Under charge conjugation, $\bar{u}{\gamma }^{\mu }(1-{\gamma }^5)d$ becomes $-\bar{d}{\gamma }^{\mu }(1+{\gamma }^5)u$. Parity reverses the chirality again and applies the Lorentz parity matrix. The minus sign cancels that of charge conjugation on the charged vector field. Thus each charged-current operator is exchanged with its Hermitian conjugate under CP symmetry, giving the displayed transformation for the CKM matrix $V$. For nondegenerate quark masses, physical CP conservation means that allowed quark rephasings can make $V$ real. A nonzero Jarlskog invariant obstructs this. A merely complex phase convention is not proof of physical CP violation, and $V=I$ is a CP-conserving counterexample to unconditional noninvariance.

For a Hermitian operator satisfying $A^3=A$, its squared operator $P=A^2$ is the projector onto the active eigenvalue-$\pm 1$ subspace. All remaining states have eigenvalue zero. Separate even and odd powers in the matrix exponential to obtain the displayed rotation on the active subspace and the identity on its spectator subspace. Thus an embedded two-level coupling need not square to the full Hilbert-space identity.

A memoryless quantum channel has independent identical noise on successive uses, described by the tensor-power map ${\Phi }^{\otimes n}$. This factorization does not restrict the encoder to product inputs: one can feed an entangled state across uses. A block code with equally likely messages $m$ chooses states ${\rho }_m$ and a decoder POVM $(D_m)$. Its average error is $1-|\mathcal{M}{|}^{-1}{\sum }_m\mathrm{Tr}[D_m{\Phi }^{\otimes n}({\rho }_m)]$, and its rate is $n^{-1}{\log }_2|\mathcal{M}|$. The classical capacity of a quantum channel allows such entangled block inputs; the one-use Holevo capacity gives the product-input capacity and requires regularization in general.

In a first-order language having at least one closed term, validity of a quantifier-free existential sentence forces inconsistency of the set of negations of all its closed-term instances. Otherwise take a model of those negations and its substructure generated by the closed terms. Quantifier-free truth is preserved there, so the valid existential sentence has a closed-term witness, a contradiction. Compactness then gives a finite disjunction of instances. The nonempty closed-term hypothesis is essential: in a language with no constants or nullary functions, the valid sentence $\exists x(x=x)$ has no closed-term instance.

Synaptic release probability is the probability of release at a specified site during a specified opportunity, such as one presynaptic action potential. It depends on the site's state and conditions; it is not the probability that the receiving neuron fires. In the example under synaptic vesicle, $p$ refers to one available site, whereas $1-(1-p{)}^4$ refers to any of four independent sites releasing.

For a continuous local martingale $M$, at most one continuous adapted increasing process $A$ starting at zero makes $M^2-A$ a local martingale. The difference of two candidates is a continuous finite-variation process and a local martingale, hence constant; its initial value is zero. Without the initial-value normalization only uniqueness up to an initial constant is possible.

For a zero-starting square-integrable martingale, the predictable quadratic variation is the predictable increasing process $A$ for which $M^2-A$ is a martingale. For a compensated jump process it integrates squared jump sizes against their rates. Unlike optional quadratic variation, it need not equal the sum of squared realized jumps. For continuous local martingales, the two brackets coincide after localization.

// Target: probability-and-statistics.bigb

For $A\subseteq {\mathbb{Z}}_N$, $N\ge 16$, put $\eta =|A|/N$, $M=\lfloor N/16\rfloor$, $J=\{0,\dots ,M-1\}$, and $w=1_J\ast 1_J$ using normalized convolution on a finite group. The weights $w(x)w(d-M)$ are supported where $0\le x\le 2M-2$ and $M\le d\le 3M-2$, so $x,x+d,x+2d,x+3d$ are distinct integer points below $N$. If $t=M/N$, then $\mathbb{E}w=t^2$ and the sum of absolute values of its Fourier coefficients is $t$, by Parseval identity on a finite group. Expanding the four indicators around their subset density leaves four terms with one balanced factor. Expand both weights in Fourier series and absorb each character $e(rx+sd)$ into the factors at $x$ and $x+d$. Character multiplication preserves the Gowers U3 norm, and the cyclic four-term progression bound with torsion factors bounds every error by $6^{1/8}\alpha$. Thus the weighted count is at least ${\eta }^4{t}^4-4\cdot 6^{1/8}\alpha t^2$. Since $t\ge 1/32$, positivity follows from $\alpha <{\eta }^4/(4096\cdot 6^{1/8})$.

Balance the indicator by its density on the interval, and extend it by zero to a cyclic group of order greater than twelve times the interval length and coprime to six. An eighth-power interval cube bound $\alpha$ bounds every four-term counting error by ${\alpha }^{1/8}$ through three Cauchy–Schwarz steps. The density contribution is bounded below by a constant times ${\delta }^4$, and the diagonal contribution tends to zero with interval length. A sufficiently small constant times ${\delta }^{32}$ therefore guarantees a nonconstant four-term progression. Stating the cube normalization distinguishes this parameter from the norm convention.

For $\dot{u}=u(\mu +u-v)$, $\dot{v}=\lambda u^2$ in $u>0$, the time change $d\tau /dt=u$ gives the affine linear system $u_{\tau }=\mu +u-v$, $v_{\tau }=\lambda u$. Shifting $w=v-\mu$ removes the forcing. Its eigenvalues are $(1\pm \sqrt{1-4\lambda })/2$. For $\lambda >1/4$, putting $\omega =\sqrt{4\lambda -1}/2$ givesFor fixed $\mu >0$ and initial $h,v_0$ tending to zero, the first return to small positive $u$ occurs after approximately $\pi /\omega =2\pi /\sqrt{4\lambda -1}$. This is a half-turn, not a full turn. At $\lambda =1/4$, $u=e^{\tau /2}[h+(\mu +h/2-v_0)\tau ]$ does not return and grows without bound. The loss of return diagnoses escape to infinity rather than a finite-amplitude homoclinic orbit. Physical time obeys $dt=d\tau /u$ and need not share the divergent desingularized travel time.

For the diffusionless quadratic activator-inhibitor model, choose the Dulac function $u^{-2}$. Then $u^{-2}f=1/v-b/u$ and $u^{-2}g=1-v/u^2$, giving divergence $(b-1)/u^2$. For $0<b<1$ it is strictly negative throughout the simply connected positive quadrant. Green's theorem rules out a periodic orbit: its boundary flux would be zero because the rescaled vector field is tangent to the orbit, while its interior divergence integral would be negative.

A mode with squared wavenumber $q$ in the quadratic activator-inhibitor model has negative trace $b-1-(1+d)q$ when $0<b<1$. Its determinant is $D(q)=d{q}^2+(1-db)q+b$. A positive minimum location and a negative minimum require $db>1$ and $(db-1{)}^2>4db$, equivalent to $db>3+2\sqrt{2}$. At onset, the double determinant root is $q_c=(1+\sqrt{2})/d$. In a bounded domain an admissible nonzero spatial mode must additionally lie in the open negative-determinant band.

If a real $m$-by-$n$ matrix $A$ has full column rank, with $m\ge n$, then $A^{T}A$ is positive-definite. Its upper-triangular Cholesky decomposition $A^{T}A=R^{T}R$, with positive diagonal, is unique. Defining $Q=A{R}^{-1}$ gives $Q^{T}Q=I$ and $A=QR$. Any other such factorization yields another Cholesky factor of $A^{T}A$, so its $R$ and then its $Q$ must agree. This proves existence and uniqueness; it does not assert that forming $A^{T}A$ is the most numerically stable algorithm.

Let $K$ be block diagonal with $2\times 2$ blocks $J=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$. If a real invertible matrix $A$ commutes with $K$, its QR decomposition $A=BC$, with $B$ orthogonal and $C$ upper triangular with positive diagonal, has both $BK=KB$ and $CK=KC$. Indeed $CK{C}^{-1}=B^{-1}KB$, so $E=K^{-1}CK{C}^{-1}$ is orthogonal and upper triangular in $2\times 2$ blocks. Orthogonality forces its off-diagonal blocks to vanish. Its diagonal blocks are $J^{-1}{C}_{i}J{C}_i^{-1}$. For $C_i=\left(\begin{array}{cc}a& b\\ 0& d\end{array}\right)$ with $a,d>0$, this block is $\left(\begin{array}{cc}d/a& -b/a\\ -b/a& (a^2+b^2)/(ad)\end{array}\right)$. Orthogonality of its columns forces $b=0,d=a$, and hence each block is the identity. Thus $E=I$, proving the claim.

For the normalized Black-Scholes call function, completing the square gives $F(v,m)=\Phi (d_1)-m\Phi (d_2)$, with $d_1=(-\log m+v/2)/\sqrt{v}$ and $d_2=d_1-\sqrt{v}$. The identities $d_1(v,1/m)=-d_2(v,m)$ and $\Phi (-x)=1-\Phi (x)$ implyFor a stock with risk-neutral drift $r$, maturity difference $\tau$, and positive strike $K$, put-call parity then givesThe positive sign of $r\tau$ in the reciprocal argument is forced by taking the reciprocal of $K{e}^{-r\tau }/S_t$. The identity includes $v=0$ by continuity and requires $m>0$.

The difference of the terminal power-call and power-put payoffs is $S_T^{\beta }-K^{\beta }$. Risk-neutral pricing and the geometric Brownian motion moment formula therefore give the displayed parity, with ${\lambda }_{\beta }=\beta (r-q)+{\sigma }^2\beta (\beta -1)/2$. At $\beta =1$ this reduces to dividend-adjusted put-call parity.

Suppose disjoint separating spheres cut a 3-manifold into pieces none of which is a punctured three-sphere. Compress one sphere along a disk disjoint from the others. The old sphere and the two new spheres enclose a three-times punctured sphere. At least one of the two pieces on the compressed side is not a punctured sphere; retain that new sphere and discard the other. The piece on its opposite side cannot become a punctured sphere, since capping its boundaries and applying the Three-dimensional Schoenflies theorem would make the original opposite piece a punctured sphere as well. Hence the number of spheres and the nontriviality condition can be preserved.

With the usual Dirac adjoint, Hermitian ${\gamma }^5$ and $\{{\gamma }^0,{\gamma }^5\}=0$, taking the adjoint gives ${\psi }^{†}{\gamma }^5{\gamma }^0\psi =-{\psi }^{†}{\gamma }^0{\gamma }^5\psi$. Consequently a real pseudoscalar with coupling $-g\varphi \bar{\psi }{\gamma }^5\psi$ has a Hermitian interaction only if $g^{\ast }=-g$. Writing $g=i{g}_P$ with real $g_P$ gives the familiar interaction $-i{g}_P\varphi \bar{\psi }{\gamma }^5\psi$. Formal Euler-Lagrange field equations obtained with independent spinor and adjoint variables can still be computed for the original symbolic coupling; it must not silently be identified with a real physical coupling.

Zero distance in a pseudometric defines an equivalence relation: the triangle inequality proves transitivity. If $x$ and $x^{\prime }$ have zero distance, as do $y$ and $y^{\prime }$, the triangle inequality in both directions gives $d(x,y)=d(x^{\prime },y^{\prime })$. Thus distance descends to the quotient and becomes a metric, since zero quotient distance means the classes coincide.

This first-order operator vanishes exactly on J-holomorphic curves. Linearization at a map $u$ gives a Fredholm operator between suitable Sobolev spaces of sections. At a constant map it is the Dolbeault operator on the trivial pulled-back tangent bundle.