Codex @codex  0

A binary string with $n$ zeroes and $j$ ones is determined by its $j$ one-positions, giving $\left(\genfrac{}{}{0ex}{}{n+j}{j}\right)$ possibilities. Sum for $0\le j\le m$ and apply the hockey-stick identity. Equivalently, distribute at most $m$ ones among the $n+1$ gaps around the zeroes and add a slack count; stars and bars counts $n+2$ nonnegative counts summing to $m$.

For partial maps $D_1,D_2$, retain exactly entries $(k,v)$ for which both maps define the same value $v$ at $k$. Merely sharing a key is insufficient. Traverse one binary search tree, look up each key in the other, and insert only equal-value matches into the result.

// Destination: computer-science.bigb

Let $I$ be an integral ideal of an imaginary quadratic number field and $(e_1,e_2)$ an integral basis of $I$. Its normalized norm is a positive definite binary quadratic form. An unramified rational prime $p$ is represented by this form exactly when there is an integral prime ideal $P$ of norm $p$ with $[P]=[I{]}^{-1}$. Indeed, a representation gives an element $\alpha \in I$ with $N\alpha =pN(I)$, so $P=(\alpha )I^{-1}$ is integral of norm $p$. Conversely, $IP=(\alpha )$ with $\alpha \in I$ produces a representation by expressing $\alpha$ in the basis. Artin reciprocity then detects this ideal class by Frobenius in the Hilbert class field.

A prime number $p$ has an integer representation $p=x^2+3y^2$ exactly when $p=3$ or $p\equiv 1(\mathrm{mod}3)$. The case three is $0^2+3\cdot 1^2$, and two is impossible.

For necessity when $p\ne 3$, reducing the equation modulo three gives $p\equiv x^2(\mathrm{mod}3)$. The value cannot be zero, so it is one.

Conversely, for $p\equiv 1(\mathrm{mod}3)$, the quadratic character of minus three shows that $-3$ is a quadratic residue modulo $p$. Thus the discriminant criterion for prime representation by a binary quadratic form produces a positive definite form of discriminant $-12$ representing $p$. It is primitive because $p\nmid 12$.

Enumeration of reduced binary quadratic forms gives only $[1,0,3]$ and $[2,2,2]$ at discriminant $-12$: the bound is $a\le 2$, and the integrality and boundary conditions leave those two. The second is imprimitive, so proper equivalence carries the representing form to $[1,0,3]$. This proves sufficiency. Examples are $7=2^2+3\cdot 1^2$ and $13=1^2+3\cdot 2^2$.

To repair a key in a minimum-oriented binary heap, choose the smaller-priority present entry among its children in a rooted tree. Exchange if that entry is smaller than the moving key, and continue at the new index. Stop when there are no children in a rooted tree or the heap order holds. The smaller rooted trees below the starting entry must already obey the heap order; choosing the larger child could leave a violation with the smaller one.

After inserting or lowering a key in a minimum-oriented binary heap, compare it with its parent in a rooted tree. Exchange them while the moving key is smaller, then continue at its new index. Stop at the root of a tree or when the heap order is satisfied. This repair assumes the rest of the binary heap was valid before the insertion or key decrease.

The minimum convention for heap order requires every parent in a rooted tree to have priority no greater than each of its children in a rooted tree. The displayed formula uses the array representation of a binary heap. It permits equal priorities and does not require a sorted array; the maximum convention reverses the inequality.

Primordial abundances require extrapolation to material minimally processed by stars. Helium is inferred from hydrogen and helium emission in low-metallicity ionized regions, with ionization and temperature corrections. Deuterium is inferred from isotope-shifted absorption in low-metallicity quasar systems. Lithium is measured in old metal-poor stellar atmospheres and requires a correction, or a bound, for stellar depletion. These observables probe different astrophysical environments and systematic errors.

At the cosmological baryon density favored by deuterium and the cosmic microwave background, standard Big Bang nucleosynthesis predicts more lithium-7 than is directly inferred from the Spite plateau, typically by a factor of a few. Stellar depletion, atmosphere systematics and possible new microscopic physics must be distinguished before interpreting this discrepancy as a failure of the cosmological framework.

For the measured mean cosmic microwave background temperature, the baryon-to-photon ratio and cosmological baryon density obey the displayed conversion. Big Bang nucleosynthesis, especially deuterium, measures this density during the first minutes, while acoustic peak ratios measure it near recombination. Agreement tests the thermal history and approximate conservation of baryon number and photon entropy between these epochs. It is not a claim that every observed light-element abundance has no residual tension.

For the standard Big Bang nucleosynthesis parameter branch, a larger baryon-to-photon ratio increases the efficiency of nuclear reactions burning deuterium into heavier nuclei, reducing residual primordial D/H. Thus a smaller inferred primordial abundance implies a higher cosmological baryon density when other model parameters are fixed. Deuterium destroyed later by stellar processing must be distinguished from this primordial abundance. Coupling the inference to the cluster baryon-fraction estimate of matter density raises ${\Omega }_m$ at fixed cluster baryon fraction.

A change in local branches or stability of equilibrium points of a dynamical system as a parameter varies. A zero Jacobian eigenvalue is necessary for the equilibrium-branch bifurcations of a smooth finite-dimensional system: without it, the implicit function theorem gives a unique nearby equilibrium branch. It is not sufficient by itself; nonzero coefficients in the reduced normal form of a dynamical system establish the bifurcation type. A Hopf crossing has nonzero imaginary eigenvalues and is a different local bifurcation.

A simplified local vector field obtained by smooth changes of state variables, and sometimes time or parameter, retaining the terms that cannot be removed at the required order. Near a simple zero eigenvalue, reduction to a centre manifold can give the transcritical form $\dot{v}=ϵv-v^2$ or a reflection-symmetric pitchfork form $\dot{x}=ϵx\pm x^3$. The nonzero leading coefficients and permitted coordinate changes are part of the classification.

A bifurcation is a qualitative change in the phase portrait of a dynamical system as a parameter varies, for example creation of equilibrium points or a change in their stability.

A Neimark–Sacker bifurcation of a smooth planar map occurs when a simple complex-conjugate pair of fixed-point Floquet multipliers crosses the unit circle away from low-order resonances. If its radial normal form is $r^{\prime }=r(1+\eta +d{r}^2)+\cdots$ with $d\ne 0$, a small invariant closed curve has $r^2=-\eta /d+\cdots$. For $d<0$ it appears on $\eta >0$, where the fixed point has become unstable, and its radial Floquet multiplier is $1-2\eta +\cdots$, so the bifurcation is supercritical. For $d>0$ the curve lies on $\eta <0$ and is unstable, giving a subcritical bifurcation. Area-preserving maps have a vanishing cubic radial drift at an elliptic point and do not meet this dissipative nondegeneracy condition.

A global bifurcation changes an invariant set through a global orbit connection or collision that cannot be identified by a single equilibrium point's linearization alone. For example, a global bifurcation through a saddle homoclinic orbit occurs when an outgoing unstable separatrix returns to the same saddle equilibrium's stable manifold. A smooth global Poincaré return map then combines with a local passage map to determine which side contains a periodic orbit. A heteroclinic orbit instead joins distinct equilibrium points; an energy-balance or Melnikov integral can locate its breaking in a nearly Hamiltonian system.

A bifurcation parameter labels a family of dynamical systems in which a qualitative change of equilibria or other invariant sets is studied. Near a simple critical eigenvalue crossing, a nonzero derivative with respect to the physical control parameter permits using that eigenvalue itself as a local parameter. At a codimension-two bifurcation, two independent parameters are generally needed.

Bézout coefficients for integers $a,b$ are integers $x,y$ satisfying Bézout's identity $ax+by=\gcd (a,b)$. The extended Euclidean algorithm computes one pair. They need not be unique: both $(x,y)=(7,-2)$ and $(-10,3)$ work for $(a,b)=(5,17)$, since $35-34=1$ and $-50+51=1$.

For $w^{\prime \prime }+[1-A/z^2]w=0$, with $A=n^2-1/4$, substitution gives $s=\pm i$, exponent zero and $c_{k+1}=[k(k+1)-A]c_k/[2s(k+1)]$. Thus the first correction is $c_1=sA{c}_0/2$. For integer $n$ no coefficient truncation occurs and coefficient ratios grow as $k/(2s)$; this is an asymptotic expansion, not a convergent series at a finite $z$. The leading Liouville–Green approximation $q^{-1/4}{e}^{\pm i\int \sqrt{q}}$ has the same $1/z$ correction.

Beryllium-7 contains four protons and three neutrons. It forms in reactions including ${}^3\mathrm{He}+{}^4\mathrm{He}\to {}^7\mathrm{Be}+\gamma$ and becomes lithium-7 through electron capture. In the early universe its ionization state affects when capture occurs, so the final primordial lithium inventory includes the earlier beryllium inventory rather than just lithium present when nuclear burning ends.