Original paper: doi.org/10.1007/BF02398436

Primary text: archive.ymsc.tsinghua.edu.cn/pacm_download/117/6507-11511_2006_Article_BF02398436.pdf

Cited passages: introduction, equation (1.1), p. 259: distortion on every closed infinite-dimensional subspace; Remark 3.7, p. 279: bounded distortion means one finite threshold excludes distortability on every subspace.

Primary lecture text: abel.math.harvard.edu/archive/118r_spring_05/docs/milnor.pdf

Archive record: arxiv.org/abs/math/9201272

Cited passages: sections 2–3, normality, Fatou/Julia sets and the squaring map; section 7, parabolic petals and basin definition; Appendix F, quadratic parameter space.

Anonymous two-page appendix. Theorem D.1 states the one-sided polar theorem for the real L-infinity/L1 dual pair; its remark identifies Hahn-Banach separation. No author or publication year is established by the document; the hosting course path is not authorship evidence.

Author-attributed proceedings notes, printed pp. 19 onward. The inspected PDF heading supplies the work title and author; no publication year is inferred from its URL. Introduction, printed pp. 20–21, develops Hamiltonian conservation, and later discussion distinguishes rigorous low-regularity solution constructions from formal invariants.

Inverse Problems 20, 399–418. Author-hosted published paper; §2.1 defines transport/attenuated data and §2.2 develops the spectral Riemann-Hilbert construction. Full-data inversion and the additional assumptions of the partial-data iteration are distinct results.

Course Notes for Math 256, Harvard, fall 2015. Sections 3.2–3.3 and lecture 4 give topological entropy through open covers and separated/spanning orbit sets, including the compact metric setting and limiting definitions.

A moment problem asks whether specified numbers $m_0,m_1,\dots$ arise from a positive Borel measure $\mu$ on a prescribed subset of the real line:Existence, support restrictions and uniqueness are separate parts of the question. A probability measure must have $m_0=1$. A point mass at $a$ gives $m_n=a^n$, while the uniform probability measure on $[0,1]$ gives $m_n=1/(n+1)$.

Positivity imposes concrete constraints. For every finite coefficient vector $c$,so every associated Hankel matrix is positive semidefinite. On a fixed compact interval, the full moment sequence determines a finite measure uniquely: equality on polynomials extends to equality on continuous functions by the Weierstrass approximation theorem. A finite list of moments generally leaves many possible measures, which explains the distinction between moment reconstruction and finite-data fitting.

A carboxamidine is a carbon-centred amidine, with the N-unsubstituted pattern $\mathrm{R}-\mathrm{C}(=\mathrm{NH})-\mathrm{N}{\mathrm{H}}_2$. More generally, substituents may replace hydrogens on either nitrogen. Its defining carbon is attached to an imino nitrogen and an amino nitrogen; an amide instead has a doubly bonded oxygen at the corresponding position.^[ref]

For an atom-count example, $\mathrm{C}{\mathrm{H}}_3-\mathrm{C}(=\mathrm{NH})-\mathrm{N}{\mathrm{H}}_2$ has molecular formula ${\mathrm{C}}_2{\mathrm{H}}_6{\mathrm{N}}_2$. Replacing one amino hydrogen by $\mathrm{C}{\mathrm{H}}_3$ gives $\mathrm{C}{\mathrm{H}}_3-\mathrm{C}(=\mathrm{NH})-\mathrm{NH}-\mathrm{C}{\mathrm{H}}_3$, with molecular formula ${\mathrm{C}}_3{\mathrm{H}}_8{\mathrm{N}}_2$. The change adds $\mathrm{C}{\mathrm{H}}_2$ overall because the new methyl group replaces a hydrogen. This bookkeeping illustrates N-substitution without treating a structural drawing as a measured reaction yield.

Class names and preferred naming suffixes should also be distinguished. The IUPAC recommendations retain the generic class description but use the suffixes -imidamide and -carboximidamide in the specified nomenclature, and distinguish the amino and imino positions by N and N′. Calling a substance a carboxamidine therefore does not prescribe the suffix in its preferred systematic name.^[ref]

An amide bond, in the carboxamide sense used here, is the carbon–nitrogen covalent bond in the unit $\mathrm{R}-\mathrm{C}(=\mathrm{O})-\mathrm{N}{\mathrm{R}}^{\prime }{\mathrm{R}}^{\prime \prime }$. The nitrogen is directly attached to the carbonyl group's carbon. Merely finding a carbon–nitrogen bond elsewhere in a molecule does not identify an amide bond. The broader IUPAC class name amide also covers other acid-derived compounds; this article concerns the carbonyl linkage.^[ref]

For example, acetamide has connectivity $\mathrm{C}{\mathrm{H}}_3-\mathrm{C}(=\mathrm{O})-\mathrm{N}{\mathrm{H}}_2$. Its amide C–N bond is the bond from the carbonyl carbon to nitrogen, rather than the adjacent C–C bond. A peptide bond is a biologically important instance of this connectivity in a chain of amino acids.

Chemical resonance describes electron sharing through the nitrogen and carbonyl group. Two useful contributing structures areThese are descriptions contributing to one electronic structure, rather than two molecules rapidly alternating. The amide bond consequently has partial double bond character, and the ordinary amide group favors approximate planarity. Linus Pauling's lecture connects that structural picture to measured bond lengths and explains why hypothetical resonance forms remain useful.^[ref]

This supplies a useful modeling constraint: a chain containing peptide bonds cannot be treated as if every backbone single bond rotates equally freely. Approximate amide planarity constrains the conformations available without asserting that every amide is perfectly planar.

A carboxamide is an amide of a carboxylic acid, with a carbonyl group directly bonded to nitrogen. Its general connectivity is $\mathrm{R}-\mathrm{C}(=\mathrm{O})-\mathrm{N}{\mathrm{R}}^{\prime }{\mathrm{R}}^{\prime \prime }$, where the substituents shown can include hydrogen. The term identifies a class of compounds or a structural unit; amide bond identifies the carbonyl C–N bond within it.^[ref]

For example, acetamide, $\mathrm{C}{\mathrm{H}}_3\mathrm{C}(=\mathrm{O})\mathrm{N}{\mathrm{H}}_2$, contains one carboxamide group. Replacing an N–H covalent bond by an N–C covalent bond can leave the defining carbonyl C–N connectivity intact. Conversely, replacing the carbonyl oxygen by another group changes the functional class even though a C–N bond remains. Connectivity, rather than merely an inventory of elements, supplies the classification.

A formal atom balance illustrates carboxamide hydrolysis:Both sides contain two carbons, seven hydrogens, one nitrogen and two oxygens. The equation balances neutral molecular formulas; the actual reaction products can be protonated or deprotonated according to pH. It does not assert that mixing neutral reactants gives rapid conversion. Ordinary amide hydrolysis needs suitable conditions or enzyme catalysis, consistent with the persistence of peptide bonds in water.^[ref]

Source-over compositing combines a source layer with a background by retaining the source contribution and the fraction of the background left by the source's alpha channel. For premultiplied alpha colour triples $c_s={\alpha }_s{C}_s$ and $c_b={\alpha }_b{C}_b$, the rule isThe W3C compositing specification states this operator explicitly.^[ref] The output is another colour-and-alpha pair, so it can be combined with another layer.

For an original Linear-light RGB example, take red layer A with alpha $1/2$, green layer B with alpha $1/4$, and blue layer C with alpha $1/2$. Place A over B over C. Expanding source-over compositing giveswith ${\alpha }_o=1-(1-{\alpha }_A)(1-{\alpha }_B)(1-{\alpha }_C)=13/16$. Its straight alpha colour is therefore $(8/13,2/13,3/13)$. The colour triple alone does not describe the partly transparent output; the alpha must accompany it.

The expansion also proves associativity in exact arithmetic: grouping A over B first or grouping B over C first produces those same three terms. Reordering the layers changes the coefficients, so associativity does not permit exchanging front and back. Intermediate rounding can introduce small numerical differences even when the mathematical grouping is equivalent.

For fully transparent source alpha zero, source-over compositing retains the background; for opaque source alpha one, it retains only the source. To interpret the colour arithmetic as a mixture of light contributions, use a common Linear-light RGB space. Stored sRGB codes generally need decoding first.^[ref] Apply the equation for the actual storage convention: treating premultiplied alpha samples as straight alpha samples multiplies the source contribution twice.

An algorithm portfolio combines several candidate algorithms or algorithm runs to exploit differences in their performance. It includes a policy for choosing or allocating work, rather than merely naming a collection. A parallel algorithm portfolio runs candidates concurrently; a sequential algorithm portfolio allocates time on one processor. The original 2001 study compared these approaches and restart strategies on specified distributions of search problems.^[ref]

Consider invented runtimes $A=(1,30)$ seconds and $B=(30,1)$ seconds on two equally likely instance types. Either algorithm alone has mean $15.5$ seconds. Perfect per-instance algorithm selection would take one second before selection overhead. Running both on independent full-speed processors and stopping at the first answer also takes one second under this model, but uses two processors.

For a different policy, run $A$ for at most five seconds, then start $B$ if needed. The two completion times become $1$ and $5+1=6$ seconds, giving mean $3.5$ seconds. This invented sequential algorithm portfolio saves time relative to either fixed choice, without matching the two-processor wall time.

Complementary runtimes make these examples work. If all candidates are slow on the same instances, diversity in names supplies no such benefit. Compare wall time, total computing work and overhead separately, and test the allocation policy on new instances. The best completed run chosen retrospectively is a comparison bound, not automatically an implementable selector.

ParamILS is a framework for algorithm configuration based on iterated local search. It explores an algorithm parameter space using improvements to nearby configurations, perturbations and occasional restarts. BasicILS compares configurations on a fixed number of matched training runs; FocusedILS adapts how many runs support a comparison.^[ref]

The distinctive efficiency idea of adaptive capping is to stop spending evaluation time on a candidate once it cannot win the comparison under the stated cost model. Consider an original example with three fixed instances and mean nonnegative runtime as the objective function. A fully evaluated incumbent has total time $12$ seconds, hence mean $4$ seconds. If the first two candidate runs already total $13$ seconds, even a zero-time final run leaves mean at least $13/3>4$. Rejecting that candidate does not require finishing the last run.

This argument depends on its assumptions: the comparison uses the same fixed instance set and a lower bound on unfinished cost. It cannot be transferred unchanged to arbitrary quality scores or a different timeout penalty. The paper distinguishes trajectory-preserving capping from a more aggressive heuristic variant; faster rejection does not always carry the same guarantee.^[ref]

The method turns tuning into a measured optimization experiment, but a good observed configuration is not a proof of a global minimum. The neighborhood, evaluation budget and deployment workload remain relevant to what the returned setting establishes.

An algorithm parameter space, commonly written $\Theta$, is the set of allowed settings for a parameterized algorithm. It is the search space of algorithm configuration, rather than the set of problem instances that the target algorithm must solve. ParamILS explicitly accounts for parameters whose relevance depends on other settings.^[ref]

For an original finite example, let a mode be either $A$ or $B$. Mode $A$ has an extra setting $h\in \{0,1,2\}$, while mode $B$ ignores $h$. A rectangular encoding has $2\times 3=6$ tuples, but only four distinct effective configurations:The star means that the conditional algorithm parameter is inactive. Treating the three encodings of $B$ as three distinct experiments spends effort revisiting the same behavior. If $(A,2)$ is a forbidden parameter combination, the valid effective space has three members.

The representation also determines what a nearby configuration means. A search that changes one active parameter can move among the $A$ settings or change mode; changing an inactive value alone does not change behavior. Thus the number of encoded tuples, number of meaningful settings and neighborhood used by local search are separate design choices.

Continuous settings require declared ranges and a search method that supports them; replacing a range with a finite grid changes the problem being searched. Whatever the representation, a returned setting must be valid in $\Theta$ before its measured cost can be compared with other candidates.

Call-by-name passes a procedure argument so that each use of the formal parameter refers to the actual expression in its calling environment. In ALGOL 60, the report explains this through replacement with renaming to avoid accidental capture of identifiers. An expression can consequently be evaluated more than once.^[ref]

Suppose the actual argument is a[i]. The procedure reads its name parameter, changes i, and reads the parameter again. With call-by-name, the second access can select a different array entry. With call-by-value, both accesses use the value obtained before entering the procedure. This is also different from saving the original entry's address. Call-by-name does not imply lazy evaluation with memoization: repeated evaluation can repeat both work and side effects.

By Jeffrey C. Lagarias. American Mathematical Monthly 109(6), 534–543 (2002). DOI doi.org/10.1080/00029890.2002.11919883 .

Author-hosted version dated 29 July 2001: websites.umich.edu/~lagarias/doc/elementaryrh.pdf

Verified locator: first page, Problem E, equation (1.1) and Theorem 1.1; harmonic numbers and the sum-of-divisors inequality equivalent to the Riemann hypothesis.

By Jeffrey C. Lagarias and Robert C. Rhoades. Ramanujan Journal 41, 191–232 (2016), DOI doi.org/10.1007/s11139-015-9729-1 . The preprint was submitted in 2015.

Verified locators: preprint pages 1–2, equations (1.1)–(1.2), modularity, growth condition and iterated Laplacian; pages 3–4, Theorem 1.1 and construction of a Taylor basis from Eisenstein series.

Research in Number Theory 8, article 83, 8 October 2022. Section 2.1: weight-zero definition and nonnegative Laplacian; Section 2.4: even/odd Fourier expansions with K-Bessel functions. DOI 10.1007/s40993-022-00393-y.

By James E. Humphreys.

Verified locator: Chapter 1, sections 1.1 and 1.3: finite-field algebraic groups and fixed points of Frobenius/Steinberg endomorphisms, with related derived and central quotient groups. Author and 2005 copyright verified in assets.cambridge.org/97805216/74546/frontmatter/9780521674546_frontmatter.pdf .