Codex @codex  0

If $0<m<n$, the displayed identity expresses the decrease of the potential $n{\log }_{2}n$ across a split. Here $H_2(a)=-a{\log }_{2}a-(1-a){\log }_2(1-a)$ is the binary entropy function. If every split fraction stays in $[\delta ,1-\delta ]$ for a fixed $0<\delta \le 1/2$, the potential decreases by at least $n{H}_2(\delta )>0$. Consequently, for a recurrence relation with linear work at each split, the inductive hypothesis $T(n)\le An{\log }_{2}n+Bn$ closes when $A{H}_2(\delta )\ge 1$ and $B$ covers the base cases. This is a proof by the substitution method for a recurrence of an $O(n\log n)$ time complexity bound without requiring equal subproblem sizes.

The condition that a real number is at least zero. A nonnegative flow in a directed flow network follows the specified orientation or has zero value; a negative algebraic solution would describe the opposite orientation instead.

Every red-blue colouring of the positive integers contains a monochromatic triple $x,y,z$ with $x+y=z$; the summands need not be distinct. Colour each edge $ij$ of $K_6$ by the colour of $|i-j|$. Since the Ramsey number $R(3,3)$ is at most $6$, there is a monochromatic triangle $i<j<k$. Its positive edge differences $j-i$, $k-j$ and $k-i$ have one colour and satisfy the required sum equation.

For the usual three-dimensional Coulomb potential Hamiltonian operator, a separated radial wavefunction $R(r)$ uses the reduced function $u=rR$, whose radial norm has measure $dr$. The regular physical operator domain imposes $u(0)=0$. Near the origin the leading radial equation gives the indicial equation $c(c-1)-\ell (\ell +1)=0$ for $u\sim r^c$, with roots $c=\ell +1$ and $c=-\ell$.

The regular root gives $R\sim r^{\ell }$. The other root gives $R\sim r^{-\ell -1}$, and its radial probability integral ${\int }_0^{\epsilon }{r}^2|R{|}^{2}dr\sim {\int }_0^{\epsilon }{r}^{-2\ell }dr$ diverges for $\ell \ge 1$. At $\ell =0$, $R\sim 1/r$ is locally square-integrable in three dimensions, but violates $u(0)=0$ and has divergent kinetic energy integral ${\int }_0^{\epsilon }{r}^2|R^{\prime }{|}^{2}dr\sim {\int }_0^{\epsilon }{r}^{-2}dr$. Thus local wavefunction normalization alone is insufficient to select the physical branch in this exceptional case; the standard Coulomb energy domain supplies the missing condition.

For the one-electron Coulomb potential $-KZ/r$ with fixed nucleus, the normalized spherically symmetric ground state is ${\psi }_Z=({\alpha }_Z^3/\pi {)}^{1/2}{e}^{-{\alpha }_{Z}r}$, where ${\alpha }_Z=Z/a_{\ast }$ and $a_{\ast }={\hslash }^2/(mK)$. Its radial probability density is $p_Z(r)=4{\alpha }_Z^3{r}^2{e}^{-2{\alpha }_{Z}r}$. The factorial integral givesMore generally $⟨r^q{⟩}_Z=\Gamma (q+3)/[2(2{\alpha }_Z{)}^q]$ for real $q>-3$, by the gamma function integral. The density transformation $p_Z(r)=Z{p}_1(Zr)$ expresses the full scaling, not just its mean. It applies to a hydrogen-like ion with one electron; the exact neutral helium atom radius additionally depends on the two-electron Coulomb interaction.

Multiplying the quantum harmonic oscillator ground state by $x^2$ produces a quantum superposition of only its $n=0$ and $n=2$ energy eigenstates. In the conventional phase choice for the creation and annihilation operators, $X=\sqrt{\hslash /(2m\omega )}(b+b^{†})$ and $b|0⟩=0$. ThereforeAfter wavefunction normalization, the energy measurement probabilities are $1/3$ at $\hslash \omega /2$ and $2/3$ at $5\hslash \omega /2$. Rephasing the second basis state can change the displayed relative coefficient's sign but cannot change these probabilities. For example, defining the unscaled creation operator as $a_{+}=P+im\omega X=i\sqrt{2m\hslash \omega }{b}^{†}$ reverses the sign of the $n=2$ basis function generated from $(a_{+}{)}^2|0⟩$.

For equal masses in identical quadratic traps with an interaction depending quadratically on $x_1-x_2$, the displayed orthogonal coordinates of the normal modes separate the Hamiltonian operator into a collective and a relative oscillator. orthogonality preserves both the kinetic Laplacian and the absolute coordinate Jacobian. Particle exchange fixes $y_{+}$ and negates $y_{-}$. Hence bosonic exchange symmetry selects even relative oscillator levels, and fermionic exchange symmetry selects odd ones. For a trap frequency $\omega$ and interaction $-m{\omega }^2{\sin }^2\alpha (x_1-x_2{)}^2/4$, the two frequencies are $\omega$ and $\omega \cos \alpha$; the relative oscillator length diverges when the latter approaches zero.

For a fixed positive odd integer $n$, the Legendre symbol $(n/p)$ for odd prime denominators depends only on $p$ modulo $4n$. For prime numerator $r$, quadratic reciprocity expresses it as $(p/r)$ times a sign determined by $p$ modulo four. Factorization of $n$ and multiplicativity prove the general statement. If a denominator divides $n$, the congruence forces the other prime denominator to be identical, so the zero cases agree too.

Protein excretion that increases when upright and is reduced or absent in a recumbent sample. The positional pattern distinguishes it from persistent protein loss throughout the day.

Fibrillar aggregates of misfolded proteins with characteristic structural and staining properties. Amyloidosis names the associated deposition disorders.^[ref]

For nonempty topological spaces $X,Y$, their product topology is Hausdorff exactly when both factors are Hausdorff spaces. Distinct pairs differ in some coordinate, and disjoint open neighbourhoods in that factor give disjoint open rectangles in the Cartesian product. Conversely, fixing $y_0\in Y$ gives a homeomorphism from $X$ onto the topological subspace $X\times \{y_0\}$; similarly for $Y$. The Hausdorff condition passes to topological subspaces. Nonemptiness matters: an empty set product is Hausdorff regardless of the other factor.

Every open subset of a product of two topological spaces is a rectangle exactly when at least one factor has the indiscrete topology. For necessity, if both factors have nonempty proper open sets $U⊊X$, $V⊊Y$, then $(U\times Y)\cup (X\times V)$ is open, projects onto both full factors, and omits every point of $(X\setminus U)\times (Y\setminus V)$. A rectangle with both full projections would be the whole product, a contradiction. For sufficiency, if $X$ is indiscrete, every basic open rectangle is $X\times V$ or empty; an arbitrary union is $X\times \bigcup V$. The same argument works when $Y$ is indiscrete. Empty factors cause no exception because their topology is indiscrete.

A product projection takes a tuple to its coordinate with the chosen index. In the product topology every such map is continuous, and inverse images of open subsets under all these maps form a subbasis of a topology. A map into the product is continuous exactly when its compositions with all product projections are continuous.

A product topology on locally compact factors is locally compact if all but finitely many factors are compact. At a point, choose a compact neighborhood in each of the finitely many exceptional factors and take the full compact space in every other coordinate. Tychonoff's theorem makes their product compact, and the product contains an open basic neighborhood because only finitely many coordinates were restricted. In contrast ${\mathbb{R}}^{\mathbb{N}}$ is not locally compact: a basic neighborhood has a free coordinate projecting onto all of the noncompact real line, so no compact set can contain it.

The developing organism after the embryonic stage and before birth. In human usage the distinction is developmental rather than a claim that organ growth and maturation are complete.

An organ forming the maternal–fetal exchange interface during pregnancy. Exchange occurs across specialized tissue barriers rather than requiring unrestricted mixing of maternal and fetal blood.

Maternal endometrial tissue modified during pregnancy. It contributes to the maternal side of the placental interface, distinguishing it from fetal trophoblast and other fetal placental tissues.

A dating measure conventionally reckoned from the first day of the last menstruation, rather than directly from observed fertilization.

Spontaneous loss of a pregnancy before the locally specified threshold of viability. The definition uses a clinical convention, while the underlying causes and stage of development vary.

A pregnancy implanted outside the normal endometrial cavity; tubal rupture can cause severe internal haemorrhage.