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The constant minimizing ${\int }_{-1}^1{x}^{2n}(x^2-c{)}^{2}dx$ is the weighted meanThis follows by differentiating the quadratic error or requiring the residual to be orthogonal to constants. The normalized weight $(2n+1)x^{2n}/2$ assigns mass $r^{2n+1}$ to $[-r,r]$ for every $0<r<1$. As $n$ increases it concentrates at the endpoints, where $x^2=1$, explaining the limiting best constant. Although the weight vanishes at zero, its integral against the square of a nonzero continuous function is positive, so the associated weighted inner product is positive definite.

Consider the squared error of approximating $x^n$, $n\ge 1$, by a polynomial $q$ under a measure containing $dx/x$ on $(0,1)$. This error is finite exactly when $q(0)=0$. A nonzero constant term makes the integrand asymptotic to $q(0{)}^2/x$, which is not Lebesgue integrable; a vanishing constant term makes $x^n-q(x)=O(x)$, giving an integrable $O(x)$ integrand.

Consequently the full space of polynomials of degree at most $n-1$ is not contained in the weighted L2 space. Its admissible subspace is $\mathrm{span}\{x,\dots ,x^{n-1}\}$, or zero when $n=1$. This finite-dimensional subspace has a unique orthogonal projection of $x^n$, hence a unique best approximation. Indeed its Gram matrix is positive definite: a nonzero polynomial cannot vanish on an interval, so its squared norm is positive. Thus a singular weight can invalidate the proposed full approximation space while still allowing a unique finite-error minimizer in its admissible part. As usual, weighted L2 spaces identify functions equal almost everywhere.

For positive mean density, this average weights a velocity by local mass rather than volume. It satisfies the exact identity $⟨\rho u⟩=⟨\rho ⟩\tilde{u}$, so it naturally represents mass flux in averaged conservation laws. The ordinary volume mean generally fails that identity when density and velocity are correlated. Applied inside a phase, the same construction uses density times that phase's indicator as its weight.

The Weierstrass elliptic function satisfies the displayed meromorphic function identity. To prove it, the Riemann-Hurwitz formula shows that its derivative has exactly three simple zeros modulo its period lattice, at the nonzero two-torsion points of a complex torus. At each such half-period $h$, evenness around $h$ gives ${\wp }^{\prime \prime }/{\wp }^{\prime }=1/(z-h)+O(z-h)$, so the right side has principal part $1/(4(z-h{)}^2)$, matching $\wp (2z)$. At zero, $\wp =z^{-2}+O(z^2)$ and ${\wp }^{\prime \prime }/{\wp }^{\prime }=-3/z+O(z^3)$ give the same principal part and zero constant term. The difference is a holomorphic elliptic function, hence constant by compactness, and its limit at zero is zero.

For the square period lattice, multiplying the defining series by the lattice symmetry $z\mapsto iz$ gives $\wp (iz)=-\wp (z)$. At $h=(1+i)/2$, one has $ih=h-1$, so periodicity gives $\wp (h)=-\wp (h)$ and hence the displayed zero. This is one of the three finite branch values of the degree-two Weierstrass elliptic function on its torus. The three values are always distinct but need not all be nonzero. Distinctness is enough for the nonsingularity of the associated elliptic curve $Y^{2}Z=4{\prod }_i(X-e_{i}Z)$.

On its period torus, the derivative has one triple pole, so degree three. The Riemann-Hurwitz formula gives total ramification six, counting each ramification point with weight $e_p-1$. The number of distinct ramification points is five when $g_2\ne 0$ and three when $g_2=0$. Indeed ${\wp }^{\prime \prime }=6{\wp }^2-g_2/2$ by the Weierstrass elliptic differential equation. If $g_2\ne 0$, each of its two possible $\wp$-values has two distinct unramified preimages for $\wp$, because coincidence with a cubic branch root would force a repeated root. These four simple zeros of ${\wp }^{\prime \prime }$ give local degree two for ${\wp }^{\prime }$, and its triple pole contributes local degree three. If $g_2=0$, nonsingularity forces $g_3\ne 0$; the two simple zeros of $\wp$ are double zeros of ${\wp }^{\prime \prime }$ and each gives local degree three for ${\wp }^{\prime }$. Along with its pole these are three distinct points, each of weight two. A hexagonal period lattice realizes this second case by its sixth-root rotational symmetry.

Thinning enlarges the assumptions without changing a conclusion. In a predicate-calculus presentation with context-sensitive generalization, a generalization valid before adding an unused assumption may not be replayable with that assumption open. An explicit weakening step nevertheless imports the already proved conclusion. In propositional calculus with only axiom schemes and modus ponens, weakening is admissible by replaying the proof. The need for a separately stated rule depends on how proof dependencies are represented.

Identify the initial endpoints of countably many closed intervals to a vertex $v$, using the quotient topology. This is the geometric realization of the complex whose only edges are $\{v,u_n\}$. A neighbourhood of $v$ may contain a separately chosen initial interval $[0,{\epsilon }_n)$ on each arm, with no positive lower bound on the ${\epsilon }_n$.

The space is not first countable. Given any proposed countable neighbourhood basis $(U_n)$ at $v$, choose a point $x_n$ at coordinate $t_n>0$ on arm $n$ with $x_n\in U_n$. This is possible by the interval topology on that arm. The set whose intersection with arm $n$ is $[0,t_n/2)$ is an open neighbourhood $V$ of $v$, by the simplexwise definition. But $x_n\in U_n\setminus V$ for every $n$, so no $U_n$ lies inside $V$. Consequently this weak star cannot embed in the plane, or in any metric space.

The terminal vertices form an infinite closed discrete subset: on each closed arm they meet only its terminal endpoint. Hence the star is not compact, by closed discrete subsets of compact spaces are finite.

Suppose the variances of a sequence of random variables are bounded by $C$ and $\mathrm{Cov}(X_i,X_j)=0$ whenever $|i-j|>r$, for a fixed nonnegative integer $r$. The Cauchy-Schwarz inequality bounds each remaining covariance in absolute value by $C$. In the variance of a sum, there are at most $rn$ off-diagonal pairs, proving the displayed bound. The Chebyshev inequality makes the average minus its mean converge to zero in Probability. If all means equal $\mu$, the average therefore converges in Probability to $\mu$. independence of every summand is unnecessary.

The weak form of $(L^{\ast }-\lambda )u=f$ includes the same scalar shift as the differential equation. The unshifted identity $B^{\ast }[u,v]=(f,v)$ instead represents $L^{\ast }u=f$ and cannot imply the shifted equation for nonzero $\lambda$ unless $u=0$. On a smooth bounded domain choose any nonzero compactly supported smooth $u$ and take $L=-\Delta +1$, $f=Lu$ for a direct counterexample. For constant drift coefficients, integration by parts reverses the drift sign in $L^{\ast }$ without changing this spectral term.

For sufficiently regular coefficients and a function $u$, this boundary quantity is the outward flux appearing when a divergence-form principal operator is integrated by parts. Its zero condition is the natural Neumann boundary condition for the corresponding bilinear form. With only bounded coefficients and a Sobolev solution, the weak formulation can encode that natural condition without asserting that a pointwise boundary derivative exists.

An element that is way-below itself.

A water potential gradient is a spatial difference in water potential, the potential of water relative to a reference state, commonly expressed in pressure units. Water tends to move toward lower water potential when a permeable route exists. Solute concentration and hydrostatic pressure both contribute: concentrated cell sap can favour entry into a plant cell, while rising turgor pressure opposes further entry. Thus concentration difference alone need not determine the final direction. ^[ref]

For nonnegative integrable $f$, Tonelli's theorem applied on the triangle $0<y<x<1$ gives the exact identityFor signed or complex integrable $f$, first apply the same calculation to $|f|$ and use the triangle inequality for an integral. The defining integral then exists absolutely for almost every $x$, and the same L1 norm bound holds. The constant $1/\beta$ is sharp: use $f_n=n{1}_{(0,1/n)}$, whose L1 norm is one and whose weighted integrals above tend to $1/\beta$.

Finiteness at every point is stronger than this L1 space assertion. For $0<\beta <1$, take $x_0=1/2$, $\gamma =(1+\beta )/2$, and $f(y)=(x_0-y{)}^{-\gamma }$ on $(0,x_0)$, zero otherwise. It is integrable because $\gamma <1$, but $(V_{\beta }f)(x_0)=\infty$ because the power near $y=x_0$ is $\beta -1-\gamma =(\beta -3)/2<-1$. This is compatible with an integrable output after changing values on a null set.

For a target positive set consisting of two disjoint intervals of total probability $p$, choosing its heavier component has misclassification risk at most $p/2$. Choosing the spanning interval has misclassification risk at most $1-p$. The smaller bound is at most $1/3$. The constant is sharp: on three ordered atoms with equal masses and labels $1,0,1$, every single interval makes a mistake on at least one atom. Together with the VC uniform convergence theorem, this gives weak PAC learning using an interval hypothesis class.

An upper set generated by a finite set $A$ is represented by the minimal elements of $A$. Removing a nonminimal generator preserves the upward closure. If each pair has a least upper bound, then $\uparrow A\cap \uparrow B=\uparrow \{a\vee b:a\in A,b\in B\}$, so finite generators support intersection without enumerating the potentially infinite represented sets.

A universal formula has a block of universal quantifiers followed by a quantifier-free formula. Empty blocks are usually permitted. Truth transfers from a first-order structure to a substructure of a first-order structure for parameters in that substructure: every tuple in the smaller domain is available in the larger domain, and quantifier-free truth is preserved and reflected by the inclusion. This is the dual preservation direction to the forward preservation of an existential formula under a structure embedding.

Quantifier-free logical formulas have the same truth value in a substructure and its ambient first-order structure on tuples from the smaller carrier. First-order term evaluation agrees; atomic relations agree by restriction; Boolean connectives preserve agreement. A universally quantified quantifier-free matrix true on all ambient tuples is therefore true on all substructure tuples.

If a universal first-order theory in a first-order language with a constant entails an existential sentence with quantifier-free matrix, it entails a finite disjunction of closed-term instances of that matrix. Otherwise a first-order model of the first-order theory and all negated ground instances would have a ground-term substructure satisfying the first-order theory. An existential witness there is a tuple of closed-term values, contradicting its negated ground instance. Compactness theorem extracts finitely many instances.

For a finite-dimensional complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$, the shifted Harish-Chandra map identifies the centre of the universal enveloping algebra with the Weyl group-invariant polynomial functions on ${\mathfrak{h}}^{\ast }$. To normalize the shift, let $M^{\chi }$ have highest weight $\chi -\rho$, where $\rho$ is the Weyl vector. A central element $z$ acts by a scalar $j_z(\chi )$, and $z\mapsto j_z$ is the displayed algebra isomorphism. The Poincaré-Birkhoff-Witt theorem gives polynomial dependence on $\chi$ by projecting $z$ onto its pure Cartan component and evaluating at $\chi -\rho$. The associated central characters agree exactly when their shifted parameters lie in the same Weyl orbit. Without incorporating the shift into the parameter, the invariance is expressed using the dot action $w\cdot \lambda =w(\lambda +\rho )-\rho$.