A palindromic number in binary representation, with no leading zeros. Every positive binary palindrome is an odd number, because its first and last numerical digits must both be one.

Decide whether a finite string is a palindrome by comparing corresponding symbols from opposite ends. The test needs at most half as many symbol comparisons as the string length and can use constant auxiliary space.

Two photons meeting at angle $\alpha$ can produce an electron and a Positron when their combined four-momentum has sufficient invariant mass. For a head-on collision the threshold is $E_1{E}_2=m_e^2{c}^4$. Threshold means zero relative momentum in the centre-of-momentum frame, not necessarily in the laboratory.

For $p>0$ and $n\ge 2$, integrating the decreasing derivative magnitude of $x^{-p}$ gives $(n-1{)}^{-p}-n^{-p}=p{\int }_{n-1}^n{x}^{-p-1}dx>p{n}^{-p-1}$. Put $p=k-1$ and sum this telescoping series to obtain the bound in the title, strict for $N>1$. The partial sums are increasing and bounded, proving convergence of the p-series when $k>1$. This gives a direct comparison test for series proof and a finite upper bound without evaluating the sum.

Fix an integer $N>1$ and a real exponent $p>1$, and put $S_p={\sum }_{n=N}^{\infty }{n}^{-p}$. There is a unique shift $k_p<N$ such thatIntegration evaluates the left side as $(N-k_p{)}^{1-p}/(p-1)$, yielding the displayed formula. Comparing the decreasing function $x^{-p}$ with unit rectangles provesThe integral is a strictly increasing function of its shift, so $0<k_p<1$. This is an exact reparametrization of a tail, not a claim that its shift is independent of $p$; interpolating shifts for different exponents is an additional approximation.

The 2-adic exponential is the power series $\exp (x)={\sum }_{n\ge 0}{x}^n/n!$ interpreted with the 2-adic absolute value. Over ${\mathbb{Q}}_2$, it converges for $v_2(x)>1$, equivalently $x\in 4{\mathbb{Z}}_2$. The general p-adic exponential function has convergence condition $|x{|}_p<p^{-1/(p-1)}$.

On $4{\mathbb{Z}}_2$, the 2-adic exponential and 2-adic logarithm give inverse group homomorphisms between the additive group $4{\mathbb{Z}}_2$ and the multiplicative group $1+4{\mathbb{Z}}_2$. The stronger domain restriction compared with odd primes comes from the powers of two dividing the factorial denominators.

For $m\ddot{x}+\gamma \dot{x}+kx=0$, let $r_s$ and $r_f$ be the slow and fast negative roots. The coefficient of the slow normal mode is exactlyWith $ϵ=km/{\gamma }^2\ll 1$ and ${\tau }_m=m/\gamma$, expansion gives $A_s=x_0+{\tau }_m{v}_0+O(ϵ)(|x_0|+|{\tau }_m{v}_0|)$. Thus the initial velocity can leave a finite displacement after its own fast decay. In particular, $x_0=0$ and $v_0\ne 0$ produce a nonzero slow tail. A first-order overdamped approximation needs an adjusted initial position after the boundary layer; setting it to $x_0$ requires the additional smallness of ${\tau }_m{v}_0$.

For one orthonormal localized orbital per simple-cubic site and real equal nearest-neighbour hopping integral $t$, the six neighbours give the displayed energy band. Each cosine ranges from minus one to one, so the positive bandwidth is $12|t|$, independently of the sign convention for $t$. Near the zone centre, $E=E_0+6t-t{a}^2{k}^2+O(k^4)$ and the band effective mass is $-{\hslash }^2/(2t{a}^2)$. It is positive there for $t<0$; for $t>0$ the zone centre is a maximum instead.

Suppose real symmetric matrices $E,F$ satisfy $E^2=F^2=I$ and $EF=-FE$. The involutions are orthogonal matrices, and $J=EF$ satisfies $J^T=-J$, $J^2=-I$. The matrix exponential therefore reduces to $R(t)=e^{tJ}=\cos tI+\sin tJ$. Multiplication gives $R(t{)}^{T}R(t)=I$ and the conjugation identitiesIndeed $JE=-F$, $EJ=F$, $JF=E$, and $FJ=-E$, which give both identities by expansion. The angle in this two-dimensional space of matrices is twice the parameter in the orthogonal matrix. These formulas hold in any finite dimension where the hypotheses can be satisfied.

An obstacle calculation produces a resolved shear-stress increment. A single crystal needs its Schmid factor, while a polycrystal needs a deformation-geometry or Taylor factor. The relation $\Delta \sigma =M\Delta \tau$ therefore requires an explicit orientation assumption rather than a universal tensile prefactor.

The modulation transfer function is the magnitude of the normalized optical transfer function in a linear, shift-invariant incoherent imaging model. It describes how strongly the system transfers sinusoidal image contrast at each spatial frequency, while discarding the transfer function's phase.

For an input $I(x)=I_0[1+m\cos (2\pi \nu x)]$ and a unit-integral point spread function, the output modulation magnitude is $mM(\nu )$. Fine alternating bars contain higher spatial frequencies, so their diminishing visibility illustrates loss of transferred modulation. Measuring image contrast across a calibrated pattern tests this response, but detector sampling, noise and nonlinear image processing can alter the result. A transfer magnitude alone does not locate image features or specify their phase shifts.

For a periodic intensity distribution, $C=(I_{\max }-I_{\min })/(I_{\max }+I_{\min })$. It should not be confused with a resist contrast parameter.

For an isotropic elastic amplitude normalized by an incident unit plane wave and outgoing $f{e}^{ikr}/r$, total cross section is $4\pi |f{|}^2$. The optical theorem gives the displayed identity. At a resonance where the real part of $1/f$ vanishes, $f=i/k$ and the cross section is $4\pi /k^2$.

For a real elastic scattering potential the first Born forward amplitude is real, while the leading total scattering cross-section is quadratic in the potential. The second Born amplitude supplies an imaginary part from its outgoing Green function. Symmetrizing the two interaction positions makes its imaginary part equal to $k/(4\pi )$ times the integrated squared first Born amplitude. The identity therefore compares quantities at the same perturbative order; a real first Born amplitude alone cannot test it against a nonzero scattering cross-section.

An optical parametric oscillator places an optical parametric amplifier inside an optical resonator that feeds back at least one generated wave. Pump energy is converted into signal and idler waves satisfying the frequency relation of the underlying three-wave mixing process.

Oscillation begins when round-trip gain overcomes loss at an allowed resonator mode. Below that threshold, weak generated emission can still occur, but sustained oscillation is absent. Tuning the phase matching and resonator together gives a tunable source at wavelengths difficult to obtain directly from a particular laser. Pump power alone does not select the output: cavity resonances, conversion gain and optical losses also matter.

A beam's transverse intensity changes its own refractive index. For positive nonlinear index and a bright centre, the accumulated phase has the curvature of a converging lens. For $I(r)=I_0{e}^{-2r^2/w^2}$ over a short length $L$, the in-medium thin-lens approximation gives $1/f=4n_2{I}_{0}L/(n_0{w}^2)$. Negative $n_2$ instead produces self-defocusing.

An ideal two-mode intensity-intensity coupling makes the phase of each mode depend on the other's photon number. With this sign convention, evolution for time $t$ sends a mode-1 coherent amplitude to $\alpha e^{-i\chi t{n}_2}$ when mode 2 has definite occupation. This single-mode unitary model does not by itself establish a strong, noiseless interaction between real travelling single photons.

Under stable negative feedback and within its linear operating range, an ideal voltage-feedback operational amplifier has equal input voltages. This relation is a consequence of large open-loop gain, and does not imply current flow between the inputs.

A voltage-feedback operational amplifier with its noninverting input grounded and input and feedback resistances $R_i,R_f$ has ideal gain $-R_f/R_i$. Virtual short and negligible input current give the result by node-current balance.

A voltage-feedback operational amplifier driven at its noninverting input, with a feedback divider $R_f,R_g$ at its inverting input, has ideal gain $1+R_f/R_g$ under stable negative feedback.