University of Wisconsin–Milwaukee

Mathematical Sciences

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The expanded mixed finite element method for generalized Forchheimer flows in porous media. (arXiv:1409.8274v1 [math.NA])

Authors: Akif Ibragimov, Thinh T. Kieu

We study the expanded mixed finite element method applied to degenerate parabolic equations with the Dirichlet boundary condition. The equation is considered a prototype of the nonlinear Forchheimer equation, a inverted to the nonlinear Darcy equation with permeability coefficient depending on pressure gradient, for slightly compressible fluid flow in porous media. The bounds for the solutions are established. In both continuous and discrete time procedures, utilizing the monotonicity properties of Forchheimer equation and boundedness of solutions we prove the optimal error estimates in $L^2$-norm for solution. The error bounds are established for the solution and divergence of the vector variable in Lebesgue norms and Sobolev norms under some additional regularity assumptions. A numerical example using the lowest order Raviart-Thomas ($RT_0$) mixed element are provided agreement with our theoretical analysis.

Stochastic Subgradient Algorithms for Strongly Convex Optimization over Distributed Networks. (arXiv:1409.8277v1 [cs.NA])

We study diffusion and consensus based optimization of a sum of unknown convex objective functions over distributed networks. The only access to these functions is via stochastic gradient oracles, each of which is only available at a different node, and a limited number of gradient oracle calls is allowed at each node. In this framework, we introduce a strongly convex optimization algorithm based on the stochastic gradient descent (SGD) updates. Particularly, we use a certain time-dependant weighted averaging of the SGD iterates, which yields an optimal convergence rate of $O(\frac{N\sqrt{N}}{T})$ after $T$ gradient updates for each node on a network of $N$ nodes. We then show that after $T$ gradient oracle calls, the SGD iterates at each node achieve a mean square deviation (MSD) of $O(\frac{\sqrt{N}}{T})$. We provide the explicit description of the algorithm and illustrate the merits of the proposed algorithm with respect to the state-of-the-art methods in the literature.

Superintegrability of (generalized) Calogero models with oscillator or Coulomb potential. (arXiv:1409.8288v1 [hep-th])

We deform N-dimensional (Euclidean, spherical and hyperbolic) oscillator and Coulomb systems, replacing their angular degrees of freedom by those of a generalized rational Calogero model. Using the action-angle description, it is established that maximal superintegrability is retained. For the rational Calogero model with Coulomb potential, we present all constants of motion via matrix model reduction. In particular, we construct the analog of the Runge-Lenz vector.

Energy Landscape of the Finite-Size Mean-field 2-Spin Spherical Model and Topology Trivialization. (arXiv:1409.8303v1 [cond-mat.stat-mech])

Motivated by the recently observed phenomenon of topology trivialization of potential energy landscapes (PELs) for several statistical mechanics models, we perform a numerical study of the finite size $2$-spin spherical model using both numerical polynomial homotopy continuation and a reformulation via non-hermitian matrices. The continuation approach computes all of the complex stationary points of this model while the matrix approach computes the real stationary points. Using these methods, we compute the average number of stationary points while changing the topology of the PEL as well as the variance. Histograms of these stationary points are presented along with an analysis regarding the complex stationary points. This work connects topology trivialization to two different branches of mathematics: algebraic geometry and catastrophe theory, which is fertile ground for further interdisciplinary research.

Kaczmarz Algorithm and Frames. (arXiv:1409.8310v1 [cs.IT])

Authors: Wojciech Czaja, James Tanis

Sequences of unit vectors for which the Kaczmarz algorithm always converges in Hilbert space can be characterized in frame theory by tight frames with constant 1. We generalize this result to the context of frames and bases. In particular, we show that the only effective sequences which are Riesz bases are orthonormal bases. Moreover, we consider the infinite system of linear algebraic equations $A x = b$ and characterize the (bounded) matrices $A$ for which the Kaczmarz algorithm always converges to a solution.

The LS-category of the product of lens spaces. (arXiv:1409.8316v1 [math.GT])

Authors: Alexander Dranishnikov

We reduced Rudyak's conjecture that a degree one map between closed manifolds cannot raise the Lusternik-Schnirelmann category to the computation of the category of the product of two lens spaces $L^n_p\times L_q^n$ with relatively prime $p$ and $q$. We have computed $cat(L^n_p\times L^n_q)$ for values of $p,q>n/2$. It turns out that our computation supports the conjecture.

For spin manifolds $M$ we establish a criterion for the equality $cat M=dim M-1$ which is a K-theoretic refinement of the Katz-Rudyak criterion for $cat M=dim M$. We apply it to obtain the inequality $cat(L^n_p\times L^n_q)\le 2n-2$ for all $n$ and odd relatively prime $p$ and $q$.

Classification of transversal gates in qubit stabilizer codes. (arXiv:1409.8320v1 [quant-ph])

This work classifies the set of diagonal gates that can implement a single or two-qubit transversal logical gate for qubit stabilizer codes. We show that individual physical gates on the underlying qubits that compose the code are restricted to have entries of the form $e^{i \pi c/2^k}$ along their diagonal, resulting in a similarly restricted class of logical gates that can be implemented in this manner. Moreover, we show that all diagonal logical gates that can be implemented transversally by individual physical diagonal gates must belong to the Clifford hierarchy. Furthermore, we can use this result to prove a conjecture about transversal gates made by Zeng et al. in 2007.

RF Energy Harvesting Enabled Power Sharing in Relay Networks. (arXiv:1409.8325v1 [cs.IT])

Authors: Xueqing Huang, Nirwan Ansari

Through simultaneous energy and information transfer, radio frequency (RF) energy harvesting (EH) reduces the energy consumption of the wireless networks. It also provides a new approach for the wireless devices to share each other's energy storage, without relying on the power grid or traffic offloading. In this paper, we study RF energy harvesting enabled power balancing within the decode-and-forward (DF) relaying-enhanced cooperative wireless system. An optimal power allocation policy is proposed for the scenario where both source and relay nodes can draw power from the radio frequency signals transmitted by each other. To maximize the overall throughput while meeting the energy constraints imposed by the RF sources, an optimization problem is formulated and solved. Based on different harvesting efficiency and channel condition, closed form solutions for optimal joint source and relay power allocation are derived.

Mathematics News -- ScienceDaily

Adding uncertainty to improve mathematical models
Mathematicians have introduced a new element of uncertainty into an equation used to describe the behavior of fluid flows. While being as certain as possible is generally the stock and trade of mathematics, the researchers hope this new formulation might ultimately lead to mathematical models that better reflect the inherent uncertainties of the natural world.
At the interface of math and science: Using mathematics to advance problems in the sciences
In popular culture, mathematics is often deemed inaccessible or esoteric. Yet in the modern world, it plays an ever more important role in our daily lives and a decisive role in the discovery and development of new ideas -- often behind the scenes. In new research, scientists have developed new mathematical approaches to gain insights into how proteins move around within lipid bilayer membranes.
Taking advantage of graphene defects: Security screening
Scientists have discovered a potential application for graphene in security screening. A new theoretical model estimates electric current rectification in graphene. Electronic transport in graphene contributes to its characteristics. Now, a Russian scientist proposes a new theoretical approach to describe graphene with defects-in the form of artificial triangular holes-resulting in the rectification of the electric current within the material. Specifically, the study provides an analytical and numerical theory of the so-called ratchet effect.
Recreating the stripe patterns found in animals by engineering synthetic gene networks
Researchers are trying to understand how networks of genes work together to create specific patterns like stripes. They have gone beyond studying individual networks and have created computational and synthetic mechanisms for a whole 'design space' of networks in the bacteria Escherichia coli. The system proves to be more efficient and powerful than building networks one-by-one, they report.
Could suburban sprawl be good for segregation?
Racially and economically mixed cities are more likely to stay integrated if the density of households stays low, finds a new analysis of a now-famous model of segregation. By simulating the movement of families between neighborhoods in a virtual 'city,' mathematicians show that cities are more likely to become segregated along racial, ethnic or other lines when the proportion of occupied sites rises above a certain critical threshold -- as low as 25 percent.
Video games could dramatically streamline educational research
Scientists have figured out a dramatically easier and more cost-effective way to do research on science curriculum in the classroom -- and it could include playing video games. Called 'computational modeling,' it involves a computer 'learning' student behavior and then 'thinking' as students would. It could revolutionize the way educational research is done.
Math model designed to replace invasive kidney biopsy for lupus patients
Mathematics might be able to reduce the need for invasive biopsies in patients suffering kidney damage related to the autoimmune disease lupus. The model could also be used to monitor the effectiveness of experimental treatments for inflammation and fibrosis, researchers say.
Ebola outbreak 'out of all proportion' and severity cannot be predicted, expert says
A mathematical model that replicates Ebola outbreaks can no longer be used to ascertain the eventual scale of the current epidemic, finds new research.