The place value is the weight of a position in positional notation. A numeral digit $d_j$ at that position contributes $d_j{b}^j$ to the represented number.

The radix is the base of positional notation. In the ordinary integer-base case it is an integer $b\ge 2$, and successive place values differ by a factor of $b$.

A numerical digit is one symbol in a positional notation string, denoting an integer from zero through one less than the radix.

A parallelepiped is a polyhedron with six parallelogram faces in three opposite pairs. Three generating vectors determine its edges; the absolute value of their determinant gives its volume in orthonormal Cartesian coordinates.

A growing extension from a pollen grain. In a compatible flower, the pollen tube provides a route for sperm to the angiosperm female gametophyte.^[ref]

A polarimeter uses polarizers to compare the direction of linear polarization before and after light passes through a sample; polarimetry explains the apparatus, readout and controls.

Take the Gaussian function $f_a(x)=e^{-\pi a{x}^2}$ with $a>0$. Every derivative is a polynomial times the same Gaussian function, so $f_a$ is a Schwartz function. The Fourier transform of a Gaussian gives, in the cycles convention,Thus the Poisson summation formula gives the exact identityThe Gaussian Fourier transform is given in scaled form in Exercise 41 of Terence Tao's 245C, Notes 2: The Fourier transform.^[ref] The identity is also the imaginary-axis transformation of the Jacobi theta function. At $a=1$ the Gaussian function is its own Fourier transform; at reciprocal values of $a$ its spatial and spectral widths exchange.

For a short derivation of the Gaussian Fourier transform, write $H(\xi )={\hat{f}}_a(\xi )$. Differentiation under the integral and integration by parts, using $f_a^{\prime }(x)=-2\pi ax{f}_a(x)$, giveThe Gaussian integral gives $H(0)=a^{-1/2}$, so this ordinary differential equation yields the claimed formula. The decay of the Gaussian function justifies these steps and removes the boundary term.^[ref]

For $a=0.04$, the physical Gaussian function has width parameter $1/\sqrt{a}=5$, while its Fourier transform has width parameter $\sqrt{a}=0.2$. Many physical samples matter, but the dual zero sample already givesFor $a=25$, use the physical side instead:These are bounds on the full positive tails, obtained below, rather than estimates based only on the first omitted term. Ordinary floating-point arithmetic rounds these sums to $5$ and $1$; the exact real numbers are slightly larger. A Gaussian sum truncation bound lets us choose the cheaper side and certify its truncation error before evaluating it.

Encircled energy is the fraction of total image radiant flux within a circle of radius $R$ around the centre of a point spread function.

PCR extension is template-directed construction of DNA from an annealed PCR primer by a DNA polymerase. Incoming deoxynucleoside triphosphates supply the new nucleotides. Each addition lengthens the growing strand at its $3^{\prime }$ end, so the product grows $5^{\prime }\to 3^{\prime }$ while the template is read $3^{\prime }\to 5^{\prime }$. The direction refers to the nucleic acid backbone, not to left or right on a page.^[ref]

For a complete worked copying step, start with template $3^{\prime }\text{-}\mathrm{TCAGGGAC}\text{-}{5}^{\prime }$ and paired PCR primer $5^{\prime }\text{-}\mathrm{AGTC}\text{-}{3}^{\prime }$. The next four template letters are GGAC, encountered in the template's $3^{\prime }$-to-$5^{\prime }$ direction. The base pairing rule therefore selects CCTG in order, givingThe PCR primer supplies four nucleotides, and PCR extension adds four more; this copying event consumes four deoxynucleoside triphosphates. On the oppositely oriented template $5^{\prime }\text{-}\mathrm{AGTCCCTG}\text{-}{3}^{\prime }$, the other PCR primer is $5^{\prime }\text{-}\mathrm{CAGG}\text{-}{3}^{\prime }$. It pairs at the right and extends leftward. Reading that template from right to left encounters CTGA after the paired region, so addition of GACT gives $5^{\prime }\text{-}\mathrm{CAGGGACT}\text{-}{3}^{\prime }$. Both finished products are the reverse complements of their respective templates.

The direction argument is short: the PCR primer and its template are antiparallel nucleic acid strands; extension takes place at the PCR primer's $3^{\prime }$ end; hence forward progress along the new strand is backward progress along the template. Reflecting a drawing reverses its arrows on the page without changing either chemical direction. This is why a leftward arrow can represent $5^{\prime }\to 3^{\prime }$ synthesis.

The time needed depends on how much DNA remains to be copied and the chosen DNA polymerase, while temperature must support that enzyme's activity.^[ref] Separating PCR annealing and PCR extension into different holds is convenient, but some compatible conditions allow a combined hold. The three chemical jobs therefore need not always require three distinct temperatures.^[ref]

PCR annealing is association of PCR primers with exposed complementary DNA regions by base pairing. It supplies the paired starting ends required for PCR extension. Cooling can permit this association without itself joining new nucleotides onto the PCR primers. The two templates can also pair with one another again, so exposure after PCR denaturation is an opportunity for binding, not a guarantee of productive copying.^[ref]

In the example below, $5^{\prime }\text{-}\mathrm{AGTC}\text{-}{3}^{\prime }$ pairs with the aligned template segment $3^{\prime }\text{-}\mathrm{TCAG}\text{-}{5}^{\prime }$. The opposite end labels show antiparallel nucleic acid strands. Pairing the right letters in the wrong directional display does not supply a correctly depicted starting end. These four-nucleotide strands are a teaching example, not a laboratory PCR primer recommendation.

The binding temperature balances enough productive pairing against unwanted binding. Its appropriate value depends on the PCR primers and reaction conditions; increasing it can improve PCR specificity but can also prevent useful binding. A predicted binding setting therefore needs experimental assessment rather than being inferred from the label PCR.^[ref]

PCR denaturation is the high-temperature stage that separates a DNA duplex into single-stranded templates for the next copying step. It disrupts the pairing and stacking arrangement of nucleobases; it does not require cutting the phosphodiester bonds along either strand. The distinction matters: breaking the backbone would damage the ordered nucleotide sequence that is meant to serve as a template.

For an original eight-base pair duplex, complete PCR denaturation exposes two eight-nucleotide strands. It leaves sixteen nucleotides altogether: separation has made the templates accessible without yet making more DNA. PCR extension, rather than heating alone, adds the new material. Already copied duplexes also need to separate at the beginning of subsequent cycles.

Separation is a condition-dependent process. Template composition and the buffer solution affect the required temperature and duration; heating longer is not automatically better because prolonged heating can reduce DNA polymerase activity. Incomplete separation leaves fewer accessible templates even when the PCR primers themselves are well chosen.^[ref]

PCR controls are scientific controls that separate different explanations for a readout. A no-template reaction contains the PCR primers and copying reagents but replaces the added template with water or sample buffer. Amplification there can indicate carried-over DNA or a primer dimer; a clean result does not exclude contamination confined to one sample.^[ref]

Different comparisons answer different questions. A known amplifiable template tests whether the PCR primers, DNA polymerase and thermal cycler can produce the expected amplicon. An extraction blank follows sample preparation without adding the sampled material. A separate amplifiable interval in the sample can test whether recovered DNA supports copying, although success there does not guarantee success at the target. A length standard calibrates gel electrophoresis; it is not a control for the copying reaction. These comparisons are an analytical design extension to the teaching example below, not a claim that its student protocol includes every comparison.

A primer dimer is an amplification artifact in which PCR primers provide extendible partners for one another. Its accumulation consumes substrates and may produce a signal even without the intended template. It is not evidence that the intended amplicon was present.^[ref]

A PCR plateau is the late phase in which further cycles add little product. Depletion of substrates and accumulation of reaction by-products can reduce PCR efficiency. Consequently, similar final amplicon amounts need not mean similar initial template amounts. In the stipulated green model, $0<N_n<K$ implies $N_n<N_{n+1}<K$, and growth approaches the imposed ceiling. That illustrates loss of starting-amount information without claiming to identify the limiting chemical process in an actual tube.^[ref]

The p-adic integers form a complete metric space and a discrete valuation ring. Every nonzero $x$ is uniquely $p^{k}u$ with $k=v_p(x)\ge 0$ and $u$ a p-adic unit. The prime $p$ is a uniformizer, and reduction modulo $p$ gives the residue field ${\mathbb{F}}_p$. The element $p$ has no inverse inside ${\mathbb{Z}}_p$, so the ring is not a field.

For a short ideal-classification outline, take a nonzero ideal and choose an element of smallest p-adic valuation $k$, which exists by the well-ordering principle. Its unit factor is invertible, so the ideal contains $p^k$. Every other element is divisible by $p^k$, proving that the ideal is $p^k{\mathbb{Z}}_p$. Completeness follows because ${\mathbb{Z}}_p$ is a closed subset of the complete metric space ${\mathbb{Q}}_p$. The zero ideal is separate from this nonzero-ideal classification.

Let $a$ be an integer not divisible by $p$. Its modular inverse modulo $p^n$ has a unique least nonnegative representative $r_n$. Reducing the inverse equation modulo $p^n$ shows that the representatives are compatible. Since $a$ is a p-adic unit,so the sequence converges to the inverse $a^{-1}$ in ${\mathbb{Z}}_p$. This computes a fraction through finite integer operations.

For $a=3$ and $p=2$, the first eight representatives are $1,3,3,11,11,43,43,171$. Each satisfies $3r_n\equiv 1(\mathrm{mod}{2}^n)$. The resulting expansion isThe geometric series has ratio $4$, whose 2-adic absolute value is $1/4$, and its sum is $1+2/(1-4)=1/3$. In the ordinary real number absolute value, the representatives grow. Different metrics impose different meanings of a good approximation. The periodic digits here are a specific instance of the rational expansions studied in Conrad's authored note.^[ref]