An infinite conducting cylinder of radius $a$ at centre height $d>a$ above a grounded plane has the same exterior electric potential as opposite line charges $\pm \lambda$ at heights $\pm c$. If $R_{+}$ and $R_{-}$ are distances from the positive and negative lines, $V=\lambda \ln (R_{-}/R_{+})/(2\pi {\epsilon }_0)$. The plane has $R_{-}=R_{+}$; the cylinder has $R_{-}/R_{+}=(d+c)/a$. Its voltage is therefore $\lambda \mathrm{arcosh}(d/a)/(2\pi {\epsilon }_0)$, giving the displayed capacitance per unit length. The Uniqueness of the Dirichlet problem fixes the exterior field. A Gaussian contour surrounding the cylinder encloses the positive image line, so $\lambda$ is also its physical charge per unit length.

For $b>0$, the Poisson equation $\Delta V=q{\delta }^{(2)}(x,y)$ in $-b<x<b$ with homogeneous Neumann boundary conditions on both edges admits the even-in-$y$ method of images solution in the title, where $r_n^2=(x-2nb{)}^2+y^2$. All image strengths are $q$. Pairing the images at $\pm 2nb$ and subtracting their constant logarithms makes each summand $O(n^{-2})$, giving local uniform convergence away from the source. The unrenormalized sum of logarithms does not converge.

Differentiating givesThe corresponding $x$-derivatives cancel in reflected pairs on $x=\pm b$. For $y>0$, integration across the strip yields a telescoping arctangent sum:One closed form, up to an additive constant, is $V=(q/4\pi )\log [\cosh (\pi y/b)-\cos (\pi x/b)]$. Its near-source logarithmic singularity has coefficient $q/(2\pi )$. Conditions at large $|y|$ or a symmetry class must be specified for a uniqueness claim; side Neumann boundary conditions alone allow additional homogeneous harmonic functions.

A cancer arising in the cervix. Persistent infection with high-risk human papillomaviruses is central to most cases, connecting a chronic viral process with later malignant transformation.

The study of the immune system, including recognition, regulation, protection and pathological responses. A strong measured response need not equal better protection if the same activity damages host tissue.

A device that regulates a fluid flow by opening, closing or restricting a passage.

A piston is a moving component that fits within a cylinder and transmits force between a fluid pressure and a mechanical linkage. For an effective face area $A$ and pressure difference $\Delta p$, the ideal axial pressure force is $F=A\Delta p$. Sliding seals reduce leakage but introduce friction, and connecting rods can add side loads. In an engine the piston converts pressure work into motion; in a compressor mechanical work instead raises fluid pressure. The swept volume is face area times stroke for a constant-bore cylinder.

A mechanical engineer applies mechanical engineering to components and systems involving motion, loads or energy conversion. The work can include modeling stresses and vibration, sizing thermal systems and specifying tolerances or manufacturing processes. A rotating shaft, for example, requires reasoning about torque, deflection, fatigue and bearing support together. The resulting design depends on load histories and material properties as well as the geometry of the part.

A gasket is a deformable component that closes a leakage path between contacting surfaces. In a diamond anvil cell, its hole also confines the sample laterally.

By Georges Matheron. Economic Geology 58(8), 1246–1266; published 1 December 1963. DOI 10.2113/gsecongeo.58.8.1246.

Cited passages: p. 1246, abstract and introduction, on mining estimation. Primary text was inspected through a scan mirror; the mirror is not the author or publisher.

By Skip Thompson. Scholarpedia 2(3):2367, 2007; DOI 10.4249/scholarpedia.2367. Publication year is distinguished from later online revisions.

Cited passages: opening definition and equation (1); Initial History Function; Derivative Discontinuities. Relevant text was inspected through indexed Scholarpedia results when the direct page was unavailable.

By Paul Lévy. Compositio Mathematica 7, 283–339; Numdam catalogs the volume as 1940. Received 29 March 1939 (p. 339).

Digitized paper: www.numdam.org/article/CM_1940__7__283_0.pdf

Cited passage: §5, pp. 301–302, especially the circle-projection paragraph on p. 302.

By Scott Kirkpatrick, Charles Daniel Gelatt Jr. and Mario P. Vecchi. Science 220(4598), 671–680 (13 May 1983).

DOI: doi.org/10.1126/science.220.4598.671

Inspected scan: hedibert.org/wp-content/uploads/2013/12/1983KirkpatrickGelattVecchi.pdf

Cited passages: pp. 672–673 and 675.

University of California, Berkeley course. James Demmel is the instructor; this role does not establish authorship of each note. Spring 1996 is the course term.

Course homepage: people.eecs.berkeley.edu/~demmel/cs267/

Collected papers by James Joseph Sylvester, edited by Henry Frederick Baker, credited as H. F. Baker in the prefatory note dated April 1904. Cambridge University Press published Volume I in 1904. The collection date is distinct from the original dates of its individual papers. Internet Archive hosts the inspected digitized volume and OCR text.

Digitized volume: archive.org/details/collectedmathem01sylvrich

Inspected OCR: archive.org/stream/collectedmathem01sylvrich/collectedmathem01sylvrich_djvu.txt

Software reference manual issued and hosted by the National Institute of Standards and Technology. No personal whole-manual authorship or separate manual publication date is established. Individual entries distinguish webpage dates from implementation dates.

Official manual, Volume II: www.itl.nist.gov/div898/software/dataplot/refman2/homepage.htm

By David Gale and Lloyd Shapley, credited as D. Gale and L. S. Shapley. Published in The American Mathematical Monthly 69(1), 9–15 (January 1962). The 2018 publisher online date concerns digitization, not original publication. The University of Toronto hosts the inspected scan; hosting does not establish authorship.

Institutional scan: iqua.ece.toronto.edu/baochun/ece1771f/papers/Gale-Shapley-1962.pdf

Cited passages: §3, printed p. 11, deferred acceptance; §5, printed p. 14, Theorem 2 and rejection induction.

By Mark Newman, credited as M. E. J. Newman. Published in Contemporary Physics 46, 323–351 (2005). The inspected arXiv version 3 was revised 29 May 2006; version 1 was submitted 1 December 2004. These preprint dates are distinct from the journal publication year.

Preprint: arxiv.org/pdf/cond-mat/0412004

Cited passages: §II, pp. 4–5, equations (3)–(5), density and survival exponents and fitting caveat; §III.D, pp. 10–11, equations (24)–(28), fraction of a total held by an upper population fraction. Its density exponent is one greater than the positive survival exponent used for the Pareto distribution.

Graduate course issued by the Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, and hosted by MIT OpenCourseWare. Course instructor: Pablo A. Parrilo; this teaching role does not establish authorship of every lecture note. Spring 2006 is the course term, not an online publication date.

Official course: ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/

Online manual of the SciPy project, identified as version 1.18.0 by the inspected page title. No publication date is inferred from its copyright footer.