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math updates on arXiv.org

+ Comment on "Darboux transformation and classification of solution for mixed coupled nonlinear Schr\"odinger equations". (arXiv:1407.7852v1 [nlin.SI])

Authors: Takayuki Tsuchida

In the recent paper quoted in the title (arXiv:1407.5194), Ling, Zhao and Guo presented a multisoliton formula for two coupled nonlinear Schr\"odinger equations with a mixed focusing-defocusing nonlinearity on a general plane-wave background and proved nonsingularity of the solutions. Unfortunately, these results were established in the landmark paper of Dubrovin, Malanyuk, Krichever and Makhan'kov in 1988. In addition, I also express an objection to some critical comments made by them on my paper arXiv:1308.6623.

+ Competition driven cancer immunoediting. (arXiv:1407.7853v1 [q-bio.TO])

Authors: Irina Kareva

It is a well-established fact that tumors up-regulate glucose consumption to meet increasing demands for rapidly available energy by switching to purely glycolytic mode of glucose metabolism. What is often neglected is that cytotoxic cells of the immune system also have increased energy demands and also switch to pure glycolysis when they are in an activated state. Moreover, while cancer cells can revert back to aerobic metabolism, rapidly proliferating cytotoxic lymphocytes are incapable of performing their function when adequate resources are lacking. Consequently, in the tumor microenvironment there must exist competition for the common resources between cancer cells and the cells of the immune system, which may drive a lot of the tumor-immune dynamics. Proposed here is a model of tumor-immune-glucose interactions, formulated as a predator-prey-common resource type system, which allows to investigate possible dynamical behaviors that may arise as a result of competition for glucose, including tumor elimination, tumor dormancy and unrestrained tumor growth.

+ The Bishop-Phelps-Bollob\'as property for operators from $\mathcal{C}(K)$ to uniformly convex spaces. (arXiv:1407.7872v1 [math.FA])

Authors: Sun Kwang Kim, Han Ju Lee

We show that the pair $(C(K),X)$ has the Bishop-Phelps-Bolloba\'as property for operators if $K$ is a compact Hausdorff space and $X$ is a uniformly convex space.

+ The Density Tur\'an problem. (arXiv:1407.7873v1 [math.CO])

Authors: Péter Csikvári, Zoltán Lóránt Nagy

Let $H$ be a graph on $n$ vertices and let the blow-up graph

$G[H]$ be defined as follows. We replace each vertex $v_i$ of $H$ by a cluster

$A_i$ and connect some pairs of vertices of $A_i$ and $A_j$ if $(v_i,v_j)$ was

an edge of the graph $H$. As usual, we define the edge density between $A_i$ and $A_j$ as $d(A_i,A_j)=\frac{e(A_i,A_j)}{|A_i||A_j|}.$ We study the following problem. Given densities $\gamma_{ij}$ for each edge $(i,j)\in E(H)$. Then one has to decide whether there exists a blow-up graph $G[H]$ with edge densities at least $\gamma_{ij}$ such that one cannot choose a vertex from each cluster so that the obtained graph is isomorphic to $H$, i.e, no $H$ appears as a transversal in $G[H]$. We call $d_{crit}(H)$ the maximal value for which there exists a blow-up graph $G[H]$ with edge densities $d(A_i,A_j)=d_{crit}(H)$ $((v_i,v_j)\in E(H))$ not containing $H$ in the above sense. Our main goal is to determine the critical edge density and to characterize the extremal graphs.

+ Genera of non-algebraic leaves of polynomial foliations of $\mathbb C^2$. (arXiv:1407.7878v1 [math.CV])

Authors: Nataliya Goncharuk, Yury Kudryashov

In this article, we prove two results. First, we construct a dense subset in the space of polynomial foliations of degree $n$ such that each foliation from this subset has a leaf with at least $\frac{(n+1)(n+2)}2-4$ handles. Next, we prove that for a generic foliation invariant under the map $(x, y)\mapsto (-x, y)$ all leaves have infinitely many handles.

+ Occupation times of long-range exclusion and connections to KPZ class exponents. (arXiv:1407.7888v1 [math.PR])

Authors: Cédric Bernardin, Patrícia Gonçalves, Sunder Sethuraman

With respect to a class of long-range exclusion processes on $\mathbb{Z}^d$, with single particle transition rates of order $|\cdot|^{-(d+\alpha)}$, starting under Bernoulli invariant measure $\nu_\rho$ with density $\rho$, we consider the fluctuation behavior of occupation times at a vertex and more general additive functionals. Part of our motivation is to investigate the dependence on $\alpha$, $d$ and $\rho$ with respect to the variance of these functionals and associated scaling limits. In the case the rates are symmetric, among other results, we find the scaling limits exhaust a range of fractional Brownian motions with Hurst parameter $H\in [1/2,3/4]$.

However, in the asymmetric case, we study the asymptotics of the variances, which when $d=1$ and $\rho=1/2$ points to a curious dichotomy between long-range strength parameters $0<\alpha\leq 3/2$ and $\alpha>3/2$. In the former case, the order of the occupation time variance is the same as under the process with symmetrized transition rates, which are calculated exactly. In the latter situation, we provide consistent lower and upper bounds and other motivations that this variance order is the same as under the asymmetric short-range model, which is connected to KPZ class scalings of the space-time bulk mass density fluctuations.

+ Cooperation and Storage Tradeoffs in Power-Grids with Renewable Energy Resources. (arXiv:1407.7889v1 [cs.IT])

Authors: Subhash Lakshminarayana, Tony Q.S. Quek, H. Vincent Poor

One of the most important challenges in smart grid systems is the integration of renewable energy resources into its design. In this work, two different techniques to mitigate the time varying and intermittent nature of renewable energy generation are considered. The first one is the use of storage, which smooths out the fluctuations in the renewable energy generation across time. The second technique is the concept of distributed generation combined with cooperation by exchanging energy among the distributed sources. This technique averages out the variation in energy production across space. This paper analyzes the trade-off between these two techniques. The problem is formulated as a stochastic optimization problem with the objective of minimizing the time average cost of energy exchange within the grid. First, an analytical model of the optimal cost is provided by investigating the steady state of the system for some specific scenarios. Then, an algorithm to solve the cost minimization problem using the technique of Lyapunov optimization is developed and results for the performance of the algorithm are provided. These results show that in the presence of limited storage devices, the grid can benefit greatly from cooperation, whereas in the presence of large storage capacity, cooperation does not yield much benefit. Further, it is observed that most of the gains from cooperation can be obtained by exchanging energy only among a few energy harvesting sources.

+ Picard curves over Q with good reduction away from 3. (arXiv:1407.7892v1 [math.NT])

Authors: Beth Malmskog, Christopher Rasmussen

Inspired by methods of N. P. Smart, we describe an algorithm to determine all Picard curves over Q with good reduction away from 3, up to Q-isomorphism. A correspondence between the isomorphism classes of such curves and certain quintic binary forms possessing a rational linear factor is established. An exhaustive list of integral models is determined, and an application to a question of Ihara is discussed.

Mathematics News -- ScienceDaily

+ Numerical learning disability: Dyscalculia linked to difficulties in reading and spelling
Between three and six percent of schoolchildren suffer from an arithmetic-related learning disability. Researchers now show that these children are also more likely to exhibit deficits in reading and spelling than had been previously suspected.
+ Creating sustainable STEM teacher programs
Faculty members who choose to champion physics teacher education, in combination with institutional motivation and commitment, ensure that STEM teacher education programs remain viable after initial funding ends.
+ Atmosphere of Titan, Saturn's largest moon
An astronomer has published the results of the comparison of his model of Titan's atmosphere with the latest data.
+ Philosopher uses game theory to understand how words, actions acquire meaning
Why does the word "dog" have meaning? If you say "dog" to a friend, why does your friend understand you? A philosopher aims to address these types of questions in his latest research, which focuses on long-standing philosophical questions about semantic meaning. Philosophers and a mathematician are collaborating to use game theory to analyze communication and how it acquires meaning.
+ Size and age of plants impact their productivity more than climate
The size and age of plants has more of an impact on their productivity than temperature and precipitation, according to a landmark study. They show that variation in terrestrial ecosystems is characterized by a common mathematical relationship but that climate plays a relatively minor direct role. The results have important implications for models used to predict climate change effects on ecosystem function and worldwide food production.
+ Math can make the Internet 5-10 times faster
Mathematical equations can make Internet communication via computer, mobile phone or satellite many times faster and more secure than today. A new study uses a four minute long mobile video as an example. The method used by the Danish and US researchers in the study resulted in the video being downloaded five times faster than state of the art technology. The video also streamed without interruptions. In comparison, the original video got stuck 13 times along the way.
+ Fair cake cutting gets its own algorithm
A mathematician and a political scientist have announced an algorithm by which they show how to optimally share cake between two people efficiently, in equal pieces and in such a way that no one feels robbed.
+ 'Game theory' model reveals vulnerable moments for metastatic cancer cells' energy production
Cancer’s no game, but researchers are borrowing ideas from evolutionary game theory to learn how cells cooperate within a tumor to gather energy. Their experiments, they say, could identify the ideal time to disrupt metastatic cancer cell cooperation and make a tumor more vulnerable to anti-cancer drugs.