For a centered record of length $T$, the periodogram is the squared modulus of its discrete Fourier transform, with normalizationUnder suitable short-memory assumptions its expectation approaches the spectral density of a stationary process, but its variance generally does not vanish. Smoothing nearby frequencies can provide an estimate with statistical consistency.

A centered weakly stationary process has a representation $X_t={\int }_{-1/2}^{1/2}{e}^{2\pi it\omega }dZ(\omega )$ with orthogonal random increments. Their variance measure is the spectral measure. When it has density $f$, the autocovariance is $\gamma (h)=\int e^{2\pi ih\omega }f(\omega )d\omega$.

In cycles per observation, frequency lies in $[-1/2,1/2]$ and $\gamma (h)=\int e^{2\pi ih\omega }{f}_c(\omega )d\omega$. Compared with angular-frequency density $f_a$, one has $f_c(\omega )=2\pi f_a(2\pi \omega )$. In particular, white noise of variance ${\sigma }^2$ has $f_c={\sigma }^2$.

The specific heat capacity at constant volume is the temperature derivative of specific internal energy at fixed density and composition: $c_V=(\partial u/\partial T{)}_{\rho }$. It is the corresponding extensive heat capacity at constant volume divided by material mass.

In the free-slip Stokes flow temperature model, let the critical temperature be $g=\sin \pi z$ and vertical velocity $f=sg$, $s=k_c^2+{\pi }^2$. A forced positive second harmonic $(W,\Theta )e^{2i{k}_{c}x}$ couples to the negative critical harmonic. Its temperature solvability condition has integrandThe identity follows by substituting $f=sg$ and differentiating. Its integral is zero because $g$ vanishes on both plates, regardless of the forced boundary value of $\Theta$. Thus the first resonant forcing coefficient vanishes. A generic nonzero conjugate-amplitude term cannot be inferred from wave-number matching alone.

For real $\mu ,\beta$ and $\lambda >0$, writing $A=x+iy$ gives a gradient flow with potential $V=-(\mu +\beta )x^2/2-(\mu -\beta )y^2/2+\lambda (x^2+y^2{)}^2/4$. The origin has exponential asymptotic stability for $\mu <-|\beta |$, is algebraically attracting at equality, and is unstable above it. Nonzero stable equilibria are the real pair $\pm \sqrt{(\mu +\beta )/\lambda }$ for $\beta >0$, or the imaginary pair $\pm i\sqrt{(\mu -\beta )/\lambda }$ for $\beta <0$, when their squared amplitudes are positive. Differentiating the two real equations gives eigenvalues $-2(\mu +\beta ),-2\beta$ on the real branch and $-2(\mu -\beta ),2\beta$ on the imaginary branch. For $\beta =0$, $\mu >0$ gives a radially attracting circle with neutral phase, so individual equilibrium points have Lyapunov stability but are not individually asymptotically attracting.

For $\mathrm{Re}p>0$, let $r_{-}$ and $r_1,r_2$ be the characteristic roots of $r^3+r+p=0$, split by the signs of their real parts. Put $d_j=3r_j^2+1$. The real-line Green function of $p+{\partial }_x+{\partial }_x^3$ isThe reciprocal-polynomial root identities imply continuity of $R_p,R_p^{\prime }$ and jump $R_p^{\prime \prime }(0+)-R_p^{\prime \prime }(0-)=1$. The distributional derivative therefore gives $(p+{\partial }_s+{\partial }_s^3)R_p={\delta }_0$. The transport term changes the roots: the pure Airy resolvent kernel with $r^3+p=0$ cannot be substituted unchanged.

For a unit timelike normal $n$, the induced metric is $h_{\mu \nu }=g_{\mu \nu }+n_{\mu }{n}_{\nu }$. The spatial covariant derivative projects the derivative and all free indices with the spatial projection tensor $P^{\mu }{}_{\nu }={\delta }^{\mu }{}_{\nu }+n^{\mu }{n}_{\nu }$. Metric compatibility gives ${\nabla }_{\lambda }{h}_{\mu \nu }=({\nabla }_{\lambda }{n}_{\mu })n_{\nu }+n_{\mu }{\nabla }_{\lambda }{n}_{\nu }$; projecting kills the normal factor in each term. Thus the spatial covariant derivative is compatible with the induced metric.

The projection $[f{]}_z\mapsto z$ is a local biholomorphism for the sheet topology on the space of germs of holomorphic functions. It need not be a topological covering map: branches may end at singularities even when another branch remains over that base point.

This stronger source condition in variational regularization requires a subgradient $w=K^{\ast }Kv$ at $u^{†}$. For each $\alpha >0$ it is equivalent to making $u^{†}$ minimize $\Vert Ku-K\bar{u}{\Vert }^2/2+\alpha J(u)$ for artificial input $\bar{u}=u^{†}+\alpha v$. The subgradient optimality condition proves both implications. The source vector may be zero: $K=I$ and $J\equiv 0$ satisfy the condition but admit no nonzero source vector.

A dictatorship always selects the top alternative of one fixed agent $d$. Other agents cannot alter the selection against that agent's preference. This is the social-choice meaning of dictatorship, rather than a political classification.

With an odd number of agents and two alternatives, select the alternative ranked first by a strict majority. The rule is onto, strategyproof, and for at least three agents has no dictator in social choice. A pivotal agent already gets its preferred alternative by reporting truthfully; a nonpivotal agent cannot change the outcome alone.

For a continuously differentiable function $F$ on an interval $I=[t-\delta /2,t+\delta /2]$,To prove it, average $|F(t){|}^2-|F(u){|}^2$ over $u\in I$ using the fundamental theorem of calculus. On either half of $I$, the resulting derivative kernel has absolute value at most $1/2$. Since $|(|F{|}^2)\prime |\le 2|FF\prime |$, the displayed bound follows.

Assume the scaling function Fourier transform and the MRA low-pass filter are $C^{p-1}$, and $m^{(k)}(\pi )=0$ for $k<p$. For every nonzero integer $j$, write $j=2^r\ell$ with $\ell$ odd. Iterating the scaling refinement equation at $2\pi j+t$ supplies a factor $m(\pi \ell +t/2^{r+1})$. All its derivatives of order less than $p$ vanish at $t=0$, by periodicity. The product rule gives ${\hat{\phi }}^{(k)}(2\pi j)=0$ for $k<p$.

For Schramm–Loewner evolution with $\kappa =4$ and $Z_t=g_t(z)-2W_t$, the Itô formula for the complex logarithm gives $d\log Z_t=-2Z_t^{-1}d{W}_t$ before swallowing. The drift cancels exactly at this parameter. Its real part is $\log |Z_t|$ and its imaginary part is the SLE4 angle martingale. The real-part quadratic variation is $4\int X_t^2/|Z_t{|}^{4}dt$, which also gives a direct non-swallowing argument.

A point on $F=0$ where all first partial derivatives vanish. For a reduced hypersurface this is the Jacobian criterion; a nondegenerate quadratic leading part defines an ordinary double point.

The singular chain group is the free abelian group on all continuous maps ${\Delta }^q\to X$, the singular simplices. In particular it is free even when the set of simplices is infinite. This freeness makes applying $\mathrm{Hom}(C_q(X),-)$ to a short exact coefficient sequence exact.

The singular cardinals hypothesis asserts that every infinite singular cardinal $\kappa$ satisfiesEquivalently, $\gimel (\kappa )=\max \{{\kappa }^{+},2^{\mathrm{cf}(\kappa )}\}$ for every infinite singular cardinal. Here $\gimel$ is the gimel function. The Generalized continuum hypothesis implies this principle, but the principle only constrains the indicated singular-cardinal exponentiation.

Write ${\sigma }_{\alpha }$ for the $\alpha$th uncountable singular cardinal in increasing order. It begins with ${\sigma }_0={\aleph }_{\omega }$ and ${\sigma }_1={\aleph }_{\omega +\omega }$. At a nonzero limit ordinal index it is continuous exactly when the supremum of its earlier values is singular. It can jump when that supremum is a weakly inaccessible cardinal. The first member of uncountable cofinality occurs at index ${\omega }_1$, with value ${\aleph }_{{\omega }_1}$.

For the bounded continuous solutions in the sine-mode expansion for a quadratically degenerate elliptic equation, every surviving mode tends to zero at $y=0$. The estimate $p_n^{+}>n$ bounds the absolute sum by a constant times $y/(1-y)$, uniformly in $x$. The lower boundary trace must therefore be zero. Continuous boundary data prescribing a nonzero lower trace and zero vertical sides are incompatible with the equation.