For a three-dimensional zero-temperature parabolic Fermi gas, let $\zeta =(N_{\uparrow }-N_{\downarrow })/N$ and $E_F={\hslash }^2(3{\pi }^{2}N/V{)}^{2/3}/(2m)$. Suppose the interaction energy per electron is $U{N}_{\uparrow }{N}_{\downarrow }/N^2=U(1-{\zeta }^2)/4$. The total energy per electron isIts Taylor expansion has quadratic coefficient $E_F/3-U/4$ and positive quartic coefficient $E_F/81$. Hence the unpolarized state loses stability at $U_c=4E_F/3\propto (N/V{)}^{2/3}$. This $U$ has units of energy. It differs from a contact coupling $g$ defined by interaction-energy density $g{n}_{\uparrow }{n}_{\downarrow }$, for which $g=U/n$ and $g_c=4E_F/(3n)$. Normalization must be fixed before comparing critical couplings. For $U\le 4E_F/3$, convexity of $x^{-1/3}$ implies $f^{\prime \prime }(\zeta )=(E_F/3)[(1+\zeta {)}^{-1/3}+(1-\zeta {)}^{-1/3}]-U/2\ge 0$, so the unpolarized minimum is a global minimum; a first-order polarized minimum does not pre-empt this threshold in the stated model.

The dimensionless factor $C$ compares a corrected linear drag resistance with the continuum Stokes's law: dividing by $C>1$ reduces the resistance. The oil drop charge readout and calibration article explains why this matters for electric charge inference.

The positive sequence $a_n=\sqrt{n}(n/e{)}^n/n!$ has logarithmic incrementsThe rational lower bound for the natural logarithm proves $u_n>0$. For $n>1$, the Taylor series givesSince $0<(k-1)/[2k(k+1)]<1/2$, a geometric series bounds $u_n<1/[2n(n-1)]\le n^{-2}$ for $n\ge 2$. The comparison test for series therefore proves $\sum u_n<\infty$. The finite telescoping sum $\ln a_{n+1}=-1+{\sum }_{k=1}^n{u}_k$ shows that $a_n$ increases to a finite positive limit. This argument establishes the existence of the constant in Stirling's formula; its value $1/\sqrt{2\pi }$ requires an independent identification.

A star used in the Galactic-centre stellar-orbit experiments. The two designations refer to the same object in the SIMBAD catalogue.^[ref]

Rainer Schödel, Reinhard Genzel and collaborators combined NTT imaging with VLT adaptive-optics infrared observations. Silicon-monoxide masers registered stellar images to the radio position of Sagittarius A*. Their 2002 observations included S2's closest approach; a fitted Kepler orbit constrained a concentrated central mass. Rejecting a particular extended cluster also required its density and survival assumptions.^[ref]

Andrea Ghez and collaborators combined Keck speckle imaging and laser-guide-star adaptive optics with infrared spectra. Image fitting supplied angular positions; spectral lines supplied velocities. The 2008 analysis treated optical distortion, coordinate alignment and blended sources: a shifting photocentre can imitate motion. Joint angular and spectral fitting constrained both distance and mass, with assumptions about central radial motion affecting the result. Apparatus calibration and confusion tests are therefore part of the dynamical inference.^[ref]

Consider a negligible-mass star moving in the stationary Newtonian gravitational potential of a point mass. Let its semi-major axis be $a$, orbital eccentricity $0\le e<1$, and orbital period $P$. Write $\mu =GM$ for the gravitational parameter. The eccentric anomaly $E$ is determined by Kepler's equation,where $\tau$ is the time of periapsis. Coordinates in the orbital plane, with the point mass at the origin, areThe focus is the central mass; the ellipse's geometric centre is displaced by $-ae$ along the first coordinate. The varying denominator makes the star move faster near periapsis, even though the mean anomaly increases uniformly.

Searching many possible signal locations increases the chance of finding an apparently unusual fluctuation somewhere. A local p-value for a specified location differs from the global p-value for the search over locations.^[ref]

For a known background rate $b$ observed for exposure $T$, independent events give a Poisson distribution with mean ${\mu }_b=bT$. An excess-count test of the background-only null hypothesis uses the upper-tail P-value in the title. The tail, rather than the probability of exactly $n$, counts all outcomes at least as discrepant in the chosen direction.

This probability is conditional on the null model; it is not the posterior probability that the null model is true. A stipulated discovery threshold is met only when the tail is below that threshold. Uncertainty in the background rate or selecting among many searches needs additional modelling rather than being absorbed into the same known-mean formula.

Under the Jeans swindle, longitudinal density disturbances in a static uniform gas have ${\omega }^2=c_s^2{k}^2-4\pi G{\rho }_0$. For $0<k<k_J=\sqrt{4\pi G{\rho }_0}/c_s$, one solution grows as $e^{s(k)t}$ with the displayed rate. It decreases monotonically with wavenumber, so the longest available wavelengths grow fastest and the growth-rate supremum is $\sqrt{4\pi G{\rho }_0}$ as $k\to 0^{+}$. There is no preferred finite fastest wavelength in this model. The exact uniform $k=0$ density change is not an ordinary finite-potential Fourier perturbation of the fixed background, and finite-cloud boundaries restrict the available long modes.

For a standard linear solid model with $0<G_R<G_U$, put $x=\omega \tau$. Its storage modulus is $G^{\prime }=G_R+(G_U-G_R)x^2/(1+x^2)$ and its loss modulus is $G^{\prime \prime }=(G_U-G_R)x/(1+x^2)$. The latter peaks at $x=1$, whereas its loss tangent isDifferentiation puts the loss-tangent peak at $x=\sqrt{G_R/G_U}$, with height $(G_U-G_R)/(2\sqrt{G_R{G}_U})$. A loss-modulus peak and a loss-tangent peak therefore need not occur at the same frequency. For weak relaxation, $G_U/G_R$ is close to one and their frequencies are approximately equal.

An ideal linear torsion spring exerts a restoring torque proportional to the relative angle between its two ends. Integrating the torque gives its elastic potential energy. A spring connecting two rotating bodies contributes $\kappa ({\theta }_2-{\theta }_1{)}^2/2$ to the potential energy; a spring connected to a fixed frame is obtained by setting one angle to zero. The coefficient $\kappa$ is the torsional spring constant.

Spinodal decomposition is spontaneous growth of composition fluctuations in a homogeneous mixture whose local mixing free energy has negative composition curvature. For a mean $\varphi$, equal regions at $\varphi \pm \delta \varphi$ change the mean free energy by $f^{\prime \prime }(\varphi )(\delta \varphi {)}^2/2+O((\delta \varphi {)}^4)$, which is negative inside the spinodal. Interfacial costs suppress sufficiently short wavelengths, while conservation of composition constrains the dynamics. Outside the spinodal but inside the binodal, the uniform state is instead metastable and requires nucleation to begin separation.

A magnon is one quantum of a spin wave, a collective oscillation of magnetic order. Exchange-coupled spins do not fluctuate independently: a disturbance spreads through the ordered system as a mode with a definite wavevector and energy.

In a simple collinear Heisenberg ferromagnet, a single magnon reduces the total spin projection along the ordered axis by one unit of $\hslash$ while distributing the disturbance across many sites. At low excitation density, magnon modes can be treated approximately as bosonic oscillators, motivating the Holstein–Primakoff transformation and linear spin-wave approximation. Their population contributes to the reduction of magnetization with increasing temperature. Strong interactions, noncollinear order and large excitation densities can require corrections to the simple independent-magnon picture.

Even a shared field does not cancel if it is sampled at different times. Write $f_A(t)=g_{A}C(t)$ and $f_B(t)=g_{B}C(t)$, where $C(t)={\mu }_{B}B(t)/h$. As an original proposed diagnostic, bracket the $B$-state reading at $t=0$ with $A$-state readings at $t=\pm \tau$:If $C(t)=C_0(1+bt)$, the denominator is exactly $g_A{C}_0$, so $R_{\mathrm{bracket}}=g_B/g_A$. A single earlier $A$ reading instead gives $R_{\mathrm{unpaired}}=(g_B/g_A)/(1-b\tau )$. For the preceding numerical model and $b\tau =0.001$, this unpaired comparison falsely gives approximately $a_e=0.00186649$; interpolation recovers $0.0012$. The sequence illustrates cancellation and is not asserted to be Kusch and Foley's exact historical timing protocol.

For a curved field $C(t)=C_0(1+bt+c{t}^2)$, the same calculation givesThe linear drift cancels but the curvature remains. Different fields sampled by the two beams, additive frequency offsets and incorrect atomic corrections also survive. For a small ratio error $\delta R$, direct differentiation of the inverse relation givesRepeated readings can reduce random measurement uncertainty; agreement among genuinely different state comparisons is what tests shared systematic errors in their interpretation.

A one-way speed of light measurement assigns a travel time between spatially separated emission and reception events and therefore requires a distant-clock synchronization convention. In an inertial frame, Einstein synchronisation assigns the vacuum value $c$ in either direction. A completed-loop comparison such as the Sagnac effect is a different observable.

A spectrophotometer measures radiant power in selected wavelength bands. Comparing a chemical solution in a cuvette with a reference blank gives transmittance and absorbance.

A sample holder providing a defined geometric light path through a chemical solution for spectrophotometry, as in the trypsin colorimetric assay.

The transverse optical coherence length summarizes the separation over which equal-time spatial optical coherence remains appreciable. It depends on a contrast criterion and on the source brightness profile. For a distant incoherent source of angular diameter $\Theta$, its characteristic scale is $\lambda /\Theta$. A uniform circular source gives uniform-disk stellar visibility $2J_1(x)/x$, with $x=\pi b\Theta /\lambda$ and baseline $b$. Its first zero is $b=1.21967\lambda /\Theta$. A uniformly bright one-dimensional strip instead has its first zero at $\lambda /\Theta$; the numerical factor is a source-model choice, not a universal constant.

Take a star graph $G$ with center $c$ and four leaves $1,2,3,4$. Let $L_G$ be its Laplacian matrix and $I$ the identity matrix. In the order $(c,1,2,3,4)$, the original example isFor a nonzero vector $u$, its quadratic form isThus $A$ is a symmetric positive-definite matrix and either ordering below admits a Cholesky decomposition. Changing the ordering changes the work, rather than the mathematical solvability.

Center first. Eliminating $c$ with pivot element $5$ leaves the Schur complementEvery pair of leaves now has coupling $-1/5$: six new edges, hence six undirected fill-in positions. The four remaining vertices form a clique. In the Cholesky decomposition, the first column has $C_{cc}=\sqrt{5}$ and $C_{ic}=-1/\sqrt{5}$. The rest of the lower triangular matrix is dense. To check that later cancellation does not remove its entries, after $r$ leaves have also been eliminated, the active leaf block isIndeed a leaf step sends $\alpha$ to $\alpha +{\alpha }^2/(2-\alpha )$. All active off-diagonal entries remain negative. The successive pivot elements in ordinary symmetric Gaussian elimination, equivalently the squared diagonals in the Cholesky decomposition, areThe Cholesky decomposition therefore stores $5+4+3+2+1=15$ nonzero entries, including its diagonal.

Leaves first. Permute to $(1,2,3,4,c)$. Each leaf has only one remaining neighbor, so its elimination changes only the center's diagonal, reducing it by $(-1{)}^2/2=1/2$. The successive pivot elements are $2,2,2,2,3$, and the exact Cholesky decomposition isThere is no fill-in, and only $5+4=9$ nonzero entries in $C$. This is a concrete instance of leaf elimination of a tree-pattern positive-definite matrix; its general proof explains why the same idea works on every tree pattern. The minimum degree algorithm also chooses a leaf at every nontrivial stage of this particular example, but that success gives no general optimality guarantee.

The input has 13 nonzero matrix elements when both symmetric halves are counted, or nine in its lower triangle. The six added undirected edges correspond to six added positions in one triangular factor, or 12 positions if both symmetric halves of the filled pattern are displayed. Comparing all 13 input entries directly with the 15 entries of $C$ would mix counting conventions.

Solar neutrino detection asks how an invisible electron neutrino from the Sun becomes a measurable laboratory record. The Homestake experiment stores a reaction product for later counting; Kamiokande records light promptly. The former measures an exposure-integrated rate, whereas the latter can retain event times and statistical source-direction information. These are complementary ways to connect stellar nuclear fusion with neutrino astronomy.

A detector samples a weighted part of a neutrino flux, rather than counting every passing neutrino. For a target population $N_T$, an ideal reaction rate iswhere ${\varphi }_e(E)$ is the differential electron neutrino neutrino flux and ${\sigma }_e(E)$ is the relevant cross section. A detection efficiency belongs inside the integral if it depends on energy. Two detectors can therefore disagree in raw count rate while observing the same source. Interpreting a deficit requires both a source prediction and a detector response.