Discrete Gaussian white noise is a jointly Gaussian sequence with expected value zero and covariance ${\sigma }^2$ at lag zero and zero at other lags. Joint normality makes the components independent. This is a sequence of ordinary random variables, unlike the generalized continuous-time white-noise process.

If $Q$ is a well-quasi-ordering, its finite words form a well-quasi-ordering under subsequence embedding with coordinatewise increase of letters. A minimal bad sequence proof removes the last letter of selected words whose last letters form a nondecreasing subsequence, then contradicts minimality. The result implies finite-subset lifting of a well-quasi-order under Hoare domination preorder.

A quasi-order is better-quasi-ordered if every array into it from every barrier in better-quasi-order theory is good: some barrier members $s◃t$ satisfy $f(s)\le f(t)$. Singleton barriers recover the usual infinite-sequence condition for a well-quasi-ordering. The stronger condition is useful for infinitary closure operations for which well-quasi-ordering alone fails. Equivalently one may use continuous maps from infinite subsets of an infinite subset of the natural numbers, with the discrete topology on the range, and compare $X$ with $X\setminus \{\min X\}$.

A finite Cartesian product of well-quasi-orderings is well-quasi-ordered in the componentwise order. Every infinite sequence in one well-quasi-order has an infinite nondecreasing subsequence: colour pairs by whether they are ordered, apply Ramsey's theorem, and exclude an all-bad subsequence. Restrict successively to such a subsequence in each coordinate. The same argument, or the infinite pigeonhole principle, proves closure under finite disjoint unions with no cross-component comparisons.

Every infinite sequence in a well-quasi-ordering has an infinite subsequence all of whose earlier terms precede its later terms. Some term must have infinitely many later terms above it: otherwise choose each new index outside the finitely many upper-cone index sets of the terms already chosen, producing a bad sequence. Apply the same observation successively inside the infinite upper-cone subsequence. Each restriction preserves all earlier inequalities, proving the assertion.

A locale is regular when every open is the join of opens well inside it. This expresses regularity without reference to points.

A weighted projective space with positive integer weights $q_0,\dots ,q_n$ is well-formed when the greatest common divisor of the weights with any one weight omitted is one. The quotient toric lattice $N={\mathbb{Z}}^{n+1}/\mathbb{Z}(q_0,\dots ,q_n)$ is free and the images of the coordinate vectors are primitive toric rays. Its fan contains the cones on all proper subsets of these rays.

For a finite list of real numbers $x_i$ and nonnegative weights of positive total $W$, the weighted mean uniquely minimizes the weighted sum of squared deviationsExpanding around ${\bar{x}}_w$ and using ${\sum }_i{w}_i(x_i-{\bar{x}}_w)=0$ givesEvery displacement from the weighted mean increases the objective. This is the simplest one-parameter weighted least squares problem: represent all entries by one common value.

For the weighted values $90$ and $60$ with weights $2$ and $8$,Thus $S(66)=1440$, while $S(75)=2250$. The minimum is defined by this squared-error criterion; a different loss can select a different summary, such as a median under unweighted absolute deviations.

Suppose disjoint nonempty groups have sizes $n_j$ and arithmetic means ${\bar{x}}_j$. The arithmetic mean of all observations isEach group contributes the sum $n_j{\bar{x}}_j$; add those totals before dividing by the overall count. This is exactly a weighted mean of group means with weights equal to group sizes. ^[ref]

Take a synthetic example with two observations equal to $90$ and eight equal to $60$. The two group means are $90$ and $60$, and the combined mean is $66$. Giving the two groups equal weights instead produces $(90+60)/2=75$, which summarizes the two group means with equal influence. To summarize all ten individuals with equal influence, use their group counts.

The group means and counts determine the combined arithmetic mean, but do not generally determine the combined variance. Variation within the groups is additional information.

A canonical product builds a prescribed discrete zero set from Weierstrass elementary factors. Choose the orders $p_n$ so that the logarithms of tail factors converge uniformly on each compact subset. Infinite product convergence from logarithmic tails then gives an entire function with exactly those zeros, counted with the prescribed multiplicities. When ${\sum }_n|a_n{|}^{-1}<\infty$, factors of order zero suffice.

The determinant of the columns $(1,\wp (z),{\wp }^{\prime }(z))$, $(1,\wp (w),{\wp }^{\prime }(w))$ and $(1,\wp (-z-w),{\wp }^{\prime }(-z-w))$ vanishes identically. Its apparent cubic poles at $z=0,-w$ have order at most two because the coefficient multiplying the cubic-pole entry vanishes linearly. Column coincidences give six zeros, unless $3w$ is a lattice point, when they give four zeros with one multiple zero. In the half-period case, the two apparent zero locations at poles are removable zeros: ${\wp }^{\prime }(w)=0$ and the two $1/z$ terms cancel, leaving $O(z)$. Thus the zero count exceeds possible pole order in every case. The argument principle proves the identity, which expresses collinearity of the three points on the Weierstrass equation of an elliptic curve.

For $2a\notin \Lambda$, the apparent double poles at $z=\pm a$ on the left are canceled by its squared zero factor. Subtract the right side: the order-three principal parts at lattice points cancel by the even Laurent expansion of $\wp$, leaving at most a simple pole per period cell. An elliptic function has total residue zero in a period cell, so that possible pole is removable. The resulting entire periodic function is constant, and its oddness makes the constant zero. This proves the identity without presupposing the elliptic differential equation.

The Normal convergence of the Weierstrass elliptic-function series allows its derivative to be written as an absolutely locally uniformly convergent sum of inverse cubes. For a period lattice $\Lambda$, this derivative sum is invariant under translation by any lattice vector, by reindexing. Consequently $\wp (z+\omega )-\wp (z)$ is constant; its apparent poles cancel. The Weierstrass elliptic function is even, so for a primitive lattice generator $\omega$, evaluation at $z=-\omega /2$ makes that constant zero. Applying this to two generators proves that this is a periodic function, namely $\wp$. This argument avoids separating the original convergent series into two divergent inverse-square sums.

The Weierstrass elliptic function has degree two on its period torus. Its three distinct half-period values $e_j$ have double local preimages, and its pole is double, so these four values are totally ramified. This also follows from ${\wp }^{\prime 2}=4{\prod }_j(\wp -e_j)$ and the principal part at the pole. A Möbius transformation with pole away from these values moves all four to finite values without changing local degrees.

For a period lattice, the displayed canonical product converges normally because its logarithmic tails are bounded by a constant times $|\omega {|}^{-3}$. It defines an odd entire function with simple zeros exactly at the lattice and derivative one at zero. Its logarithmic derivative is the Weierstrass zeta function, and the derivative of that logarithmic derivative is minus the Weierstrass elliptic function. Sigma is quasi-periodic rather than elliptic; balanced products of its translates construct meromorphic functions with prescribed divisors on a complex torus.

For generic fixed $w$, regard the right side as an elliptic function of $z$. The apparent pole at zero cancels, giving limiting value $\wp (w)$. At $z=w$ the slope ratio is finite. At $z=-w$, its squared slope has the same double principal part as $\wp (z+w)$ and no simple pole term. These exhaust the possible poles because the Weierstrass elliptic function has degree two. The difference is therefore an entire elliptic function, hence constant by Liouville theorem, and its value at zero is zero. Meromorphic continuation extends the identity to the exceptional parameters.

For a symplectic left Lie group action, use actual fundamental fields ${\xi }_P$ and ${\iota }_{X_f}\omega =df$. They satisfy $[{\xi }_P,{\eta }_P]=-[\xi ,\eta {]}_P$. Cartan's magic formula then gives ${\iota }_{[\xi ,\eta {]}_P}\omega =d[\omega ({\xi }_P,{\eta }_P)]$. Thus every commutator generator has a globally exact contraction. A semisimple Lie algebra is perfect, so every generator has a global Hamiltonian. The separate semisimple moment-map equivariance result removes the constant two-cocycle obstruction and gives an equivariant moment map for a connected group.

Suppose linear operators $A_t$ and $A_0$ on a normed space satisfy $|A_{t}f|,|A_{0}f|\le Sf$, where $S$ has weak type with a normed domain, and all relevant suprema are measurable. If $A_{t}g\to A_{0}g$ almost everywhere on a dense subspace, the same holds for every input. For a fixed approximation $g$, the limiting error is at most $2S(f-g)$. The measure where it exceeds $a$ is at most $(2C\Vert f-g\Vert /a{)}^q$, which tends to zero as the approximation improves. This is a convergence-to-a-specified-limit version of almost-everywhere convergence from weak-type domination.

Suppose a family of linear operators, including its limiting operator, is pointwise dominated by a sublinear operator of weak type (1,1). Convergence almost everywhere on an $L^1$-dense subspace then extends to every $L^1$ input. Approximate the input by errors of norm at most $2^{-2n}$. The weak estimate and the Borel-Cantelli first lemma make the corresponding domination errors tend to zero almost everywhere. Linearity bounds the difference between the full and approximating convergence errors by twice that domination error. This argument does not need an uncountable supremum to be measurable.

The integral expression for the weak Lq space is homogeneous, positive definite up to equality almost everywhere, and obeys the triangle inequality by integrating $|f+g|\le |f|+|g|$ over each set. Its exponent is $-1/q^{\prime }$, not $-1/q$. The sharp comparison constant with the distribution-function quasi-norm is $q^{\prime }$; the power function $x^{-1/q}$ on $(0,\infty )$ attains that ratio on initial intervals.