For left Haar measure, $T_{x}f(t)=f(x^{-1}t)$ is an isometry on $L^p(G)$, $1\le p<\infty$. For compactly supported continuous functions, uniform continuity on a slightly enlarged compact support proves norm continuity. Approximation by those functions and the triangle inequality extend it to all $L^p$. The continuous image of a compact set of translation parameters is consequently compact in $L^p$.

On $[0,1]$, the continuous functions $f_n(x)=\sin (nx)$ do not form a Cauchy sequence in the Lp space with $p=1$. Put $d_n=f_n-f_{2n}$. The identity ${\int }_0^1{d}_n^{2}dx=1-\sin (2n)/(4n)-\sin (4n)/(8n)-\sin n/n+\sin (3n)/(3n)$ tends to $1$. Since $|d_n|\le 2$, one has ${\int }_0^1|d_n|dx\ge \frac{1}{2}{\int }_0^1{d}_n^{2}dx$, which stays bounded away from zero. Thus arbitrarily late pairs are separated in the integral norm; in particular the sequence cannot converge in that norm.

The sum consists of measurable functions $f=g+h$ with $g\in L^{p_0}$ and $h\in L^{p_1}$, with norm ${\inf }_{f=g+h}(\Vert g{\Vert }_{p_0}+\Vert h{\Vert }_{p_1})$. It is a Banach space. One construction identifies it with the quotient of $L^{p_0}{\oplus }_1{L}^{p_1}$ by the closed kernel of the addition map. Bounded endpoint operators are therefore bounded on the corresponding sum spaces.

The Lp space for $p=1$ consists of equivalence classes of integrable functions, with L1 norm $\Vert f{\Vert }_1=\int |f|d\mu$. Conditional expectation is a contraction in this norm. Convergence in this space is L1 convergence, which controls the change in every integral over a measurable subset.

Assign a real unit vector $u_v$ to each graph vertex, with $u_v\perp u_w$ whenever distinct vertices are nonadjacent. A handle is a further unit vector $h$. In the handle formulation of the Lovász number, one minimizes ${\max }_v|h\cdot u_v{|}^{-2}$ over representations and handles. Orthonormal here does not mean every pair of assigned vectors is orthogonal.

Let $B_v$ be independent Bernoulli random variables on a rooted $d$-ary tree, and put $A_v=\{B_v=1,B_w=0\text{ for every child }w\}$. Events at nonadjacent vertices use disjoint variables, giving a dependency graph of events of maximum degree $d+1$. Assign leaf success probability $p$ and successive probabilities $r_{j+1}=p/(1-r_j{)}^d$. If $p>{\max }_{0\le r\le 1}r(1-r{)}^d=p_d$, the recursion eventually reaches one. Set the final root success probability to one. Each event has probability at most $p$, but following success vertices down from the root must reach an occurring event. Thus the bad events cover the sample space. This gives an upper bound for uniform avoidance thresholds without assuming a converse to the Lovász local lemma.

In an independent random two-colouring of $K_N$, a given $k$-vertex set is monochromatic with probability $2^{1-\left(\genfrac{}{}{0ex}{}{k}{2}\right)}$. Events on sets intersecting in at most one vertex use disjoint edges. Thus dependency degree is at most $\left(\genfrac{}{}{0ex}{}{k}{2}\right)\left(\genfrac{}{}{0ex}{}{N-2}{k-2}\right)$. The Lovász local lemma gives a colouring avoiding all monochromatic $K_k$ when $e{2}^{1-\left(\genfrac{}{}{0ex}{}{k}{2}\right)}(1+(\genfrac{}{}{0ex}{}{k}{2}\left)\right(\genfrac{}{}{0ex}{}{N-2}{k-2}))\le 1$. For $N=ck{2}^{k/2}$ the expression is $O(k^{3/2}(ec/\sqrt{2}{)}^k)$, by Stirling's approximation. Taking $c$ tending sufficiently slowly to $\sqrt{2}/e$ proves $R(k)\ge (\sqrt{2}/e+o(1))k{2}^{k/2}$.

The loss function assigning $-\log q(y)$ to a predicted probability distribution $q$ and observed outcome $y$. It is the negative logarithmic scoring rule; its expected value under the true probability distribution is cross-entropy. Predicting zero probability for an observed outcome gives infinite loss.

For homogeneous Stokes flow in a smooth bounded volume, use $J_{ij}(r)=({\delta }_{ij}/|r|+r_i{r}_j/|r{|}^3)/(8\pi \mu )$ and $K_{ijk}(r)=-3r_i{r}_j{r}_k/(4\pi |r{|}^5)$. The Lorentz reciprocal theorem with a point-force Stokeslet gives the displayed representation, with $c=1$ inside, $0$ outside and $1/2$ on a smooth boundary. The double-layer integral at the boundary has its limiting principal-value meaning. In an exterior problem the body's outward normal is opposite to the fluid-domain normal, reversing both boundary terms; a nonzero ambient flow adds its far-boundary contribution.

Use a rotlet as the auxiliary Stokes flow, with a rigid sphere at its center. Its exterior-domain boundary torque is the imposed auxiliary couple. In the Lorentz reciprocal theorem, the actual sphere's zero torque removes the other surface term, while the volume term is the rotlet dotted with the body force. Varying the auxiliary couple gives the displayed angular velocity, assuming the integrals and the far boundary limit converge.

For $r\ge 3$, the path-loop Serre fibration has contractible total space. The relative homology of a fibration over a sphere gives $H_j(\Omega S^r)\cong H_{j-r+1}(\Omega S^r)$ for $j>0$, with negative homology zero and $H_0=\mathbb{Z}$. This proves the displayed groups. The unit map $S^{r-1}\to \Omega S^r$, adjoint to the identity of $S^r$, represents the generator of the first nonzero homotopy group. Naturality of the Hurewicz theorem shows that it induces an integral homology isomorphism in all degrees through $2r-3$.

The based path space $P_{x_0}X$ consists of continuous paths beginning at $x_0$, with variable endpoint and the compact-open topology. Endpoint evaluation is a Serre fibration with fiber the based loop space. Its total space is contractible by $\gamma (t)\mapsto \gamma (st)$ as $s$ decreases to zero. This differs from a space of paths with both endpoints fixed, which need not be contractible. The long exact sequence of homotopy groups of a fibration gives ${\pi }_i(\Omega X)\cong {\pi }_{i+1}(X)$ for connected $X$ in the usual based ranges.

For based continuous maps $\alpha :S^p\to G$ and $\beta :S^q\to G$ into a topological group, take the pointwise commutatorIt is exactly the identity on $S^p\vee S^q\subseteq S^p\times S^q$, so it descends to the smash product $S^p\wedge S^q\cong S^{p+q}$. Its homotopy class is the Samelson product $⟨\alpha ,\beta ⟩\in {\pi }_{p+q}(G)$. This measures failure of the two maps to commute; it vanishes for an abelian group.^[ref]

For an ordinary loop space, reversal is an inverse up to homotopy, and a retraced loop need not be the constant loop. Literal descent therefore requires a strict topological group model or the corresponding coherent contractions, rather than treating loop reversal as a strict inverse. Equivalently, suspension-loop adjunction defines the operation through the sign-adjusted Whitehead product in the original space.

In positive degrees these operations give the graded antisymmetry and Jacobi identity of the loop-space homotopy Lie algebra. Under the Hurewicz homomorphism, the product becomes the graded commutator in the Pontryagin ring.

In a scalar field path integral with weight $\exp [i(S+\int J\varphi )/\hslash ]$, each internal propagator contributes $\hslash$, each interaction vertex contributes ${\hslash }^{-1}$, and each source insertion contributes ${\hslash }^{-1}$. A connected graph with $I$ internal lines and $V$ interaction vertices has loop order $L=I-V+1$. Its amputated amplitude, or its coefficient in $\log Z[J]$ with the source insertions included, is proportional to ${\hslash }^{I-V}={\hslash }^{L-1}$. An ordinary unamputated connected $n$-field correlation function instead has the additional $n$ external propagator factors and scales as ${\hslash }^{L+n-1}$. Multiplication by $-i\hslash$ to form the connected generating functional shifts the source-kernel counting to ${\hslash }^L$. These are compatible normalization conventions.

For a connected Feynman diagram with external valence $n$ and cubic, quartic and sextic vertices, counting half-edges gives $2I+n=3V_3+4V_4+6V_6$. Using $L=I-(V_3+V_4+V_6)+1$ yields $L-1+n/2=V_3/2+V_4+2V_6$. Since each summand lies between half and twice its vertex count, the displayed bounds follow. More generally vertices of valence between $r_{\min }$ and $r_{\max }$ give $(r_{\min }/2-1)V\le L-1+n/2\le (r_{\max }/2-1)V$.

For $u=U\cos (\omega t)\sin (\pi y/L)$ on $|y|<L/2$, positive diffusivity $\kappa$, nonzero frequency and insulating walls, a transversely uniform zeroth moment remains constant. Its first moment is proportional to $g(t)\sin (\pi y/L)$, where $g^{\prime }+ag=U\cos (\omega t)$. The periodic solution is $g=U[a\cos (\omega t)+\omega \sin (\omega t)]/(a^2+{\omega }^2)$. The longitudinal moment hierarchy for shear transport then gives $m_2=2Kt{m}_0+O(1)$ after both transverse and periodic transients. The spatial average of ${\sin }^2(\pi y/L)$ is $1/2$ and the time average of ${\cos }^2(\omega t)$ is $1/2$, producing the factor $1/4$. At exactly zero frequency the steady enhancement is instead $U^2/(2a)$; averaging indefinitely over slow cycles does not commute with setting the frequency to zero.

For zero shift vector, negative-expansion extrinsic curvature $K=-3H$ and $S^i{}_j={\tilde{K}}^i{}_j$, retain the leading spatial-gradient equations of a minimally coupled scalar. They give $\dot{\varphi }=N\Pi$, $\dot{\Pi }=-N(3H\Pi +V_{,\varphi })$, $\dot{H}=-4\pi GN{\Pi }^2-N{S}^i{}_j{S}^j{}_i/2$, and ${\dot{S}}^i{}_j=-3NH{S}^i{}_j$. The first-gradient momentum constraint is ${\partial }_{i}H=-4\pi G\Pi {\partial }_i\varphi -{\nabla }_j{S}^j{}_i/2$. Shear decays as $a^{-3}$ during expansion. Its neglect after inflationary damping yields the equations underlying conservation of the nonlinear curvature covector.

For the constant leading cell-concentration mode normalized to one, the leading vertical velocity mode in shallow long-wave bioconvection solves $W_1^{(4)}=-R_0{\alpha }^2$ and $W_1=W_1^{\prime }=0$ at both $z=0,1$. The displayed quartic satisfies all four conditions and is unique because a homogeneous cubic with those conditions is zero. Its integral is $-R_0{\alpha }^2/720$. The next cell-conservation solvability condition is ${\alpha }^2+{\sigma }_2=-\int W_{1}dz$, hence ${\sigma }_2={\alpha }^2(R_0/720-1)$. The leading threshold is independent of the distinguished leading gyrotactic coefficient; a horizontally uniform conservation mode does not select it.

For the logistic map $F(u)=\lambda u(1-u)$, factor $F(F(u))-u$ and remove its two fixed points. The remaining quadratic equation is ${\lambda }^2{u}^2-\lambda (\lambda +1)u+\lambda +1=0$. Its distinct roots form a periodic orbit of period two when $\lambda >3$. The periodic-orbit multiplier is $F^{\prime }(u_{-})F^{\prime }(u_{+})=4+2\lambda -{\lambda }^2$, so this orbit is asymptotically stable for $3<\lambda <1+\sqrt{6}$. It is born at a period-doubling bifurcation at $\lambda =3$.

For a positive response, the model $\log Y=\eta +\epsilon$ with $\epsilon \sim N(0,{\sigma }^2)$ gives conditional median $e^{\eta }$ and conditional mean $e^{\eta +{\sigma }^2/2}$. The latter follows from the moment-generating function of a normal distribution. Additive factor effects in $\eta$ become multiplicative effects on these response summaries. Its constant log-scale variance corresponds to original-scale variance proportional to the square of the mean; direct exponentiation of fitted log means incurs retransformation bias.