| Nama Pengarang | : | Takeaki Yamazaki |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Nomor Edisi | : | 133 (No. 9) |
| Nomor Halaman | : | 879-882 |
| Abstraksi Narasi | : | In this paper, we prove that the power mean (or the quasi-arithmetic mean) of two positive real numbers satisfies logarithmic concavity. The power mean is a one-parameter mean which interpolates arithmetic, geometric and harmonic means. Our main result is an extension of the inequality by Jameson. Then we give counterexamples for which we cannot extend our main result to weighted power means case and multivariate power means. |
| Nama Pengarang | : | Byungchang So & Eunsoo Lee |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Nomor Edisi | : | 133 (No. 9) |
| Nomor Halaman | : | 866-878 |
| Abstraksi Narasi | : | This paper presents a new proof of Archimedes’ volume propositions for conoids and spheroids that unifies the treatment of all three conic sections. While Archimedes established that solids generated by rotating conic sections maintain specific volume ratios to cones with the same base and axis, his original arguments—both the exhaustion method in On Conoids and Spheroids and the mechanical approach in Method—require distinct arguments for each type of conic section. Our approach transforms both the conic solid and its reference cone into combinations of cone-related figures through an axis-stretching technique, then establishes volume ratios using only elementary proportional relationships. This unified geometric treatment reveals the underlying structural unity between different conic sections, while employing only tools from Euclid’s Elements and techniques found throughout Archimedes’ works. |
| Nama Pengarang | : | Marc Chamberland & Armin Straub |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Nomor Edisi | : | 133 (No. 9) |
| Nomor Halaman | : | 855-865 |
| Abstraksi Narasi | : | The world of primes has many gaps between evidence and theorems. Here, we review Legendre’s conjecture on primes between consecutive squares and recent progress on the weaker question of primes between consecutive larger powers. Assuming the Riemann hypothesis (RH), we observe that a recent result of Emanuel Carneiro, Micah Milinovich and Kannan Soundararajan, combined with a large-scale computation by Jonathan Sorenson and Jonathan Webster, implies the existence of primes between ????2+???? and (????+1)2+???? for all real ????≥1 when ????≥1/4. For smaller values of ????>0, we provide an explicit bound ????0=????0?(????) such that primes exist in these intervals whenever ????≥????0 (again assuming RH). We conclude with an application to Mills-type prime-generating constants. |
| Nama Pengarang | : | Peter Selinger |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Nomor Edisi | : | 133 (No. 9) |
| Nomor Halaman | : | 842-853 |
| Abstraksi Narasi | : | We answer the question: what is the longest winning path on a Hex board of size ????×????? |
| Nama Pengarang | : | Akira Iino & Masashi Sakiyama |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Nomor Edisi | : | 133 (No. 9) |
| Nomor Halaman | : | 831-841 |
| Abstraksi Narasi | : | Let m be a square-free integer with ????>1. For the Cartesian plane over a real quadratic field ??(√????) to contain an equilateral pentagon, ????≡3 (mod?4) is necessary. If ????<200, then ????≡3 (mod?4) is also sufficient. |
| Nama Pengarang | : | Mel Currie & Dan Kalman |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Nomor Edisi | : | 133 (No. 9) |
| Nomor Halaman | : | 811-830 |
| Abstraksi Narasi | : | he remarkable equationlim????→∞?2?????√2−√2+√2+?+√2=????,where n is the number of radicals, inspires an obvious question. What other similar sorts of equations hold? This leads first to an entire family of related sequences, and then to a curious function ?????(????) that encompasses them all. Numerical and graphical investigations suggest many conjectures about ?????(????). Some of these we have managed to prove, but most have stubbornly resisted our efforts. |
| Nama Pengarang | : | A. Bura, C. Good & T. Samuel |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Nomor Edisi | : | 133 (No. 8) |
| Nomor Halaman | : | 784-791 |
| Abstraksi Narasi | : | The property of shadowing is fundamental in both the theory of symbolic dynamics as well as continuous dynamical systems. A quintessential class of discontinuous dynamical systems are those driven by transitive piecewise monotone interval maps and in particular ????-transformations, namely transformations of the form ????????,????:??????????????+???? (mod 1) acting on [0, 1]. We provide a short elegant proof showing that this class of dynamical systems do not possess the property of shadowing, complementing and extending the work of Liang Chen and Raquel Portela. |
| Nama Pengarang | : | Jonathan Lenchner & Rik Sengupta |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Nomor Edisi | : | 133 (No. 8) |
| Nomor Halaman | : | 776-783 |
| Abstraksi Narasi | : | In 1893, James Joseph Sylvester posed the following problem: given n not all collinear points in the plane, must there be a line determined by two of the points that does not pass through any of the other points? In 1941, Eberhard Melchior studied the equivalent dual problem in the projective plane: given a set of n lines in the (real) projective plane, not all passing through a common point, must there be a point where exactly two of the lines intersect? Such a point of intersection is called an ordinary point. Via a clever double counting combinatorial argument, Melchior found that, in fact, there must be three such points. Given the many simple “visual” proofs of what is today known as the Sylvester-Gallai Theorem—the theorem that states there must be at least one ordinary point—a natural question is whether there is a simple visual proof that recovers all three of Melchior’s ordinary points. This paper provides such a proof. |
| Nama Pengarang | : | Mert Çarbo?a & Yusuf Yayl? |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Nomor Edisi | : | 133 (No. 8) |
| Nomor Halaman | : | 758-774 |
| Abstraksi Narasi | : | This study provides a new perspective on the relationship between circulant matrices and umbrella matrices from the viewpoint of Lie theory, working with real-valued matrices. For the first time, we prove that orthogonal circulant matrices whose characteristic value differs from −1 are precisely umbrella matrices, and we define the associated matrix group as the Circulant Umbrella Matrix (CUM) group. Furthermore, we show that the Lie algebra of this group consists of skew-symmetric circulant matrices and we establish a direct connection between the Lie group and its Lie algebra via the Cayley transformation. To visualize these theoretical findings, we construct a four-dimensional hypersurface by selecting an appropriate parametric orbit curve and analyzing its projections in three-dimensional subspaces. In particular, we emphasize how the combination of orthogonality, circulant structure, and zero-row-sum skew-symmetry yields rich geometric and algebraic properties. Finally, we evaluate circular convolution from an orthogonality viewpoint and demonstrate that the elements of the CUM Lie group offer an alternative solution to the Yang–Baxter equation (YBE) while supporting applications in signal processing, cryptography, and other fields that depend on specialized orthogonal transformations. |
| Nama Pengarang | : | Gabriel Nagy |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Nomor Edisi | : | 133 (No. 8) |
| Nomor Halaman | : | 748-757 |
| Abstraksi Narasi | : | We provide a proof of the Cayley-Hamilton Theorem, based on the arithmetic of minimal polynomials, combined with a gentle introduction to spectral theory. |
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