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Annotated bibliography of 18th, 19th, and early 20th century works involving Lambert series. A tour of 19th and early 20th century analytic number theory., Comment: 42 pages

There is not much that can be said for all $x$ and for all $n$ about the sum \[ \sum_{k=1}^n \frac{1}{|\sin k\pi x|}. \] However, for this and similar sums, series, and products, we can establish results for almost all $x$ using the tools of continued fractions. We present in detail the appearance of these sums in the singular series for the circle method. One particular interest of the paper is the detailed proof of a striking result of Hardy and Littlewood, whose compact proof, which delicately uses analytic continuation, has not been written freshly anywhere since its original publication. This story includes various parts of late 19th century and early 20th century mathematics., Comment: 83 pages; thanks for correspondence with Christoph Aistleitner

In this paper we prove asymptotic formulas for the $L^p$ norms of $P_n(\theta)=\prod_{k=1}^n (1-e^{ik\theta})$ and $Q_n(\theta)=\prod_{k=1}^n (1+e^{ik\theta})$. These products can be expressed using $\prod_{k=1}^n \sin\Big(\frac{k\theta}{2}\Big)$ and $\prod_{k=1}^n \cos\Big(\frac{k\theta}{2}\Big)$ respectively. We prove an estimate for $P_n$ at a point near where its maximum occurs. Finally, we give an asymptotic formula for the maximum of the Fourier coefficients of $Q_n$., Comment: 18 pages

This is a translation of Euler's 1773 "Variae observationes circa angulos in progressione geometrica progredientes", E561 in the Enestr{\"o}m index. I translated this paper as a result of my study of Euler's work on the infinite product $\prod_{k=1}^\infty (1-z^k)$. If one instead considers the finite product $\prod_{k=1}^n (1-z^k)$, one can study its behavior on the unit circle. The absolute value of $\prod_{k=1}^n (1-e^{ik\theta})$ is $2^n |\prod_{k=1}^n \sin k\theta/2|$. My interest in the product $\prod_{k=1}^n \sin k\theta/2$ has inspired me to become acquainted with Euler's papers on trigonometric identities, in particular E447, E561, and E562. E561 says nothing about the product $\prod_{k=1}^n \sin k\theta/2$, but it has identities which I had not seen before. The identities have a form similar to Vi\`ete's infinite product $\prod_{k=1}^\infty \cos \theta/2^k=\frac{\sin\theta}{\theta}$., Comment: 9 pages, E561

Translation (by J.B.) from the original Latin of Euler's "De infinities infinitis gradibus tam infinite magnorum quam infinite parvorum" (1780). E507 in the Enestr\"om index. Euler discusses orders of infinity in this paper. In other words this paper is about how different functions approach infinity or 0 at different rates. I was not certain about what Euler means by "infinities infiniti". Probably he means that $x,x^2,x^3$, etc. are infinitely many orders of infinity, and $\log x,(\log x)^2, (\log x)^3$, etc. are infinitely many orders of infinity, so the combinations of them are an infinity of infinities of orders of infinity. In fact Euler mentions other orders of infinity in this paper. It would be worthwhile to study this paper more to figure out exactly what Euler means here. Another translation of the title is "On the infinitely infinite orders of the infinitely large and infinitely small". Here's another place Euler uses the phrase "infinities infiniti". The phrase "infinities infiniti" from the title is used by Euler also in section 21 of E302, "De motu vibratio tympanorum". Truesdell translates this phrase on p. 333 of "The rational mechanics of flexible or elastic bodies" as "infinity of infinities". I'd like to thank Martin Mattmueller for clearing up some questions., Comment: 13 pages; E507 in the Enestroem index. Corrected a few typos

``In this paper we give the history of Leonhard Euler's work on the pentagonal number theorem, and his applications of the pentagonal number theorem to the divisor function, partition function and divergent series. We have attempted to give an exhaustive review of all of Euler's correspondence and publications about the pentagonal number theorem and his applications of it.'' This paper gives an exhaustive summary of Euler's work on the pentagonal number theorem. I have gone through all of Euler's published correspondence (except with du Maupertuis and Frederic II) and his papers to find each time he discusses the pentagonal number theorem or applications of it. I have translated from the Latin many sections of his correspondence and papers that are not available in the English, and also sections in the French and German with Dr. Paul Mezo and Christian Leger., Comment: 21 pages, revised

Translation from the Latin of Euler's "Demonstratio theorematis circa ordinem in summis divisorum observatum" (1760). E244 in the Enestroem index. In his previous paper E243, Euler stated the pentagonal number theorem and assuming it proved a recurrence relation for the sum of divisors function. In E244 Euler proves the pentagonal number theorem, and then repeats the proof of the recurrence. In Proposition 1, Euler proves that $\prod_{n=1}^\infty (1-a_n)=1-\sum_{n=1}^\infty a_n(1-a_1)...(1-a_{n-1})$. In Proposition 2 he explicitly states the case $a_n=x^n$. Then in Proposition 3, Euler applies Proposition 2 then collects terms: each time this leaves two terms from the series $\sum_{n=-\infty}^\infty (-1)^n x^{n(3n-1)/2}$. This is explained in detail in a paper I am working on about Euler's work on the pentagonal number theorem., Comment: 8 pages; E244

Translated from the Latin original, "Observationes circa bina biquadrata quorum summam in duo alia biquadrata resolvere liceat" (1772). E428 in the Enestroem index. This paper is about finding A,B,C,D such that $A^4+B^4=C^4+D^4$. In sect. 1, Euler states his "quartic conjecture" that there do not exist any nontrivial integer solutions to $A^4+B^4+C^4=D^4$. I do not know whether he stated this conjecture previously. In sect. 3, Euler sets A=p+q, B=p-q, C=r+s,D=r-s. Taking r=p and s=q gives the trivial solution of C=A and B=D, but this gives Euler the idea of making p and r multiples of each other and q and s multiples of each other. If k=ab this again gives the obvious solution, so in sect. 5: Perturb k to be $k=ab(1+z)$. Euler works out two solutions. One is A=2219449, B=-555617, C=1584749, D=2061283. Hardy and Wright, fifth ed., p. 201 give a simpler parametric solution of $A^4+B^4=C^4+D^4$. Thomas Heath in his Diophantus, second ed., pp. 377-380 gives a faithful explanation of Euler's solution., Comment: 6 pages; E428

This is a translation from the Latin of Euler's "Problema algebraicum de inveniendis quatuor numeris ex datis totidem productis uniuscuiusque horum numerorum in summas trium reliquorum", Opera Postuma 1 (1862), 282-287, reprinted in Leonhardi Euleri Opera omnia I.6. This paper is E808 in the E808 in the Enestr\"om index. We are given the four products $a=v(x+y+z), b=x(v+y+z), c=y(v+x+z), d=z(v+x+y)$. We want to determine $v,x,y,z$ from this data. Euler wants to do this symmetrically: instead of successively eliminating variables and finding out, say, z in terms of a,b,c and then finding y, and then finding x, and then finding v, Euler introduces a new variable that has the same relation with each of the unknowns v,x,y,z and works with that. Euler finds that there will be up to eight choices of v,x,y,z for each choice of a,b,c,d., Comment: 8 pages; E808 in the Enestr\"om index

Translated from the Latin original, "Theorema arithmeticum eiusque demonstratio", Commentationes arithmeticae collectae 2 (1849), 588-592. E794 in the Enestroem index. For m distinct numbers a,b,c,d,...,\upsilon,x this paper evaluates \[ \frac{a^n}{(a-b)(a-c)(a-d)...(a-x)}+\frac{b^n}{(b-a)(b-c)(b-d)...(b-x)} +...+\frac{x^n}{(x-a)(x-b)(x-c)...(x-\upsilon)}. \] When $n \leq m-2$, the sum is 0, which Euler had already shown in sect. 1169 of his Institutiones calculi integralis, vol. I, E366. When $n=m-1$ the sum is $=1$. When $n \geq m$, then using Newton's identities Euler gets expressions in terms of sums of powers of the numbers a,b,c,d,...,\upsilon,x. The notation used in this paper is confus(ing/ed), and indeed the editors of the Opera omnia made several corrections to the original publication. Near the end of the paper, "... for each of them one searches for what the character will be..." is the only reading of the sentence I can come up with. Examples of a work that refers to this theorem is J.-P. Serre, Corps locaux, 2nd ed., ch. III, sect. 6 (for $n \leq m-1$), and D. E. Knuth, The Art of Computer Programming, vol. 1, sect. 1.2.3, exercise 33., Comment: 7 pages; E794

Translation from the Latin of Euler's "Observatio de summis divisorum" (1752). E243 in the Enestroem index. The pentagonal number theorem is that $\prod_{n=1}^\infty (1-x^n)=\sum_{n=-\infty}^\infty (-1)^n x^{n(3n-1)/2}$. This paper assumes the pentagonal number theorem and uses it to prove a recurrence relation for the sum of divisors function. The term "pentagonal numbers" comes from polygonal numbers. Euler takes the logarithmic derivative of both sides. Then after multiplying both sides by $-x$, the left side is equal to $\sum_{n=1}^\infty \sigma(n) x^n$, where $\sigma(n)$ is the sum of the divisors of $n$, e.g. $\sigma(6)=12$. This then leads to the recurrence relation for $\sigma(n)$. I have been studying in detail all of Euler's work on the pentagonal number theorem, and more generally infinite products. I would be particularly interested to see if anyone else worked with products and series like these between Euler and Jacobi, and I would enjoy hearing from anyone who knows something about this., Comment: 13 pages; E243

Translated from the Latin original "Evolutio producti infiniti $(1-x)(1-xx)(1-x^3)(1x^4)(1-x^5)(1-x^6)$ etc. in seriem simplicem" (1775). E541 in the Enestroem index. In this paper Euler is revisiting his proof of the pentagonal number theorem. He gives his original proof explained a bit differently, and then gives a different proof. However this second proof is still rather close to his original proof. To understand the two proofs, I wrote them out using subscript notation and sum/product notation. It would be a useful exercise to try to really understand the proofs without using any modern notation. The right notation takes care of a lot for us, which we would otherwise have to keep active in our minds., Comment: 7 pages; E541

Translated from the Latin original, "De numeris amicabilibus" (1747). E100 in the Enestroem index. Euler starts by saying that with the success of mathematical analysis, number theory has been neglected. He argues that number theory is still a worthwhile subject. First, because there are very elegant and difficult questions in it. Second, because some of the greatest figures in mathematics so far had spent time working on problems in number theory. In particular, Descartes spent time looking for (and finding) a pair of amicable numbers that previously in Europe had been unknown. Euler also mentions the work of Frans van Schooten. Then Euler explains the form of the amicable numbers that were then known, which were $2^n xy$ and $2^n z$, where $x,y,z$ are prime, and the two conditions i) $z=xy+x+y$ and ii) $2^n(x+y+2)=xy+x+y+1$ are satisfied. Euler says that to find more amicable numbers different forms should be tried. He gives 30 pairs of amicable numbers but does not explain how he found them. This is discussed in Ed Sandifer's November 2005 "How Euler did it". One of the pairs (pair XIII) Euler lists is incorrect., Comment: 3 pages; E100

Floodwater mapping is an important remote sensing process that is used for disaster response, recovery, and damage assessment practices. Developing a system to read in Synthetic Aperture Radar (SAR) data and perform land cover classification will allow for the production of near real-time inundation mapping, enabling government and emergency response entities to get a preliminary idea of the situation. SAR is a unique remote sensing tool. Data in this project was obtained by NASA Jet Propulsion Laboratory’s Uninhabited Aerial Vehicle SAR (UAVSAR), an L-band radar mounted to a Gulfstream III jet. Data collected by UAVSAR is similar to what will be available from the NASA-Indian Space Research Organization (NISAR) mission starting in early 2022. Using Python and ArcGIS applications, a model was developed using training samples taken from NOAA post-event aerial photography and UAVSAR data gathered in the aftermath of Hurricane Florence in September 2018.

Remote sensing datasets, particularly acquired by Synthetic Aperture Radar (SAR) sensors, have become increasingly important in severe weather disaster impact studies given their ability to observe the Earth largely irrespective of weather and sunlight conditions. The reliability of existing SAR change detection products applied on a pair of SAR images is constrained by the limitation of current methods to differentiate and classify disaster specific changes from anthropogenic surface alterations. Moreover, the inherent properties of the sensors, variations in SAR backscatter due to changes in surface conditions, and other factors exacerbate these limitations. We proposed a novel procedure expanding on earlier SAR-based change detection methods to exclude anthropogenic alterations and other sources of ambiguity that might lead to inaccurate mapping of the impacts of severe weather disasters. We applied the proposed procedure that is based on long term interferometric and amplitude-based change detection analyses of Sentinel- 1 SAR imagery to two study sites recently impacted by severe weather disasters (Flooding post severe weather events in urban centers; Hailstorm damage on crops). For the first case study, Sentinel-1 SLC scenes from two flood events in the Houston area (April 2016 flooding event and Hurricane Harvey of August-September 2017) were used to construct a flood map depicting areas repeatedly affected by the flood. Pixels with consistent coherence values in the pre-disaster coherence stack were retained for comparison with the pre- and post-disaster coherence stack and pixels with significant decline (greater than 60%) in coherence values were retained in the final flood map. The findings of the applied technique were calibrated and validated through datasets from NOAA/NWS Service storm reports, aerial imaging (NOAA and Civil Air Patrol), Federal Emergency Management Agency (FEMA) reporting, and targeted collections of NASA’s L-band UAVSAR data. Findings and products derived from the adopted methodology can be useful in disaster response and mitigation activities.