Choose an oriented embedded closed disc in a closed oriented positive-dimensional manifold and collapse its complement to a point. A target point away from the collapsed value has one preimage with local degree $+1$, proving global degree one by the degree as a sum of local degrees.

For $k>0$, an epimorphism ${\pi }_1({\Sigma }_2)\to \mathbb{Z}/k$ gives a connected $k$-sheeted cover. Its Euler characteristic is $-2k$, hence its genus is $k+1$, and the lifted orientation gives degree $k$. For negative $k$ compose the $|k|$-sheeted cover with an orientation-reversing homeomorphism of the target. For $k=0$ take a constant map from the torus.

Let $M,N$ be closed connected oriented manifolds of the same dimension, and suppose $a^r$ generates the top integral cohomology of $N$. If every class of degree $|a|$ on $M$ has zero rth cup power, every continuous map $f:M\to N$ has mapping degree zero. Indeed, $f^{\ast }(a^r)=(f^{\ast }a{)}^r=0$, and evaluation on the fundamental class gives $\mathrm{deg}f=0$. For $S^2\times S^4\to {\mathbb{CP}}^3$, the degree-two class is a multiple of the sphere class $u$, whose square vanishes.

A negligible-mass radiative envelope surrounding a core of mass $M_c$, carrying constant luminosity and having constant opacity, obeys $dP/dT=16\pi acG{M}_c{T}^3/(3{\kappa }_{0}L)$. Neglecting the outer pressure integration constant gives $\Lambda =4\pi acG{M}_c/(3{\kappa }_{0}L)$. The gas pressure fraction $\beta$ is constant because $P_{\mathrm{rad}}=a{T}^4/3$, and $\rho =\mu \beta \Lambda T^3/\mathcal{R}$. Hydrostatic balance gives $T=T_s+\mu \beta G{M}_c(1/r-1/R_s)/(4\mathcal{R})$; dropping this outer-boundary constant gives the displayed deep-envelope approximation. A positive gas-pressure fraction requires $L<4\pi cG{M}_c/{\kappa }_0$.

For a one-dimensional motion with velocity $v(t)$, total distance traveled is ${\int }_a^b|v(t)|dt$, while net displacement is ${\int }_a^{b}v(t)dt$. In several dimensions, replace absolute value by the Euclidean norm of the velocity vector; this is the arc length of the trajectory.

If $v(t)=t-1$ for $0\le t\le 2$, the body reverses direction at $t=1$. Its displacement is zero, but its total distance is ${\int }_0^1(1-t)dt+{\int }_1^2(t-1)dt=1$. Splitting at reversals makes the physical meaning of the integral visible.

The lower and upper integration limits $a,b$ specify the ordered endpoints of a definite integral ${\int }_a^{b}f(x)dx$. Interchanging the integration limits reverses its sign: ${\int }_b^{a}f(x)dx=-{\int }_a^{b}f(x)dx$.

A block has finite representation type exactly when its defect group is cyclic. In the forward implication, the block bimodule restricted to its defect group on both sides contains the regular defect-group bimodule as a summand; tensoring with arbitrary defect-group modules transfers finiteness to the defect group. Conversely, vertices are contained in the defect group and a cyclic $p$-group has only finitely many indecomposable modules, so there are only finitely many possible induced sources and their summands.

For a block idempotent $e$, an $e$-Brauer pair consists of a $p$-subgroup $P$ and a block idempotent $b_P$ of $k{C}_G(P)$ for which $b_P{\mathrm{Br}}_P(e)\ne 0$. The subgroup of a maximal pair is a defect group. A maximal pair's stabilizer defines the inertial quotient of the block.

The defect of a block of a group algebra is the integer $d$ for which a defect group of a block has order $p^d$. Conjugacy of defect groups of a block makes this independent of the choice. A block of a group algebra has defect zero when its defect group of a block is trivial.

For a maximal Brauer pair $(D,b_D)$ of a block of a group algebra, the inertial quotient is the displayed quotient of its pair stabilizer by $D{C}_G(D)$. It has order coprime to $p$, and its order is the inertial index of a block. For abelian $D$, conjugation embeds it in $\mathrm{Aut}(D)$.

Let $k$ be an algebraically closed field and let a block of a group algebra $B$ have a central defect group of a block $D$. Its quotient by $J(kD)B$ is a defect-zero block, hence a full matrix algebra over $k$. The algebra $B$ is free over the central local ring $kD$. Lift a primitive idempotent from this matrix quotient; its corner is free of rank one over $kD$, and is therefore $kD$. Lifting matrix units gives $B\cong {\mathrm{Mat}}_r(kD)$.

For a block of a group algebra with identity $e$ and defect group of a block $D$, the trace criterion for defect groups writes $e={\mathrm{Tr}}_D^G(a)$ with $a\in (RG{)}^D$. On any module in the block, multiplication by $a$ is $D$-linear and its relative trace is the identity. The D. Higman criterion therefore makes the module relatively $D$-projective.

Let $P$ be any Sylow subgroup containing a defect group of a block $D$. Regard the block of a group algebra as a permutation module summand for $G\times G$, acting by left and right multiplication. Its vertex is $\Delta D$. Restricting to $P\times P$ retains a summand with this vertex. Transitive permutation modules for a finite p-group are indecomposable, with their point stabilizers as vertices. Thus some orbit in $G$ has stabilizer $\Delta D$ at a suitably chosen point $c$. This means $c$ centralizes $D$ and $P\cap {}^{c}P=D$. Choosing $P$ to contain a Sylow subgroup of $N_G(D)$ shows that $D=O_p(N_G(D))$; every normal p-subgroup of $G$ is contained in both Sylow factors and hence in $D$.

For a block idempotent $b$ with defect $D$, the Brauer quotient of a module of its modular block bimodule at $\Delta D$ is nonzero: on the fixed basis $C_G(D)$, the block projection acts by ${\mathrm{Br}}_D(b)$. Restriction to $D\times D$ splits into transitive permutation modules; their stabilizers have order at most $|D|$. A nonzero quotient at $\Delta D$ therefore forces a summand with stabilizer exactly $\Delta D$, which is the regular $kD$ bimodule. Tensoring on the right with any $kD$-module $U$ gives $U\mid {\mathrm{Res}}_D^G(b{\mathrm{Ind}}_D^{G}U)$. Consequently finite representation type of the modular block forces finite representation type of $kD$.

Rows of the symmetric-group decomposition matrix are all ordinary shapes, columns are regular partitions, and $d_{\mu \lambda }=[S^{\mu }:D^{\lambda }]$. A nonzero column antisymmetrizer of shape $\lambda$ on $M^{\mu }$ forces dominance: the first $r$ rows contain at most ${\sum }_j\min (r,{\lambda }_j^{\prime })$ column entries. For a regular shape its diagonal simple appears once, giving a rectangular triangular matrix and a unitriangular regular-row submatrix.

If a finite group $G$ is generated by $h$ elements, send the $a$-generators of the genus-$h$ surface group to those generators and its $b$-generators to the identity. The surface relation is satisfied, so this is a surjective group homomorphism. Its kernel defines a connected regular covering space, on which the deck transformation group $G$ acts freely. The lifted orientation and multiplicativity of the Euler characteristic give genus $g=1+|G|(h-1)$.

A terminating decimal is a decimal representation with only finitely many nonzero fractional numerical digits. For a reduced fraction $a/q$ with $q>0$, this occurs exactly when every prime factor of $q$ is $2$ or $5$.

If the representation terminates after $k$ fractional digits, write $a/q=M/10^k$ for an integer $M$. Since $a,q$ are coprime integers, $q$ divides $10^k=2^k{5}^k$. This proves necessity. Conversely, if $q=2^u{5}^v$, choose $k=\max \{u,v\}$. Then $q\mid 10^k$, making $M=a10^k/q$ an integer and giving a finite representation.

For example, $3/40=0.075$ because $40=2^3\cdot 5$ divides $10^3$. The reduced fraction $1/6$ has a denominator factor $3$, so its decimal representation cannot terminate. A zero tail is distinct from a nonzero repeating block; the latter is measured by decimal period from multiplicative order.

Let $m=(f(b)-f(a))/(b-a)$ for a function differentiable on a neighborhood of $[a,b]$. If $m=k$, use the whole interval. If $m>k$, the continuous extension of $(f(x)-f(a))/(x-a)$ takes values $f^{\prime }(a)$ and $m$ at the endpoints, so the intermediate value theorem supplies a secant of slope $k$ starting at $a$. If $m<k$, use the continuous extension of $(f(b)-f(x))/(b-x)$ and obtain a secant ending at $b$. The mean value theorem then produces an interior derivative equal to $k$, without requiring a continuous derivative.

For underdamping, let $x_n,x_{n+1}$ be successive same-sign extrema of displacement relative to equilibrium, one full oscillation period apart. The logarithmic decrement isComparing extrema $N$ full cycles apart instead gives $\delta =N^{-1}\ln (|x_n|/|x_{n+N}|)$. Adjacent extrema of opposite sign are only half a cycle apart and give half this decrement. This distinction matters when inferring a damping ratio. Near critical damping, a full second same-sign extremum may be too small to resolve above measurement noise; a full-trace fit with uncertainty is then more informative than a peak ratio.

An ultrasonic motion sensor estimates distance from the return delay of an acoustic wave pulse reflected by a target. The teaching setup places a flat flag on the cart facing the detector above the air track, and specifies a minimum range of about $0.3\mathrm{m}$ for its detector. That apparatus-specific limit should not be assumed for every sensor model. ^[ref]