Georg Cantor's first significant work on set theory is often considered to be his 1874 article titled "Über eine Eigenschaft der reellen Zahlen" (translated as "On a Property of the Real Numbers"). In this paper, Cantor introduced the concept of sets and laid the groundwork for later developments in set theory, including his work on different types of infinities and cardinality.

Carleman's inequality is a mathematical result in the field of functional analysis and approximation theory. It provides a bound on the norms of a function based on the norms of its derivatives. Specifically, it is often used in the context of the spaces of functions with certain smoothness properties. One of the most common forms of Carleman's inequality is related to the Sobolev spaces and is used to show the equivalence of certain norms.

A piecewise linear function is a function composed of multiple linear segments. Each segment is defined by a linear equation over a specific interval in its domain. Essentially, the function "pieces together" several lines to create a graph that can take various forms depending on the specified intervals and the slopes of the lines.

The Poincaré–Miranda theorem is a result in topology that relates to the existence of continuous choices of functions under certain conditions. It is often used in the context of multiple variables and can be seen as a generalization of the intermediate value theorem for higher-dimensional spaces.

"Reverse Mathematics: Proofs from the Inside Out" is a book by Jonathan E. Goodman and Mark W. Johnson, published in 2018. It is an exploration of the field of reverse mathematics, which is a branch of mathematical logic concerned with classifying axioms based on the theorems that can be proved from them. Reverse mathematics typically investigates the connections between various mathematical theorems and the foundational systems necessary to prove them.

Charles Scott Sherrington (1857–1952) was a British neurophysiologist and a key figure in the field of neuroscience. He is best known for his discoveries related to the functioning of the nervous system and for his pioneering work on reflexes, which helped to lay the groundwork for our understanding of how the nervous system processes information.

Continued fractions are a way of expressing numbers through an iterative process of fractions, where a number is represented as a whole number plus a fraction, and that fraction can itself be expressed in a similar manner.

In mathematics, the term "commensurability" often refers to a relationship between two or more mathematical objects, indicating that they share a common measure or can be expressed in terms of one another in a way that involves a rational relationship. Here are a few contexts in which commensurability is often discussed: 1. **Geometry**: In the context of geometry, two line segments are said to be commensurable if their lengths can be expressed as a ratio of two integers.

The extended real number line is a concept in mathematics that extends the usual set of real numbers to include two additional elements: positive infinity (\(+\infty\)) and negative infinity (\(-\infty\)). This extension is useful because it allows for a more comprehensive way to handle limits, summations, integrals, and other mathematical constructs.

A number line is a straight horizontal or vertical line that represents numbers in a linear format. It is used to visualize numerical values and their relationships. Here are some key features and uses of a number line: 1. **Representation of Numbers**: The number line usually has evenly spaced intervals along its length, each representing a specific number. The midpoint is often labeled as zero (0), with positive numbers extending to the right and negative numbers extending to the left.

The Rational Zeta series, often denoted as \( \zeta(s) \) when discussing rational functions, is a generalization of the Riemann Zeta function, which traditionally applies to the natural numbers. The Rational Zeta function can be defined for rational numbers or more generally for other complex numbers.

Adam Sedgwick (1785–1873) was a prominent English geologist and a significant figure in the early development of geology as a scientific discipline. He is best known for his work in stratigraphy and for his contributions to the understanding of the geological time scale. Sedgwick was a professor at the University of Cambridge and played a key role in the establishment of a systematic approach to classifying rock layers and understanding Earth's history.

Richard Chenevix (1790-1830) was an Irish chemist known for his contributions to the field of chemistry, particularly in the study of chemical reactions and the properties of various elements. He conducted important research during a time when chemistry was rapidly developing as a science, and he was among the early figures to explore the nature of chemical substances systematically. Chenevix is notable for his investigations into the behaviors and characteristics of metals and other compounds.

Arthur Evans (1851–1941) was a British archaeologist best known for his work on the ancient Minoan civilization of Crete. He is most famous for his excavation of the Palace of Knossos, which he began in 1900. Evans's discoveries at Knossos, including elaborate frescoes, pottery, and architectural features, significantly advanced the understanding of Minoan culture.

Cyril Norman Hinshelwood (1897–1967) was a British physical chemist known for his significant contributions to the field of chemical kinetics and reaction mechanisms. He was awarded the Nobel Prize in Chemistry in 1956, along with Nikolay Semenov, for their work on the study of extremely fast reactions, particularly those that occur in gases. Hinshelwood's research helped to deepen the understanding of how chemical reactions proceed and the factors that influence reaction rates.

Henry Baker (naturalist) was an English naturalist known for his contributions to the study of natural history in the 18th century. He was born in 1698 and died in 1774. Baker is particularly noted for his work on the study of insects and his writings, which contributed to the understanding of entomology during his time. He was a member of various scientific societies and communicated his findings through publications that were significant in the field of natural history.

Jacques Charles François Sturm (1803–1855) was a notable French mathematician and physicist recognized for his contributions to various fields, including mathematics, celestial mechanics, and mathematical physics. He is particularly known for his work in the development of the Sturm-Liouville theory, an important area in the theory of differential equations. Sturm's work laid the groundwork for many concepts in analysis and applied mathematics, particularly in the context of eigenvalue problems.

John Scott Haldane (1860–1936) was a prominent Scottish physiologist, biochemist, and philosopher known for his significant contributions to the field of respiratory physiology and gas exchange. He is particularly recognized for his research on the effects of gases on human health, including the study of carbon dioxide and oxygen in the blood. One of his notable achievements was the development of the Haldane effect, which describes how the binding of oxygen to hemoglobin affects its affinity for carbon dioxide.

Joseph Dalton Hooker (1817–1911) was a prominent British botanist, explorer, and one of the founders of modern plant geography. He was a key figure in the study of plant taxonomy and biogeography, and he was the son of William Jackson Hooker, a notable botanist and director of the Royal Botanic Gardens, Kew.

Marcellin Berthelot (1827–1907) was a prominent French chemist and politician known for his significant contributions to the fields of organic chemistry and chemical thermodynamics. He is particularly recognized for his work on the synthesis of organic compounds and the study of thermochemical processes.