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Cauchy’s Residue Sore Thumb
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| Pengarang | : | Harold P. Boas |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 1) |
| Halaman | : | 16-28 |
| Abstrak | : | Cauchy's method from two centuries ago for computing integrals along the real axis by passing into the complex plane is not rigorous by present-day standards. Yet when properly formulated, his original approach is simpler than modern presentations of the residue calculus. |
Partnership, Partition, and Proof: The Path to the Hardy–Ramanujan Partition Formula
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 1) |
| Halaman | : | 3-15 |
| Abstrak | : | The year 2018 marks 100 years since the publication of one of the most startling results in the history of mathematics: Hardy and Ramanujan’s asymptotic formula for the partition function. To celebrate the centenary, this paper looks at the creation of their remarkable theorem: where it came from, how it was proved, and how the assistance of a third contributor helped to influence its ultimate form. |
A Resolvent Criterion for Normality
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 2) |
| Halaman | : | 149-156 |
| Abstrak | : | 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 normal matrix? 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. |
Entropy of Subordinate Shift Spaces
| Pengarang | : | Marcin Kulczycki |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 2) |
| Halaman | : | 141-148 |
| Abstrak | : | We introduce a new family of shift spaces—the subordinate shifts. Using subordinate shifts, we prove in an elementary way that for every nonnegative real number t there is a shift space with entropy t. |
A New Extension of the Riemann Integral
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 2) |
| Halaman | : | 130-140 |
| Abstrak | : | Using an equivalence relation on the hyperreals called approximation, a new extension of the Riemann integral is motivated and introduced in which every bounded function is integrable and for which there exists a function g: [0, 1] → R simultaneously satisfying (1) g is integrable, (2) g is unbounded on every subinterval of [0, 1], and (3) g is identical to its average value function. |
Pascal Functions
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 2) |
| Halaman | : | 115-129 |
| Abstrak | : | We define Pascal functions by adapting the arithmetic rule that creates the Pascal triangle. By developing and applying properties of Pascal functions, we discover new identities and find new perspectives of old identities. The identities all involve binomial coefficients, with some also involving Stirling numbers, Stirling polynomials, associated Stirling numbers of the second kind, or Bell numbers. |
The Circle Problem of Gauss and the Divisor Problem of Dirichlet—Still Unsolved
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 2) |
| Halaman | : | 99-114 |
| Abstrak | : | Let r2(n) denote the number of representations of the positive integer n as a sum of two squares, and let d(n) denote the number of positive divisors of n. Gauss and Dirichlet were evidently the first mathematicians to derive asymptotic formulas for ∑n ? xr2(n) and ∑n ? xd(n), respectively, as x tends to infinity. But what is the error made in such approximations? Number theorists have been attempting to answer these two questions for over one and one-half centuries, and although we think that we essentially “know” what these errors are, progress in proving these conjectures has been agonizingly slow. Ramanujan had a keen interest in these problems, and although, to the best of our knowledge, he did not establish any bounds for the error terms, he did give us identities that have been used to derive bounds, and two further identities that might be useful, if we can figure out how to use them. In this paper, we survey what is known about these two famous unsolved problems, with a moderate emphasis on Ramanujan's contributions. |
A Class of Simple Rearrangements of the Alternating Harmonic Series
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 3) |
| Halaman | : | 245-256 |
| Abstrak | : | We present an easily defined countable family of permutations of the natural numbers for which explicit rearrangements (i.e., the sums induced by the permutations) can be computed. The digamma function proves to be the key tool for the computations found here for the alternating harmonic series. The permutations φ under consideration are simple in the sense that φ?φ is the identity function. We show that the countable set of rearrangements obtained from the permutations considered below is dense in the reals. |
The Continuous Binomial Coefficient: An Elementary Approach
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 3) |
| Halaman | : | 231-244 |
| Abstrak | : | In this article, we use elementary methods to investigate continuous binomial coefficients: functions of the real variable x defined by way of the gamma function with y a fixed real number. We begin with a brief qualitative description of these functions and then derive several interesting representations of them including an infinite product and Taylor series. We also prove various integral formulas involving continuous binomial coefficients, many of which remarkably mirror summation formulas of the familiar binomial coefficients. We conclude by proving a continuous analog of the binomial theorem. |
Multiplying and Factoring Matrices
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 3) |
| Halaman | : | 223-230 |
| Abstrak | : | All of us learn and teach matrix multiplication using rows times columns. Those inner products are the entries of AB. But to go backward—to factor a matrix into triangular or orthogonal or diagonal matrices—outer products are much better. Now AB is the sum of columns of A times rows of B: rank one matrices. Our goal is to produce those columns and rows as simply as possible for A = LU (elimination) and A = CE (echelon form) and A = QR (Gram–Schmidt). Diagonalization by eigenvectors and by singular vectors is also expressed by columns times rows. |