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Applications in physics of the multiplicative anomaly formula involving some basic differential operators 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 ${}_1ψ_1 ( a; b; q, z )$)$
...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 $\hbar$
...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
...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
...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
...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
...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
...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
...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
...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


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



p momentum

Kinetic energy


T kinetic energy , v particle velocity

Static fields (Electrostatics)

Electrostatic potential


E electric field , electrostatic potential

Static fields (electrostatics)

Electrostatic force between two static charges: Coulombs law


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


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

Newtons field equations


G constant of gravitation , g gravitational field strength

Propagation of light

Doppler effect


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

Special relativity

mass energy relation


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


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.


F: force, p momentum

Newtons second law


F: force, p momentum

Charges in the Magnetic Field (Magnetism)

Magnitude of the Magnetic Force on a Moving Charge


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


Magnetic Force on a Current


Magnetic Field Created by a Current


Magnetic Force on a Moving Charge, also called lorentz force



Frequency of an Electromagnetic Wave


Law of Reflection


Index of Refraction


Snell’s Law


Critical Angle for total reflection


Focal Length for a Spherical Concave Mirror


Mirror and Lens Equation




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


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


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


DC Circuits with Capacitor and Resistance

two Resistors in Parallel


Power Dissipated in a Resistor


Heat Dissipated in a Resistor


two Resistors in Series


Stored Energy of a Capacitor


two Capacitors in Series


two Capacitors in Parallel





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


Parallel circuit rules






Voltage division

Voltage division


Current division

Current division


Ohms law

Ohms law


Joules law

Joules law


Series circuit rules







Permittivity; ε0 is the permittivity in vacuum




εr is the relative permittivity or dielectric constant


Current of capacitor


Voltage of capacitor


Voltage of capacitor


Energy of capacitor



Current of inductor


Energy of inductor


Definition of Work


F is the force along the way s

Compton Scattering


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

Strength of the homogenous E-Field


F is the force on a charge Q

Definition of Power


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.


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



m: mass, V: volume



pressure is defined by force per area

Underwater Pressure


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

The ideal gas

Gay-Lussacs law


Boyle-Mariottes law


Amontons law


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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