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The domain of ${\exp }_p$ is the star-shaped open setSmooth dependence for ordinary differential equations makes $(p,v)\mapsto {\exp }_p(v)$ smooth on its open domain in $TM$.

If $N$ is geometric on $\{1,2,\dots \}$ with parameter $p$, independently of unit-rate exponential variables $S_i$, thenis exponential with rate $p$. Its moment-generating function is $p/(p-\theta )$ for $\theta <p$.

For independent exponential variables $S_i$ of rates $q_i$ and $Q={\sum }_i{q}_i$, let $T={\min }_i{S}_i$ and let $K$ be the minimizing index. ThenConsequently $T$ is exponential with rate $Q$, $\mathbb{P}(K=k)=q_k/Q$, and $K$ and $T$ are independent.

If $A$ and $B$ are independent exponential variables with the same rate, then $A/(A+B)$ is uniform on $(0,1)$ and is independent of $A+B$.

If $T\sim \mathrm{Exp}(\mu )$, thenThus an exponential remaining lifetime has the same law regardless of the elapsed lifetime.

For $S_t=S_0{e}^{\mu t+\sigma W_t}$, shifting Brownian drift bymakes $S_t=S_0{e}^{(r-{\sigma }^2/2)t+\sigma W_t^Q}$ under an equivalent martingale measure $Q$. The discounted stock is then a martingale.

Suppose a transient nearest-neighbour jump chain on $\mathbb{N}$ has uniformly bounded expected visits to each state and the holding rate at $i$ grows geometrically. Thenso the continuous-time chain explodes almost surely.

Conditional on a jump path $(Y_n)$, if ${\sum }_{n}1/q_{Y_n}<\infty$, then the sum of exponential holding times has finite expectation and is finite almost surely.

For $p=n^{-2/3}\log n$, the expected number of pairs of $K_4$ copies sharing a vertex isThus, with probability tending to one, all the $K_4$ copies are vertex-disjoint. Together with sparse clique-count concentration, this gives arbitrarily many vertex-disjoint copies with probability tending to one.

For $p=n^{-2/3}\log n$, the number $X$ of copies of $K_4$ satisfiesand $\mathrm{var}(X)=o((\mathbb{E}X{)}^2)$. Two distinct copies have dependent indicators only when they share at least two vertices; pairs sharing two or three vertices contribute respectively $O(n^6{p}^{11})$ and $O(n^5{p}^9)$ to the variance.

The $p$th de Rham cohomology group is the quotient of closed $p$-forms by exact $p$-forms.

When the converted P-wave is evanescent, its amplitude decays away from the boundary and the reflected SV amplitude has modulus one. The P field stores reactive energy near the boundary but carries zero mean normal power.

For a finite simplicial complex $K$, let $f_i$ count its $i$-dimensional simplexes. Its Euler characteristic iswhere $b_i={\dim }_{\mathbb{Q}}{H}_i(|K|;\mathbb{Q})$ is its rational Betti number.

Work over $\mathbb{Q}$, so the chain groups, cycle modules and boundary modules are finite-dimensional vector spaces. Choosing vector-space complements gives $C_i\cong B_i\oplus H_i\oplus B_{i-1}$. The boundary dimensions occur twice in the alternating sum, with opposite signs, and cancel. This proves the formula; it does not require splitting off possibly torsion integral homology as a direct summand of an integral cycle module. The rank cancellation in the Euler-Poincare formula gives the same argument without choosing complements.

For a primitive cube root of unity $\omega$, the Eisenstein integers form the triangular latticeTheir multiplicative norm is $N(a+b\omega )=a^2-ab+b^2$.

A Gaussian integer has the form $a+bi$ with $a,b\in \mathbb{Z}$. Its multiplicative norm is $N(a+bi)=a^2+b^2$, which makes $\mathbb{Z}[i]$ a Euclidean domain.

A Euclidean function on an integral domain $R$ assigns a nonnegative integer to every nonzero element so that, for $a,b\in R$ with $b\ne 0$, there are $q,r\in R$ satisfying $a=bq+r$ and either $r=0$ or $d(r)<d(b)$.

The two variances areSince $S_2^2\le S_1{S}_3$ by Cauchy-Schwarz, maximum likelihood has no larger variance, with equality when all positive exposures are equal.

Katona's circle method places a finite ground set in a uniformly counted cyclic order, proves a bound for the members of a set family that appear as cyclic intervals, and double-counts pairs of a member and a compatible cyclic order.

Two grammars over a common terminal alphabet are isomorphic when a bijection between their nonterminal alphabets maps one start symbol to the other and maps their productions bijectively after being extended to fix every terminal symbol. Isomorphic grammars are equivalent.

For two distinct nonsingular Gaussian class distributions, vary the class-zero prior continuously. The class-zero and class-one error probabilities of the corresponding Bayes classifier cross. At a crossing they are equal, making the classifier an equalizer rule, a minimax decision rule, and the associated prior a least favorable prior.