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The gauge-covariant kinetic term for two massless quarks separates into left- and right-handed parts. Independent constant unitary quark flavor rotations preserve these terms, giving classical $U(2{)}_L\times U(2{)}_R$ internal symmetry in addition to local color $SU(3)$. The singlet axial $U(1{)}_A$ has a chiral anomaly, while the nonsinglet chiral symmetry and vector baryon number remain exact quantum symmetries, up to their shared discrete identifications. Equal nonzero quark masses reduce the continuous flavor symmetry to $SU(2{)}_V\times U(1{)}_V$.

An ancilla qubit is a two-dimensional quantum ancilla. It is the quantum version of an ancilla bit; an auxiliary quantum register can contain several such qubits.

For a binary relation with no symmetry or loop restrictions, an extension axiom demands a fresh element realizing any prescribed choices of incoming edges, outgoing edges and its own loop relative to a finite list of distinct parameters. All such axioms together eliminate quantifiers: the quantifier-free type of a parameter tuple determines which witness patterns can occur. These axioms differ from the undirected loop-free theory of the random graph.

For a positive nonsquare integer $D$ and positive integers $a,q$, the integer $D{q}^2-a^2$ is nonzero. Therefore $|\sqrt{D}-a/q|=|D{q}^2-a^2|/(q^2(\sqrt{D}+a/q))$. If $a/q\le \sqrt{D}+1$, the denominator factor is at most $2\sqrt{D}+1$; otherwise the approximation error already exceeds one. This proves that a quadratic irrational of this form is a badly approximable number. For $D=2$, splitting instead at $a/q=2$ gives the strict bound $|\sqrt{2}-a/q|>1/(4q^2)$.

// Target: analytic-number-theory.bigb

A real quadratic form is indefinite when it takes both positive and negative values. For a real symmetric matrix representing it, this means that its eigenvalues include both signs. For example, $q(x,y)=x^2-y^2$ has $q(1,0)=1$ and $q(0,1)=-1$; a zero value at the nonzero vector $(1,1)$ does not imply positive semidefiniteness.^[ref]

Let $Q:V\to {\mathbb{F}}_2$ be a quadratic form whose polar bilinear form of a quadratic form $B$ is nondegenerate, so $\dim V=2m$. For a symplectic basis $e_1,f_1,\dots ,e_m,f_m$, defineThis does not depend on the symplectic basis. The hyperbolic plane $Q(x,y)=xy$ has Arf invariant zero; $Q(x,y)=x^2+xy+y^2$ has Arf invariant one and vanishes only at zero. Every such quadratic form is an orthogonal direct sum of hyperbolic planes, with one copy of the latter plane replacing a hyperbolic plane when the Arf invariant is one.^[ref]

A useful basis-independent test isOn one two-dimensional summand with symplectic basis $e,f$, write $Q(xe+yf)=ax+by+xy$; the four terms sum to $2(-1{)}^{ab}$. Multiplication over the orthogonal direct sum proves the formula and the invariance of its sign. Consequently the number of zeros is $2^{2m-1}+(-1{)}^{\mathrm{Arf}(Q)}{2}^{m-1}$. For a fixed $B$ and $m\ge 1$, the numbers of binary quadratic refinements with Arf invariant zero and one are respectively $2^{m-1}(2^m+1)$ and $2^{m-1}(2^m-1)$.

Its symmetric matrix is $11^T-I$. The line spanned by $1$ has eigenvalue $n-1$ and its orthogonal complement has eigenvalue $-1$. Thus for $n>1$ it is nondegenerate with inertia $(1,n-1,0)$; for $n=1$ it vanishes. This gives a useful explicit example for Sylvester's law of inertia.

An integral quadratic lattice is a finite-rank free abelian group with a nondegenerate integral symmetric bilinear form. Its norm convention here is $q(x)=B(x,x)$. A basis gives an integral Gram matrix; basis changes preserve its determinant up to multiplication by the square of a unit.

The displayed bilinear form is the polar form of a quadratic form. In odd characteristic of a field, $Q(x)=B_Q(x,x)/2$ recovers the quadratic form; in characteristic of a field two, different forms can have the same alternating bilinear form. Over ${\mathbb{F}}_2$, two such forms differ by a linear map to the field, giving the affine parametrization of binary quadratic refinements.

For an ice mass in still water, approximate the wind stress by a fixed vector $a{\mathbf{e}}_x$ while retaining quadratic drag in the water. With mass per water-drag area $\mu =m/A_w$, water coefficient $b={\rho }_w{C}_w$ and $f=2\Omega \sin \varphi$, the steady balance is $a{\mathbf{e}}_x=bU\mathbf{u}+\mu f\hat{\mathbf{z}}\times \mathbf{u}$. Resolving along the drift and perpendicular to it gives $a\cos \theta =b{U}^2$, $a\sin \theta =\mu |f|U$. Thus $b^2{U}^4+(\mu f{)}^2{U}^2=a^2$ and $\tan \theta =\mu |f|/(bU)$. The direction is right of the wind in the northern hemisphere and left in the southern hemisphere. This angle neglects intrinsic boundary-layer turning, ocean currents and contact forces; it is not a universal observed sea-ice turning angle.

For positive downward speed, gravity and quadratic drag give $m\dot{v}=mg-\alpha v^2$, $v(0)=0$. Separation and the integral of $(1-s^2{)}^{-1}$ give the displayed approach to terminal velocity. The fallen distance is $(m/\alpha )\log \cosh (\sqrt{\alpha g/m}t)$. The limit as $\alpha \to 0$ recovers $v=gt$ and distance $g{t}^2/2$.

For each facility $i$ and location $k$, independently minimize its row contribution over bijections from the other facilities to the other locations, giving ${\ell }_{ik}$. Every complete assignment has row contribution at least its corresponding ${\ell }_{ik}$, so minimizing their sum is a lower bound on the quadratic objective. Fixed assignments are retained as constraints in this assignment problem, which the Hungarian algorithm solves. The independently optimal rows need not be simultaneously realizable, explaining why the bound can be strict.

If subsystem $S$ has pure state $|u⟩⟨u|$ in a normalized joint pure vector $|\Omega {⟩}_{SE}$, then $|\Omega {⟩}_{SE}=|u{⟩}_S\otimes |e{⟩}_E$. Indeed, for an orthonormal basis $\{v_j\}$ of $u^{\perp }$, each squared norm $\Vert (⟨v_j|\otimes I)|\Omega ⟩{\Vert }^2=⟨v_j|{\rho }_S|v_j⟩$ is zero. Therefore the joint vector has no component outside $\mathbb{C}u\otimes {\mathcal{H}}_E$. In particular, if Alice and Bob share a known pure entangled state, an adversary holding a purifying system has no correlations with it. This does not make every pure bipartite state useful for key distribution: a pure product state has no shared random correlations.

Suppose a perturbed polynomial has the local form $A(x-x_0{)}^2+ϵB+O((x-x_0{)}^3,ϵ(x-x_0),{ϵ}^2)$ with $AB\ne 0$. Dominant balance gives $x-x_0={ϵ}^{1/2}z$ and $A{z}^2+B=0$. The two resulting Puiseux series split a double root; their leading displacements can be real or a complex conjugate pair. Ordinary Taylor series in $ϵ$ cannot describe this splitting.

Choose integers $M>1$, $a$, $c$ and a seed $S_0$. Iterate $S_{n+1}=(a{S}_n+c)\mathrm{mod}M$ and return $U_n=S_n/M$. This gives a deterministic finite-state pseudorandom number generator with output in $[0,1)$. A nonzero increment gives a mixed generator; $c=0$ gives a multiplicative generator. Its period is at most $M$, but even a full period does not guarantee good multidimensional uniformity or statistical independence.

A pseudorandom number is an output of a deterministic algorithm chosen to imitate samples from a specified probability law. For example, a linear congruential generator returns fractions $S_i/M$ on a finite grid in $[0,1)$. Their intended uniform distribution is an approximation, and their serial dependence must be assessed separately from the generator's period.

A pseudo-Riemannian metric is a smooth nondegenerate symmetric bilinear form on every tangent space, with locally constant signature. Its positive-definite special case is a Riemannian metric. Unlike a positive-definite metric, an indefinite metric admits nonzero null vectors. In either signature its Levi-Civita connection is uniquely determined by torsion-freeness and metric compatibility.

Geometric splitting theorems give hypotheses under which a manifold decomposes as a metric product. The Cheeger-Gromoll splitting theorem gives a Euclidean line factor for a complete connected Riemannian manifold with nonnegative Ricci curvature containing a globally minimizing geodesic line. Other splitting theorems use different curvature or causal hypotheses.

For a normal subgroup $N$ with Prüfer rank $r$ and quotient of rank $s$, let $H=⟨h_1,\dots ,h_k⟩\le G$. Choose at most $s$ elements $x_j\in H$ generating the image of $H$. Write $h_i=w_i(x_1,\dots ,x_s)n_i$ with $n_i\in N\cap H$. The finitely generated subgroup $⟨n_1,\dots ,n_k⟩$ needs at most $r$ generators, so $H$ needs at most $r+s$. The full intersection $H\cap N$ need not be finitely generated.

Identify classical propositional formulae when their equivalence is provable from a fixed theory. Their equivalence classes form the Lindenbaum algebra: meet and join are induced by conjunction and disjunction, complementation by negation, and the order by provable implication. Provable equivalence is preserved by each connective, so these operations are well-defined; the familiar classical propositional identities make this a Boolean algebra. A homomorphism of lattices to $\{0,1\}$ preserving zero and one also preserves complements and therefore specifies a truth valuation of the atomic propositions. All axioms of the fixed theory represent one. Separating the class of an unprovable formula from one consequently gives a countervaluation, proving the completeness theorem for propositional logic.