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Potpourri of Conjectures and Open Questions in Nonlinear Analysis and Optimization
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| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 49 (No. 2) |
| Halaman | : | 255-273 |
| Abstrak | : | We present a collection of fourteen conjectures and open problems in the fields of nonlinear analysis and optimization. These problems can be classified into three groups: problems of pure mathematical interest, problems motivated by scientific computing and applications, and problems whose solutions are known but for which we would like to know better proofs. For each problem we provide a succinct presentation, a list of appropriate references, and a view of the state of the art of the subject. |
A Hybrid Approach for Efficient Robust Design of Dynamic Systems
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 49 (No. 2) |
| Halaman | : | 236-254 |
| Abstrak | : | We propose a novel approach for the parametrically robust design of dynamic systems. The approach can be applied to system models with parameters that are uncertain in the sense that values for these parameters are not known precisely, but only within certain bounds. The novel approach is guaranteed to find an optimal steady state that is stable for each parameter combination within these bounds. Our approach combines the use of a standard solver for constrained optimization problems with the rigorous solution of nonlinear systems. The constraints for the optimization problems are based on the concept of parameter space normal vectors that measure the distance of a tentative optimum to the nearest known critical point, i.e., a point where stability may be lost. Such normal vectors are derived using methods from nonlinear dynamics. After the optimization, the rigorous solver is used to provide a guarantee that no critical points exist in the vicinity of the optimum, or to detect such points. In the latter case, the optimization is resumed, taking the newly found critical points into account. This optimize?and?verify procedure is repeated until the rigorous nonlinear solver can guarantee that the vicinity of the optimum is free from critical points and therefore the optimum is parametrically robust. In contrast to existing design methodologies, our approach can be automated and does not rely on the experience of the designing engineer. A simple model of a fermenter is used to illustrate the concepts and the order of activities arising in a typical design process. |
Dynamical Bias in the Coin Toss
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 49 (No. 2) |
| Halaman | : | 211-235 |
| Abstrak | : | We analyze the natural process of flipping a coin which is caught in the hand. We show that vigorously flipped coins tend to come up the same way they started. The limiting chance of coming up this way depends on a single parameter, the angle between the normal to the coin and the angular momentum vector. Measurements of this parameter based on high?speed photography are reported. For natural flips, the chance of coming up as started is about .51. |
Mathematical Models of Avascular Tumor Growth
| Pengarang | : | Tiina Roose |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 49 (No. 2) |
| Halaman | : | 179-208 |
| Abstrak | : | This review will outline a number of illustrative mathematical models describing the growth of avascular tumors. The aim of the review is to provide a relatively comprehensive list of existing models in this area and discuss several representative models in greater detail. In the latter part of the review, some possible future avenues of mathematical modeling of avascular tumor development are outlined together with a list of key questions. |
Domino Waves
| Pengarang | : | C. J. Efthimiou |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 49 (No. 1) |
| Halaman | : | 111-120 |
| Abstrak | : | Motivated by a proposal of Daykin [Problem 71?19*, SIAM Rev., 13 (1971), pp. 569–570], we study the wave that propagates along an infinite chain of dominoes and find the limiting speed of the wave in an extreme case. |
A Matrix Perturbation View of the Small World Phenomenon
| Pengarang | : | Desmond J. Higham |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 49 (No. 1) |
| Halaman | : | 91-108 |
| Abstrak | : | We use techniques from applied matrix analysis to study small world cutoff in a Markov chain. Our model consists of a periodic random walk plus uniform jumps. This has a direct interpretation as a teleporting random walk, of the type used by search engines to locate web pages, on a simple ring network. More loosely, the model may be regarded as an analogue of the original small world network of Watts and Strogatz [Nature, 393 (1998), pp. 440–442]. We measure the small world property by expressing the mean hitting time, averaged over all states, in terms of the expected number of shortcuts per random walk. This average mean hitting time is equivalent to the expected number of steps between a pair of states chosen uniformly at random. The analysis involves nonstandard matrix perturbation theory and the results come with rigorous and sharp asymptotic error estimates. Although developed in a different context, the resulting cutoff diagram agrees closely with that arising from the mean?field network theory of Newman, Moore, and Watts [Phys. Rev. Lett., 84 (2000), pp. 3201–3204]. |
Thermodynamic Analysis of Interacting Nucleic Acid Strands
| Pengarang | : | |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 49 (No. 1) |
| Halaman | : | 65-88 |
| Abstrak | : | Motivated by the analysis of natural and engineered DNA and RNA systems, we present the first algorithm for calculating the partition function of an unpseudoknotted complex of multiple interacting nucleic acid strands. This dynamic program is based on a rigorous extension of secondary structure models to the multistranded case, addressing representation and distinguishability issues that do not arise for single?stranded structures. We then derive the form of the partition function for a fixed volume containing a dilute solution of nucleic acid complexes. This expression can be evaluated explicitly for small numbers of strands, allowing the calculation of the equilibrium population distribution for each species of complex. Alternatively, for large systems (e.g., a test tube), we show that the unique complex concentrations corresponding to thermodynamic equilibrium can be obtained by solving a convex programming problem. Partition function and concentration information can then be used to calculate equilibrium base?pairing observables. The underlying physics and mathematical formulation of these problems lead to an interesting blend of approaches, including ideas from graph theory, group theory, dynamic programming, combinatorics, convex optimization, and Lagrange duality. |
Generalized Chebyshev Bounds via Semidefinite Programming
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 49 (No. 1) |
| Halaman | : | 52-64 |
| Abstrak | : | A sharp lower bound on the probability of a set defined by quadratic inequalities, given the first two moments of the distribution, can be efficiently computed using convex optimization. This result generalizes Chebyshev’s inequality for scalar random variables. Two semidefinite programming formulations are presented, with a constructive proof based on convex optimization duality and elementary linear algebra. |
A Classification of Spikes and Plateaus
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 49 (No. 1) |
| Halaman | : | 35-51 |
| Abstrak | : | In this paper we classify local maxima into spikes and plateaus. We give analytic definitions for spikes and plateaus in terms of a nonlocal gradient and a fourth order derivative. In higher dimensions the Hesse matrix of is of relevance. This classification is applied to pattern formation models in mathematical physics and mathematical biology, including Cahn–Hilliard equations, chemotaxis equations, reaction?diffusion equations, Gierer–Meinhardt models, and Gray–Scott models. We show for some of these examples that the stability of spatial patterns depends on the spike versus plateau type of the solution. We prove, for example, that scalar reaction?diffusion equations in any spatial dimension cannot have stable spike steady states. |
The Mathematics of Phylogenomics
| Pengarang | : | Lior Pachter |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 49 (No. 1) |
| Halaman | : | 3-31 |
| Abstrak | : | The grand challenges in biology today are being shaped by powerful high?throughput technologies that have revealed the genomes of many organisms, global expression patterns of genes, and detailed information about variation within populations. We are therefore able to ask, for the first time, fundamental questions about the evolution of genomes, the structure of genes and their regulation, and the connections between genotypes and phenotypes of individuals. The answers to these questions are all predicated on progress in a variety of computational, statistical, and mathematical fields. The rapid growth in the characterization of genomes has led to the advancement of a new discipline called phylogenomics. This discipline results from the combination of two major fields in the life sciences: genomics, i.e., the study of the function and structure of genes and genomes; and molecular phylogenetics, i.e., the study of the hierarchical evolutionary relationships among organisms and their genomes. The objective of this article is to offer mathematicians a first introduction to this emerging field, and to discuss specific mathematical problems and developments arising from phylogenomics. |