A bound on ${\sup }_x\mathbb{P}(|Z_n(x)|>t)$ does not give the same bound on $\mathbb{P}({\sup }_x|Z_n(x)|>t)$. Even independent standard normal random variables each have uniformly bounded tails while their Gaussian maximum over $N$ indices grows at scale $\sqrt{\log N}$. This distinction matters in nonparametric statistics when a shrinking bandwidth permits many spatial windows.

Write a positive continuous local martingale as $M=\exp (X-⟨X⟩/2)$ using its stochastic logarithm. If $M_t\to 0$, its logarithmic bracket must diverge. Otherwise the finite-bracket convergence lemma makes $X$ converge finitely and the exponential has a positive limit.

For independent Bernoulli distribution upper-triangular entries of a symmetric zero-diagonal adjacency matrix of a graph, put $W=A-\mathbb{E}A$. For a unit vector $x$, $x^{\top }Wx=2{\sum }_{i<j}{x}_i{x}_j{W}_{ij}$ has sub-Gaussian variance proxy at most ${\sum }_{i<j}{x}_i^2{x}_j^2\le 1/2$, by the Hoeffding lemma. Hence $\mathbb{P}(|x^{\top }Wx|>u)\le 2e^{-u^2}$. The volumetric bound for Euclidean metric nets gives a $1/4$-metric net of the unit sphere with at most $9^n$ points. The quadratic form net bound gives $\Vert W{\Vert }_{\mathrm{op}}\le 2{\max }_x|x^{\top }Wx|$ on that net. The union bound gives $\mathbb{P}(\Vert W{\Vert }_{\mathrm{op}}>2\sqrt{n\log 9+\log 2+s})\le e^{-s}$; integrating this tail proves the displayed expectation bound.

If every counted halo hosts one counted galaxy and the baryon fraction and conversion efficiency are mass-independent without scatter, galactic stellar mass is a constant multiple of halo mass. The galaxy mass function is obtained by changing variable with $dM/d{M}_{\ast }=1/{ϵ}_{\ast }$. Satellites, efficiency variation and scatter prevent this mapping from being generally exact.

Small-angle gravitational encounters occur at rate $2\pi bdbnu$ and supply kicks $2Gm/(bu)$. Independent kick variances add to $d⟨\Delta v^2⟩/dt=8\pi G^2{m}^{2}n\ln \Lambda /u$. Setting the accumulated variance to $u^2$ and using virial scaling gives a stellar relaxation time of order $0.1N/\ln N$ stellar crossing times, with the coefficient dependent on profile and crossing convention.

For an isolated quasistatic cluster with near-zero escape energies, the bound negative energy is approximately fixed. The virial theorem and gravitational radius then imply $R_g\propto M^2$. A homologous mean density scales as $M^{-5}$ and increases as mass evaporates. Tidal limitation or finite positive escape energies modifies this simple model.

For independent Steinhaus random variables, condition on their acute angles and use the sharp Rademacher second-moment inequality on the $2d$ sign coefficients $a_i\cos {\theta }_i,i{a}_i\sin {\theta }_i$. Their squared moduli sum to ${\sum }_i|a_i{|}^2$ independently of the angles, so the conditional first moment is at least the same fixed square-root bound. Averaging proves the assertion.

For odd prime $p$, the reduced powers are natural stable cohomology operations. They satisfy $P^0=1$, the Cartan formula, and the instability conditions $P^{i}a=0$ for $2i>|a|$ and $P^{i}a=a^p$ for $|a|=2i$. In particular, a degree-two class has $P^{1}x=x^p$ and no higher nonzero reduced power. Stability lets these operations detect distinctions between suspensions whose additive cohomology and cup products agree.

The Cartan product formula says that total Steenrod squares and total Steenrod reduced powers preserve the cup product:It is named after Henri Cartan and is distinct from Cartan's magic formula for the Lie derivative of a differential form, named after Élie Cartan.

These mod-two identities rewrite nonadmissible products of Steenrod squares. For example, ${\mathrm{Sq}}^1{\mathrm{Sq}}^1=0$, ${\mathrm{Sq}}^1{\mathrm{Sq}}^2={\mathrm{Sq}}^3$, and ${\mathrm{Sq}}^1{\mathrm{Sq}}^3=0$. In general the sum ranges over $0\le t\le \lfloor a/2\rfloor$ and coefficients are reduced modulo two.

An index sequence obeying the displayed inequalities represents an admissible product ${\mathrm{Sq}}^{i_1}\cdots {\mathrm{Sq}}^{i_r}$. Its degree increment is the sum of the indices. The Adem relations rewrite every square monomial in terms of admissible ones, whose Steenrod excess controls universal instability.

For positive $q,K,g^{\prime }$ in a steady depositing gravity current, the function $K(h)=q^2/h+g^{\prime }{h}^2/2$ has a unique minimum at $h_c=(q^2/g^{\prime }{)}^{1/3}$, where $\mathrm{Fr}=1$ and $K_c=3q^{4/3}(g^{\prime }{)}^{1/3}/2$. When $K>K_c$ there are two positive depth roots: $h_{-}<h_c$ is supercritical flow and $h_{+}>h_c$ is subcritical flow. Differentiating the momentum invariant and using $g_x^{\prime }=-v_s{g}^{\prime }/q$ givesThe supercritical depth decreases towards $q^2/K$ as $g^{\prime }\downarrow 0$, whereas the subcritical depth increases like $\sqrt{2K/g^{\prime }}$. Because $K_c$ decreases downstream, an initially strict branch never reaches a finite-distance critical point. A critical inlet instead gives a singular depth derivative, so a smooth critical continuation is not justified by this model.

In a bidisperse particle suspension transported by a steady depositing gravity current, species with concentrations $c_i$ and downward settling velocity magnitudes $v_i$ satisfy $c_i(x)=c_{i0}{e}^{-v_{i}x/q}$. Their local particle deposition fluxes are $v_i{c}_i$. The local deposited particle-volume fraction belonging to species one isFor $v_1>v_2>0$ and both species present, $f_1$ decreases strictly downstream, since $f_1^{\prime }=(v_2-v_1)f_1(1-f_1)/q$. If $g_i^{\prime }=g_{0i}{c}_i$, use $c_i=g_i^{\prime }/g_{0i}$ rather than buoyancy ratios unless $g_{01}=g_{02}$. A number fraction additionally divides each species volume flux by its single-particle volume. Integrating deposition over $[0,x]$ instead gives cumulative weights $q{c}_{i0}(1-e^{-v_{i}x/q})$.

For $\Phi (s,t,\theta )=s\theta$, take a smooth symbol that turns on increasingly high frequencies in disjoint spatial bumps centered at $t_j\to 0$. Superpolynomially small bump coefficients keep all symbol class estimates valid. The amplitude vanishes near every finite-frequency point over $t=0$, yet its oscillatory integral has delta singularities at $(0,t_j)$. Closedness of singular support forces a singularity at $(0,0)$ as well. A bound using only ordinary, rather than closed conic, amplitude support can consequently fail.

For a complete-market investment-consumption problem with constant relative risk aversion utility, the nonlinear wealth-homogeneity coefficient equation may contain $f^{\prime 2}/f$. Writing $f=g^R$ cancels this gradient square against the one from $f^{\prime \prime }$. In the index-driven correlation model the result is ${\sigma }_0^2{g}^{\prime \prime }/2+B{g}^{\prime }-\delta (m)g+1=0$, a linear differential equation; the positive economic solution gives consumption $w/g$.

Swapping consecutive labels $i,i+1$ preserves a standard Young tableau exactly when their cells are incomparable in the row-and-column order. Their contents then differ by more than one in absolute value. These admissible swaps connect all standard tableaux of any fixed shape, as adjacent swaps connect linear extensions of a partially ordered set.

A Keplerian accretion disk with a zero-torque inner boundary condition and local blackbody cooling has $T_{\mathrm{eff}}^4=3GM\dot{M}(1-\sqrt{r_{\mathrm{in}}/r})/(8\pi \sigma r^3)$. The Stefan–Boltzmann law uses the flux through one face; the standard thin-disk dissipation flux includes both faces. The scale $T_{\mathrm{in}}=[3GM\dot{M}/(8\pi \sigma r_{\mathrm{in}}^3){]}^{1/4}$ is a normalization, not the actual temperature at the stress-free inner edge.

If an outer expansion contains $-(\epsilon /2)\log x$ and the layer coordinate is $x=\sqrt{\epsilon }z$, its overlap contains $-(\epsilon /2)\log z-(\epsilon /4)\log \epsilon$. Matching forces an inner expansion term $-(\epsilon /4)\log \epsilon \mathrm{erf}z$ when that homogeneous profile tends to one in the overlap and vanishes at the boundary. The logarithm multiplies the matching profile, rather than an arbitrary constant.

For closed hyperbolic surfaces of genus $g\ge 2$, there is a closed proper real-analytic exceptional subset of Teichmüller space such that a surface outside it is determined up to isometry by its unmarked length spectrum, equivalently its Laplace-Beltrami operator spectrum. The analytic part of the proof controls possible spectral matchings using finite determining length data; the geometric part recognizes persistent matchings as changes of marking, using the collar lemma and variations of Fenchel–Nielsen coordinates. Orientation reversal remains invisible to the spectrum.

For pure blackbody radiation, $P=a{T}^4/3$ depends only on temperature. Constant pressure therefore fixes temperature, so a usual finite constant-pressure temperature derivative is unavailable. In the material gas-radiation limit the specific-heat ratio diverges, although all three stellar adiabatic exponents approach $4/3$. The radiation index $4/3$ is an adiabatic pressure-volume response, not a finite heat-capacity ratio.