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For a Keller--Segel model with $c_t=-kb$, a positive-speed travelling wave $B(z),C(z)$, $z=x-vt$, satisfies $v{C}^{\prime }=kB$ and $D{B}^{\prime }+vB-\alpha B{C}^{\prime }/C=0$ when the integrated bacterial flux constant is zero. Writing $\mu =\alpha /D>1$ and $\xi =v(z-z_0)/D$, its positive band solution isTo derive it, integrate $B^{\prime }/B=\mu C^{\prime }/C-v/D$ to obtain $B=K{C}^{\mu }{e}^{-vz/D}$, then separate $v{C}^{\prime }=kB$. The positive integration constant is a translation of the wave. Concentration rises monotonically from zero behind to $C_{\infty }$ ahead; $B$ vanishes at both ends and has a unique maximum at $\xi =-\log (\mu -1)$. The condition $\mu >1$ describes this exponential-tail branch; it is not asserted to exclude every limiting or weak travelling wave at other parameter values.

In a locally small category, the morphisms from $A$ to $B$ form the set $\mathcal{C}(A,B)$. Postcomposition and precomposition give the corresponding covariant and contravariant representable functors.

A morphism is a continuous map $f:X\to Y$ together with a homomorphism ${\mathcal{O}}_Y\to f_{\ast }{\mathcal{O}}_X$ of sheaves of rings whose induced map on every stalk is a local homomorphism. Applied to schemes, this is exactly a morphism of schemes.

An exhaustion of a countable graph is an increasing sequence of finite connected graph vertex sets whose union is the whole graph vertex set. In a locally finite graph, every prescribed finite graph path together with all graph neighbours of its visited graph vertices lies in a sufficiently large exhaustion set. This permits coupling finite-volume and infinite-volume simple random walks up to a finite stopping time.

Here the set is a nonempty compact metrizable convex subset of a Hausdorff locally convex space. The ambient space can be infinite-dimensional. Continuous affine functions separate its points, and weak-star compact dual balls with separable preduals provide examples. This general setting is needed for Choquet theorem.

The fraction relation defining localization of a module gives this criterion. For localization at powers of a single element $f$, a fraction vanishes exactly when some power of $f$ annihilates its numerator. Finitely many such vanishing relations can be cleared with one common exponent. This is distinct from detecting global zero by localizing at every prime ideal.

For a square-integrable discounted claim martingale $dM=hdW$ and discounted stock $d\tilde{S}=\tilde{S}\sigma dW$, choose stock holdings $h/(\tilde{S}\sigma )$. The remaining wealth lies in the bond. A nonnegative claim makes this replication admissible; discounted admissible wealth is a supermartingale, establishing the minimal initial cost.

For a non-dividend-paying stock and constant interest rate $r$, discounted call prices satisfy $C_T=\frac{1}{2}{K}^2\sigma (T,K{)}^2{C}_{KK}-rK{C}_K$. Strike differentiation recovers the discounted terminal density as $C_{KK}$; differentiating the discounted payoff identity supplies the maturity derivative. Put-call parity implies the same equation for put prices.

The exact centered quadratic Gaussian transformation has a connected three-quadratic-vertex contribution $8{\alpha }^3\int d^{3}qP(q)P(|{\mathbf{k}}_1-\mathbf{q}|)P(|{\mathbf{k}}_2+\mathbf{q}|)/(2\pi {)}^3$. It is higher order than the usual tree-level local primordial bispectrum. Infrared or ultraviolet regularization may be needed.

Uniformly sample one positive local term and measure the effect $h_j$. The averaged energy-flag effect on a witness is $H/M$, where $M$ is the number of terms. Repetition with a threshold halfway between the promised flag probabilities uses $O(M^2/(b-a{)}^2)$ witness registers for constant error. A spectral tensor-product analysis, as in QMA parallel repetition with entangled witnesses, proves soundness against entangled witnesses. Constant-locality positive operator-valued measures are efficiently implementable using one ancilla qubit.

A smooth vector field is complete if its maximal integral curves of a vector field exist for every real time. Its local flow is then a global one-parameter group of diffeomorphisms; uniqueness of ordinary differential equations proves the group law and the inverse-time identity. On a compact manifold without boundary, every smooth vector field is complete: compactness gives a uniform positive local existence interval through every point, so a curve can be successively extended past any proposed finite endpoint. On $\mathbb{R}$, $x^2{\partial }_x$ is not complete because its solution $x(t)=x_0/(1-t{x}_0)$ has finite-time blowup for $x_0>0$.

An infinite field with the trivial absolute value on a field is discrete, hence locally compact, but its residue field is the original infinite field. Thus local compactness forces a finite residue field and a discrete nontrivial value group only when the valuation is assumed nontrivial.

For a locally compact field with a nontrivial Non-Archimedean absolute value, scaling a compact neighborhood makes its valuation ring compact. Its open maximal ideal gives a finite discrete residue field, so that ideal is also closed and compact. A nonzero element of maximal absolute value in the ideal generates it. Successive division by this uniformizer makes the value group discrete.

In a locally compact nontrivially valued field, the valuation ring is compact. A Cauchy sequence has a tail inside a translate of this compact set, hence a convergent subsequence. The Cauchy property forces the full sequence to converge.

An algebra for a monad for the list monad is precisely a monoid. For an action $a$, the identity is $a([])$ and multiplication is $a([x,y])$. The algebra laws force the unit and associativity laws and determine $a$ as ordered multiplication. Conversely ordered multiplication defines a list action. Algebra morphisms are exactly monoid homomorphisms, giving an isomorphism of categories ${\mathbf{Set}}^{\mathrm{List}}\cong \mathbf{Mon}$.

Suppose $f$ is a differentiable function on an open convex set $D\subseteq {\mathbb{R}}^n$ and has an $L$-Lipschitz gradient there, in the Euclidean norm. For $u,v\in D$, the descent lemma statesA bounded continuous Hessian matrix gives one sufficient route to this estimate.^[ref]

For a short proof, put $d=v-u$. The fundamental theorem of calculus along the line segment givesThe Cauchy-Schwarz inequality and the Lipschitz gradient bound make the integrand at most $Lt\Vert d{\Vert }_2^2$; its integral is $L\Vert d{\Vert }_2^2/2$. This proof needs no assumption that $f$ is a convex function.

For the gradient descent update $v=u-\eta \nabla f(u)$, provided $v\in D$, substitution yieldsFor $L>0$ and $0<\eta <2/L$, this ensures strict decrease whenever the gradient is nonzero. At $\eta =2/L$ it guarantees only nonincrease; the quadratic example has a nonshrinking alternating component at this endpoint. At $\eta =1/L$ the guaranteed decrease is $\Vert \nabla f(u){\Vert }_2^2/(2L)$. A bound valid only on $D$ cannot justify a step leaving $D$.

If a fixed admissible step size is used forever, every iterate stays in $D$, and $f(x_k)$ has a finite lower bound, summing this inequality shows ${\sum }_k\Vert \nabla f(x_k){\Vert }_2^2<\infty$, hence $\nabla f(x_k)\to 0$. That conclusion alone gives neither convergence of the iterates nor a global minimum. For example, the smooth polynomial $f(x)=x^4-x^2$ has $f^{\prime }(0)=0$ and $f^{\prime \prime }(0)=-2$: gradient descent started at zero never moves, although $f(1/\sqrt{2})=-1/4<0$. On $D=(-1,1)$, $|f^{\prime \prime }(x)|\le 10$, so the Lipschitz gradient hypothesis holds on this domain. Convex functions and strongly convex functions provide additional structure; the exact quadratic convergence result above uses a positive-definite matrix.

For a transport equation on a domain, prescribe boundary data only where the velocity points into the domain. On $x>0$ with speed $a>0$, a bounded weak solution with initial data $u_0$ and inflow data $f$ satisfiesThe integrals are over the quadrant and its two boundary rays. Backtracking characteristic curves gives $u_0(x-at)$ when $x\ge at$ and $f(t-x/a)$ otherwise. Corner values need not match for a bounded weak solution.

A linear system is base-point-free if its sections have no common zero. Their ratios then define an everywhere-defined morphism to projective space whose hyperplane pullbacks are members of the original system.

For a resting midlatitude layer, the slow Rossby wave equation is $({\nabla }_h^2-a^2){\eta }_t=-\beta {\eta }_x$, where $a=f_0/\sqrt{gH}$. With the plane wave convention $e^{i(kx+ly-\omega t)}$,The zonal phase velocity is westward for $\beta >0$, but the zonal group velocity changes sign at $k^2=l^2+a^2$.

Suppose a piecewise finite difference uses two translated evaluations divided by $\alpha$. If the pulled-back evaluation intervals cover almost every input point at most $m$ times, then $|a-b{|}^2\le 2(|a{|}^2+|b{|}^2)$ gives $\Vert R_{\alpha }f{\Vert }_2^2\le 2m\Vert f{\Vert }_2^2/{\alpha }^2$. This converts an interval-overlap count into an operator norm estimate without requiring pointwise values of an L2 space equivalence class outside the almost-everywhere evaluation rule.