Authored technical account: www.scholarpedia.org/article/General_linear_methods . Scholarpedia2(4):2852; DOI:10.4249/scholarpedia.2852 . Cited definition section, equations(1)–(2), internal/external stages and inclusion of Runge-Kutta and linear multistep methods. Search-indexed primary text verified; one subsequent direct fetch timed out.

Bulletin of the American Mathematical Society 49(1), 1–23. Primary PDF: www.ams.org/bull/1943-49-01/S0002-9904-1943-07818-4/S0002-9904-1943-07818-4.pdf . Cited for variational approximation by finite-parameter trial functions, including piecewise-linear constructions; no claim that Courant alone originated every finite-element method. Direct PDF access returned 403.

By Matthias Heymann, Eric Vanden-Eijnden.

Communications on Pure and Applied Mathematics 61, 1052–1117. DOI: doi.org/10.1002/cpa.20238 . Author-hosted PDF. Sections 1.1–1.3: action admissibility, noise convention, free-time quasipotential and geometric reformulation. Wiki uses drift a and covariance D to avoid the paper's a/b notation ambiguity; the time-reparameterization and gradient examples are explicitly derived.

Archive: arxiv.org/abs/1801.09922

Primary text: arxiv.org/pdf/1801.09922

Consulted Definition5.1 and equation(47), printed pp19–20; section5.1 and Theorem5.1, printed pp21–22, compactness/lower-semicontinuity hypotheses for existence. PDF title page displays18December2017, whereas archive version1 was posted30January2018; these are distinct dates.

Publisher: doi.org/10.1287/mnsc.13.7.492

Management Science13(7), pp492–498, published1March1967. Consulted publisher abstract, identifying the algorithm and its relationship between fractional and parametric programming; full paper not accessed. The wiki separately supplies its elementary compact-domain convergence proof.

Verified author list and archive history: arxiv.org/abs/1001.1549

Primary text: arxiv.org/pdf/1001.1549

Consulted passages: section1, equation(5), printed p2, symmetric second-order splitting; equation(11) and following negative-step discussion, printed pp3–4. Archive abstract records submission10January2010 and journal reference Bol.Soc.Esp.Mat.Apl.50(2010),47–61. The fetched PDF separately displays25April2022 on its title page; do not equate that displayed typesetting date with first publication.

Author-hosted text: web.stanford.edu/~boyd/papers/pdf/prox_algs.pdf

Consulted passages: section1.1, proper closed convex definition, printed p124; section1.2, projection onto a nonempty closed convex set, pp125–126; section3.2, resolvent interpretation, pp137–138; sections4.2–4.3, projected/proximal gradients and acceleration, pp148–152. The consulted PDF title page labels itself Foundations and Trends in Optimization1(3)(2013),123–231 and has a placeholder DOI; this draft label is not asserted as final journal-publication metadata.

Publisher chapter: link.springer.com/chapter/10.1007/978-3-319-55456-3_3

Author-hosted text: hplgit.github.io/fdm-book/doc/pub/diffu/html/._diffu-solarized001.html

Consulted passage: Chapter3, Diffusion Equations, section3.1, explicit finite-difference discretization of one-dimensional diffusion and Forward Euler update. Publisher lists chapter pp207–322, first online22June2017, both authors. This is a whole-book source card; cite the chapter in its locator.

Official text: netlib.org/lapack/lug/node38.html

Cited passages: Computational Routines, Linear Equations: symmetric indefinite factorization uses symmetric diagonal blocks of order one or two.

Published paper, BIT 3, 27–43; author byline Germund G. Dahlquist. Section 1 gives the constant-coefficient, first-derivative and coprimality assumptions; §2 defines A-stability and proves the order barrier. The university hosts the scan and is not its author.

Online text: ftp.fftw.org/burrus-notes.html

Cited passage: prime factor algorithm discussion, identifying the coprime-factor index map and absence of twiddle-factor multiplication. Search indexing exposed the relevant passage; direct fetching returned a gateway error during this pass. No revision date is asserted.

Transcendental number theory studies numbers that satisfy no nonzero polynomial equation with integer coefficients, and stronger independence properties among numbers and special-function values. The distinction is algebraic: $\sqrt{2}$ is irrational but algebraic because it satisfies $x^2-2=0$.

A useful example is the Liouville constant $L={\sum }_{n\ge 1}10^{-n!}$. Truncating after $N$ terms gives a rational $p_N/q_N$ with $q_N=10^{N!}$ andFor any fixed algebraic degree, these approximations eventually contradict the rational-approximation lower bound in Liouville approximation theorem. Thus exceptionally good approximation can certify transcendence.

Other results, such as the six exponentials theorem, constrain collections of exponential values using linear independence of their inputs. Proving each number irrational is weaker than proving it transcendental, and proving separate transcendence is weaker than proving algebraic independence of a collection.

Computational number theory studies algorithms for arithmetic objects, including integers, finite fields, polynomials and algebraic number fields. Typical tasks are greatest-common-divisor computation, modular inversion, primality testing and integer factorization. The output may be an answer together with a certificate that can be checked independently.

For example, the extended Euclidean algorithm finds $7\cdot 15-26\cdot 4=1$, certifying that the inverse of seven modulo twenty-six is fifteen. Modular exponentiation by repeated squaring computes $a^e\mathrm{mod}m$ with a number of modular multiplications proportional to the bit length of $e$, instead of multiplying $e$ copies.

Input size is measured in bits, not by the numerical magnitude alone. An integer near $2^k$ has roughly $k$ bits, so an algorithm using time proportional to that integer can be exponential in its input length. Primality testing and complete factorization are distinct tasks; solving the former does not automatically supply the latter.

Geometry of numbers uses the geometry of lattices in real vector spaces to study arithmetic questions. A full-rank lattice $\Lambda =A{\mathbb{Z}}^n\subset {\mathbb{R}}^n$ has covolume $|\det A|$, the volume of a fundamental parallelotope. Geometric size can force a region to contain a nonzero lattice point.

Minkowski convex body theorem states that a centrally symmetric convex measurable body $K$ of volume greater than $2^n\mathrm{covol}(\Lambda )$ contains a nonzero point of $\Lambda$. The symmetry and convexity let two lattice-congruent points in $K/2$ produce a difference in $K$.

For $\Lambda ={\mathbb{Z}}^2$, a centered disk of radius $R$ has area $\pi R^2$, so the theorem guarantees a nonzero integer point when $R>2/\sqrt{\pi }$. This sufficient bound need not be the exact first radius where a point occurs: the unit coordinate vectors already have length one. Such estimates connect shortest-vector problems with ideal classes, Diophantine approximation and arithmetic volume bounds.

Deoxyadenosine triphosphate, usually abbreviated dATP, is a nucleotide containing adenine, a deoxyribose sugar and three phosphate groups. The “deoxy” distinguishes its sugar from the ribose in adenosine triphosphate. The usual molecular formula for the neutral acid is ${\mathrm{C}}_{10}{\mathrm{H}}_{16}{\mathrm{N}}_5{\mathrm{O}}_{12}{\mathrm{P}}_3$; its ionization state in solution depends on conditions.^[ref]

A DNA polymerase uses dATP as the substrate for adding an adenine-containing unit to a growing DNA strand opposite a template thymine. The strand's terminal $3^{\prime }$ hydroxyl attacks the incoming nucleotide's alpha phosphate; the other two phosphates leave as pyrophosphate. The incorporated unit is therefore a monophosphate residue within the chain, not a triphosphate hanging from the completed backbone. This distinction connects nucleotide chemistry with the directional $5^{\prime }\to 3^{\prime }$ growth of DNA.^[ref]

A polymerase assay can supply the template, primer, enzyme and nucleotide substrates, then compare product lengths with and without dATP. A failure to extend past a template position requiring adenine tests substrate dependence; it does not by itself determine every enzymatic step.

3'-Phosphoadenylyl sulfate, abbreviated PAPS, is a nucleotide used as an activated donor in sulfation. Its adenosine unit bears phosphate groups at the ribose 3′ and 5′ positions; a sulfo group is attached to the 5′ phosphate. That covalent arrangement distinguishes PAPS from free sulfate dissolved alongside an unrelated nucleotide.^[ref]

A concrete example is heparan sulfate glucosamine N-sulfotransferase. The enzyme transfers the sulfur-containing group from PAPS to a glucosamine nitrogen within heparan sulfate, leaving adenosine 3',5'-bisphosphate. Abbreviating that product as PAP, the transfer can be summarized schematically asThis schematic omits proton and charge bookkeeping; the IUBMB reaction specifies the actual polymer-associated acceptor.^[ref]

The example separates donor availability from product specificity. PAPS supplies a transferable group, while the enzyme determines which acceptor and attachment site participate. Supplying sulfate is therefore not, by itself, a specification of which position in a glycosaminoglycan becomes modified.

Adenosine 3′,5′-bisphosphate, often abbreviated PAP, is a nucleotide with separate phosphate groups attached at the 3′ and 5′ positions of the ribose in adenosine. The positions are part of its identity: two phosphate groups on different sugar positions do not make the same structure as the linked phosphate chain of adenosine diphosphate.^[ref]

An important reaction example is sulfation using PAPS. In the reaction catalyzed by heparan sulfate glucosamine N-sulfotransferase, PAPS transfers a sulfur-containing group to a glucosamine unit in heparan sulfate, leaving adenosine 3',5'-bisphosphate. The product retains the two phosphate groups but lacks the transferred sulfo group. Counting phosphorus atoms alone would therefore fail to distinguish donor from product.^[ref]

This is a useful example of why a molecular formula or a count of functional groups is less informative than a chemical structure: attachment positions and connectivity determine which molecule is present.

UDP-N-acetylgalactosamine is a nucleotide sugar combining a uridine diphosphate moiety with N-acetylgalactosamine. It serves as a donor for glycosyltransferases: the sugar-derived residue is incorporated into an acceptor, and uridine diphosphate is released. The nucleotide-linked donor and the free N-acetylgalactosamine molecule are therefore different chemical participants.^[ref]

A structural example is chondroitin synthesis. Addition of an N-acetylgalactosamine residue to an appropriate growing chain uses UDP-N-acetylgalactosamine, while formation of the alternating glucuronic acid portion requires a different nucleotide sugar. The donor is not itself the finished polysaccharide.^[ref]

A primary characterization of a chondroitin synthase tested donor specificity as well as product identity. Structurally related nucleotide sugars did not simply replace the required donors in that assay. This illustrates why the shared nucleotide portion does not make every nucleotide sugar interchangeable: the sugar identity and the accepting enzyme both matter.^[ref]

Uridine diphosphate, conventionally abbreviated UDP in chemistry, is a nucleotide consisting of uridine with a linked diphosphate group at the ribose 5′ position. The uridine portion contains uracil and ribose. ChEBI records the neutral acid representation with molecular formula ${\mathrm{C}}_9{\mathrm{H}}_{14}{\mathrm{N}}_2{\mathrm{O}}_{12}{\mathrm{P}}_2$ and distinguishes it from its deprotonated forms.^[ref]

A useful reaction example is glycosyltransferase action. UDP-N-acetylgalactosamine contains an additional sugar-derived portion; transferring that portion to an acceptor can release uridine diphosphate. The free nucleotide product and the complete nucleotide sugar donor are therefore not interchangeable names for one molecule.^[ref]

For a simple connectivity check, count the phosphorus atoms in the free product and in the linked donor: both contain two. That count alone cannot reveal whether the transferable sugar is still attached. Distinguishing the chemical structures explains why observing a nucleotide product can inform a transfer reaction, whereas measuring total phosphorus cannot establish the same conclusion.

A nucleoside diphosphate is a nucleotide whose nucleoside carries a diphosphate chain: two phosphate groups joined through a phosphoanhydride bond. For the usual 5′ species, the phosphate nearest the sugar is attached at its 5′ position. Adenosine diphosphate is an example, with adenine as its nucleobase and ribose as its sugar. ChEBI records its acid structure as ${\mathrm{C}}_{10}{\mathrm{H}}_{15}{\mathrm{N}}_5{\mathrm{O}}_{10}{\mathrm{P}}_2$.^[ref]

Counting two phosphorus atoms is necessary for this example but does not establish that connectivity. A molecule bearing two separate phosphate esters on different sugar positions would have different connectivity from a single diphosphate chain. The chemical structure, including the bridging oxygen, distinguishes the alternatives.

The adenylate kinase reaction makes the relationship with nucleoside monophosphates and nucleoside triphosphates explicit: one adenosine triphosphate and one adenosine monophosphate interconvert with two adenosine diphosphates.^[ref] The number of nucleotide molecules stays two while their phosphate counts change from three plus one to two plus two.

An original comparison also shows why identifying a nucleoside diphosphate is more informative than measuring total phosphorus. A sample containing $1\mathrm{mmol}$ of pure adenosine diphosphate contains $2\mathrm{mmol}$ of phosphorus atoms. A different sample containing $1\mathrm{mmol}$ each of adenosine monophosphate and free phosphate has that same phosphorus total, but different molecules and reaction possibilities. Total phosphorus alone cannot discriminate the samples.