More generally, two field extensions inside a common overfield are linearly disjoint field extensions when the multiplication map from their tensor product of fields into that overfield is injective. The following basis criterion is the finite-degree case.

Finite extensions $L_1/K$ and $L_2/K$ are linearly disjoint when products of basis elements form a basis of their compositum. In particular,Extensions of coprime finite degrees are linearly disjoint.

The field automorphism group fixing $K$ consists of all field automorphisms of $L$ fixing $K$ pointwise, with composition. It is defined without assuming that $L/K$ is Galois. For $L=K(\alpha )$ with algebraic degree $d$, the action on the roots of the minimal polynomial inside $L$ is faithful, so this group embeds in $S_d$ and is finite.

For all finite-dimensional density operators, including pure states and mixed states, the unsquared quantum fidelity and trace distance obeyIn the squared quantum fidelity convention this is $1-\sqrt{F_{\mathrm{sq}}}\le D\le \sqrt{1-F_{\mathrm{sq}}}$. The bounds translate a high quantum fidelity into a small operationally meaningful trace distance. If $F\ge 1-\epsilon$ with $0\le \epsilon \le 1$, then $D\le \sqrt{2\epsilon -{\epsilon }^2}$; if $D\le \delta$, then $F\ge 1-\delta$. The upper bound is attained by the pure-state trace distance and fidelity identity. For the fidelity of commuting mixed states example, the bounds read $0.0652\lesssim 0.3\lesssim 0.3551$. ^[ref]

A POVM $\{E_i\}$ gives outcome probability distributions $p_i=\mathrm{Tr}(\rho E_i)$ and $q_i=\mathrm{Tr}(\sigma E_i)$ by the Born rule. Their classical overlap, the Bhattacharyya coefficient, is $B(p,q)={\sum }_i\sqrt{p_i{q}_i}$. Fuchs and Caves showed that minimizing this overlap over all POVMs gives the unsquared quantum fidelity:The same POVM is applied to either possible input quantum state. Each measurement in quantum mechanics has $B(p,q)\ge F$; some measurement attains equality, including when the density operators have zero eigenvalues. Thus quantum fidelity quantifies the smallest classical overlap that observation can reveal. For fidelity of commuting mixed states, measurement in a common eigenbasis attains the minimum. This operational interpretation compares statistics without assigning a probability that two mixed states are equal. ^{[ref] [ref]}

When two density operators are commuting elements, choose a common orthonormal eigenbasis and write $\rho =\mathrm{diag}(p_i)$, $\sigma =\mathrm{diag}(q_i)$. Their quantum fidelity becomes the classical overlap of the two probability distributions, while their trace distance becomes total variation distance:This follows by computing the positive square root of an operator entry by entry in the common eigenbasis. For the qubit mixed states $\rho =\mathrm{diag}(0.9,0.1)$ and $\sigma =\mathrm{diag}(0.6,0.4)$,The two contributions to quantum fidelity are about $0.734847$ and $0.2$. In contrast, the chance that two independent computational basis measurements, one on each preparation, print the same label is $0.9(0.6)+0.1(0.4)=0.58$. That particular event has a specified probability; neither $F$ nor $F_{\mathrm{sq}}$ is that probability.

Here trace distance $0.3$ means an equal-prior quantum state discrimination success of $0.65$ by the Holevo–Helstrom theorem. Measuring in the common computational basis and guessing $\rho$ on outcome zero and $\sigma$ on outcome one achieves $[0.9+0.4]/2=0.65$. Also $\sqrt{1-F^2}\approx 0.3551>D$: the pure-state trace distance and fidelity identity is an equality for pure states, rather than a formula for arbitrary mixed states.

For any unitary $U$, the unique positive square root of an operator obeys $\sqrt{UA{U}^{†}}=U\sqrt{A}{U}^{†}$. Conjugating the positive operators appearing in trace distance and quantum fidelity therefore conjugates their square roots. Cyclicity of the trace removes the unitary factors, proving invariance of both quantities under simultaneous unitary transformations of the states.

// Target: quantum-information-theory.bigb

For normalized pure states $\rho =|\psi ⟩⟨\psi |$ and $\sigma =|\phi ⟩⟨\phi |$, the trace distance and quantum fidelity satisfyThe last two expressions distinguish the unsquared quantum fidelity from the squared quantum fidelity. Identical pure states give $D=0$, and orthogonal pure states give $D=1$. The equality need not hold for mixed states; the Fuchs-van de Graaf inequalities give bounds in that case.

Brief proof: for two distinct pure states, their two-dimensional span contains the nonzero part of $\rho -\sigma$. This restriction has trace zero and determinant $-(1-F^2)$, hence eigenvalues $\pm \sqrt{1-F^2}$. Its trace norm is twice the positive eigenvalue, giving trace distance $D=\sqrt{1-F^2}$. For identical pure states, $\rho -\sigma =0$ directly.

This is the square of the unsquared quantum fidelity, another common convention for fidelity of quantum states. If one state is the normalized pure state $|\psi ⟩⟨\psi |$, thenBy the Born rule, the squared quantum fidelity here is the probability that a measurement in quantum mechanics with acceptance orthogonal projector $|\psi ⟩⟨\psi |$ accepts $\sigma$. For two pure states it is $|⟨\psi |\phi ⟩{|}^2$: the comparison of $|0⟩$ with $|+⟩$ has acceptance probability $1/2$ and unsquared quantum fidelity $1/\sqrt{2}$.

For two general mixed states, neither quantum fidelity nor squared quantum fidelity is a probability that the states are equal. Equality of two specified density operators is a definite mathematical condition. The measurement characterization of quantum fidelity instead describes overlap between outcome probability distributions. The ensemble average used in reliable quantum source compression is normally the average squared quantum fidelity with the emitted pure state; the convention must be stated when comparing formulas.

For a connected oriented topological surface of finite type with negative Euler characteristic, every orientation-preserving mapping class has a periodic representative, preserves an essential multicurve, or has a pseudo-Anosov map representative. In the usual disjoint classification, the reducible case means a nonperiodic class preserving such a multicurve. The assertion concerns representatives up to isotopy, rather than requiring an arbitrary homeomorphism in a periodic class to have finite order.

For a torus with one puncture, the action on first homology identifies its mapping class group with $S{L}_2(\mathbb{Z})$. For the corresponding matrix $M$, the periodic classes have $|\mathrm{tr}M|<2$ or $M=\pm I$. Nonperiodic reducible classes have $|\mathrm{tr}M|=2$ and $M\ne \pm I$; pseudo-Anosov map classes have $|\mathrm{tr}M|>2$. For example, $\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$ is a Dehn twist, while $I$ is periodic despite having the same trace.^[ref]

A puncture is distinguished from a boundary circle fixed pointwise: boundary twists give additional mapping classes invisible to the displayed $S{L}_2(\mathbb{Z})$ action. For the closed torus, the hyperbolic case has an Anosov homeomorphism representative.

The curvature form of a connection on a vector bundle is $F=dA+A\wedge A$ in a local frame. A principal connection has the corresponding Lie algebra-valued form $F=dA+\frac{1}{2}[A\wedge A]$. It measures the failure of horizontal directions to be integrable.

The curvature is $F(A)=d_A^2$ and in a local frame satisfies $F(A)=dA+A\wedge A$.

A section of a fiber bundle $\pi :E\to B$ is a continuous map $s:B\to E$ with $\pi \circ s={\mathrm{id}}_B$. It chooses a point of each fibre continuously. A section of a vector bundle is the special case for vector fibres; a projective bundle section chooses a line in each fibre of the underlying vector bundle.

The arrow on a charged-fermion line in a Feynman diagram tracks fermion-number flow and fixes the order of the gamma matrices between external Dirac spinors. For an external particle it agrees with the particle direction; for an external antiparticle it is opposite to the antiparticle direction. It therefore need not point along the assigned four-momentum of every external leg.

A Feynman integral is the momentum-space integral associated with a Feynman diagram: each independent loop momentum is integrated with measure $d^{d}k/(2\pi {)}^d$, and propagators, vertex numerators and momentum conservation determine the integrand. Its definition also requires a pole prescription and, when divergent, a regularization in quantum field theory. Feynman parameters combine propagator denominators and permit shifts of loop momenta. A massless two-propagator example is $\int d^{d}k/[(2\pi {)}^d{k}^2(p-k{)}^2]$, with its causal prescriptions understood. Subtracting local counterterms turns a regulated diagram into a finite contribution to a renormalized amplitude.

An orthogonal projection $P$ splits a one-particle Hilbert space into empty and filled modes. Take completed exterior Fock spaces of empty particles and of holes in the filled modes, with the graded tensor product enforcing the canonical anticommutation relations between both factors. The vacuum is killed by particle annihilation in $PH$ and by particle creation in $(I-P)H$. Only finite particle-hole excitations are needed for a dense basis. Normal ordering subtracts the filled-vacuum expectation from bilinear observables.

Let $\chi =\chi (r)$, $r=|x|$, and let $f(r)|0⟩$ be a normalized eigenstate of $H_0$ with $E>0$, where $|0⟩$ is the Clifford vacuum. Acting with the supercharge differentiates only its wavefunction:The normalized partner is $E^{-1/2}(f^{\prime }-{\chi }^{\prime }f){\psi }_r^{†}|0⟩$. Indeed $[H,Q]=0$, and $\Vert Qf|0⟩{\Vert }^2=E\Vert f|0⟩{\Vert }^2$; acting with $Q$ again returns $\sqrt{E}$ times the original state. The expression at $r=0$ is interpreted by the regular extension of the original state. Although its coefficient depends only on $r$, its degree-one components contain the radial unit vector.

For $n\ge 1$, Pépin's test says that the Fermat number $F_n$ is a prime number exactly when the displayed modular congruence holds. Modular exponentiation evaluates it by exponentiation by squaring without constructing the enormous ordinary power. The index restriction excludes $F_0=3$, where the base itself is divisible by the candidate.^[ref]

For a proof outline of sufficiency, any prime factor $p$ of a passing candidate gives $3$ multiplicative order exactly $F_n-1$, a power of two: its half-power is $-1$, rather than $1$. Fermat's little theorem forces $F_n-1\mid p-1$, so $p\ge F_n$, and hence $p=F_n$. Conversely, for a prime candidate, $F_n\equiv 5(\mathrm{mod}12)$; quadratic reciprocity and Euler's criterion give the required half-power. At $n=1$, the check is $3^2\equiv 4=-1(\mathrm{mod}5)$.

If $m<n$, a common divisor of $F_m$ and $F_n$ divides both $F_n-2$, by the Fermat number product identity, and $F_n$. It therefore divides two. Both Fermat numbers are odd integers, so their greatest common divisor is one. Each term's prime factors are consequently different from the prime factors of every other term.

For the normalized Feigenbaum map, put $\gamma =-g(1)$ and defineThe Feigenbaum attractor is the nested intersection $A={\bigcap }_n{\bigcup }_i{\Delta }_i^{(n)}$. Each level interval has two children, and the largest interval length tends to zero. Consequently every compatible infinite binary address selects one point of this Cantor set.^[ref]

At level $n$, the intervals are visited cyclically: the label changes from $i$ to $i+1$ modulo $2^n$. Expressing $i$ in binary therefore turns iteration on $A$ into the dyadic adding machine. One adds one with carries, rather than discarding the first symbol as in a full shift. For example, addresses written with the least significant digit first begin $000\dots \mapsto 100\dots \mapsto 010\dots \mapsto 110\dots$.

There are $2^n$ geometric pieces at depth $n$, but those pieces are visited in a cycle. Counting them does not make the time evolution a two-symbol full shift. The Feigenbaum geometric potential uses this depth to study spatial contraction and Hausdorff dimension.

The Feigenbaum geometric potential is a bounded negative symbolic potential $U$ encoding the lengths of nested intervals of the Feigenbaum attractor. In an appropriate binary coding,and changing only the distant tail of an address changes $U$ by an exponentially small amount. For odd interval labels, a sum along reversed address prefixes satisfieswith a bound independent of the depth and address. Even-label intervals have lengths comparable to neighboring odd-label intervals. These uniform estimates connect geometric lengths with thermodynamic formalism.^[ref][ref]

The bounded logarithmic error is important. For fixed real $t$, exponentiation givesAfter summing and taking $n^{-1}\log$, the error is at most $|t|C/n$, which vanishes. Thus geometric partition sums and symbolic partition sums give the same limiting topological pressure. The almost-additive partition-function limit supplies existence under the bounded-distortion hypotheses; bounded negative potentials give a strictly decreasing pressure.

This potential measures geometric contraction across partition levels. It is not an unbounded derivative potential evaluated at the critical point, where $g^{\prime }(0)=0$, and partition depth is distinct from elapsed orbit time.

Can feedback delay change correction timing even when the physical task stays the same? Susilaradeya and colleagues tested this in a human tracking experiment. Participants pressed with both index fingers against fixed force sensors; the two forces controlled a screen cursor along two diagonal axes. A target moved in slow circular motion, usually at $0.2$ rotations per second. The researchers sampled the forces at $50$ samples per second and inserted an additional delay of $0$, $100$, $200$, $300$, or $400$ milliseconds before displaying the cursor. Each trial lasted $20$ seconds; the $70$ trials included $14$ per delay, in pseudorandom order. ^[ref]

The readout was fluctuation in cursor angular velocity, analyzed by Fourier transforms over approximately $10$ seconds from the middle of each trial. The principal peak was around $2$ hertz without added delay, around $1.4$ hertz with $100$ milliseconds, and around $1$ hertz with $200$ milliseconds. Thus changing observation timing changed the timing of corrective motion. ^[ref] This is an empirical motivation for taking stale observations seriously. Human feedback control includes prediction and other dynamics; these observations do not measure $k$ in the ideal integrator model or test its $k\tau =\pi /2$ boundary.