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The Infinitude of the Primes and Some Coloring Theorems
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| Pengarang | : | Azem Berivan Ad?belli & Haydar Goral |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 132 (No. 8) |
| Halaman | : | 766-778 |
| Abstrak | : | We first prove the infinitude of the primes via a special case of Rado’s theorem whose proof is based on the infinite Ramsey theorem. In the proof, we use the colorings of the positive integers introduced by Levent Alpoge [Citation1] and Andrew Granville [Citation2]. Finally, using Rado’s theorem for integral domains, we will give another proof for the infinitude of nonassociated prime elements in any unique factorization domain R with a few units. |
JEPANG DALAM DUNIA YANG BERUBAH
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Analisa |
| Volume / Edisi | : | IX-7, JULI (No. 7) |
| Halaman | : | 577-594 |
| Abstrak | : | - |
Local Receptive Fields Based Extreme Learning Machine
| Pengarang | : | Guang-Bin Huang,Zuo Bai,Kasun, Liyanaarachchi Lekamalage Chamara,Chi Man Vong |
| Nama Majalah/Jurnal | : | IEEE Computational intelligence |
| Volume / Edisi | : | 10 (No. 2) |
| Halaman | : | 18-29 |
| Abstrak | : | Extreme learning machine (ELM), which was originally proposed for "generalized" single-hidden layer feedforward neural networks (SLFNs), provides efficient unified learning solutions for the applications of feature learning, clustering, regression and classification. Different from the common understanding and tenet that hidden neurons of neural networks need to be iteratively adjusted during training stage, ELM theories show that hidden neurons are important but need not be iteratively tuned. In fact, all the parameters of hidden nodes can be independent of training samples and randomly generated according to any continuous probability distribution. And the obtained ELM networks satisfy universal approximation and classification capability. The fully connected ELM architecture has been extensively studied. However, ELM with local connections has not attracted much research attention yet. This paper studies the general architecture of locally connected ELM, showing that: 1) ELM theories are naturally valid for local connections, thus introducing local receptive fields to the input layer; 2) each hidden node in ELM can be a combination of several hidden nodes (a subnetwork), which is also consistent with ELM theories. ELM theories may shed a light on the research of different local receptive fields including true biological receptive fields of which the exact shapes and formula may be unknown to human beings. As a specific example of such general architectures, random convolutional nodes and a pooling structure are implemented in this paper. Experimental results on the NORB dataset, a benchmark for object recognition, show that compared with conventional deep learning solutions, the proposed local receptive fields based ELM (ELM-LRF) reduces the error rate from 6.5% to 2.7% and increases the learning speed up to 200 times. |
New Trends of Learning in Computational Intelligence
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | IEEE Computational intelligence |
| Volume / Edisi | : | 10 (No. 2) |
| Halaman | : | 16-17 |
| Abstrak | : | - |
CIS Publication Spotlight
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | IEEE Computational intelligence |
| Volume / Edisi | : | 10 (No. 2) |
| Halaman | : | 14-15 |
| Abstrak | : | - |
Around the GCD of the Values of Two Polynomials
| Pengarang | : | Arnaud Bodin & Christian Drouin |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 132 (No. 8) |
| Halaman | : | 754-765 |
| Abstrak | : | We propose a mathematical walk around the gcd of the values ?????(????) and ?????(????) of two polynomials evaluated at an integer n. This is an opportunity to use a very powerful tool: the resultant. |
Integer Area Dissections of Lattice Polygons via a Non Abelian Sperners Lemma
| Pengarang | : | Aaron Abrams & Jamie Pommersheim |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 132 (No. 8) |
| Halaman | : | 737-753 |
| Abstrak | : | We give a simple and complete description of those convex lattice polygons in the plane that can be dissected into lattice triangles of integer area. A new version of Sperner’s lemma plays a central role. |
The Eighty Fifth William Lowell Putnam Mathematical Competition
| Pengarang | : | Daniel H. Ullman & Paul Zeitz |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 132 (No. 8) |
| Halaman | : | 723-736 |
| Abstrak | : | The 85th William Lowell Putnam Mathematical Competition took place on December 7, 2024. There were 3988 undergraduates who participated in the competition at 477 institutions across the United States, Canada, and Mexico. The competition is held annually, under the auspices of the Mathematical Association of America. It is supported by the William Lowell Putnam Prize Fund for the Promotion of Scholarship, an endowment established by Elizabeth Lowell Putnam in 1927 in memory of her husband. The Problems Committee for the 85th competition consisted of Brian Hunt (chair), University of Maryland; Greta Panova, University of Southern California; and George Gilbert, Texas Christian University. Additional proposed problems were contributed by Patrick Corn, Susquehanna International Group; Gregory Galperin, Eastern Illinois University; David Grabiner, Department of Defense; Karl Mahlburg, Susquehanna International Group; Michael Larsen, Indiana University; and Richard Stanley, Massachusetts Institute of Technology and University of Miami. |
Should Hydrodynamics Be Taken into Account when Calculating the Growth Rate of Microalgae in a Photobioreactor
| Pengarang | : | J. Ignacio Fierro U., Liu-Di Lu, Olivier Bernard |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 85 (No. 4) |
| Halaman | : | 1906-1925 |
| Abstrak | : | Microalgae, as photosynthetic organisms, are cultivated in photobioreactors for various industrial applications. Light intensity, a critical factor influencing their growth rate, is inherently nonuniform within photobioreactors. In regions distant from the illuminated surface, microalgae experience photolimitation due to insufficient photon availability, hindering optimal activation of the photosynthetic machinery. Conversely, near the illuminated surface, excessive light intensity can damage key photosynthetic proteins, leading to photoinhibition. While mixing in photobioreactors does not alter the light gradient, it influences the light exposure history of cells through hydrodynamic advection. In this study, we employ Han’s mechanistic model to describe the dynamics of photon harvesting and its consequences, including photoinhibition and photolimitation. First, we calculate the time-averaged growth rate for arbitrary continuous light signals, revealing how mixing impacts growth under the assumption of periodic light signals generated by hydrodynamics. Next, we address the computational challenge of estimating growth rates in photobioreactors using computational fluid dynamics (CFD), modeling a single-phase incompressible fluid. Finally, we analyze the case of a raceway pond, evaluating errors arising when growth rate is estimated without accounting for hydrodynamics. We analytically demonstrate that the gain in growth is related to the cell movement along the light gradient. Our results show that in predominantly laminar hydrodynamic regimes, hydrodynamics has only a marginal effect on microalgal growth. Moreover, we show that the average productivity can be estimated based on a static approximation of the average growth rate taking into account the light distribution, with an error lower than 10%. |
The Learned Range Test Method for the Inverse Inclusion Problem
| Pengarang | : | Shiwei Sun, Giovanni S. Alberti |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 85 (No. 4) |
| Halaman | : | 1881-1905 |
| Abstrak | : | We consider the inverse problem consisting of the reconstruction of an inclusion ???? contained in a bounded domain Ω ⊂????? from a single pair of Cauchy data (????|????Ω,????????????|????Ω), where Δ????? =0 in Ω ?–––???? and ???? =0 on ????????. We show that the reconstruction algorithm based on the range test, a domain sampling method, can be written as a neural network with a specific architecture. We propose to learn the weights of this network in the framework of supervised learning, and to combine it with a pretrained classifier, with the purpose of distinguishing the inclusions based on their distance from the boundary. The numerical simulations show that this learned range test method provides accurate and stable reconstructions of polygonal inclusions. Furthermore, the results are superior to those obtained with the standard range test method (without learning) and with an end-to-end fully connected deep neural network, a purely data-driven method. |