Charge conjugation symmetry is invariance of the dynamics under charge conjugation. It pairs a process with its particle-antiparticle conjugate at equal rates. C violation is the failure of this invariance.

Average $1-\cos (uy)$ over $0\le u\le 1/\lambda$. Multiplication by $\lambda$ gives $1-\sin (y/\lambda )/(y/\lambda )$, with value zero at $y=0$. This is at least $1-\sin 1$ when $|y|\ge \lambda$. Tonelli's theorem then gives the displayed bound for a probability measure with characteristic function $\varphi$. The bound converts control of the characteristic function near zero into tightness of probability measures and is useful in proving the Lévy continuity theorem.

If $\mathbb{E}|X|<\infty$, the characteristic function $\varphi (u)=\mathbb{E}{e}^{iuX}$ satisfies ${\varphi }^{\prime }(0)=i\mathbb{E}X$. The bound $|(e^{iuX}-1)/u|\le |X|$ and the dominated convergence theorem justify differentiation at zero. For mean zero, $\psi (u)=|1-\varphi (u)|/|u|$ extends continuously by $\psi (0)=0$, and $|1-\varphi (u{)}^n|\le n|u|\psi (u)$ by a finite geometric series. This supplies a direct characteristic-function proof of the weak law of large numbers.

// Target: martingale.bigb

For a constant-coefficient linear recurrence relation over the complex numbers, a nonzero root $\mu$ of multiplicity $m$ in its characteristic polynomial of a linear recurrence supplies the $m$ linearly independent solutionsSubstitute $y_n={\mu }^n{z}_n$. Each factor $E-\mu$ of the shift linear operator becomes $\mu$ times the forward difference operator on $z_n$, so its $m$th power kills every polynomial in $n$ of degree less than $m$. Their independence follows from the Lagrange root bound over a field. The same argument works over a field of characteristic zero; the displayed monomials need modification in positive characteristic.

A zero root needs separate initial-data treatment: for characteristic polynomial ${\lambda }^m$, the recurrence relation is $y_{n+m}=0$, leaving the first $m$ values arbitrary.

Every complex character of a representation of a finite group $G$ is an integer linear combination of characters of induced representations from one-dimensional group representations of subgroups $H_j\le G$. Negative coefficients are allowed. Unlike Artin induction, the subgroups need not be cyclic groups. Applied to an Artin L-function, compatibility with induced representations turns this character identity into a product of one-dimensional Artin L-functions with integer exponents, reducing the behavior at $s=1$ to abelian extensions.

Every complex character of a representation of a finite group $G$ is a rational linear combination of characters of induced representations from one-dimensional group representations of cyclic subgroups $C_j$. One-dimensionality follows because the irreducible representations of a finite cyclic group over the complex numbers are one-dimensional. The rational coefficients distinguish this statement from Brauer induction, which permits integer coefficients by allowing more general subgroups. For a rational-valued character, another form of Artin induction uses rational combinations of induced representations of trivial group representations of cyclic subgroups.

A group of automorphisms of a finite group permutes both its conjugacy classes and its irreducible characters. The two permutation representations have the same character: each automorphism fixes equally many conjugacy classes and irreducible characters. The invertible character table intertwines the two permutation matrices, which have equal traces. The claim concerns fixed classes, not merely fixed elements.

Column orthogonality and the regular character give quadratic and linear equations for missing character values once their degrees are known. These equations can admit spurious rational solutions. Every character value is an algebraic integer, so a rational candidate must be an ordinary integer. Applying this extra condition can remove ambiguities that orthogonality alone leaves.

The irreducible characters of a finite group form a basis of its complex class functions. Equality of every irreducible character value therefore forces equality of every class-function value, including indicators of conjugacy classes, and hence conjugacy. The converse follows from the class-function property.

The pulsation period of a Cepheid variable is correlated with its absolute magnitude in a specified photometric band. Calibrating the relation with stars of known distance allows an observed period and corrected apparent magnitude to give a distance. Extinction and metallicity corrections matter for a precise cosmic distance ladder.

Put $c=m\kappa /{\hslash }^2$, $k_n=(n+1/2)\pi /a$, and ${\delta }_n=a({\lambda }_n-k_n)$. The quantization equation gives ${\delta }_n=\arctan (c/{\lambda }_n)$, so ${\lambda }_n{\delta }_n\to c$ and ${\delta }_n\to 0$. Therefore ${\lambda }_n^2-k_n^2\to 2c/a$, proving the displayed absolute energy shift. Its ratio to the unperturbed energy tends to zero. The normalized unperturbed even state has probability density $1/a$ at the origin, so its first-order delta-barrier expectation is also $\kappa /a$.

Set $c_n=4^{-n}\left(\genfrac{}{}{0ex}{}{2n}{n}\right)$, with $c_0=1$. Cancellation of factorials givesConsequently $c_n$ decreases to zero: $\log c_n\le -\frac{1}{2}{\sum }_{j=1}^{n}1/j\le -\frac{1}{2}\log (n+1)$. The inequality uses $\log (1-x)\le -x$ and comparison with ${\int }_1^{n+1}dt/t$. Conversely $c_n\ge 1/(n+1)$ by induction, since its ratio is at least $(n+1)/(n+2)$. Thus $\sum c_n$ diverges while $\sum (-1{)}^n{c}_n$ has conditional convergence by the alternating series test. These elementary bounds require no asymptotic formula for factorials.

Cellular reprogramming changes an established cell state so that a different developmental program becomes accessible. Experimentally induced cellular reprogramming must be distinguished from the ordinary progression of cell differentiation. An ovum's cytoplasm can reset aspects of a transferred cell nucleus; this does not mean that a specialized donor normally returns to an early developmental state in its original surroundings. ^[ref]

Terminal differentiation is advanced specialization at the end of a developmental lineage. It often includes withdrawal from the cell cycle; the term does not mean that every nuclear gene needed for another cell fate has been deleted. A mature mammalian red blood cell is an especially restrictive example because it lacks a cell nucleus; the intestinal nucleus transfer experiment used nucleated donors.

Cell fate is the developmental outcome of a biological cell or its descendants in a specified setting. What it normally becomes and what it can become after an experimental intervention are different questions. The intestinal nucleus transfer experiment tests potential after cellular reprogramming, rather than directly tracing the normal cell fate of the donor.

Wee1 is a protein kinase that inhibits CDK1 by adding a phosphate group at an inhibitory site. Purified human Wee1 phosphorylated Cdc2 at Tyr15 in the reported in vitro assay, with no Thr14 modification detected. A catalytic-site mutation abolished both the measured protein kinase activity and the ability to inhibit division. This identifies a catalytic brake rather than merely a correlated protein amount.^[ref]

Do not attribute all inhibitory CDK1 phosphorylation to Wee1: the Xenopus laevis Myt1 experiment separately identified activity toward both Thr14 and Tyr15. Nor does every phosphorylation inhibit its target: activating Thr161 modification in the studied Xenopus laevis Cdc2 has the opposite role. In cell-cycle activation feedback, suppression of Wee1 removes a brake on CDK1, reinforcing the transition without implying that the total amount of either protein instantly disappears.

Cdc25 is a protein phosphatase: it removes inhibitory phosphate groups from CDK1, relieving inhibition of an appropriately prepared cyclin B–CDK1 assembly. Its reaction is removal, whereas Wee1 is a protein kinase adding inhibitory modification. Removing the inhibitory sites is different from removing the activating phosphorylation supplied by CAK.^[ref]

Cdc25 action on CDK1 must also be distinguished from phosphorylation of Cdc25 itself. In cell-cycle activation feedback, CDK1-dependent regulation promotes Cdc25 activity, which can then activate additional inhibited CDK1. The complete regulatory path can involve intermediates. A histone H1 kinase assay of a mixture reports downstream protein kinase action; assigning its change to Cdc25 requires a perturbation or an independent protein phosphatase assay, rather than interpreting histone H1 labeling as phosphate removal.

Cyclin B is a regulatory protein partner of CDK1. The protein kinase catalytic component is CDK1; bound cyclin B helps make that component competent to act. Changes in cyclin B synthesis and destruction regulate how much assembly can form, while CDK1 phosphorylation states determine whether bound assemblies are inhibited. This separates the supply of regulatory partners from release of an already assembled brake.^[ref]

More cyclin B therefore need not give a proportional increase in CDK1 activity. Compare the matched-input cell-cycle hysteresis experiment, where different histories produced different activities, and the distinct measurements under CDK1 activity and protein abundance. Neither comparison makes cyclin B itself the catalytic protein kinase.

Cyclin-dependent kinase 1, abbreviated CDK1 and historically called Cdc2, is the catalytic protein kinase in the CDK1–cyclin B assembly. CDK1 transfers a phosphate group from ATP to a protein substrate; cyclin B supplies a regulatory binding partner. Binding and phosphorylation regulate catalytic competence, so detecting CDK1 protein alone does not establish activity. The distinction makes a durable catalytic component reusable as the cell cycle changes its regulatory inputs.^[ref]

The cell-cycle activation feedback uses CDK1 activity as both an output and a regulatory input: activity promotes Cdc25 activation and suppresses Wee1, reinforcing release from inhibitory phosphorylation. The already described cell-cycle hysteresis experiment tests how that output depends on history at matched cyclin B input; the histone H1 kinase assay below explains what its output measures. A binding diagram alone cannot establish hysteresis.

A 2003 study split a Xenopus laevis egg extract into portions with different histories. One received calcium to destroy endogenous cyclin B before receiving nondegradable cyclin B; the other received nondegradable cyclin B first, then calcium. The same sustained cyclin B dose was approached from low or high Cdc2 activity. A phosphorimager quantified histone H1 phosphorylation, sampling every $15$–$30$ minutes until an apparent stationary response. At $45$–$60\mathrm{nM}$, the dose maintained high activity but failed to induce it from the low state.^[ref]

Controls tracked labeled endogenous cyclin B disappearance after calcium treatment and persistence of nondegradable cyclin B, addressing unequal remaining input. The comparison supports hysteresis under extract conditions; finite observations cannot establish indefinite persistence. It does not isolate each reinforcing path or establish thresholds in intact biological cells.^[ref]