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An exponentially weighted consumption habit is $h_t=e^{-\lambda t}{h}_0+{\int }_0^t\lambda e^{-\lambda (t-s)}{c}_{s}ds$. Differentiation gives $d{h}_t=\lambda (c_t-h_t)dt$. The state remains positive from positive initial habit and nonnegative consumption.

When habit evolves by $dh=\lambda (c-h)dt$, increasing consumption both costs wealth and increases future habit. Consequently the consumption coefficient in the Hamilton-Jacobi-Bellman equation is $V_w-\lambda V_h$. A finite interior optimum requires this effective shadow price to be positive.

After scaling wealth by habit and applying the wealth-variable Legendre dual, the effective consumption shadow price is proportional to $\mathcal{D}=z-\lambda Rz{J}^{\prime }+\lambda (R-1)J$. Its power in the utility conjugate keeps the dual equation nonlinear. When $\lambda =0$, habit is fixed and the dual equation is a linear Euler differential equation with effective relative risk aversion $R+\alpha$.

For finite sets of positive real numbers $A,B$ with $|B|\ge 2$, $E(A,B)\le 4\lceil \log |B|\rceil |A+A||B+B|$, with natural or binary logarithm. Partition ray occupancies of $A\times B$ into $\lceil \log |B|\rceil$ classes of multiplicative width $e$, or width $2$ for the binary convention. Injectivity of sums on two distinct rays and disjoint open planar sectors bound each class's sum of squared occupancies by $4|A+A||B+B|$. A singleton class contributes at most $|A||B|$. Combining this with the energy lower bound gives a sum-product inequality.

Concentration of the prime omega function near $\log \log N$, together with $\omega (ab)=\omega (a)+\omega (b)-\omega (\gcd (a,b))$ and the bounded mean of $\omega (\gcd (a,b))$, gives $M(N)\ll N^2/\log \log N$. This elementary argument uses only distinct prime factors.

A bilinear map is a map $B:V\times W\to U$ between vector spaces that is linear in each argument when the other is fixed. The nonlinear temperature advection can be written as the diagonal of a bilinear map, $\mathcal{N}(\varphi ,\chi )=-\mathbf{u}[\varphi ]\cdot \nabla \chi$, when velocity depends linearly on temperature. Unlike a bilinear form, its output need not be a scalar.

A causal square-integrable linear time series can be written $X_t=\mu +{\sum }_{j\ge 0}{\psi }_j{\epsilon }_{t-j}$. Square summability of the coefficients ensures mean-square convergence for white noise. An innovation representation additionally identifies its driving noise as the innovation process.

For coefficients ${\theta }_0=1,{\theta }_1,\dots ,{\theta }_q$, the long-run variance of a stationary process sampled every second time is ${\sigma }^2[({\sum }_{j\text{ even}}{\theta }_j{)}^2+({\sum }_{j\text{ odd}}{\theta }_j{)}^2]$. It equals ${\sigma }^2[\Theta (1{)}^2+\Theta (-1{)}^2]/2$.

On the unit round sphere $S^n$, a geodesic of speed $L$ on $[0,1]$ has normal Jacobi equation $y^{\prime \prime }+L^{2}y=0$. Its conjugate parameters are $j\pi /L$, each with multiplicity $n-1$. If $L/\pi$ is not an integer, its index is $(n-1)\lfloor L/\pi \rfloor$. For $L=m\pi >0$, its index is $(m-1)(n-1)$ and its nullity is $n-1$.

For a classical algebraic variety $X$ and affine variety $Y$, a homomorphism on the indicated rings gives a morphism by evaluating the images of affine coordinate generators at each point of $X$. The relations defining $Y$ remain zero. Local fractions pull back to local regular fractions, proving regularity; the coordinate ring generators also prove uniqueness.

For a proper convex function with sequential lower semicontinuity $U=-\log \mu$, form $U_{\lambda }(x)={\inf }_y\{U(y)+\lambda \Vert x-y{\Vert }^2\}$ with $\lambda >0$. The Moreau envelope theorem gives $\nabla U_{\lambda }(x)=2\lambda (x-{\mathrm{prox}}_{U/(2\lambda )}(x))$. If $e^{-U_{\lambda }}$ is integrable, it defines a smooth surrogate probability density function. Infimizing the logarithm of a probability density function with a positive quadratic penalty has the opposite sign and does not generally smooth it. For the Laplace distribution, that incorrect operation leaves $-\log 2-|x|-1/(4\lambda )$, which is not differentiable at zero.

For proper lower semicontinuous convex functions, the Moreau envelope has gradient $\nabla f_{\tau }=(I-{\mathrm{prox}}_{\tau f})/\tau$. This follows by subgradient inversion under convex conjugacy and the quadratic conjugate. Firm nonexpansiveness of the proximal residual makes the gradient $1/\tau$-Lipschitz continuous.

For $r=\Vert x{\Vert }_2$, the Moreau envelope of the Euclidean norm is $r^2/(2\tau )$ when $r\le \tau$ and $r-\tau /2$ otherwise. Its gradient is $x/\tau$ in the core and $x/r$ outside. The approximation error is between zero and $\tau /2$.

Attach an $(i+1)$-cell to $S^i$ by a degree-$n$ map, with $i\ge 1$ and $n\ge 2$. The integral cellular differential is multiplication by $n$, while its modulo-$n$ reduction is zero. Lifting the degree-$i$ generator modulo $n^2$ and applying the differential gives $n$, so the Bockstein homomorphism sends that generator to the degree-$(i+1)$ generator. This supplies a nonzero example in every positive degree.

A real-valued function $f:D\to \mathbb{R}$ is a nondecreasing function if every pair $x<y$ in its domain of a function satisfies $f(x)\le f(y)$. Flat intervals are allowed, unlike for a strictly increasing function. For example,is a continuous nondecreasing function, but it repeats its output on each flat interval.

A nondecreasing function can also be discontinuous. The step function $F(x)=0$ for $x<0$ and $F(x)=1$ for $x\ge 0$ has a jump discontinuity at zero without ever decreasing. It is the cumulative distribution function of a random variable that equals zero with probability one; quantile functions handle such jumps through a generalized inverse. Nondecreasing allocation functions permit critical-value payments in single-parameter mechanisms.

For positive summable weights $w_j$ at points $q_j$, the strict cumulative function $F(x)={\sum }_{q_j<x}{w}_j$ is nondecreasing and left-continuous. A finite-head/tail argument proves left-continuity, and its right jump at an atom is the mass there. This strict convention differs at atoms from the usual right-continuous cumulative distribution function.

In the uniform-label monotone coupling of Bernoulli percolation, let $I_p$ mean that the origin belongs to an infinite percolation cluster, and let $M=\inf \{p:I_p\}$. The occurrence parameters form an upper interval. Therefore $\{M<p\}={\bigcup }_{r<p,r\in \mathbb{Q}}{I}_r$ and $\theta (p-)=\mathbb{P}(M<p)$. Since $\{M<p\}\subseteq I_p\subseteq \{M\le p\}$, the jump is $\theta (p)-\theta (p-)=\mathbb{P}(I_p\cap \{M=p\})$. Occurrence at the infimum is not presumed.

If an increasing event $A$ depends on $m$ sites or edges, the uniform-label monotone coupling of Bernoulli percolation and the union bound give $0\le {\mathbb{P}}_q(A)-{\mathbb{P}}_p(A)\le m(q-p)$ for $p\le q$. A change of the event requires a coordinate with label in $(p,q]$. For bond percolation from the origin to the boundary of $[-n,n{]}^2$, first-hit graph paths use the $4n(2n+1)$ internal edges, so the same count bounds the parameter sensitivity.

A monomorphism $m$ is extremal if every factorization $m=he$ with $e$ an epimorphism forces $e$ to be an isomorphism. A strong monomorphism has this property by its lifting property. In a complete category that is a well-powered category, intersecting all strong subobjects containing a given morphism yields an epimorphism–strong-monomorphism factorization; applying it to an extremal monomorphism proves that the extremal monomorphism is strong. The factorization definition and lifting definition should be kept distinct until the needed ambient hypotheses are supplied.

For $V=\lambda {\varphi }^n$ on $\varphi >0$, slow-roll approximation gives $ϵ\simeq n^2{M}_{\mathrm{Pl}}^2/(2{\varphi }^2)$ and $\eta \simeq 2n{M}_{\mathrm{Pl}}^2/{\varphi }^2$. The slow-roll e-fold count integrates to $N\simeq ({\varphi }_N^2-{\varphi }_e^2)/(2n{M}_{\mathrm{Pl}}^2)$. For $n=2,3,4$, the loose order-one conditions $ϵ\le 1$ and $|\eta |\le 1$ imply ${\varphi }^2\ge 2n{M}_{\mathrm{Pl}}^2$ in this approximation. Genuine slow roll requires a much stronger hierarchy. A prescribed endpoint ${\varphi }_e^2=2n{M}_{\mathrm{Pl}}^2$ gives ${\varphi }_N^2=2n(N+1)M_{\mathrm{Pl}}^2$ andThe actual end criterion for accelerated expansion is $ϵ=1$, whose leading potential estimate instead gives ${\varphi }_e^2\simeq n^2{M}_{\mathrm{Pl}}^2/2$. The prescribed endpoint is a convention, not an exact slow-roll or inflation boundary.