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Presented with appropriate formulas and illustrations are five lessons suitable for college mathematics undergraduate students, including Pi and the limit of (sin alpha)/alpha, the structure of orthogonal transformations, a simple proof of Zorn's Lemma, the converse of Liouville's Theorem, and visualizing the p-adic integers. (JJK)

"Numbers and the World: Essays on Math and Beyond" by David Mumford is a collection of essays that covers a wide range of topics, from mathematics to neuroscience, artificial intelligence, history, and social issues. Mumford, a highly acclaimed mathematician, offers his unique and creative perspectives on these subjects. The book is not a technical exposition of his mathematical work, but rather a collection of his thoughts and musings, originally posted on his blog. Mumford's writing style is clear and accessible, making this book an interesting read for both mathematicians and non-mathematicians alike. [Extracted from the article]

In 2018, the U.S. Supreme Court considered a proposed mathematical formula to help detect unconstitutional partisan gerrymandering. We show that in some cases, this formula only flags bizarrely-shaped districts as potentially constitutional. [ABSTRACT FROM AUTHOR]

The gamma function was originally discovered by Euler to solve an interpolation problem for the factorials. However, its ubiquity cannot be due to its algebraic nature alone. This article presents the close connection between the volumes of various spaces and the values of the gamma function to better understand its frequent appearance in many mathematical formulas. This connection produces a surprising similarity between Viète’s formula for π and Knar’s formula. [ABSTRACT FROM AUTHOR]

Given a normal matrix A and an arbitrary square matrix B (not necessarily of the same size),what relationships between A and B, if any, guarantee that B is also a normalmatrix? We provide an answer to this question in terms of pseudospectra and norm behavior. In doing so, we prove that a certain distance formula, known to be a necessary condition for normality, is in fact sufficient and demonstrates that the spectrum of a matrix can be used to recover the spectral norm of its resolvent precisely when the matrix is normal. These results lead to new normality criteria and other interesting consequences. [ABSTRACT FROM AUTHOR]

Updating a given a matrix Amxn by a rank-one matrix B = cdT, where c and d are appropriately sized column vectors, is a common practice throughout all applied areas of mathematics, science, and engineering. Because rank is often tied to the number of degrees of freedom or the level of independence in underlying models or data, it can be imperative to know exactly how the update term affects rank. While it is well known that a rank-one update can only increase or decrease rank by at most one, there is not a widely known formula for exactly how this occurs. This note presents an expression in simply stated terms for the exact rank of a rank-one updated matrix. [ABSTRACT FROM AUTHOR]

Cayley's tree formula is one of the most beautiful results in enumerative combinatorics with a number of well-known proofs. A fascinating and recurring theme in mathematics is the existence of rich and surprising interconnections between apparently disjoint domains. We will, hopefully, see one such instance in this note, in which we give a short proof of Cayley's tree formula for counting the number of distinct labeled trees on n vertices by employing a combinatorial argument. In fact, we deduce a nonlinear recursive relation for the number of labeled trees on n vertices and then we show that this recursion is same as the celebrated Cayley tree formula. One interesting feature of this proof is that a discrete counting problem is solved by going from a recursion, to a differential equation, to complex analysis, and then back to a discrete formula. [ABSTRACT FROM AUTHOR]

In this note, we build on recent work in [7] to establish formulas for ζ(2n) as sums over all the unrestricted integer partitions of n. [ABSTRACT FROM AUTHOR]

We examine Euclid's lemma that if p is a prime number such that p|ab, then p divides at least one of a or b. Specifically, we consider the common misapplication of this lemma to numbers that are not prime, as is often made by undergraduate students. We show that a randomly chosen implication of the form r |ab ⇒ r |a or r |b is almost surely false in a probabilistic sense, and we quantify this with a corresponding asymptotic formula. [ABSTRACT FROM AUTHOR]

We give a new solution of the famous Basel problem, inspired by Euclidean geometry. [ABSTRACT FROM AUTHOR]

Several proposed mathematical problems and solutions provided by readers are presented including one from Stefano Siboni on equilibrium length, one from Robin Oakapple on the construction of two rectangles, and one from Octavian Ganea on a skew-symmetric determinant inequality.

The Cayley–Bacharach theorem states that all cubic curves through eight given points in the plane also pass through a unique ninth point. We write that point as an explicit rational function in the other eight. [ABSTRACT FROM AUTHOR]

It is shown that if two planar convex n-gons are oppositely oriented, then the segments joining the corresponding vertices have a common transversal. A different formulation is also given in terms of two cars moving along two convex curves in opposite directions. Some possible analogs in 3-space are also considered, and an example is shown that the full analog is not true in this setting. [ABSTRACT FROM AUTHOR]

The well-known limit comparison test is only applicable for series with nonnegative terms. Thus, it can be used only for proving or disproving the absolute convergence of a series. In this note we formulate and justify a modified version of the limit comparison test for general series. The test can be used to prove the convergence of conditionally convergent series. [ABSTRACT FROM AUTHOR]

In this short article, we give a novel and elegant proof of a recursion formula related to confluent Vandermonde determinants. This leads to an easy derivation of the well-known determinant identity due to Schendel. [ABSTRACT FROM AUTHOR]

The article focuses on the solution to a mathematical problem on compositions of integers that avoid arithmetic sequence. It states that the Fibonacci sequence is used to present the number of compositions of a particular integer. It mentions that closed formulas are obtained as the Fibonacci generating function evaluates to a rational function. It highlights that bivariate versions exist for the results of the problem.

The article presents two elementary properties for entire functions and their applications. It mentions the consequence of Cauchy's theorem and Liouville's theorem. It highlights various mathematical formulas, equations and theorems regarding the same. It also mentions several integration, differentiation and polynomials characterization for the same.

The article presents a proof for a comprehensive Pythagorean theorem for all dimensions. It highlights various mathematical formulas, mathematical equations and theorems regarding the same. It also mentions several mathematical; problems depicting the use of integration, differentiation and integers to prove the Pythagoras theorem.

The article presents a proof for Euler's Theorem on series of prime reciprocals. It highlights various mathematical formulas, equations and theorems regarding the same. It also mentions several integration, differentiation and numerical calculations for set of positive integers and factorization of prime numbers.

The article discusses the derivation of lower and upper bounds for infinite series Σ∞k=n k-s. It presents the criteria of the Cauchy integral for infinite series. It also highlights various mathematical formulas, theorems and integration & differentiation for the same.

The article presents a proof for a identity involving prime numbers and its corollaries. It highlights various mathematical formulas, equations and theorems regarding the same. It mentions several integration, differentiation and prime factorization for the same. It also informs about numerical calculations for a set of integers.

The article discusses the geometry, definitions and basic properties of cardioid. It presents several mathematical problems to highlight geometry of cardioid. It highlights various mathematical formulas, mathematical equations and algorithms regarding the same. It also mentions several theorems, coordinate system and coefficients for the same.

The article presents a proof of sharper bound in a van der Corput's lemma for harmonic analysis by mathematics professor K. Rogers in 2005. It highlights various mathematical formulas, equations and integers regarding the same. It also mentions several theorems, differentiation and integration for the same.

The article discusses the measurement of ω-primarily function in commutative monoids. It informs about the importance of understanding factorization theory of monoids. It highlights various mathematical formulas, equations and algorithms regarding the same. It also mentions several trigonometric functions, theorems and coefficients for the same.

The article presents an analysis for computing monthly deposits, loans principals and monthly payments with non-negative interest rate. It highlights various mathematical formulas, mathematical equations and algorithms regarding the same. It also mentions several trigonometric functions, theorems and coefficients to calculate monthly payments.

The article discusses the derivation of lower and upper bounds for infinite series Σ∞k=n k-s. It presents the criteria of the Cauchy integral for infinite series. It also highlights various mathematical formulas, theorems and integration & differentiation for the same.

The article focuses on formulas relating angles, areas and distances in triangles and tetrahedra. Topics discussed include Heron's formula for triangles, Cayley-Menger determinant, Cayley-Menger matrix, laws of cosines, application of laws of cosines in tetrahedra such as availability of a Cayley cosine cubic for each face of a tetrahedron and trigonometric identity of the Cayley cosine cubic.

The article examines the formula for the cubic spline, which is integrated exactly, used to derive a composite integration rule. Topics discussed include a reversed approach, resulting in end conditions for the spline that are derived to produce high-order quadarature formulas. Also discussed are spline to quadarature formula, and quadarature formula to spline.

The article presents a simple proof of Keith Ball's formula for the volume of intersection of an (n-1)-dimensional hyperplane with an n-dimensional cube using the authors's previous study of sine integrals. Topics discussed include a simple proof of Ball's formula given recently for that volume by R. Frank and H. Riede.

The article presents a simple formula for the signature of a foldable triangulation of a lattice polygon in terms of its boundary. Topics discussed include the formula yielding lower bounds on the number of real roots of certain systems of polynomial equations known as Wronski systems. Also discussed are lattice polygons and Wronski polynomials.

The article discusses the application of the Pigeonhole principle in a mathematical problem. Topics include an integer of the forms 111•••111 is called a repunit, a proof that provides additional insights on the number of 1's and 0's in the multiple of k, and the assumption that the integers are expressed as decimal numbers.

The article discusses the Stirling's mathematical formula for central extended binomial coefficients. Discussion and highlights are given on the derivation of asymptomatic formulas for the coefficients using the sum distribution of independent discrete uniform random variables with a local limit variant and Central Limit Theorem.

The article discusses the asymptomatic formula for the mathematical equation (1 + 1/x)x. Highlights are given on the presentation of method to create estimations or approximations of the natural logarithmic constant e and formula based on the partition function. Also provided is information on Maclaurin series expansion.

The article discusses an explicit analytical solution for the generalization of the Fermat-Torricelli problem. It states that Torricelli proved that the circles surrounding the equilateral triangles constructed on sides and outside of the triangle. It mentions that the denominators and numerators in the right-hand sides of the formulas could be reduced by the common factor.

The article offers information on approximation formulas for the birthday problem regarding the depiction of probability of matching birthdays in a randomly selected group of people. It reports that the formulas offer computable approximation which provide close-fit solutions, easier calculations and can be analytically solved for any number of persons. It also mentions that the approximation analyses the probability of birthdays of two or three persons on same day that exceeds a given value.

The article offers information on mathematical derivation related to the basel problem, commonly known as euler identity using double integrals. It states that the derivation of basal problem was calculated by using inter-relation between a double integral and a geometric series. It also mentions that a proof of that uses a rational function with the lowest degree among a set of functions used in another mathematical theorems.

The article presents new proof of a classical formula using the technique of differentiation under the integral sign and evaluates other definite integrals as well.

The article focuses on Jacobi's four squares theorem which is a formula that expresses an infinite product as an infinite series called a product-to-sum identity. It explores the product-to-sum identities in a single complex variable q from which Jacobi's four squares formula can be deduced by equating coefficients of q" or the "parents" of the formula using some identities of Ramanujan with unique results. It says that the same is done for Legendre's four triangular numbers theorem.

In this article, the author discusses the basic properties of evolutes and involutes of spatial curves showing that evolutes and involutes are independent of dimensions. He mentions that the best method of describing an evolute relates it to osculating circles. He further mentions that an involute of a given curve is that curve in which all tangents of the given curve are normal. He also describes frenet formulas for spatial curves in an orthonormal frame.

The article presents a study which describes the hyperplane sections of the n-Dimensional Cube. The researchers have deduced an elementary formula and its application in various dimensions in a face of a cube. It reveals that situation changes when intersecting a four-dimensional cube with a three-dimensional hyperplane.

The article informs that the mathematical formulas which are used in the volume of multidimensional balls and cones are deduced based on the Cavalieri's principle and basic properties of the Lebesgue measure. According to this principle, two bodies are having equal volumes if one can place these bodies in space. Several mathematical theorems and formulas in support of this principle are presented.

The article presents several mathematical problems related to topics including triangles, integers and elements submitted by several students and professors including Eugen J. Ionascu from the Columbus State University in Columbus, Georgia, Cosmin Pohoata at the Princeton University in Princeton, New Jersey and Vlad Matei from the University of Bucharest in Bucharest, Romania. It also presents solutions to previous proposed problems.

The article presents information about proofs regarding the existence of equiangular polygons related to edge lengths in an arithmetic progression. Several mathematical theorems, hypothesis and mathematical formulas are provided for supporting these proofs. Several figures of arithmetic polygons are also presented.

The article presents information about the proof of an unrecorded evaluation of the Legendre symbols. It informs about the evaluation of Gauss' lemma, a number theory, by the use of algorithms. It mentions that a student Daniel Apin has proposed for the evaluation of the Legendre symbols by the use of Gauss' lemma. Several theorems and mathematical formulas are provided for supporting the proof.

The article discusses about the theory of functional equations in context with several mathematical theorems and formulas. It informs that the determination of homogeneous functional mean has been done through functional equations techniques. Information about the mean value theorem of calculus and homogeneity of the functional mean in context with logarithmic equations are provided.

The article informs about perfect numbers and amicable numbers. Information about the Euler theorem, Dickson's theorem and natural numbers are provided. Several mathematical theorems and formulas in support of the number systems are provided. Information about the Dickson's theorem in context with the book "Basic Number Theory" is presented.

The article reports about the beatty sequences in reference to two athletes who are running in opposite directions on a round track. It informs that this mathematical sequence interprets for partitioning criteria. Information about several mathematical theorems and formulas are provided in support of these mathematical sequences. Theorems given by mathematicians including Hisasi Morikawa, Ernst Zermelo and Abraham Fraenkel are also provided.

The article focuses on the Gauss's Cyclotomic Formula, where p in an extended formula is an odd prime. It states that determinant of a circulant matrix, and Wendt determinant are involved in the formula. It also explores the mathematical solutions for the formula, where there is integer polynomials R (x,y) and S (x,y).

The article focuses on a mathematical proof of mathematician Leonhard Euler's infinite product for sine product formula. It is stated that Gamma function and elementary probability theory prove product formula for the sine function. Some deductions or inferences of sine formula are also presented in the article. Validity of sine product formula is for the whole complex plane.