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A finite-measure set is a measurable set on which the ambient measure is finite. Restricting to such a set makes its indicator function belong to every Lp space with finite exponent, enabling local Radon-Nikodym theorem arguments even when the whole measure space is not sigma-finite.

Taking the real and imaginary parts of the finite geometric series ${\sum }_{m=1}^N{e}^{im\theta }$ yieldsThese formulas apply when $\theta \notin 2\pi \mathbb{Z}$. At multiples of $2\pi$, the sums are directly $N$ and $0$. Complex exponentials turn two real trigonometric summations into one finite geometric series.

For a finite-field Kakeya set $A\subseteq {\mathbb{F}}_q^n$, a smaller cardinality would give a nonzero polynomial of total degree of a polynomial at most $q-1$ vanishing on $A$, by the dimension of a bounded-total-degree polynomial space. Its restriction to each complete affine line in a vector space has $q$ roots of a polynomial and degree less than $q$, hence vanishes identically. The top homogeneous polynomial part then vanishes in every direction. A polynomial of degree less than $q$ in each variable cannot vanish everywhere on ${\mathbb{F}}_q^n$ unless it is zero, giving a contradiction. For the prime field, this is the polynomial nonvanishing below the field size.

A Hermitian operator on a $2^n$-dimensional Hilbert space, with $2^n$ distinct eigenvalues all belonging to $\{c/2^n:0\le c<2^n\}$, must occupy this entire grid. This follows from the pigeonhole principle: the set of possible values and the set of distinct eigenvalues have the same cardinality. Its minimum eigenvalue is zero, so the operator is necessarily singular. Distinctness therefore does not imply invertibility; it forces a one-dimensional kernel in this specific complete-grid setting.

A filtered colimit in a category is a colimit whose indexing category is a small filtered category. In the Category of sets, filtered colimits commute with finite limits in sets. A filtered colimit of modules is an example of this general categorical construction.

For a dominant morphism of irreducible varieties, generic fiber dimension is source dimension minus target dimension, and this dimension occurs over a nonempty open subset. In particular, constant fiber dimension $d$ gives source dimension equal to target dimension plus $d$. For algebraic groups the equal-dimensional components and coset fibers allow the same dimension formula componentwise.

The kinetic electron Fermi temperature is $T_F=[\sqrt{m_e^2{c}^4+p_F^2{c}^2}-m_e{c}^2]/k_B$, where the Fermi momentum $p_F=\hslash (3{\pi }^2{n}_e{)}^{1/3}$ counts the net electrons. Strong degeneracy requires $T\ll T_F$. With mean molecular weight per electron ${\mu }_e$ and $n_e=\rho /({\mu }_e{m}_u)$, $T_F=(m_e{c}^2/k_B)[\sqrt{1+(\rho /{\rho }_{\ast }{)}^{2/3}}-1]$. Its nonrelativistic and ultrarelativistic limits scale as ${\rho }^{2/3}$ and ${\rho }^{1/3}$, respectively.

For a Markov process generator and suitable real functions, this bilinear expression records the local quadratic energy. Positivity of $\Gamma (f,f)$ follows by differentiating $P_t(f^2)\ge (P_{t}f{)}^2$. For $L=D^2-xD$ it is $f^{\prime }{g}^{\prime }$, giving the Gaussian Dirichlet energy after integration against the invariant measure.

For a gauge condition $\chi [A]=0$, the linearization of $\chi [A+D\omega ]$ in the gauge parameter is the Faddeev-Popov operator. Its determinant is represented by Faddeev-Popov ghost fields.

For equal oscillation amplitudes $\delta$, frequency $\omega$, mean separation ${\ell }_0$ and phase lag $\varphi$, the leading mean total force on the fluid is $9\pi \mu a_1{a}_2{\delta }^2\omega \sin \varphi /{\ell }_0^2$. Each sphere supplies half. The large-separation calculation expands the Stokeslet interaction; reciprocal in-phase and antiphase motions have zero mean force.

The centre separation for equal-amplitude oscillations has minimum ${\ell }_0-2\delta |\sin (\varphi /2)|$. A bound ${\ell }_0\ge \delta$ alone cannot guarantee nonoverlap. An asymptotic mobility calculation needs this minimum much larger than the sphere radii.

A grain with radiation-pressure coefficient $\beta <1$ feels effective central gravitational parameter $G{M}_{\star }(1-\beta )$. Combining its mean motion with the planet's mean motion at an exterior-type $(j+1):j$ frequency commensurability gives the displayed semi-major axis. The commensurability lies geometrically exterior only for $j<\sqrt{1-\beta }/(1-\sqrt{1-\beta })$. Radiation changes the grain's attraction, so the same frequency ratio need not imply the usual radial ordering.

The nominal semi-major axis of an exterior $(p+q):p$ mean-motion resonance follows from $(p+q)n_r=p{n}_p$ and Kepler's third law. Apsidal precession displaces the actual libration center because stationarity of the resonant argument requires $(p+q)n-p{n}_p-q\dot{\varpi }=0$.

To add two extensions of $M$ by $N$, take their direct sum, form a pullback in a category along the diagonal $M\to M\oplus M$, and form a pushout in a category along addition $N\oplus N\to N$. This produces another extension of $M$ by $N$ and defines addition in ${\mathrm{Ext}}_R^1(M,N)$. In a quiver representation, vertexwise splittings express the two extensions by off-diagonal arrow matrices, and the Baer sum adds those matrices modulo the extension complex of quiver representations coboundaries.

Restriction at one arrow induces a surjection ${\mathrm{Ext}}_Q^1(X,Y)\twoheadrightarrow {\mathrm{Hom}}_k(\ker f_{\rho },\mathrm{coker}{g}_{\rho })$. Coboundaries vanish on the kernel after quotienting by the image, while every map between these spaces can be lifted and extended. Nonzero kernel and cokernel therefore force a nonsplit extension. For $X=Y$, rigidity forces every arrow to have maximal rank.

For an ideal generated by a finite regular sequence in a commutative ring, the self-Ext functor algebra is exterior on the degree-one dual generators. Contraction chain maps on the Koszul resolution represent those generators, square to zero, and anticommute. The exterior square-zero relations also apply in characteristic two.

For an adapted continuous process starting at zero, the displayed conditional characteristic function identifies every increment as $N(0,t-s)$ and makes it independent of the preceding sigma-field. Multiply by the indicator of an event in that sigma-field, then use the uniqueness theorem for characteristic functions for finite measures. Thus the process is a Brownian motion in the given filtration.

Let $K(\Delta )={\max }_t\#\{r:\Vert {\theta }_r-t\Vert \le \Delta /2\}$. Without a separation assumption,For $0<\Delta \le 1$, each point belongs to at most $K(\Delta )$ integration arcs in the Sobolev–Gallagher inequality. For $\Delta \ge 1$, $K(\Delta )$ is the total number of points, and the Cauchy-Schwarz inequality bound $|S{|}^2\le N\sum |a_n{|}^2$ suffices.

Since the differential of the Exponential map of a Lie group at zero is the identity, the inverse function theorem makes it a diffeomorphism on a sufficiently small neighborhood of zero. Its local inverse is a logarithm near the group identity. It need not extend to a global logarithm, even for the real general linear group.

A convex normal neighbourhood is an open set whose points have unique smoothly varying joining geodesics, with the joining geodesic contained in the set. The corresponding star-shaped restriction of each exponential map is a diffeomorphism onto the neighbourhood. Hence its differential is invertible and no such joining geodesic admits a nonzero Jacobi field vanishing at both endpoints. Every point of a Riemannian manifold has arbitrarily small such neighbourhoods. This is stronger than merely requiring the existence of a minimizing geodesic in the set.