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If a nonzero holomorphic function has expansionthen $m$ is the order or multiplicity of its zero at $z_0$.

This is the integral version of the Minkowski inequality.

For suitable measurable $F(x,y)$ and $1\le p<\infty$,Pair the left side with a unit vector in the dual $L^q$ space, use Tonelli's theorem, and apply Hölder's inequality in $y$.

For nonzero $y\in {\ell }^q$, a normalized sequence proportional to $\mathrm{sgn}(y_n)|y_n{|}^{q-1}$ attains equality in Hölder's inequality.

Exponents $p,q\in [1,\infty ]$ are conjugate when $1/p+1/q=1$, with the convention $1/\infty =0$.

A differential form is harmonic when it lies in the kernel of the Hodge Laplacian $\Delta =d\delta +\delta d$. On a compact manifold, this is equivalent to $d\alpha =0$ and $\delta \alpha =0$.

Its relation to cohomology is part of Hodge theory.

The codifferential is the formal $L^2$ adjoint of the exterior derivative. On $p$-forms in dimension $n$, one convention is

On an oriented Riemannian four-manifold, a two-form $\alpha$ is anti-self-dual when $\ast \alpha =-\alpha$.

On an oriented Riemannian four-manifold, a two-form is anti-self-dual when $\ast \alpha =-\alpha$. Here $\ast$ is the Hodge star operator.

On an oriented Riemannian four-manifold, a two-form $\alpha$ is self-dual when $\ast \alpha =\alpha$.

On an oriented Riemannian four-manifold, a two-form is self-dual when $\ast \alpha =\alpha$. Here $\ast$ is the Hodge star operator.

The Milliken–Taylor theorem gives monochromatic systems of separated block sums with a fixed compressed coefficient vector, such as $(1,2,3)$. Systems associated with two nonproportional compressed coefficient vectors need not have the same color: in particular, there is a finite coloring for which a finite-sums set and the $(1,2,3)$ Milliken–Taylor system cannot be monochromatic together.

For a sequence $(x_i)$, its finite-sums set is

An orthonormal sequence $(e_n)$ satisfies $⟨e_n,e_m⟩=0$ for $n\ne m$ and $\Vert e_n\Vert =1$.

This is the sequential form of orthonormality.

A sequence $x_n$ converges weakly to $x$ when $⟨x_n,y⟩\to ⟨x,y⟩$ for every $y$ in the Hilbert space.

Every bounded linear functional on a Hilbert space is inner product with one unique vector, with equal norms.

Every open cover of an affine variety has a finite subcover. Refine it by distinguished opens $D(f_i)$. Their complements have no common point, so the Nullstellensatz writes $1$ as a finite combination of the $f_i$; the corresponding finite family of distinguished opens covers.

Let $I\subseteq k[x_0,\dots ,x_n]$ be a homogeneous ideal over an algebraically closed field, and letbe the irrelevant ideal. Then $V_{+}(I)=\varnothing$ exactly when either $I=(1)$ or $\sqrt{I}=\mathfrak{m}$. For a proper homogeneous ideal this is also equivalent to ${\mathfrak{m}}^N\subseteq I$ for some $N$.

It is the homogeneous projective version of the Hilbert Nullstellensatz.

A complex affine algebraic set is either finite or has the cardinality of $\mathbb{C}$. One proof takes a positive-dimensional irreducible component and applies Noether normalization: the resulting finite dominant map to a positive-dimensional affine space has a point above every closed point.

For an ideal $I\subseteq k[x_1,\dots ,x_n]$ over an algebraically closed field,

For an algebraically closed field $k$, every maximal ideal of $k[x_1,\dots ,x_n]$ has the formfor a unique point $a\in k^n$. Equivalently, every proper ideal has a common zero.

Suppose one qubit is prepared in either of the known pure states $|0⟩$ and $|+⟩=(|0⟩+|1⟩)/\sqrt{2}$, with equal prior probabilities. Measuring in the computational basis seems natural, but a tilted projective measurement gives better quantum state discrimination. The two density operators and their difference areThe Pauli matrices give ${\Delta }^2=I/2$, and the eigenvalues of $\Delta$ are ${\lambda }_{\pm }=\pm 1/\sqrt{2}$. Their absolute sum gives trace distance $D=1/\sqrt{2}$, agreeing with trace distance between qubit states.

The positive spectral projector points along the unit Bloch vector $\mathbf{n}=(-1,0,1)/\sqrt{2}$:The outcome labels $+$ and $-$ here refer to the sign of the eigenvalue of $\Delta$; they do not denote a Hadamard basis measurement. Guess $|0⟩$ after $P_{+}$ and $|+⟩$ after $P_{-}$. This sign choice is the equal-prior Helstrom measurement for two pure states, a case of the Holevo–Helstrom theorem. ^[ref]

An explicit orthonormal basis for this projective measurement iswith $P_{\pm }=|h_{\pm }⟩⟨h_{\pm }|$. Thus the real coefficient rotation angle is $\beta =-\pi /8$ relative to the computational basis, while the Bloch vector of $P_{+}$ is tilted by $\pi /4$ from the positive $z$ axis toward negative $x$. State-vector and Bloch sphere angles differ by a factor of two.

Write $a=(1+1/\sqrt{2})/2$ and $b=(1-1/\sqrt{2})/2$, so $a+b=1$. The Born rule gives the conditional outcome probabilitiesThe event $P_{+}$ has probability gap $a-b=1/\sqrt{2}=D$, attaining the variational characterization of trace distance. Both preparations are identified correctly with conditional probability $a$, so

For comparison, a $Z$ projective measurement in the computational basis has conditional outcome probabilities $(1,0)$ on ${\rho }_0$ and $(1/2,1/2)$ on ${\rho }_{+}$. Guess zero on outcome zero and plus on outcome one. The average success isand the measured total variation distance is only $1/2$. The optimal tilted projective measurement gains about $0.103553$ in absolute success probability by balancing the two conditional error rates instead of identifying $|0⟩$ perfectly at the expense of $|+⟩$.

These are statistics for alternative measurements on independently prepared copies. Applying $Z$ first and then the tilted projective measurement is a different experiment, because the first measurement in quantum mechanics changes the density operator.

Heat capacity measures the heat supplied per small temperature increase along a specified process:The subscript $X$ records the constraint, such as fixed volume or fixed pressure. The units are $\mathrm{J}{\mathrm{K}}^{-1}$. The symbol $\delta Q$ emphasizes that heat is a transfer along a path, rather than a state function with a differential $dQ$. Consequently, an unspecified heat capacity does not determine the required heat.

For a closed system of fixed composition undergoing a quasistatic process, assume that only pressure-volume work occurs, with no friction and no change in bulk kinetic energy or potential energy. The first law of thermodynamics and $H=U+pV$, where $U$ is internal energy and $H$ is enthalpy, then giveThese partial derivatives define heat capacity at constant volume and heat capacity at constant pressure, with the amount and composition also held fixed. Integrating along the selected constraint gives $Q={\int }_{T_1}^{T_2}{C}_X(T)dT$ within a single thermodynamic phase; $Q=C_X\Delta T$ requires constant $C_X$. ^[ref]

For a computed example, take a monatomic gas in the classical ideal gas regime, with amount of substance $n=1\mathrm{mol}$ and stipulated molar gas constant $R=8.314\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1}$. The equipartition theorem gives $U=3nRT/2$, and the ideal gas law gives $pV=nRT$. ThusFor the same temperature increase $\Delta T=10\mathrm{K}$, the two processes requireHere work done by the gas is positive. At fixed pressure, the extra heat supplies expansion work; the internal energy increase is the same because internal energy of an ideal gas depends only on temperature. Their final volume values differ. ^[ref]

The practical value of heat capacity is that a measured heat input and temperature response can determine it. Once its constraint and temperature dependence are known, it predicts heating requirements and supplies entropy changes through entropy from heat capacity. ^[ref]