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With the $1/n$ Rademacher convention, an empirical risk minimizer has expected excess risk at most twice the Rademacher complexity of its loss class.

For $h_M(x)=x^{T}Mx$, $M⪰0$, $\mathrm{tr}M\le s$, and $\Vert x\Vert \le C$, the complexity is at most $C^{2}s/\sqrt{n}$.

Independent signs cancel cross terms in $\mathbb{E}\Vert {\sum }_i{\sigma }_i{A}_i{\Vert }_F^2$, leaving ${\sum }_i\Vert A_i{\Vert }_F^2$.

Composition with an $L$-Lipschitz scalar map increases Rademacher complexity by at most the factor $L$ after centering at zero.

A linear functional has the same supremum over a class and its convex hull. Thus taking convex combinations does not increase empirical Rademacher complexity.

For $|D_i|\le M$ and a binary class $H$ of VC dimension at most $p$, the vectors $(D_{i}h(X_i){)}_{i=1}^n$ have norm at most $M\sqrt{n}$ and at most $(n+1{)}^p$ distinct values. The Massart finite-class lemma gives

For a finite $A\subseteq {\mathbb{R}}^n$ with ${\max }_{a\in A}\Vert a{\Vert }_2\le R$, its empirical Rademacher complexity is at most $R\sqrt{2\log |A|}/n$.

Identify the vertical sides of $[0,1{]}^2$, then collapse each horizontal side separately to a point. The resulting quotient is homeomorphic to $S^2$ throughEquivalently, the quotient is the suspension of the circle.

Identify $x,y\in \mathbb{R}$ when $x-y\in \mathbb{Q}$. Every nonempty saturated open subset of $\mathbb{R}$ meets every other one because rational translates of any open interval meet any prescribed nonempty open interval. The quotient $\mathbb{R}/\mathbb{Q}$ has multiple points but no two nonempty disjoint open subsets, so it is not Hausdorff.

If $q:X\to X/R$ is the quotient map and a continuous map $f:X\to Y$ is constant on each equivalence class, there is a unique continuous map $F:X/R\to Y$ satisfying $Fq=f$. It is given by $F([x])=f(x)$; continuity follows becausefor every open $U\subseteq Y$.

A quotient map is a surjective continuous map $q:X\to Y$ for which $U\subseteq Y$ is open exactly when $q^{-1}(U)$ is open.

An $M/G/1$ queue has Poisson arrivals, independent identically distributed service times, and one server. If service has mean $m$ and $\lambda m<1$, its mean busy period isThis follows by decomposing the busy period into the first service and the busy-period descendants of arrivals during that service.

Poisson arrivals see time averages says that an exogenous, homogeneous Poisson process samples prearrival states with the same long-run frequencies as observation over time, provided the observed system cannot anticipate future arrivals and the relevant time averages exist. The system may depend on all past arrivals: an ordinary queue does. The required independence is between future increments of the Poisson process and the history available to the system, including its service information. Assume measurable state observations with ordinary piecewise continuous functions as sample paths; write $X(t-)$ for the state immediately before time $t$.

For a state set $B$, let $A(T)$ count offered arrivals by time $T$ and let $t_k$ be their times. With a positive finite constant arrival rate, the PASTA equality isalmost surely when the time limit exists. The indicator functions count visits on the left and time spent on the right. For a stationary process with ergodicity, the right side is the stationary probability of $B$. It matters that the arriving customer has not yet changed the observed state.

In the M/M/1 utilization and delay example, server utilization is $0.8$. Thus PASTA says that 80 percent of offered arrivals find the server busy and 20 percent find the whole queue empty. If an offered arrival seeing population $n$ is admitted with conditional probability $p(n)$, using fresh randomness independent of that state, the accepted fraction is ${\sum }_n{\pi }_{n}p(n)$, where $\pi$ is the stationary distribution. Multiplying this fraction by the offered rate gives the admitted queue throughput when every admitted customer eventually completes service. Accepted arrivals are selected by state and need not themselves form a Poisson process or see the unconditioned time frequencies.

Arbitrary arrivals need not see time averages. Suppose a customer arrives every two minutes and its service time is exactly one minute, with one server and no other customers. The server is busy half the time, but every arrival finds it idle. A uniformly random initial phase can even make the occupancy a stationary process, without repairing the disagreement. Nevertheless Little's law still gives $L=(1/2)\cdot 1=1/2$, with rate measured in customers per minute. This separates Little's law from PASTA.

Ronald W. Wolff's 1982 paper gave a general PASTA theorem under nonanticipation, rather than requiring the observed system to be independent of its driving Poisson process. Its Theorem 1 compares the two long-run fractions; stationarity is a way to establish convergence, not the central sampling condition. ^[ref]. ^[ref].

For a scalar differential operator of order $k$, the principal symbol replaces each order-$k$ derivative ${\partial }^{\alpha }$ by ${\xi }^{\alpha }$ and discards lower-order terms. For a quasilinear partial differential equation, the coefficients are evaluated at the prescribed lower-order data.

The p-Laplacian is the nonlinear divergence-form operatorIt is the Euler-Lagrange operator of the p-energy. For $p>2$ it is degenerately elliptic where $Du=0$.

A subgroup is quasiconvex when it is a quasiconvex subset of a Cayley graph. In a hyperbolic group, a finitely generated subgroup is quasiconvex exactly when it is quasi-isometrically embedded.

For sequential binding with dimensionless parameters $\alpha ,\beta ,\gamma$, the reduced product rate has the formIt is quadratic at low substrate concentration and saturates at high concentration.

A quasigeodesic is a quasi-isometric embedding of an interval into a metric space.

A bilipschitz equivalence distorts all distances by multiplicative constants bounded away from zero and infinity. Every bilipschitz equivalence is a quasi-isometry.

A finitely generated subgroup $H\le G$ is quasi-isometrically embedded when its inclusion, equipped with word metrics from finite generating sets, is a quasi-isometric embedding. This property is independent of those generating sets.