# University of Wisconsin–Milwaukee

## Mathematical Sciences

The news feeds below are not published by the Mathematical Sciences Department at UW-Milwaukee, but we hope you find them informative.

Small-energy series for one-dimensional quantum-mechanical models with non-symmetric potentials. (arXiv:1410.5813v1 [quant-ph])

Authors: Paolo Amore, Francisco M. Fernández

We generalize a recently proposed small-energy expansion for one-dimensional quantum-mechanical models. The original approach was devised to treat symmetric potentials and here we show how to extend it to non-symmetric ones. Present approach is based on matching the logarithmic derivatives for the left and right solutions to the Schr\"odinger equation at the origin (or any other point chosen conveniently) . As in the original method, each logarithmic derivative can be expanded in a small-energy series by straightforward perturbation theory. We test the new approach on four simple models, one of which is not exactly solvable. The perturbation expansion converges in all the illustrative examples so that one obtains the ground-state energy with an accuracy determined by the number of available perturbation corrections.

Rate-optimal Graphon Estimation. (arXiv:1410.5837v1 [math.ST])

Authors: Chao Gao, Yu Lu, Harrison H. Zhou

Network analysis is becoming one of the most active research areas in statistics. Significant advances have been made recently on developing theories, methodologies and algorithms for analyzing networks. However, there has been little fundamental study on optimal estimation. In this paper, we establish optimal rate of convergence for graphon estimation. For the stochastic block model with $k$ clusters, we show that the optimal rate under the mean squared error is $n^{-1}\log k+k^2/n^2$. The minimax upper bound improves the existing results in literature through a technique of solving a quadratic equation. When $k\leq\sqrt{n\log n}$, as the number of the cluster $k$ grows, the minimax rate grows slowly with only a logarithmic order $n^{-1}\log k$. A key step to establish the lower bound is to construct a novel subset of the parameter space and then apply Fano's lemma, from which we see a clear distinction of the nonparametric graphon estimation problem from classical nonparametric regression, due to the lack of identifiability of the order of nodes in exchangeable random graph models. As an immediate application, we consider nonparametric graphon estimation in a H\"{o}lder class with smoothness $\alpha$. When the smoothness $\alpha\geq 1$, the optimal rate of convergence is $n^{-1}\log n$, independent of $\alpha$, while for $\alpha\in (0,1)$, the rate is $n^{-\frac{2\alpha}{\alpha+1}}$, which is, to our surprise, identical to the classical nonparametric rate.

Weak Factorization System for Actions of Po-monoids on Posets. (arXiv:1410.5839v1 [math.CT])

The concept of a weak factorization system is related to injective objects in slice categories. Let $S$ be a pomonoid, in this paper, {\bf Pos}-$S$, the category of $S$-posets and $S$-poset maps, is considered. One of the main aims of this paper is to draw attention to weak factorization systems for {\bf Pos}-$S.$ We show that if the identity element of $S$ is the bottom element, then $(\mathcal {D}, {\mathcal{SE}}_S)$ is a weak factorization system in {\bf Pos}-$S,$ where $\mathcal D$ and ${\mathcal{SE}}_S$ are the class of down-closed embedding $S$-poset maps and the class of all split $S$-poset epimorphisms, respectively. Among other things, we show that if $\mathcal{U_{\mathcal{P}}}$ and $\mathcal{E_S}$ denote the class of unitary $S$-poset monomorphisms $f: X\to Y$ such that $Y-im (f)$ is projective and the class of all $S$-poset epimorphisms respectively, then $(\mathcal{U_P},\mathcal{E_S})$ is a weak factorization system for {\bf Pos}-$S$ if and only if all $S$-posets have discrete order.

The utmost rigidity property for quadratic foliations with an invariant line on $\mathbb{P}^2$. (arXiv:1410.5840v1 [math.CV])

Authors: Valente Ramirez

In this work we consider holomorphic foliations of degree two on the projective plane $\mathbb{P}^2$ having an invariant line. In a suitable choice of affine coordinates these foliations are induced by a quadratic vector field over the affine part in such a way that the invariant line corresponds to the line at infinity. We say that two such foliations are topologically equivalent provided there exists a homeomorphism of $\mathbb{P}^2$ which brings the leaves of one foliation onto the leaves of the other and preserves orientation both on the ambient space and on the leaves.

The main result of this paper is that in the generic case two such foliations may be topologically equivalent if and only if they are analytically equivalent. In fact, it is shown that the analytic conjugacy class of the holonomy group of the invariant line is the modulus of both topological and analytic classification. We obtain as a corollary that two generic orbitally topologically equivalent quadratic vector fields on $\mathbb{C}^2$ must be affine equivalent.

This result improves, in the case of quadratic foliations, a well-known result by Ilyashenko that claims that two generic and topologically equivalent foliations with an invariant line at infinity are affine equivalent provided they are close enough in the space of foliations and the linking homeomorphism is close enough to the identity map on $\mathbb{P}^2$.

Cooperative Non-Orthogonal Multiple Access in 5G Systems. (arXiv:1410.5846v1 [cs.IT])

Authors: Zhiguo Ding, Mugen Peng, H. V. Poor

Non-orthogonal multiple access (NOMA) has recently received considerable attention as a promising candidate for 5G systems. A key feature of NOMA is that users with better channel conditions have prior information about the messages of the other users. This prior knowledge is fully exploited in this paper, where a cooperative NOMA scheme is proposed. Outage probability and diversity order achieved by this cooperative NOMA scheme are analyzed, and an approach based on user pairing is also proposed to reduce system complexity in practice.

Profile decompositions for wave equations on hyperbolic space with applications. (arXiv:1410.5847v1 [math.AP])

The goal for this paper is twofold. Our first main objective is to develop Bahouri-Gerard type profile decompositions for waves on hyperbolic space. Recently, such profile decompositions have proved to be a versatile tool in the study of the asymptotic dynamics of solutions to nonlinear wave equations with large energy. With an eye towards further applications, we develop this theory in a fairly general framework, which includes the case of waves on hyperbolic space perturbed by a time-independent potential. Our second objective is to use the profile decomposition to address a specific nonlinear problem, namely the question of global well-posedness and scattering for the defocusing, energy critical, semi-linear wave equation on three-dimensional hyperbolic space, possibly perturbed by a repulsive time-independent potential. Using the concentration compactness/rigidity method introduced by Kenig and Merle, we prove that all finite energy initial data lead to a global evolution that scatters to linear waves in infinite time. This proof will serve as a blueprint for the arguments in a forthcoming work, where we study the asymptotic behavior of large energy equivariant wave maps on the hyperbolic plane.

A method of deforming G-structures. (arXiv:1410.5849v1 [math.DG])

Authors: Severin Bunk

We consider deformations of G-structures via the right action on the frame bundle in a base-point-dependent manner. We investigate which of these deformations again lead to G-structures and in which cases the original and the deformed G-structures define the same instantons. Further, we construct a bijection from connections compatible with the original G-structure to those compatible with the deformed G-structure and, finally, consider several examples.

A Fast Hybrid Primal Heuristic for Multiband Robust Capacitated Network Design with Multiple Time Periods. (arXiv:1410.5850v1 [math.OC])

We investigate the Robust Multiperiod Network Design Problem, a generalization of the Capacitated Network Design Problem (CNDP) that, besides establishing flow routing and network capacity installation as in a canonical CNDP, also considers a planning horizon made up of multiple time periods and protection against fluctuations in traffic volumes. As a remedy against traffic volume uncertainty, we propose a Robust Optimization model based on Multiband Robustness (B\"using and D'Andreagiovanni, 2012), a refinement of classical Gamma-Robustness by Bertsimas and Sim that uses a system of multiple deviation bands. Since the resulting optimization problem may prove very challenging even for instances of moderate size solved by a state-of-the-art optimization solver, we propose a hybrid primal heuristic that combines a randomized fixing strategy inspired by ant colony optimization, which exploits information coming from linear relaxations of the problem, and an exact large neighbourhood search. Computational experiments on a set of realistic instances from the SNDlib show that our original heuristic can run fast and produce solutions of extremely high quality associated with low optimality gaps.

## Mathematics News -- ScienceDaily

New theorem determines age distribution of populations from fruit flies to humans
The initial motivation of a new study was to estimate the age structure of a fruit fly population, the result a fundamental theorem that can help determine the age distribution of essentially any group. This emerging theorem on stationary populations shows that you can determine the age distribution of a population by looking at how long they still have to live.