A polynomial in one indeterminate; its coefficients lie in a specified ring. For example, $3x^2+1$ is univariate, whereas $x+y$ involves two indeterminates.

A cubic polynomial in one variable has polynomial degree three. Over a splitting field, its factorization of polynomials contains three polynomial factors of polynomial degree one when multiplicity of a root is counted.

Vieta's formulas connect two descriptions of a polynomial: its coefficients and its roots of a polynomial. They let us read the sum and product of the roots of a polynomial without finding the individual roots of a polynomial, and turn a proposed factorization of polynomials into checkable coefficient identities. The key is that coefficients encode elementary symmetric polynomials, which are unchanged by a permutation of the roots of a polynomial.

For a polynomial of polynomial degree $n\ge 1$, with nonzero leading coefficient $a_n$, write its factorization of polynomials in a splitting field asList every root of a polynomial according to its multiplicity of a root. ThenIn particular, ${\sum }_i{r}_i=-a_{n-1}/a_n$ and ${\prod }_i{r}_i=(-1{)}^n{a}_0/a_n$. With coefficients in the real numbers or complex numbers, the fundamental theorem of algebra ensures that a list of $n$ roots of a polynomial in the complex numbers exists when multiplicity of a root is counted.^[ref]

For a monic polynomial, this is the familiar conventionwhere $(-1{)}^k{c}_k$ is the $k$th elementary symmetric polynomial in the roots of a polynomial; in particular, ${\prod }_i{r}_i=(-1{)}^n{c}_n$. The ratio by $a_n$ in the general statement matters: multiplying every coefficient by the same nonzero scalar preserves both the roots of a polynomial and these ratios.

A short derivation explains every alternating sign. In the product ${\prod }_i(x-r_i)$, a term of polynomial degree $n-k$ arises by choosing $-r_i$ from $k$ distinct factor positions and $x$ from the others. Summing the resulting products gives $(-1{)}^k{e}_k$. Values at different positions may coincide: the selection is of positions, not distinct root values. Matching coefficients gives the formula. This is the distributive property organized by the number of chosen root factors.

A good solvent favors contact with polymer segments strongly enough that the effective excluded volume is positive. A long dilute flexible chain swells relative to an ideal chain; the three-dimensional size exponent is approximately $0.588$, often replaced by $3/5$ in a scaling estimate. This designation depends on the polymer, solvent and temperature, rather than on the solvent alone.

Flory–Huggins solution theory is an incompressible lattice description of mixing a polymer with a solvent. A chain occupies $N$ connected sites, while a solvent molecule occupies one; the displayed mixing free energy is per site. The first two terms are translational entropy of mixing contributions and the last is a mean-field excess contact interaction. A positive interaction parameter $\chi$ penalizes polymer–solvent contacts. If the site volume is $v_0$, the physical free-energy density is $f/v_0$ and the polymer volume fraction gives monomer number density $\varphi /v_0$. Composition-dependent interaction parameters require extra derivative terms in stability calculations.

The theta temperature is the temperature at which the leading effective two-segment interaction of a dilute long polymer vanishes, so its large-scale size is approximately that of an ideal chain. In the simplest Flory–Huggins solution theory this is the condition $\chi =1/2$. It is the infinite-chain limit of the upper critical temperature in the usual energetic model, not the critical temperature of every finite chain.

A poor solvent favors effective attraction between polymer segments relative to solvent contacts. A dilute flexible chain tends to contract, and sufficiently unfavorable conditions produce a compact polymer globule. Solvent quality depends on the polymer, solvent and temperature. The simple Flory–Huggins solution theory identifies the poor-solvent side of the theta condition with $\chi >1/2$; this does not guarantee that every finite-density mixture is inside its spinodal.

The overlap concentration is the concentration at which distinct dilute polymer coils first substantially interpenetrate. It corresponds to roughly one chain per coil volume. If a chain contains $N$ repeat units, has molar mass $M$ and occupies a coil volume $V_{\mathrm{coil}}$, then its chain number density is approximately $1/V_{\mathrm{coil}}$, its repeat-unit number density is $N/V_{\mathrm{coil}}$, and its mass concentration is $M/(N_A{V}_{\mathrm{coil}})$. A scaling convention uses $V_{\mathrm{coil}}\sim R^3$; a commonly chosen geometric convention uses $V_{\mathrm{coil}}=4\pi R_g^3/3$. These give different order-one prefactors because a coil has no sharp surface. For an ideal chain, $R\sim b{N}^{1/2}$ and a site-volume estimate $v_0\sim b^3$ gives overlap volume fraction ${\varphi }^{\ast }\sim N^{-1/2}$. A good solvent chain with the Flory approximation $R\sim b{N}^{3/5}$ instead gives ${\varphi }^{\ast }\sim N^{-4/5}$.

A solvent under conditions where the leading effective pair excluded volume of a long dilute polymer vanishes. In the simplest Flory–Huggins solution theory this means $\chi =1/2$. The chain is approximately an ideal chain, while three-segment interactions can still affect the osmotic pressure of a concentrated solution. A particular polymer-solvent pair often meets this condition only near its theta temperature.

A solution above the polymer overlap concentration but below the concentrated regime, with interpenetrating chains. Local polymer structure is characterized by a correlation length or concentration-blob size. Scaling formulas describe a broad semidilute regime most reliably when the concentration greatly exceeds overlap while remaining small compared with one in volume fraction.

An adsorption blob is a subchain of size comparable to the adsorbed layer thickness $\xi$. If the free-chain size law is $R\sim b{g}^{\nu }$, it contains $g\sim (\xi /b{)}^{1/\nu }$ monomers. Confinement costs of order $k_{B}T$ per blob produce $F_{\mathrm{conf}}\sim N{k}_{B}T/g$. A simple contact fraction $b/\xi$ gives $F/(N{k}_{B}T)\sim (b/\xi {)}^{1/\nu }-ϵb/\xi$. This model predicts $\xi /b\sim {ϵ}^{-\nu /(1-\nu )}$ and binding gain per monomer of order $k_{B}T{ϵ}^{1/(1-\nu )}$. It assumes a many-blob layer and a monomer-scale attraction range; it does not replace the full surface-critical theory of polymer adsorption.

Let an adsorbed Gaussian chain gain free energy $\mu >0$ per bound monomer. A pulled segment of $n$ Kuhn length steps spanning height $z$ has elastic free energy $3k_{B}T{z}^2/(2n{b}^2)$, giving $F=-(N-n)\mu +3k_{B}T{z}^2/(2n{b}^2)$. Minimizing over $n$ yields $n=(z/b)\sqrt{3k_{B}T/(2\mu )}$ and $F_{\min }=-N\mu +z\sqrt{6k_{B}T\mu }/b$. Its height derivative is the constant peeling force displayed above. The plateau lasts while $0<n<N$ and the Gaussian chain approximation remains valid. In the slab model $F_{\mathrm{ads}}=N{k}_{B}T[(b/\xi {)}^2-ϵb/\xi ]$, optimization gives $\mu =k_{B}T{ϵ}^2/4$ and $f=\sqrt{3/2}ϵk_{B}T/b$.

A nucleotide substitution replaces a nucleotide at a specified position without changing sequence length. In a fixed reading frame, a single substitution can preserve an amino acid assignment, change it or introduce a stop codon; the mutation ledger exhibits all three possibilities.

In December 1947, John Bardeen and Walter Houser Brattain obtained transistor action at Bell Telephone Laboratories. William Shockley proposed the distinct bipolar junction transistor within the following month.^[ref]

Their 1948 paper describes ground, etched germanium, initially also anodized and rinsed. Surface contacts separated by $50$–$250\mu \mathrm{m}$ share a broad base contact. Positive emitter biasing injects electron holes; negative collector biasing draws them toward the collector. An input signal changes emitter electric current, and a collector electrical load converts the resulting change into voltage.^[ref]

Early electric field control experiments gave unexpectedly weak responses. John Bardeen proposed that surface states screened the interior. Surface charge could create a barrier and a hole-rich semiconductor inversion layer. Replacing an electrolyte control with a gold contact reversed the response sign, suggesting electron hole carrier injection. Closely spaced contacts made from a gold-coated wedge split with a razor then enabled power gain. The relative importance of surface and bulk transport depended on preparation and collector forming.^[ref]

The 1948 authors found that collector-current increments could exceed emitter-current increments; their current gain depended on biasing.^[ref] A terminal-current ratio alone cannot identify the microscopic path. Surface states, the local barrier, carrier injection and contact preparation prevent a single ideal modern bipolar junction transistor model from explaining every point-contact transistor.

For the conventional electric current directions drawn in the figure, $I_E$ enters the emitter, $I_C$ leaves the collector and signed $I_B$ enters the base. Kirchhoff's current law requiresThus a local current gain $\alpha >1$ requires an accompanying base-current change. The collector increment is not simply a count of injected electron holes, and it does not violate electric charge conservation.

For an experiment following the functional arrangement, first establish each contact's separate current-voltage characteristic relative to the base. Then hold collector biasing fixed, vary a small emitter signal, and compare the collector electrical load voltage with the input. A decisive power gain check measures both input electric current and voltage and the output power in a known electrical load; voltage enlargement alone is insufficient. These are proposed measurement controls, not additional claims about the 1947 laboratory record.

Repeating a setting tests drift; reversing or removing emitter biasing tests whether the response requires carrier injection. Contact motion, surface contamination, temperature, direct electrical coupling and noise can all affect the inference. A small-signal model describes measured local terminal response without deciding every microscopic mechanism.

A fissile material produced from uranium-238 by neutron capture followed by two beta-minus decays.

A yield point marks the onset of permanent deformation according to a stated experimental criterion. In a tensile testing experiment, low loads can produce recoverable extension, while higher loads leave a residual extension after unloading. A sharp transition may be visible in the stress-strain curve, but many specimens change gradually from mainly elastic to increasingly plastic response.

When no unambiguous sharp point exists, an offset proof stress gives an operational definition: draw a line parallel to the initial elastic slope and displaced by a specified engineering strain, then identify its intersection with the loading curve. The chosen offset must be reported. The yield point is distinct from both the limit of exact proportionality and the tensile strength at maximum load. This matters because a small measurable departure from linear response does not by itself establish a particular amount of permanent plastic strain.

In his account of the 1973 work, Stanley Norman Cohen describes purified R6-5 plasmid DNA cut with EcoRI, reassociated and joined by DNA ligase. The mixture entered calcium chloride-treated Escherichia coli; plates containing antibiotics selected biological colonies showing antibiotic resistance. The recovered pSC102 carried three R6-5 fragments and kanamycin antibiotic resistance. The team then joined material from pSC102 to the carrier pSC101 providing tetracycline antibiotic resistance, recovering pSC105 with both forms of antibiotic resistance.^[ref]

Stained agarose gel electrophoresis, gradient centrifugation and electron microscope comparisons characterized recovered plasmids. Parental and mixture lanes in Figure 3 supplied fragment comparisons; pSC101 alone gave one EcoRI fragment, while pSC105 combined its carrier fragment with the resistance-bearing fragment. This supported joined, propagating genetic material beyond selection alone. The comparisons did not determine every nucleotide sequence or establish a universal construction efficiency.^[ref]

The interesting inference combines two kinds of evidence: a recoverable phenotype identifies descendants worth examining, and the extracted DNA tests a structural explanation. The fragment comparisons are molecular scientific controls; they answer a different objection from a control for background growth. The hypothetical diagrams here illustrate that logic without reproducing the historical fragment lengths or biological colony counts.

Plasmid restriction mapping compares fragment lengths after a restriction digest with the pattern predicted by specified cut positions. Use the original 6000-base pair plasmid feature map. Abstract restriction enzyme A cuts once, at 1000. Abstract restriction enzyme B cuts twice, at 2500 and 4500. Assume complete cutting of identical circular copies at exactly those sites, with no loss of DNA. These assumptions define a length-accounting exercise; the labels A and B do not stand for particular commercial enzymes.

For the combined restriction digest, successive clockwise distances areand the final fragment crosses the chosen zero:Their sum is $1500+2000+2500=6000$. The single A cut opens the circle into one full-length linear fragment. It does not produce pieces of lengths 1000 and 5000, because coordinate zero was never an end.

For contrast, stipulate a different starting substrate: a linear 6000-base pair DNA fragment with physical ends at 0 and 6000 and the same three internal cut coordinates. Its complete restriction digest givesbase pairs, summing again to 6000. There are four fragments, including two of equal length. Even an ideal length-resolving comparison would show only three distinct sizes if both 1500-base pair fragments occupy the same gel band. Fragment number and visible gel band number need not agree.

The general counting argument is short. For $k\ge 1$ distinct complete cuts of a circle, each fragment runs from one cut to the next, so there are $k$ fragments. For a linear substrate with $k$ distinct internal cuts, its two pre-existing ends give $k+1$ intervals. Summing successive distances counts every original base pair once in either case. The rule assumes internal double-strand cuts and does not treat an uncut circle as a zero-length product.

In practice, compare released linear fragments with an appropriate DNA ladder using agarose gel electrophoresis. The original uncut circular DNA need not migrate like a linear fragment of equal length; the existing agarose gel electrophoresis discussion explains that shape-dependent limitation. Partial cutting also changes the expected pattern. In this example, a circular copy cut at 1000 and 2500 but not 4500 yields lengths 1500 and 4500. A 4500-base pair product would therefore be compatible with that missed cut rather than uniquely identifying a different construction.

Matching lengths constrain an arrangement without determining every nucleotide sequence. For example, any change of letters preserving all three cut positions leaves this predicted pattern unchanged. Even the order of fragments can be ambiguous when only their unordered lengths are observed. DNA sequencing across both joins and through the plasmid insert tests additional claims that this length model cannot decide.

A plasmid insert is the DNA fragment added to a carrier plasmid during construction. In the plasmid feature map, its stipulated boundaries are 1000 and 2500, giving 1500 base pairs. The label identifies a construction role; it does not establish gene expression or functional complementation.