

m: mass, V: volume

Pressure

$\displaystyle{P}=\frac{{F}}{{A}}$

pressure is defined by force per area

Underwater Pressure

$\displaystyle{P}={P}_{{{a}{t}{m}}}+{g}\cdot{d}$

P is the pressure of the atmosphere, g=0.981 N/m, d: depth

The ideal gas


Gay-Lussacs law

$\displaystyle\frac{{V}_{{1}}}{{V}_{{2}}}=\frac{{T}_{{1}}}{{T}_{{2}}}$



Boyle-Mariottes law

$\displaystyle\frac{{p}_{{1}}}{{p}_{{2}}}=\frac{{V}_{{2}}}{{V}_{{1}}}$



Amontons law

$\displaystyle\frac{{p}_{{1}}}{{p}_{{2}}}=\frac{{T}_{{1}}}{{T}_{{2}}}$



Archimedes principle


The upward buoyant force that is exerted on a body immersed in a fluid, whether fully or partially submerged, is equal to the weight of the fluid that the body displaces.

$$





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