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The ${\pi }^{+}p$ channel has baryon number one, strangeness zero and $I_3=3/2$, matching the ordinary $uuu$ Delta baryon. The $K^{+}p$ channel has strangeness plus one, which no three-light-quark baryon can have because each strange quark contributes minus one. A resonance in that channel would require exotic content such as four quarks and an antiquark; the ordinary low-energy qqq multiplets therefore explain the contrast.

Let $\Delta (N)={\sum }_{n\le N}(\tau (n)-\log n-2\gamma )$. The Möbius divisor-sum identity and the Von Mangoldt divisor identity give $\mu \ast \tau =1$ and $\mu \ast \log =\Lambda$, soHere $\ast$ is Dirichlet convolution. The divisor summatory asymptotic bounds the terms with $d\le N/K$ in absolute value by $O(N/\sqrt{K})$. For $d>N/K$, group by $j=\lfloor N/d\rfloor <K$: the remaining sum is ${\sum }_{j<K}\Delta (j)(M(N/j)-M(N/(j+1)))=o_K(N)$. First let $N$ tend to infinity, then $K$ tend to infinity. This proves the second Chebyshev function asymptotic from cancellation of the Mertens function.

For fixed $\delta >0$, the Mertens first theorem and partial summation imply convergence of ${\sum }_{p}1/(p(\log p{)}^{\delta })$ and, more precisely, the displayed tail asymptotic with error $O((\log x{)}^{-1-\delta })$. Write $A(t)={\sum }_{p\le t}(\log p)/p=\log t+O(1)$ and integrate against $(\log t{)}^{-1-\delta }$. The boundary at infinity vanishes; combining the finite boundary with the integral gives the coefficient $1/\delta$.

The prime factorization of $N!$ gives $\log N!={\sum }_{p^j\le N}\lfloor N/p^j\rfloor \log p$. The Chebyshev estimate bounds the error on replacing each floor by its argument by $O(N)$. Dividing by $N$ and using the Stirling formula gives ${\sum }_{p^j\le N}(\log p)/p^j=\log N+O(1)$. The contribution of $j\ge 2$ is bounded by the convergent series ${\sum }_{n\ge 2}(\log n)/(n(n-1))$. This proves the Mertens first theorem without the Prime number theorem.

For fixed $\lambda >1$, integrate $(\log t{)}^{\lambda -1}$ against $A(t)={\sum }_{p\le t}(\log p)/p$. The Mertens first theorem gives $A(t)=\log t+O(1)$. Partial summation makes the main integral $(\log x{)}^{\lambda }/\lambda$; the bounded error in $A$ contributes at most a constant times $(\log x{)}^{\lambda -1}$. The restriction $\lambda >1$ ensures that integrating the derivative $(\lambda -1)(\log t{)}^{\lambda -2}/t$ has this bound.

Put $A(t)={\sum }_{p\le t}(\log p)/p$. The Mertens first theorem gives $A(t)=\log t+O(1)$. Partial summation yieldsThe main terms sum to $2\sqrt{\log x}$ up to a constant. The error integral is bounded because ${\int }_2^{\infty }dt/(t(\log t{)}^{3/2})$ converges. Thus a prime sum with a fractional logarithmic weight follows from an elementary bounded-error estimate, without the Prime number theorem.

An ordering binary relation used by a memory model to constrain observation of shared data. For the mutex example, T1's write precedes its release, that release synchronizes with T2's successful acquisition, and that acquisition precedes T2's read. Together these steps order the write before the read. Wall-clock timing alone does not establish happens-before.^[ref]

For the C++ case used here, conflicting accesses by different software threads, at least one non-atomic, with neither ordered before the other by happens-before. Such a data race has undefined behavior. A race condition is broader: the lost update with separate atomic reads and writes depends on their order while avoiding a data race.^[ref]

Execution behaves as if steps occurred in one total order preserving each participant's program order; reads observe the latest preceding writes in that order. This is the definition given in the 1979 paper How to Make a Multiprocessor Computer That Correctly Executes Multiprocess Programs, printed p. 690. It permits many thread interleavings, including the lost update under mutual exclusion; ordering individual accesses does not combine them into one atomic operation.

For $u^{\prime \prime }=u^2-\kappa -\epsilon (\lambda +u)u^{\prime }$ with $\kappa >0$, the Hamiltonian homoclinic orbit has $H=2{\kappa }^{3/2}/3$. Its energy balance is $\lambda \int v^{2}dt+\int u{v}^{2}dt=0$. Along this orbit the ratio of the second integral to the first is $-5\sqrt{\kappa }/7$, giving the displayed simple zero and hence the leading homoclinic parameter curve.

Perturb the Hamiltonian blow-up of a reflection-symmetric double-zero point by $u^{\prime }=-sv+v^3/2+\epsilon (\mu -v^2/2)u+O({\epsilon }^2)$ and $v^{\prime }=2u+\epsilon (\mu -v^2/2)v+O({\epsilon }^2)$. Then $H^{\prime }=\epsilon (\mu -v^2/2)(2u^2+s{v}^2-v^4/2)+O({\epsilon }^2)$. On the positive heteroclinic branch $dt=dv/[\sqrt{2}(s-v^2/2)]$, so its first energy change is $\epsilon M/\sqrt{2}$, where $M={\int }_{-\sqrt{2s}}^{\sqrt{2s}}(\mu -v^2/2)(s+v^2/2)dv$. A simple zero gives cancellation of the leading stable/unstable manifold splitting and continues to a nearby heteroclinic bifurcation. Since $\partial M/\partial \mu >0$, there is a unique first-order balance for fixed $s>0$.

For the imperfect soft Duffing-van der Pol oscillator at positive small $\lambda$ and $|\epsilon |\ll \sqrt{\lambda }$, the symmetric connection of the Hamiltonian system is $v=\sqrt{\lambda }\tanh (\sqrt{\lambda /2}t)$. Its damping integral is $4{\lambda }^{3/2}(\kappa -\lambda /5)/(3\sqrt{2})$. The two saddle equilibrium barrier heights differ by $-4\epsilon {\lambda }^{3/2}/3$ to first order. Equating the energy change to this difference gives the two oppositely directed connection curves in the display. These are weak-damping asymptotic curves, not exact global formulas at order-one parameters.

// Destination: astrophysical-fluid-dynamics.bigb

Perturb the first integral of the quadratic fold-Hopf amplitude flow by $u^{\prime }=-2uv+\epsilon u({\lambda }_1+v^2)$, keeping $v^{\prime }={\lambda }_2+v^2+u^2$. Then $F^{\prime }=-\epsilon u({\lambda }_1+v^2)({\lambda }_2+u^2+v^2)$. On the interior heteroclinic orbit for ${\lambda }_2=-a^2$, $u=\sqrt{3(a^2-v^2)}$ and $v^{\prime }=2(a^2-v^2)$, so the first-order change is $-\epsilon \sqrt{3}{\int }_{-a}^a({\lambda }_1+v^2)\sqrt{a^2-v^2}dv=-\epsilon \pi \sqrt{3}{a}^2({\lambda }_1+a^2/4)/2$. The boundary connection contributes zero. The simple zero gives the displayed leading persistence curve, rather than an exact all-orders parameter equality.

For $\dot{q}=u_0(q)+ϵu_1(q,t)$ with planar Hamiltonian flow $u_0=(-{\partial }_y{\psi }_0,{\partial }_x{\psi }_0)$, let $q_0$ be a heteroclinic orbit between two hyperbolic fixed points. Its first-order displacement satisfies ${\dot{q}}_1=D{u}_0{q}_1+u_1$. The covector $w=\nabla {\psi }_0(q_0)$ satisfies $\dot{w}=-(D{u}_0{)}^{T}w$, obtained by differentiating $\nabla {\psi }_0\cdot u_0=0$. Thus $(w\cdot q_1{)}^{\prime }=w\cdot u_1$. Integrating the stable correction from the future and the unstable correction from the past gives the displayed function. At a regular section the first-order normal separation is $ϵM/|\nabla {\psi }_0|$. A simple zero yields a transverse intersection of the perturbed manifolds, by the implicit function theorem.

For a bulk type-II superconductor with negligible demagnetizing field, raising the applied magnetic field intensity past the lower critical value $H_{c1}$ favors a superconducting mixed state containing quantum vortex lines; superconductivity survives until the upper critical value $H_{c2}$. The magnetic field extends around each vortex over a penetration scale, not solely inside its core. Thus a nonzero average magnetic flux density can coexist with superconductivity. Below $H_{c1}$, a type-II superconductor also supports a Meissner state; a vortex-free state above the equilibrium entry field may instead be metastable. ^[ref]

Real preparation histories may leave quantum vortex lines held in place by material inhomogeneities, retaining magnetic flux and producing hysteresis. This flux pinning restricts access to equilibrium: it is different from the arbitrary frozen interior magnetic field of the ideal perfect conductor model. Harvard's vortex-imaging work illustrates strong pinning to bulk disorder in a type-II superconductor. ^[ref] Separately, flux conservation in a superconducting loop and fluxoid and flux quantization in a superconductor explain why a hole can retain magnetic flux. Neither bore retention, vortex penetration nor weak screening in a thin sample should be silently identified with the equilibrium Planar London screening calculation.

The 1933 report Ein neuer Effekt bei Eintritt der Supraleitfähigkeit by W. Meissner and R. Ochsenfeld describes cooling tin and lead cylinders in a fixed transverse magnetic field. Two parallel cylinders were about $140\mathrm{mm}$ long and $3\mathrm{mm}$ thick, separated by $1.5\mathrm{mm}$. A roughly $10\mathrm{mm}$ rotating search coil between them drove a ballistic galvanometer. At about $5\mathrm{G}$, the reported below/above-transition induction ratios were $1.70$ for tin single crystals and $1.77$ for polycrystalline lead. The increased gap magnetic field indicates redistribution outside the cylinders, rather than an unchanged frozen field. Coil-winding geometry and noncircular tin specimens limited comparison with the ideal prediction. ^[ref]

Their second arrangement used a lead tube, about $130\mathrm{mm}$ long with $3\mathrm{mm}$ outer and $2\mathrm{mm}$ inner diameters. A rotating search coil could sit inside or alongside it. The bore retained magnetic flux after removing the applied magnetic field; the coil nearly filled the bore, preventing a test of interior field uniformity. A hole containing magnetic field is distinct from magnetic field penetrating the surrounding material exhibiting superconductivity. ^[ref]

The modern Field-cooled Meissner experiment provides a complementary readout. MIT's probe I places a sample in a brass holder, with an $810$-turn test coil of $7.1\mathrm{mm}$ inner diameter, $10.5\mathrm{mm}$ outer diameter and $12.3\mathrm{mm}$ length, inside a longer solenoid. A nearby silicon-diode thermometer tracks cooling by helium gas. With lead still normal, the guide applies about $200\mathrm{mA}$ to the solenoid and connects the test coil to a current-integrator box and oscilloscope. Cooling produces a transient; warming gives the opposite sign. Switching off the solenoid while the sample is normal checks the sign. ^[ref]

The inferred change is in linked magnetic flux, not a direct scan of $B(z)$. For a stationary test coil the pulse obeys $\int Vdt=-\Delta \Psi$, where $\Psi$ is its magnetic flux linkage. An integrator or AC-coupled oscilloscope needs its transfer response accounted for before extracting this pulse area. Keep the source electric current fixed during cooling, monitor temperature, and compare warming and cooling slowly enough to reduce sample–thermometer lag. An AC inductance change alone can reflect changed normal-state electrical conductivity; the fixed-field expulsion test distinguishes that alternative. ^[ref] Neither this coil readout nor the historical gap-field measurement resolves the microscopic London penetration depth by itself.

Compare two ideal models for a stationary, solid, simply connected sample. In the perfect conductor flux-history model the interior electric field is constrained to zero from the moment perfect conduction starts. Faraday's law of induction then givesConsequently the interior magnetic field retains its value at that moment. Cooling in an already applied magnetic field freezes a nonzero interior value; starting perfect conduction in zero magnetic field and then applying a magnetic field preserves zero. The zero initial interior magnetic field often assumed in perfect conductor wave-reflection problems is one particular preparation, not a general prediction of expulsion. ^[ref]

The Meissner state instead selects the screened equilibrium after either preparation, provided that equilibrium is accessible and the applied magnetic field lies in the Meissner state regime. The first London equation alone illustrates the missing information: at fixed ${\lambda }_L$, combining it with Faraday's law of induction givesThis permits an arbitrary time-independent integration field. The second London equation imposes the additional zero value of that bracket in the vortex-free local model.^[ref] Establishing superconductivity during cooling is not the fixed-parameter, zero-electric field model: transient electric fields accompany changing magnetic flux, and ${\lambda }_L$ itself depends on temperature. The equilibrium London equations predict final spatial screening, not the cooling rate or the time-resolved expulsion pulse.

A Meissner state is an equilibrium state of superconductivity without quantum vortex penetration, with screening electric currents responding to the applied magnetic field. In a solid, simply connected sample much thicker than the London penetration depth, the deep interior magnetic field is approximately zero. This is a statement about a phase and a geometry, not a rule that every sample exhibiting superconductivity always has $\mathbf{B}=0$. In particular, the Meissner state exists for sufficiently weak magnetic fields in both type-I superconductors and type-II superconductors. ^[ref]

A sensor bias is a systematic offset between the measured output and the physical plant output, modeled here as $y_m=y+b$ with constant $b$. Unlike zero-mean measurement noise, a sensor bias persists when repeated readouts are averaged. A sensor bias affects the physical meaning of a zero tracking error; for a temperature readout, independent thermometer calibration can detect it.

For every Lebesgue measurable set $A\subseteq {\mathbb{R}}^n$, its Lebesgue density is $1$ at almost every point of $A$ and $0$ at almost every point of its complement.