Recent Questions - History of Science and Mathematics Stack Exchange most recent 30 from hsm.stackexchange.com 2023-11-29T22:46:40Z https://hsm.stackexchange.com/feeds https://creativecommons.org/licenses/by-sa/4.0/rdf https://hsm.stackexchange.com/q/15962 8 Who are the youngest mathematicians that published an original research article in a peer-reviewed journal? User303131 https://hsm.stackexchange.com/users/19397 2023-11-28T19:55:11Z 2023-11-29T18:43:46Z <p>There is a lot of interesting information about young mathematicians, but I cannot find any information about the youngest mathematician that published an original research article in a peer-reviewed journal.</p> <p>Does anyone have this information? Thanks a lot.</p> https://hsm.stackexchange.com/q/15958 2 What is Cardano trying to say in this passage of his Ars Magna Arithmeticæ? Charles Bukowski https://hsm.stackexchange.com/users/19392 2023-11-27T19:59:43Z 2023-11-27T19:59:43Z <p>It is well known that Cardano considered the problem of &quot;dividing 10 into two parts the product of which is 40&quot; in his <em>Ars Magna</em>. This problems leads to the complex solutions <span class="math-container">$5+ \sqrt{-15}$</span> and <span class="math-container">$5- \sqrt{-15}$</span>, which Cardano would write as <span class="math-container">$5\tilde{p}R\tilde{m}15$</span> and <span class="math-container">$5\tilde{m}R\tilde{m}15$</span>, respectively.</p> <p>For this problem, Cardano supplemented his algebraic reasoning with a geometric construction of &quot;completing the square&quot; (see Richard Witmer's english translation of the <em>Ars Magna</em>, p. 219). With this construction, one has to imagine a negative area, that of substracting <span class="math-container">$40$</span> units of area to the square <span class="math-container">$AD$</span>:</p> <p><a href="https://i.stack.imgur.com/lIzw8.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/lIzw8.png" alt="" /></a></p> <p>However, Cardano seemingly thinks the problem with this construction isn't the negative area, but the fact that <span class="math-container">$AD$</span> is an area while <span class="math-container">$40$</span> is a length:</p> <blockquote> <p>Yet the nature of AD is not the same as that of 40 or of AB, since a surface is far from the nature of a number and from that of a line, though somewhat closer to the latter. This truly is sophisticated, since with it [i. e., with <span class="math-container">$\sqrt{-15}$</span>] one cannot carry out the operations one can in the case of a pure negative and other [numbers].</p> </blockquote> <p>Cardano also treats complex numbers in his other lesser-known book <em>Ars Magna Arithmeticæ</em>. He considers the problem of &quot;dividing <span class="math-container">$1$</span> in two parts whose product is <span class="math-container">$3$</span>&quot;. This is the same as solving the equation <span class="math-container">$x^2-x+3=0$</span>. The solutions are complex: <span class="math-container">$$x=\frac{1}{2}\pm \sqrt{\frac{1}{4}-3}=\frac{1}{2}\pm \sqrt{-\frac{11}{4}}.$$</span> Cardano writes this solutions as <span class="math-container">$\frac{1}{2}$</span>.<span class="math-container">$\tilde{p}$</span>.R.v.<span class="math-container">$\frac{1}{4}$</span>.m.3. and <span class="math-container">$\frac{1}{2}$</span>.<span class="math-container">$\tilde{m}$</span>.R.v.<span class="math-container">$\frac{1}{4}$</span>.m.3. (see image below). This is done on page 374, <a href="https://archive.org/details/imgmar3940alllMiscellaneaOpal/page/372/mode/2up" rel="nofollow noreferrer">volume IV of Cardano's <em>Opera omnia</em></a> (which contains the <em>Ars Magna Arithmeticæ</em>).</p> <p>My question is: <strong>what is Cardano trying to say in the following excerpt of <em>Ars Magna Arithmeticæ</em> (p. 374)?</strong> Does it have anything to do with his confusion about the different &quot;natures&quot; of <span class="math-container">$AD$</span> and <span class="math-container">$40$</span>? He adds the word &quot;quadrati&quot; to the solutions (<span class="math-container">$\frac{1}{2}$</span>.<span class="math-container">$\tilde{p}$</span>.R.v.<span class="math-container">$\frac{1}{4}$</span>quadrati.m.3.), but I don't see the reason for doing this. It seems as if he added a variable <span class="math-container">$y^2$</span> to the solutions, like <span class="math-container">$$x=\frac{1}{2}\pm \sqrt{\frac{1}{4}y^2-3}.$$</span></p> <p>I don't have a clue why Cardano does this. I have only found a paper of <a href="https://www.researchgate.net/publication/295682735_L%27Ars_magna_arithmeticae_nel_corpus_matematico_di_Cardano" rel="nofollow noreferrer">Veronica Gavagna</a> about the <em>Ars Magna Arithmeticæ</em>, but she doesn't talk much about this.</p> <p><a href="https://i.stack.imgur.com/41TjI.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/41TjI.png" alt="enter image description here" /></a></p> https://hsm.stackexchange.com/q/15948 6 How did the concept of pH originate and develop? Bhavya Jain https://hsm.stackexchange.com/users/19382 2023-11-24T19:04:41Z 2023-11-27T10:18:05Z <h2>Background &amp; My research</h2> <p>To begin I did some research to find a few articles on the history of pH namely <em>&quot;The Symbol for pH&quot;</em> - William B. Jensen, <em>&quot;One-Hundred Years of pH&quot;</em> - Rollie J. Myers (but couldn't access it) and &quot;<em>The origin and the meaning of the little p in pH&quot;</em> - Jens G Nørby (but couldn't access this also).</p> <p>So to start I did some reading on the wiki article for pH and found that S.P.L. Sørensen first introduced the term in his paper in 1909 in order to explain the acidity and basicity of the acids/bases on the &quot;Arrhenius concept&quot;. To explain to some extent the use of the logarithmic scale I found out that his experiments contained the use of electrodes with one being a calomel electrode and the other being a normal hydrogen electrode. Upon experimental results and an equation (which I think was the Nernst Equation) he found that the electric potential of the cell was proportional to −log[H+].</p> <p>To somehow explain the use of &quot;p in pH&quot; I found out S.P.L. Sørensen in his original paper used <span class="math-container">$P_H$</span> to represent this “hydrogen ion exponent”. Although as his work became more popular several more variations like Ph and pH were introduced with the latter getting officially adopted by the &quot;Journal of Biological Chemistry&quot; in 1910's and eventually gaining more recognition than the other variations. The origin of &quot;P&quot; in the original text is speculated to be related to puissance (French), potenz (German) or potens (Danish) which were the languages in which Sørensen published.</p> <p>This was all I could find from my research and still have some doubts like how the little p in &quot;pH&quot; evolved/generalized to be used as &quot;an operator&quot; like - <span class="math-container">$pK_a, pK_b, pK_w$</span> etc. which are further expressed below in the main question.</p> <hr /> <h2>Question:</h2> <p>I was recently introduced to the concept of pH in chemistry and its value being the following: <span class="math-container">$$\mathrm{pH} =-\log[\mathrm H^+]$$</span> This was followed up by the &quot;generalization of the concept of pH&quot; in <em>&quot;finding the p&quot;</em> of a base/acid (e.g., <span class="math-container">$pK_a, pK_b, pK_w$</span> etc.)</p> <p>Now I want to know how is it that the definition of pH evolved from being the &quot;hydrogen ion concentration&quot; to the &quot;activity of hydrogen ions&quot; as well some background on the experiments that he performed which led him to define these concepts the way they are? Also how is it that the concept became &quot;more generalized&quot; and evolved from being the negative common logarithm of the concentration of hydrogen ions to a more generalized &quot;operation&quot; applicable to things more than just the hydrogen ions?</p> <p>Any sort of articles and further reading links will be very much appreciated which could help me learn more about the &quot;history of pH&quot; and how it originated and evolved as a concept.</p> https://hsm.stackexchange.com/q/15946 3 What are some famous named groups of scientists? Mauricio https://hsm.stackexchange.com/users/6609 2023-11-24T10:24:54Z 2023-11-28T12:20:09Z <p>Scientists are often associated together and get famous group names. In physics, I know of</p> <ul> <li>Via Panisperna boys (Enrico Fermi and co. working in Rome in nuclear physics)</li> <li>Princeton string quartet (Gross, Harvey, Martinec, and Rohm, working in string theory)</li> <li>Oxford calculators (14th century philosophers working on early mechanics)</li> <li>The Martians (Hungarian scientists working in the Manhattan project like Szilard, von Neumann, Wigner and Teller, plus later other Hungarian mathematicians)</li> </ul> <p>Do you know of any other named famous groups? I could not find a key word on the internet to easily look up for more. Of course, I am excluding formal associations and societies intended to gather more members. Also excluding faculty teams unless they got a popular pseudonym or popular name (like Via Panisperna boys). This question is a [tag:big list] for science and math groups in general not only physics.</p> https://hsm.stackexchange.com/q/15944 0 Why is Power in an electric circuit equal to VI? [closed] potato https://hsm.stackexchange.com/users/19380 2023-11-24T05:48:29Z 2023-11-24T05:48:29Z <p>Where did this formula come from? Everyone I asked just told me to substitute values of in ohms law to derive this but no one told why is power equal to voltage * current. Part of the reason for this is because I never understood what is work done in a system and this caused everything from there to not make any sense. Can somebody explain it in layman terms so that my squishy potato brain can understand it?</p> https://hsm.stackexchange.com/q/15940 3 Dirac’s debt to Hamilton James Propp https://hsm.stackexchange.com/users/6587 2023-11-21T21:45:47Z 2023-11-24T01:54:03Z <p>According to Tobias Hurter’s popular exposition <em>Too Big for a Single Mind</em> (narrated in the present tense):</p> <blockquote> <p>Dirac makes use of an elegant mathematical tool developed by the Irish mathematician William Hamilton in the nineteenth century.</p> </blockquote> <p>This is from a passage discussing Dirac in Cambridge in the summer of 1925; see pages 127-128.</p> <p>What “elegant mathematical tool” is Hurter talking about? Was it quaternions? Unfortunately the notes at the back of the book do not say anything more about this episode.</p> https://hsm.stackexchange.com/q/15939 1 How did someone discover LCM? Steve https://hsm.stackexchange.com/users/19370 2023-11-21T10:15:16Z 2023-11-21T16:36:07Z <p>How did someone came up with an idea that if we do prime factorization of two numbers and then multiply all the prime factors but including common ones only once, we will get a number that is the least common multiple of these two numbers?</p> <p>And why do we take the repeating factors only once? I know this question might sound silly but I am a total beginner and math who don't know anything. Please help me understand this.</p> https://hsm.stackexchange.com/q/15936 6 Origin of exact and closed differential expressions Mikhail Katz https://hsm.stackexchange.com/users/604 2023-11-20T12:17:27Z 2023-11-20T21:08:32Z <p>In differential geometry and other fields, an expression involving differentials can be <em>closed</em> or <em>exact</em>. In <span class="math-container">$\mathbb R^2\setminus\{0\}$</span> for example, <span class="math-container">$dr$</span> is exact whereas <span class="math-container">$d\theta$</span> is closed but not exact. What is the origin of these terms? Note that I deliberately avoided the use of the term &quot;form&quot; because the use of <em>open</em> and <em>exact</em> in reference to expressions involving differentials may have preceded the theory of differential forms.</p> https://hsm.stackexchange.com/q/15932 0 Need a reference for Euler's velocity initial condition for the wave equation user45664 https://hsm.stackexchange.com/users/7211 2023-11-19T18:13:41Z 2023-11-20T03:44:02Z <p>In DOI: 10.4236/ahs.2020.94019 235 Advances in Historical Studies, p.234 D’Alembert and the Wave Equation: Its Disputes and Controversies, or <a href="https://www.scirp.org/pdf/ahs_2020112716312281.pdf" rel="nofollow noreferrer">https://www.scirp.org/pdf/ahs_2020112716312281.pdf</a> p.6 of 11</p> <p>A. R. E. Oliveira states</p> <blockquote> <p>Euler criticized d’Alembert’s work pointing out that the two arbitrary functions ϕ and ψ are determined by the initial conditions of the problem: ... which represent the initial form of the curve and the <strong>distributions of initial velocities</strong></p> </blockquote> <p>but he does not give a specific reference. He does list 5 Euler references--only 1 in English. My questions are: <em>Which is the source which contains this assertion by Euler and is there an English translation?</em></p> https://hsm.stackexchange.com/q/15931 0 In JJ Thomson's cathode ray experiment I need values for the electric field and magnetic field when net force on the cathode beam = 0 Saif https://hsm.stackexchange.com/users/19362 2023-11-19T16:25:43Z 2023-11-25T06:52:58Z <p><a href="https://i.stack.imgur.com/rBoI6.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/rBoI6.png" alt="JJ thomson's setup for the cathode ray tube" /></a></p> <p>I asked here as well <a href="https://chemistry.stackexchange.com/questions/177889/in-jj-thomsons-cathode-ray-experiment-why-is-effects-of-gravity-on-electron-not">https://chemistry.stackexchange.com/questions/177889/in-jj-thomsons-cathode-ray-experiment-why-is-effects-of-gravity-on-electron-not</a> <a href="https://physics.stackexchange.com/questions/789225/in-jj-thomsons-cathode-ray-experiment-why-is-effects-due-to-gravity-on-electron?noredirect=1#comment1773186_789225">https://physics.stackexchange.com/questions/789225/in-jj-thomsons-cathode-ray-experiment-why-is-effects-due-to-gravity-on-electron?noredirect=1#comment1773186_789225</a> <br><br> <span class="math-container">$${}$$</span> Explaining the setup:- <br> The experiment is described in the picture. Instead of the magnets in the picture imagine two circular coils on both the sides with current running through it, this creates a magnetic field perpendicular to the loop which can be predicted using biot savart law. Run the current in the loop in such a way so that the magnetic field is going into your screen, away from you. Without the magnets the beam is deflected up towards the positive plate proving the charge on the beam is -ve as +ve and -ve attract. q(v x B) = Magnetic force, so the magnetic force is acting downwards. q/m ratio can be determined as well by combining both the magnets and the charged plates and changing the electric and magnetic fields until there seems to be no deflection on the beam so net force acting on beam must be close to 0 and we solve taking all the forces, more details here <a href="https://scienceready.com.au/pages/thomsons-discovery-of-the-electron" rel="nofollow noreferrer">https://scienceready.com.au/pages/thomsons-discovery-of-the-electron</a> <br><br> I had a problem with how he finds the q/m ratio which is why the gravitational force on the electron is not considered? Only the magnetic force and electric force is considered. So i thought maybe gravitational force is negligible? I included gravitational force and got an expression which relates q/m to a constant value <span class="math-container">$$Magnetic force + Gravitational force = Electric force$$</span> <span class="math-container">$$qvB +m9.8 = qE$$</span> <span class="math-container">$$v = E/B - (9.8m)/(qB)$$</span> when we only have the magnetic force acting on the beam, it will always act perpendicular to the velocity due to the magnetic field going into the screen, so it will constantly change the direction of velocity but not its magnitude and the beam will move in a circle. <span class="math-container">$$centripetal force = magnetic force$$</span> <span class="math-container">$$mv^2/r = qvB$$</span> <span class="math-container">$$mv/r = qB$$</span> <span class="math-container">$$\frac{m(E/B - (9.8m)/(qB))}{r} = qB$$</span> <span class="math-container">$$\frac{mE}{Br} - \frac{9.8m^2}{Bqr} -qB = 0$$</span> this can be treated as a quadratic equation where the variable is m <br> so if we have an equation ax^2 + bx + c = 0 its roots are = <span class="math-container">$\frac{-b +- \sqrt{b^2 - 4ac}}{2a}$</span> <span class="math-container">$$m = \frac{ ( -E/Br +- \sqrt{ \frac{E^2}{B^2 r^2} - \frac{4 \cdot 9.8}{r}})Bqr }{2 \cdot 9.8}$$</span> <span class="math-container">$$\frac{m}{q} = \frac{ ( -E/Br + \sqrt{ \frac{E^2}{B^2 r^2} - \frac{4 \cdot 9.8}{r}})Br }{2 \cdot 9.8}$$</span></p> <p><br>but I dont know at what magnetic and electric fields so i need the values of electric field of the charged plates and magnetic field of the coils at which there is no net deflection and the radius of the circle the beam goes through when only the magnetic field acts on it, provide the source as well. Maybe there is some other way to understand why gravitational force is not included if so could you explain that way.</p> https://hsm.stackexchange.com/q/15929 3 Seeking Comprehensive References on the History of Scientific Notation Humberto José Bortolossi https://hsm.stackexchange.com/users/12383 2023-11-18T21:46:47Z 2023-11-23T10:27:36Z <p>I am on a quest to uncover the rich tapestry of history surrounding <a href="https://en.wikipedia.org/wiki/Scientific_notation" rel="nofollow noreferrer">scientific notation</a> as a way of expressing numbers. Specifically, I'm interested in scholarly books, peer-reviewed articles, and historical documents that delve into:</p> <ul> <li>The scientific context and historical era during which scientific notation was first conceptualized and utilized.</li> <li>The key scientists and mathematicians who played pivotal roles in the development and refinement of scientific notation.</li> <li>The reasons for the widespread adoption and enduring popularity of scientific notation in the scientific community. My research so far, including a review of the available Wikipedia content, has not yielded the depth of information I require. I would greatly appreciate any leads on academic or historical sources that provide a thorough analysis of these points.</li> </ul> https://hsm.stackexchange.com/q/15927 0 How did Schrödinger do quantum mechanics with wave functions? user19358 https://hsm.stackexchange.com/users/19358 2023-11-18T19:16:14Z 2023-11-19T20:11:37Z <p>On my way to learn about the very beginning of quantum mechanics and its different formulations, starting with Heisenberg infinite matrices and Schrödinger's wave functions, I can really not find till now a single reference in which it is explained how Heisenberg and Schrödinger were doing quantum mechanics i.e. determining probabilities about measurements for positions and general observables within their own formulation framework of, respectively, infinite matrices and wave functions (i.e. without talking about Heisenberg and Schrödinger's pictures inside a Hilbert space 𝐿2(ℝ) for example). I mean for example for Schrödinger, given a wave function 𝜓(𝑥) of a fixed system, say an electron, what was exactly his interpretation of 𝜓(𝑥)? (Before Born's interpretation came I mean) and how he did to do computations with it in order to predict the probabilities for the position, the momentum the energy and so on?</p> https://hsm.stackexchange.com/q/15926 0 Euclid's use of antenaresis and Heath's commentary zeynel https://hsm.stackexchange.com/users/16575 2023-11-18T13:55:11Z 2023-11-18T21:29:40Z <p>In Book 7, Prop. 1. Euclid uses repeated subtraction to prove that two numbers are relatively prime. As explained <a href="http://aleph0.clarku.edu/%7Edjoyce/java/elements/bookVII/propVII1.html" rel="nofollow noreferrer">here</a> the Greek word for repeated subtraction is &quot;antenaresis&quot;. There isn't much information about Antenaresis online. I found <a href="https://youtu.be/Egm6AydHguI?si=BwtFHNmVaamFyLmd" rel="nofollow noreferrer">this video</a> (and <a href="https://drive.google.com/file/d/1Io8GnIm-qegGZQwJlbNtb35O8FeLHzEW/view" rel="nofollow noreferrer">the text mentioned in the video</a>).</p> <p>I have two questions. (1) Do you have more information on antenaresis and (2) What does Heath do in his commentary to the mentioned proposition?</p> <p>Heath's commentary can be found <a href="https://drive.google.com/file/d/1Io8GnIm-qegGZQwJlbNtb35O8FeLHzEW/view" rel="nofollow noreferrer">on page 234 of this link</a>:</p> <p>This is the screen shot:</p> <p><a href="https://i.stack.imgur.com/sm8kb.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/sm8kb.png" alt="enter image description here" /></a></p> <p>I understand what Euclid does but I don't understand Heath's notation.</p> <p>For instance, how does Heath's notation work for a=85 and b=54?</p> https://hsm.stackexchange.com/q/15925 -2 Did Emmy Noether know quaternion? [closed] pokssin https://hsm.stackexchange.com/users/18258 2023-11-18T11:05:29Z 2023-11-20T10:10:12Z <p>I am looking at the quaternion, and it is said that this concept was invented by a man named 'William Rowan Hamilton' while taking a walk. And I looked the properties of this concept, it is quite similar to a ring.</p> <p>So, the question that suddenly occurred to me was whether, from a historical perspective, it could be said that Emmy Noether was inspired by this quaternion and was able to create a new algebraic system called a ring.</p> https://hsm.stackexchange.com/q/15924 2 How did Emmy Noether become interested in abstract algebra? pokssin https://hsm.stackexchange.com/users/18258 2023-11-18T06:56:03Z 2023-11-21T02:52:07Z <p>Emmy Noether was initially interested in invariant theory. But how did she become interested in abstract algebra? And why did she become particularly interested in ring and ideal theory?</p> https://hsm.stackexchange.com/q/15922 0 First recorded use of oxymel to treat wounds Adrien Hingert https://hsm.stackexchange.com/users/18091 2023-11-17T13:47:01Z 2023-11-17T13:47:01Z <p>What is the first recorded use of a mixure of honey and vinegar to treat wounds?<br /> <a href="https://en.wikipedia.org/wiki/Oxymel" rel="nofollow noreferrer">Cato the Elder</a> indicates it was used during Roman times, but it's unclear if this was as a tonic or if it was used to clean wounds</p> https://hsm.stackexchange.com/q/15921 4 Is there existing footage of Stanislaw Mazur giving Per Enflo a live goose for solving the approximation problem? James Hanson https://hsm.stackexchange.com/users/19348 2023-11-17T05:27:48Z 2023-11-24T16:45:21Z <p>There is a famous incident in the history of mathematics involving the mathematician Per Enflo being awarded a live goose by Stanislaw Mazur for solving problem 153 in the Scottish Book by constructing the first known example of a Banach space failing to have the approximation property. The incident is somewhat amusing because the award was offered in 1936 but not claimed until 1972, at which point the prize of a live goose was decidedly anachronistic.</p> <p>There is a commonly used picture of the event, which seems to be more or less the only one online:</p> <p><a href="https://i.stack.imgur.com/E5Mge.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/E5Mge.jpg" alt="Stanislaw Mazur awarding a live goose to Per Enflo." /></a></p> <p>I was told recently that this was actually broadcast on Polish television. <a href="https://perenflo.com/math" rel="nofollow noreferrer">This statement is also corroborated by Enflo's personal website.</a> Was a recording of this made, and if so does it still exist today?</p> https://hsm.stackexchange.com/q/15920 1 Whence Whitehead's essence? Frode Alfson Bjørdal https://hsm.stackexchange.com/users/8094 2023-11-17T02:33:31Z 2023-11-17T13:02:58Z <p>In the article <em>Quine’s New Foundations</em> of The Stanford Encyclopedia of Philosophy (Summer 2019 Edition), Thomas Forster writes:</p> <blockquote> <p>In  Hailperin gave the first of a number of finite axiomatisations of NF now known. Many of them exploit the function <span class="math-container">$x\mapsto \{y|x\in y\}$</span> which is injective and total and is an <span class="math-container">$\in$</span>-isomorphism. This function was known to Whitehead, who suggested to Quine that <span class="math-container">$\{y|x\in y\}$</span> should be called the “essence” of <span class="math-container">$x$</span> (a terminology clearly suggested by a view of sets as properties-in-extension).</p> </blockquote> <p>How did Alfred North Whitehead communicate the essential point of view to Quine?</p> https://hsm.stackexchange.com/q/15918 0 Asymptotically similar functions with opposite parity, were they considered, are they useful? Quasi-parabolas [closed] Anixx https://hsm.stackexchange.com/users/2203 2023-11-16T03:43:08Z 2023-11-16T03:43:08Z <p>So, can we transform an even function into an odd function and vice versa? Let's consider this method:</p> <p>Transformation even-&gt;odd:</p> <p>Suppose <span class="math-container">$f_{even}(x)$</span> is a function which satisfies the following condition:</p> <p><span class="math-container">$$f_{even}(x)=\sum_{k=0}^\infty G(2k)\frac{x^{2k}}{(2k)!}$$</span></p> <p>Where the coefficient function <span class="math-container">$G^*(x)=G(2s)$</span> is equal to its Newton series expansion:</p> <p><span class="math-container">$$G^*(s) = \sum_{k=0}^\infty \binom{s}k \Delta_s^k G^* \left (0\right)$$</span></p> <p>What we are doing here, is Newtonian interpolation of consecutive derivatives over even points so to get the values at odd points.</p> <p>The function <span class="math-container">$f_{even}(x)$</span> is evidently even. Now the operator</p> <p><span class="math-container">$$\operatorname{oddify} f_{even}(x)=\sum_{k=0}^\infty G(2k+1)\frac{x^{2k+1}}{(2k+1)!}$$</span></p> <p>transforms an even function to an odd counterpart. The operator is linear.</p> <p>The opposite process is similar, for odd function:</p> <p><span class="math-container">$$f_{odd}(x)=\sum_{k=0}^\infty G(2k+1)\frac{x^{2k+1}}{(2k+1)!}$$</span></p> <p>and <span class="math-container">$G^*(s)=G(2s+1)$</span>, satisfying the same Newton series condition,</p> <p>the following operator gives an even counterpart:</p> <p><span class="math-container">$$\operatorname{evenize} f_{odd}(x)=\sum_{k=0}^\infty G(2k)\frac{x^{2k}}{(2k)!}$$</span></p> <p>Examples.</p> <hr /> <p><span class="math-container">$f_{even}(x)=\cosh x$</span>;</p> <p><span class="math-container">$f_{odd}(x)=\sinh x$</span>;</p> <p><span class="math-container">$G(s)=1$</span></p> <p><a href="https://i.stack.imgur.com/Pc8hz.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Pc8hz.png" alt="enter image description here" /></a></p> <hr /> <p><span class="math-container">$f_{even}(x)=x \coth \left(\frac{x}{2}\right)$</span></p> <p><span class="math-container">$f_{odd}(x)=x$</span></p> <p><span class="math-container">$G(s)=-2s\zeta(1-s,1)=2 B_s(1)$</span></p> <p><a href="https://i.stack.imgur.com/ZoWDe.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/ZoWDe.png" alt="enter image description here" /></a></p> <hr /> <p><span class="math-container">$f_{even}(x)=\csc ^2(x)-\frac{1}{x^2}$</span></p> <p><span class="math-container">$f_{odd}(x)=\frac{\psi ^{(1)}\left(1-\frac{x}{\pi }\right)}{\pi ^2}-\frac{\psi ^{(1)}\left(\frac{x}{\pi }+1\right)}{\pi ^2}$</span></p> <p><span class="math-container">$G(s)=\frac{2 (-1)^s \psi ^{(s+1)}(1)}{ \pi ^{s+2}}$</span></p> <p><a href="https://i.stack.imgur.com/BBAZB.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/BBAZB.png" alt="enter image description here" /></a></p> <hr /> <p>What one can notice in these examples is that one counterpart function can be much more complicated than the other, even one can be elementary while the other is not.</p> <p>Regarding the second example. Obviously, the method cannot work into the direction from <span class="math-container">$f(x)=x$</span>. First thought I had when thinking about an even counterpart of this function was &quot;what function can have non-zero even values while all dd values except at 1 equal to zero?@, and, oops, the obvious solution is Bernoulli numbers (in this case, <span class="math-container">$B_n(1)$</span> so to be equal to Newton's expansion).</p> <hr /> <p>Now, can we find the counterparts to polynomials? For some, yes! We just have to integrate the solution for <span class="math-container">$f(x)=x$</span>. We also divide the result by 2 so to make a better plot.</p> <p><span class="math-container">$f_{even}=\frac{x^2}{4}+\frac{\pi^2}6$</span></p> <p><span class="math-container">$f_{odd}=\text{Li}_2\left(e^x\right)+ x \log \left(1-e^x\right)-\left(\frac{x^2}{4}+\frac{\pi ^2}{6}\right)$</span></p> <p><span class="math-container">$G(s)=B_{s-1}(1)$</span></p> <p>(here, Bernoulli polynomials of negative order should be understood as their generalization via Hurwitz Zeta)</p> <p><a href="https://i.stack.imgur.com/r2n5i.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/r2n5i.png" alt="enter image description here" /></a></p> <p><span class="math-container">$f_{even}=2 \text{Li}_3\left(e^x\right)-x \text{Li}_2\left(e^x\right)-\left(\frac{x^3}{12}+\frac{\pi ^2 x}{6}\right)$</span></p> <p><span class="math-container">$f_{odd}=\frac{x^3}{12}+\frac{\pi ^2 x}{6}$</span></p> <p><span class="math-container">$G(s)=B_{s-2}(1)$</span></p> <p><a href="https://i.stack.imgur.com/gckT9.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/gckT9.png" alt="enter image description here" /></a></p> <p>And so on. The both functions are infinitely differentiable, so this gives us a family of functions, asymptotically equal to the polynomials, but with the opposite parity.</p> <p>The pairs of &quot;polynomials&quot; can be constructed this way:</p> <p><span class="math-container">$$f_n(x)=(n-1)\operatorname{Li}_n\left(e^x\right)-x\operatorname{Li}_{n-1}\left(e^x\right)$$</span></p> <p><span class="math-container">$$feven_n(x)=\frac{f(x)+f(-x)}2$$</span></p> <p><span class="math-container">$$fodd_n(x)=\frac{f(x)-f(-x)}2$$</span></p> <p>At even <span class="math-container">$n$</span> the even part is polynomial, the odd part is &quot;pseudo-polynomial&quot;, at odd <span class="math-container">$n$</span> the even part is pseudo-polynomial, the odd part is polynomial.</p> <p>Via differentiation we can also construct an odd counterpart of constant function:</p> <p><span class="math-container">$f_{even}=1$</span></p> <p><span class="math-container">$f_{odd}=\frac{\sinh (x)-x}{\cosh (x)-1}$</span></p> <p><span class="math-container">$G(s)=2 B_{s+1}(1)$</span></p> <p><a href="https://i.stack.imgur.com/VIlfz.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/VIlfz.png" alt="enter image description here" /></a></p> <hr /> <p>That said, I wonder whether anyone ever considered this family of functions that asymptotically behave as polynomial, but have opposite parity?</p> <p>Were, say, even counterparts of cubic parabola ever used in engineering? Do they describe motion of any bodies in mechanics?</p> https://hsm.stackexchange.com/q/15915 3 Who postulated the first Lagrangian for electrodynamics? Mauricio https://hsm.stackexchange.com/users/6609 2023-11-15T11:14:13Z 2023-11-17T18:03:05Z <p>I am trying to find who first translated Maxwell's equations and Lorentz's force into the Lagrangian formalism. It seems a very straightforward thing to do if you know enough of electromagnetism and Lagrangians so maybe nobody cared to give too much credit to the scientist who first published it.</p> <p>Maybe there were some attemps to do it even before Maxwell's equation?</p> <p>Who was the first to formulate the Lagrangian of electromagnetism?</p> https://hsm.stackexchange.com/q/15913 11 What was the motivation for the choice of the subset symbol? Paul Tanenbaum https://hsm.stackexchange.com/users/10975 2023-11-15T01:35:06Z 2023-11-15T14:07:22Z <p>I gather that the symbols <span class="math-container">$\subset$</span> and <span class="math-container">$\supset$</span> <a href="https://www.math.ucdavis.edu/%7Eanne/WQ2007/mat67-Common_Math_Symbols.pdf" rel="noreferrer">were introduced</a> by Ernst Schröder in his 1890 <em>Vorlesungen über die Algebra der Logik</em>. This account also appears—attributed to good old Cajori—in <a href="https://math.stackexchange.com/questions/1253991/bourbaki-and-the-symbol-for-set-inclusion">an answer on the Mathematics SE site</a> along with a quotation from Peano’s 1889 <em>Arithmetices Principia Nova Methodo Exposita</em> explaining use of a left-right-reflected upper-case C to mean contained in (or, as he put it in the Latin original, <em>continetur</em>).</p> <p>It seems safe to hypothesize that Peano chose the reflected C <strong>because</strong> C is the first letter of <em>continetur</em>. Indeed, I recall being taught in grade school in the 1960s that such was the motivation for the symbol. But Nathan Moore, though a fine fifth-grade teacher, cannot be presumed to have been an authority on either the history of mathematical notation or the influences of Latin—through Norman French—on the lexicon of Modern English. And anyway, maybe my memory has corrupted itself over the past half century and all he really said was that the resemblance between <span class="math-container">$\subset$</span> and “C for <em><strong>c</strong>ontained in</em>” can serve as a handy mnemonic. But back to Peano. Even though the hypothesis</p> <blockquote> <p>Peano’s choice stemmed from <em>continetur</em>’s starting with C</p> </blockquote> <p>does seem reasonable, that apparent reasonableness does not constitute sufficient grounds for asserting that the hypothesis is veridical.</p> <p>Apparently relevant too is the fact that in the same work, Peano also uses <span class="math-container">$\cap$</span> and <span class="math-container">$\cup$</span> as we do today, but defines them as <em>et</em> and <em>vel</em>, respectively (in English, <em>and</em> and <em>or</em>). Although I relied in grade school on the anglophone-obvious mnemonic “<span class="math-container">$\cup$</span> is for <em><strong>u</strong>nion</em>,” the Peano-reasonable hypothesis (analogous to “mirrored C is for <em><strong>c</strong>ontinetur</em>”) is “<span class="math-container">$\cup$</span> is for <em><strong>v</strong>el</em>.”</p> <p>But here’s a major wrinkle that sings out for anyone working in order theory. There are enchanting parallels between <span class="math-container">$\subset$</span> and <span class="math-container">$&lt;$</span>. Consider:</p> <ul> <li>We denote the reflexive closure of <span class="math-container">$\subset$</span> by <span class="math-container">$\subseteq$</span>. Likewise, we denote the reflexive closure of <span class="math-container">$&lt;$</span> by <span class="math-container">$\leq$</span>.</li> <li><span class="math-container">$\leq$</span> is the canonical, the prototypical partial order on <span class="math-container">$\bf R$</span>. Likewise, the canonical partial order on any power set <span class="math-container">$2^S$</span> is <span class="math-container">$\subseteq$</span>.</li> <li>We denote the order duals of <span class="math-container">$&lt;$</span> and <span class="math-container">$\leq$</span> by the left-right-reflected symbols <span class="math-container">$&gt;$</span> and <span class="math-container">$\geq$</span>, respectively. Likewise, the duals of <span class="math-container">$\subset$</span> and <span class="math-container">$\subseteq$</span> we denote by the reflections <span class="math-container">$\supset$</span> and <span class="math-container">$\supseteq$</span>, respectively.</li> <li>Of the symbols <span class="math-container">$&lt;$</span> and <span class="math-container">$&gt;$</span>, we use the convex-left one for the relation of increasing magnitude: <span class="math-container">$(x,y)\in &lt;$</span> means that that <span class="math-container">$x$</span> is the number with the lesser magnitude. Likewise, we use the convex-left <span class="math-container">$\subset$</span> for the relation of increasing cardinality: <span class="math-container">$(X,Y)\in\subset$</span> means that <span class="math-container">$X$</span> is the set with the lesser cardinality. This is consistent with the way that the notational similarity between <span class="math-container">$0$</span> and <span class="math-container">$\emptyset$</span> reflects the mathematical similarity between the objects that those two symbols represent.</li> </ul> <p>I note in passing that because of all these order-theoretic parallels, I am among the authors who use <span class="math-container">$\subset$</span> only to mean the reflexive reduction of <span class="math-container">$\subseteq$</span>, which is to say, <span class="math-container">$\subsetneq$</span>. Of course, that usage of <span class="math-container">$\subset$</span> is not the more common, so I always point it out explicitly. And sometimes I simply use <span class="math-container">$\subsetneq$</span> instead.</p> <p>But more importantly, it seems to me that the parallels are just too good to have happened entirely by accident. Indeed, in his <a href="https://math.hawaii.edu/%7Etom/history/set.html" rel="noreferrer"><em>Earliest Uses of Symbols of Set Theory and Logic</em></a>, Tom Craven states that before Schröder introduced <span class="math-container">$\subset$</span> et al., “the symbols <span class="math-container">$&lt;$</span> and <span class="math-container">$&gt;$</span> had been used.&quot; Craven may have found this in Cajori, to which I don’t have access.</p> <p>So, was Schröder influenced by Peano’s reflected C for <em>continetur</em>? And more generally, what can we say for certain about all the considerations that did lead Schröder to his choice… and, perhaps to different degrees, led to his notation’s now universal acceptance?</p> https://hsm.stackexchange.com/q/15911 1 Was "potency set" used for power set? Frode Alfson Bjørdal https://hsm.stackexchange.com/users/8094 2023-11-14T16:37:37Z 2023-11-16T02:02:14Z <p>Cross posted at <a href="https://mathoverflow.net/q/457909/37385">Math Overflow</a></p> <p>For historical reasons, the English term &quot;power set&quot; in set theory is a translation of the German &quot;Potenzmenge&quot;, which is still in use in German mathematical literature.</p> <p>Did some English language mathematics use the term &quot;potency set&quot; for power set, and if so, what are good references.</p> https://hsm.stackexchange.com/q/15897 23 What scientists and mathematicians were afraid to publish their findings? Max Muller https://hsm.stackexchange.com/users/4296 2023-11-12T11:49:03Z 2023-11-16T07:35:25Z <p><strong>Background</strong></p> <p>I am interested in scientists and mathematicians that were afraid to publish their findings during their lifetime, and to what degree such fears hinder scientific progress.</p> <p>So far, I've identified three famous scientists that were afraid to publish their important findings:</p> <ol> <li>Nicolas Copernicus. He delayed the publication of his work on a heliocentric model of the solar system for fear of either astronomical objections, or objections based on religious grounds.</li> <li>Carl Friedrich Gauss. Though fear might not always have been the main reason for him to delay the publication of his work -- he often waited quite some time before he published anything. He was a perfectionist, so perhaps one might describe it as a fear for not publishing something that was up to his high standards.</li> <li>Dan Shechtman. He was afraid of <strike>publishing his findings on quasicrystals</strike> publishing alone, because he needed the expertise of his coauthors to explain his observations regarding quasicrystals. Moreover, he faced strong opposition from one of the most celebrated scientists of his time: Linus Pauling.</li> </ol> <p>Some also believe Charles Darwin was afraid to publish his theory of evolution and avoided doing so for 20 years, but this has been <a href="https://www.cam.ac.uk/research/news/darwins-delay-the-stuff-of-myth#:~:text=theory%20of%20evolution.-,New%20Cambridge%20research%20shows%20Darwin%20had%20no%20fears%20about%20publishing,by%20a%20Cambridge%20University%20academic." rel="noreferrer">refuted</a>.</p> <p>I am also aware of cases of scientists and mathematicians that were scorned for their ideas (Ludwig Boltzmann, Ignaz Semmelweis, Georg Cantor), or ignored (Gregor Mendel, George Zweig). Although these are interesting (albeit tragic) cases as well, for the purposes of this particular question I would like to restrict the examples to those that were afraid of publishing -- for whatever reason.</p> <p><strong>Question</strong></p> <blockquote> <p>What other well-known scientists were afraid to publish their findings, and why?</p> </blockquote> https://hsm.stackexchange.com/q/15852 4 How did Scott and Amundsen detect the South Pole? Ritesh Singh https://hsm.stackexchange.com/users/10548 2023-10-25T11:38:23Z 2023-11-23T09:59:47Z <p>How did Scott and Amundsen detect the direction to the South Pole during their expedition? How did they determine the exact South Pole on reaching there?</p> <p><a href="https://en.wikipedia.org/wiki/Comparison_of_the_Amundsen_and_Scott_expeditions" rel="nofollow noreferrer">Comparison of the Amundsen and Scott expeditions</a> doesn't give this information.</p> https://hsm.stackexchange.com/q/15839 2 Why aren't Nobel nomination archives updated more often? Mauricio https://hsm.stackexchange.com/users/6609 2023-10-20T21:32:52Z 2023-11-15T12:39:52Z <p>I am not sure if this question is on topic here but I will give it a try. According to the Nobel Nomination archive <a href="https://www.nobelprize.org/nomination/archive/" rel="nofollow noreferrer">official website</a>, nomination data cannot be &quot;revealed until 50 years later&quot; <em>at least</em>. Peace and Literature Prizes are updated up to 1971. Physics Prize nomination archives are revealed up to 1970. Why don't we have information on 1971, 1972 and 1973? Medicine is even worse because it is stuck at 1953 for no clear reason.</p> <p>Per <a href="https://www.nobelprize.org/frequently-asked-questions/#h-how-nobel-prize-laureates-are-chosen" rel="nofollow noreferrer">Nobel frequently-asked questions</a>:</p> <blockquote> <p>For prizes older than 50 years old, you can browse the nomination archive. Note that in some cases the archives are sealed as long as people mentioned are still alive.</p> </blockquote> <p>At first I though that the nominees should be deceased in order for the information to be released but people like Jeffrey Goldstone (nominated in 1968) are still alive. Maybe this condition is about the nominators?</p> <p>What restricts a new year of nominations to be revealed each year? Do we know how often the archives get updated?</p> https://hsm.stackexchange.com/q/15810 13 Who coined the term "signal-to-noise ratio" and when did statisticians start using the term "noise" to describe randomness? vy32 https://hsm.stackexchange.com/users/6112 2023-10-08T15:48:52Z 2023-11-16T03:17:16Z <p>I'm writing about the history of the concept of noise and am having trouble tracking down references from when the term &quot;noise&quot; started being associated with statistical noise such as Gaussian Noise and not just things like people talking in the background and RF noise that prevents the interception of radio signals.</p> <p>An <a href="https://doi.org/10.1126/science.os-1.25.304.c" rel="nofollow noreferrer">article in Science</a> describing a new telephone device invented by Tuft’s College professor A. E. Dolbear noted that the words sounded “clear without the sputtering and confused noises” of Alexander Bell’s system. By the 1920s, <a href="https://doi.org/10.1126/science.os-2.53.310" rel="nofollow noreferrer">engineers at AT&amp;T</a> are measuring and comparing the noise between trans-Atlantic radio links and cables.</p> <p>But I can't find references that go from these early uses of the term &quot;noise&quot; to &quot;Gaussian Noise.&quot; The earliest reference I've found so far is <a href="https://apps.dtic.mil/sti/tr/pdf/AD0035947.pdf" rel="nofollow noreferrer">Technical Report 189</a> from the Cruft Laboratory, June 1, 1954.</p> https://hsm.stackexchange.com/q/14928 5 Early helium spectrum measurements and their challenge for Bohr's quantum mechanics David Bailey https://hsm.stackexchange.com/users/6504 2022-11-13T16:19:40Z 2023-11-25T13:31:38Z <p>My understanding is that explaining ortho- and para- helium spectral lines was a key motivation for Heisenberg's new quantum theory. For example, Birthwistle's 1928 <a href="https://books.google.ca/books?id=N-c3AAAAIAAJ&amp;pg=PA215&amp;lpg=PA215&amp;dq=%22HEISENBERG%27S%20RESONANCE%20THEORY%20OF%20THE%20ORTHO%20AND%20PARA%20HELIUM%20SPECTRA%22&amp;source=bl&amp;ots=m4mBNidvft&amp;sig=ACfU3U3C9P0TJmbMiTNN7snjAZIaPU-AQg&amp;hl=en&amp;sa=X&amp;ved=2ahUKEwij0IKyvKv7AhVojIkEHZHvCmcQ6AF6BAgHEAM#v=onepage&amp;q=%22HEISENBERG%27S%20RESONANCE%20THEORY%20OF%20THE%20ORTHO%20AND%20PARA%20HELIUM%20SPECTRA%22&amp;f=false" rel="nofollow noreferrer">&quot;The New Quantum Mechanics&quot;, Chapter XXVI</a> states:</p> <blockquote> <p>It is well known that the spectral terms of helium can be divided into two sets such that no term of the one will combine with a term of the other to produce a spectral line. … One set by its transitions gives the 'para helium' lines … the other set gives the 'ortho helium' lines …</p> </blockquote> <p>Since it is &quot;well known&quot;, Birthwistle give no sources. He further goes on to say</p> <blockquote> <p>The obvious failure of the classical mechanics and the correspondence principle to solve the problem of a nucleus with two outer electrons was one of the factors which compelled Heisenberg to seek for a new quantum mechanics …</p> </blockquote> <p>When and how was the helium spectrum measured sufficiently to establish the existence of two sets of line and the challenge for Bohr's quantum mechanics recognized? Are there any good articles or books that discuss this?</p> https://hsm.stackexchange.com/q/14667 9 History of greater-than symbol used in reverse? Joseph O'Rourke https://hsm.stackexchange.com/users/606 2022-08-07T20:01:26Z 2023-11-15T12:05:06Z <p>I was surprised to find that Oliver Byrne's 1847 marvelous <em>The Elements of Euclid</em> (color version)<sup>1</sup> uses <span class="math-container">$\sqsubset$</span> to mean &quot;greater than&quot; and <span class="math-container">$\sqsupset$</span> to mean &quot;less than,&quot; in contrast our current <span class="math-container">$&gt;$</span> and <span class="math-container">$&lt;$</span> (p. xxvii). This is especially puzzling because:</p> <blockquote> <p>&quot;The symbols &lt; and &gt; first appear in Artis Analyticae Praxis ad Aequationes Algebraicas Resolvendas (The Analytical Arts Applied to Solving Algebraic Equations) by Thomas Harriot (1560-1621).&quot; <a href="https://mathshistory.st-andrews.ac.uk/Miller/mathsym/relation/" rel="noreferrer">link</a></p> </blockquote> <p>Any speculation on Byrne's motivation? To me it seems so non-intuitive that <span class="math-container">$\sqsubset$</span> should mean <span class="math-container">$&gt;$</span> that I am surprised it was used in a book that tries to make the math easier to grasp. Perhaps <span class="math-container">$\sqsubset$</span> was in common use in parallel with <span class="math-container">$&gt;$</span>?</p> <hr /> <p>   <a href="https://i.stack.imgur.com/3GQdN.jpg" rel="noreferrer"><img src="https://i.stack.imgur.com/3GQdN.jpg" alt="Cover" /></a></p> <hr /> <p><sup>1</sup> Byrne, Oliver. <em>The first six books of the Elements of Euclid</em>: in which coloured diagrams and symbols are used instead of letters for the greater ease of learners. William Pickering, 1847.</p> https://hsm.stackexchange.com/q/11344 3 Who wrote "projective geometry" for the first time? ANACLETO 77 https://hsm.stackexchange.com/users/10985 2020-01-17T11:05:57Z 2023-11-28T17:50:22Z <p>I guess Desargues did not use that term. Anyone could help me know where did it appear for the first time? Thanks</p> https://hsm.stackexchange.com/q/8265 10 How did Eratosthenes determine that Alexandria and Syene were on the same meridian? Chaim https://hsm.stackexchange.com/users/9545 2019-01-31T22:45:11Z 2023-11-25T03:44:54Z <p>As discussed <a href="https://hsm.stackexchange.com/questions/6295/how-did-eratosthenes-knew-the-exact-time-of-the-day">over here</a>, Eratosthenes measured the earth’s circumference by comparing shadows cast at apparent noon at two locations separated by a known distance.</p> <p>Although accounts of the event (like the one cited above, and <a href="http://www.classichistory.net/archives/eratosthenes-circumference-of-earth" rel="noreferrer">this one</a>, and <a href="https://www.nationalgeographic.org/thisday/jun19/eratosthenes-measures-circumference-earth/" rel="noreferrer">this one</a>) do not emphasize the issue, it seems that Eratosthenes’ method depended on an assumption that the two measurements were on the same longitude. (When I think through the claim, I don't understand how any inference would be possible if we did not make such an assumption.)</p> <p>How was this known?</p>