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For three distinguishable spin one-half systems, the Clebsch-Gordan decomposition first combines two spins into $0\oplus 1$, then couples the third: $0\otimes 1/2=1/2$ and $1\otimes 1/2=1/2\oplus 3/2$. Thus the tensor product has one quartet and two doublets. A Hamiltonian operator depending only on the total $S^2$ and $S_z$ acts identically on the two doublet copies. In particular $H={ϵ}_t{S}^2/{\hslash }^2+{ϵ}_z{S}_z/\hslash$ has energy eigenvalues ${ϵ}_{t}S(S+1)+{ϵ}_{z}M$, generically once for each quartet projection and twice for each doublet projection. Equal numerical energies have their multiplicities added.

The small displacement of a fluid parcel from its undisturbed position in a linear acoustic model. Its time derivative is the acoustic particle velocity.

In an isothermal atmosphere with constant density scale height $H$, the equation $\delta {\rho }_{tt}-c^2\delta {\rho }_{zz}-(c^2/H)\delta {\rho }_z=0$ loses its first spatial derivative under $\delta \rho =e^{-z/(2H)}\Psi$. Direct differentiation gives ${\Psi }_{tt}-c^2{\Psi }_{zz}+c^2\Psi /(4H^2)=0$. Thus a plane wave has dispersion relation ${\omega }^2=c^2{k}^2+{\omega }_c^2$, with real acoustic cutoff frequency ${\omega }_c=c/(2H)$. The cutoff term is positive; a minus sign would instead predict growing sufficiently long spatial modes. Vertical propagation requires $|\omega |\ge {\omega }_c$ within this ideal constant-coefficient model.

The 1998 High-Z Supernova Search Team paper by Adam Riess, Brian Schmidt and collaborators presented supernova evidence for acceleration.^[ref]

The 1999 Supernova Cosmology Project paper by Saul Perlmutter and collaborators independently analysed 42 distant events. Alexei V. Filippenko coauthored both papers.^[ref]

The High-Z Supernova Search Team search used the CTIO Blanco 4-m telescope and a $2048\times 2048$ charge-coupled device, comparing fields across dark runs. Difference images located rising candidates; Keck, MMT and ESO spectroscopy classified them and measured host redshifts. Follow-up photometry used filters approximating emitted B and V bands. Calibrated field stars transferred the magnitude zero point; late host images removed galaxy light, point spread function fits measured brightness, and inserted artificial stars tested recovery.^[ref]

The Supernova Cosmology Project likewise discovered rising events with CTIO 4-m charge-coupled device imaging, then followed their light curves with several telescopes. Keck LRIS and ESO spectroscopy supplied classification and redshift, preferably from narrow host lines. Cross-filter K corrections connected received R/I observations to emitted B/V light.^[ref]

Why does fainter mean accelerating? First it means a larger inferred luminosity distance only at fixed standardized luminosity and corrected attenuation. Next it favours acceleration only after a comparison family is specified: an FLRW metric, representative propagation of light, common nearby and distant standardization, and controlled measurement calibration and selection. For the usual general relativity models containing nonnegative pressureless matter, spatial curvature of an FLRW universe and a cosmological constant, the shape of the Hubble diagram constrains ${\Omega }_{m,0},{\Omega }_{\Lambda ,0}$ and thus $q_0={\Omega }_{m,0}/2-{\Omega }_{\Lambda ,0}$. A positive residual relative to one decelerating model rejects that prediction; it does not alone prove $q_0<0$ across arbitrary expansion histories or arbitrary opacity laws.

The essential advance is that a nearby sample fixes an intercept while a range of distant redshifts measures the shape. The two teams supplied independent observations and analysis, although common nearby calibrators and Type Ia supernova physics can produce shared uncertainties. The dark-energy acceleration criterion explains the interpretation within general relativity; the worked supernova distance comparison shows the size and sign of the effect without fitting any survey data.

A 2-core can contain bridges in a graph, and even vertices that lie on no cycle in a graph. Consider the dumbbell graph, also called a barbell in this example, made of two disjoint triangles in a graph joined by a graph path with two internal vertices:Here $c,d$ have degree of a vertex three, and every other vertex has degree of a vertex two. No deletion is possible, so the whole graph is its 2-core. Yet $cx,xy,yd$ are bridges in a graph: deleting any one separates the two triangles. None belongs to a cycle in a graph, and $x,y$ belong to no cycle in a graph either. Keeping only edges on graph cycles would incorrectly discard the entire joining graph path.

Every cycle in a graph does survive pruning, because each of its vertices keeps its two cycle graph neighbours. The barbell shows that the converse fails: the 2-core can be larger than the union of all graph cycles. Its condition is local minimum degree, not protection against a single connection failing.

A biconnected component is a different object: a maximal connected subgraph with no cut vertex of its own. Under the block convention used here, the barbell has five biconnected components: its two triangles and its three individual joining edges, each with its endpoints. Its cut vertices are $c,x,y,d$. A 2-connected component with at least three vertices must remain connected after any single vertex is deleted; the whole barbell fails that test. An informal phrase such as "biconnected core" therefore needs its own definition and must not be substituted for 2-core. Even an isolated edge is a block under this convention, although its 2-core is empty.

The usefulness of the 2-core is this precise separation of local persistence from global robustness. In a connected graph with nonempty 2-core, the deleted vertices form a forest, with each of its connected components of a graph attached by exactly one edge to the 2-core. Otherwise a deleted graph cycle, or a graph path joining two surviving attachment points, would itself have survived. Stripping these attached trees exposes a smaller structure for studying graph cycles, graph excess and the graph kernel, while preserving the joining bridges in a graph that organize that structure.

Suppose $T$ is any vertex set whose induced subgraph has every degree of a vertex at least two. No vertex of $T$ can be the first one deleted from $T$: before such a deletion it still has its two graph neighbours within $T$, contradicting eligibility for deletion. Consequently every qualifying $T$ survives every legal deletion sequence. Once deletion stops, its remainder $R$ itself qualifies, because every surviving vertex has at least two surviving graph neighbours. Therefore $T\subseteq R$ for every qualifying $T$: $G[R]$ is the unique maximum and the final remainder is independent of deletion order. The intermediate FIFO queues and deletion orders can differ.

There is also a direct existence argument. If $A$ and $B$ qualify, then $A\cup B$ qualifies: each vertex retains at least the two graph neighbours it already had in $A$ or in $B$. Taking the union of all qualifying vertex sets produces the maximum. This rules out a collection of competing, incomparable inclusion-maximal answers. It also explains why restricting the search to connected subgraphs would change the question.

Use adjacency lists and a FIFO queue to implement the deletion rule in time complexity $O(|V|+|E|)$. Compute each initial degree of a vertex once, mark every vertex alive, and put all vertices of degree of a vertex zero or one in the FIFO queue. Each vertex is inserted at most once. Repeatedly remove the front vertex $v$, mark it dead, and traverse its original adjacency list. For each still-alive graph neighbour $u$, decrease the stored degree of a vertex of $u$ by one; if it has just become one, append $u$. A vertex already queued can drop from one to zero before its turn, but it stays queued and requires no second insertion. Stop when the FIFO queue is empty; the alive vertices induce $C_2(G)$.

For the displayed example, initially enqueue $(f,g,z)$ in that order. The successive FIFO queues, after each deletion and all its updates, areThis explains the delayed deletion of $d$: its initial degree of a vertex was three, but the relevant degree of a vertex is always its current one in the surviving induced subgraph. Removing only the original vertices of degree of a vertex one would give the wrong answer.

Initialization inspects every vertex and every adjacency list entry once. During deletion each original adjacency list is traversed at most once, so at most $2|E|$ entries are inspected in an undirected graph. All insertions, removals and stored-degree updates cost constant time each. Thus total time complexity is $O(|V|+|E|)$ and additional storage, beyond the adjacency lists, is $O(|V|)$. The cost includes isolated vertices; repeatedly rescanning the whole surviving graph after each deletion would lose this bound. The single-threshold FIFO queue method is enough for the 2-core; the full k-core of a graph decomposition computes all thresholds.

Sphalerite is the cubic mineral form of zinc sulfide, $\mathrm{ZnS}$. Its zinc-blende structure has tetrahedral coordination around the two types of lattice sites.

The same structural pattern occurs in other compound semiconductors, so the mineral name also labels a reusable crystal-structure type. Chemical composition and structural type are related descriptions, not interchangeable ones.

A free scalar Klein-Gordon equation with rest mass $m$ isHere $c$ is the speed of light and $\hslash$ the reduced Planck constant. In the static limit, comparison with the static Yukawa Green function identifiesThis $\lambda$ is the reduced Compton wavelength, rather than $h/(mc)$; the two differ by $2\pi$. It is an exponential decay length of this exchange field, not the geometric size of a nucleon or an exact radius of a bound state. This article uses $\lambda$ for a length; the 1935 paper used that letter for its inverse.

For a deliberately rounded worked example, stipulate $\hslash c=197.3\mathrm{MeV}\mathrm{fm}$ and $m{c}^2=140\mathrm{MeV}$. ThenA femtometre is $10^{-15}\mathrm{m}$. The chosen rest energy is a convenient pion-scale input; neither it nor the resulting Yukawa potential is asserted to be an exact measured nuclear interaction. Holding $\hslash c$ fixed, chosen rest energies $70$, $140$ and $280\mathrm{MeV}$ give lengths $2.819$, $1.409$ and $0.7046\mathrm{fm}$. Doubling the exchange rest mass halves the decay length.

Range alone does not guarantee binding: the potential energy must overcome the kinetic energy cost of localizing a wavefunction. The existing exponential variational bound for the Yukawa potential makes this precise. For the same toy $V$, with reduced mass $M$, choosing a normalized exponential wavefunction proportional to $e^{-\alpha r}$ with $\alpha =\kappa /2$ yieldsFor additional stipulated inputs $M{c}^2=470\mathrm{MeV}$ and $A=100\mathrm{MeV}\mathrm{fm}$, this gives $⟨H⟩\approx -3.66\mathrm{MeV}$. A negative value proves a bound state in this scalar model by the quantum variational principle; it is an upper bound on its ground-state energy, not a prediction for the deuteron. A nonnegative trial value would not prove the absence of binding.

Let $\kappa >0$ be an inverse length. In three spatial dimensions, seek the Green's function that decays at infinity and satisfieswhere ${\nabla }^2$ is the Laplacian and ${\delta }^{(3)}$ the three-dimensional Dirac delta function. The point source and the differential operator have spherical symmetry. For $r>0$, the radial Laplacian givesWriting $u(r)=r{G}_{\kappa }(r)$ reduces this to $u^{\prime \prime }-{\kappa }^{2}u=0$. Decay at infinity removes the growing exponential function, leaving $G_{\kappa }(r)=C{e}^{-\kappa r}/r$.

The Dirac delta function fixes the coefficient. Integrating over a ball $B_{ϵ}$ and using the divergence theorem givesThe first term tends to $4\pi C$, while the second is $O({ϵ}^2)$. Hence $C=1/(4\pi )$ andThis solves the source equation in the sense of distributions, including at the origin. Merely verifying the equation for $r>0$ would leave the normalization undetermined. The limit $\kappa \to 0$ at fixed $r>0$ is $1/(4\pi r)$, the Green function of the Laplacian for the convention $-{\nabla }^2$.

For an attractive scalar toy interaction, absorb constants into $A>0$ and setHere $A$ has units of energy times length, and negative $F_r$ points inward. At equal $A$, comparison with the unscreened model $V_0=-A/r$ givesThus the potential energy and force have the same exponential scale but different suppression factors. At $x=1$, these ratios are approximately $0.368$ and $0.736$; at $x=3$, they are $0.0498$ and $0.199$. Finite range means rapid decay, not an abrupt cutoff.

Kai Siegbahn's early instrument used iron-free magnetic double focusing and a thin-window Geiger-Müller counter. His later instrument, compatible with ultra-high vacuum, used a spherical electrostatic electron energy analyzer and monochromatic aluminum X-rays. Argon mixed with methane supplied simultaneous $2p$ reference lines.^[ref]

Two sulfur lines in sodium thiosulfate distinguished chemical environments within one compound. A separate gold comparison showed a fingerprint obscuring substrate lines.^[ref]

For an unknown sample, first document its preparation and transfer, then compare survey and detailed scans under reproducible illumination, contact and acquisition conditions. Repeat a region to test change during exposure. If a cleaning treatment is used, compare before and after rather than assuming that cleaning preserves the original chemical bonds. Report the photoelectron binding energy reference, reference lines, charge-compensation conditions and instrumental settings with the inferred chemical shift in photoelectron spectroscopy. A chemical interpretation should survive these checks as well as explain the line separation.

A chemical shift in photoelectron spectroscopy is a difference in binding energy for the same kind of core electron in different chemical surroundings, using comparable energy references. Identifying a chemical element assigns its family of lines; resolving chemical shifts within that family asks how its chemical atoms are bonded. For example, distinguishing a metal from its chemically modified surface requires more than detecting that element in both places.

The removal energy compares the initial electronic state with the state left after emission. Redistribution of the remaining electrons changes that final state's energy; a measured shift therefore includes relaxation as well as initial bonding effects.^[ref]

This makes chemical shift in photoelectron spectroscopy useful but conditional. Compare the same electron shell, the same photoelectron binding energy reference, and compatible sample conditions. Agreement with a chemically appropriate reference supports an assignment; a line position by itself need not uniquely determine an oxidation state. An unresolved sum of two chemical components can even have a centroid where neither component lies. The following Gaussian mixture distribution shows exactly how that happens without claiming a material identification.

Let a conducting sample and conducting analyzer be in good electrical contact, unbiased, at equilibrium, and without appreciable charging during illumination. Their Fermi levels then coincide. Choose that common Fermi level as zero. Define $E_B>0$ as the binding energy relative to this zero: removing an electron into the electron reservoir at the Fermi level costs $E_B$. It is not the cost of removing the electron all the way to the sample's vacuum level.

Write the sample and analyzer work functions as ${\varphi }_s$ and ${\varphi }_a$. Their local vacuum levels lie at ${\varphi }_s$ and ${\varphi }_a$ above the common Fermi level. Conservation of energy for an electron emitted without additional losses gives total outgoing energy $h\nu -E_B$, where $h$ is the Planck constant and $\nu$ the photon frequency. Immediately outside the sample its kinetic energy is $K_s=h\nu -E_B-{\varphi }_s$. The contact-related electric potential changes this toThus the sample work function cancels when the measured kinetic energy is referenced to the analyzer. Substituting ${\varphi }_s$ for ${\varphi }_a$ in the final conversion would mix two different energy references. An instrument's calibrated binding energy scale can incorporate its effective analyzer work function, so the correction need not appear as a separate operator-entered number.

As an original numerical example, stipulate $h\nu =1486.6\mathrm{eV}$, ${\varphi }_a=4.5\mathrm{eV}$ and a line at $E_B=284.0\mathrm{eV}$. The predicted analyzer kinetic energy is $1198.1\mathrm{eV}$. If ${\varphi }_s=5.0\mathrm{eV}$, the surface value is instead $K_s=1197.6\mathrm{eV}$; the contact-related increase is $0.5\mathrm{eV}$. A second line at $286.0\mathrm{eV}$ gives $1196.1\mathrm{eV}$ at the analyzer. A $+2.0\mathrm{eV}$ chemical shift in photoelectron spectroscopy therefore appears as a $-2.0\mathrm{eV}$ shift in measured kinetic energy. These are chosen values, not a measured carbon spectrum.

Charging breaks the assumed reference. If a uniformly charging sample rises by electric potential $V>0$ relative to the analyzer, an escaping electron loses an additional energy $eV$, with $e$ the positive elementary charge. Applying the unbiased conversion consequently returnsTwo features from the same stable potential retain their separation. Features from regions at different potentials acquire different offsets, which can mimic chemical differences or broaden a line. A conducting holder alone does not establish electrical contact through an insulating surface layer. Compare repeat scans, electrical contact and charge-compensation settings before interpreting an offset as a change in chemical bonds.

Measurement calibration is a separate issue: a known reference line tests the scale, while charge compensation controls the specimen's electric potential. In an original offset model $E_{B,\mathrm{read}}=E_{B,\mathrm{true}}+b$, one known line determines $b$. An unknown scale factor in $E_{B,\mathrm{read}}=a{E}_{B,\mathrm{true}}+b$ requires two distinct known lines to determine both $a$ and $b$. Re-reading a reference after a sample checks drift. Treating an unidentified contamination peak as a universally fixed reference would assume the chemical identity and electrical behavior that the measurement should test.