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math updates on arXiv.org

+ Erratum to "Full and reduced C*-coactions". Math. Proc. Camb. Phil. Soc. 116 (1994), 435--450. (arXiv:1410.7767v1 [math.OA])

Authors: S. Kaliszewski, John Quigg

Proposition 2.5 of the named article states that a full coaction of a locally compact group on a C*-algebra is nondegenerate if and only if its normalization is. While that result is correct, unfortunately the proof in that article has a gap: it only proves the easier forward implication. The error is significant because this result underpins much of the literature on crossed-product duality, which ultimately relies on nondegeneracy. In this note, we provide a correct proof of a somewhat expanded version of the result.

+ On the capacity functional of excursion sets of Gaussian random fields on $\R^2$. (arXiv:1410.7786v1 [math.PR])

Authors: Marie Kratz, Werner Nagel

When a random field $(X_t, \ t\in {\mathbb R}^2)$ is thresholded on a given level $u$, the excursion set is given by its indicator $~1_{[u, \infty)}(X_t)$. The purpose of this work is to study functionals (as established in stochastic geometry) of these random excursion sets, as e.g. the capacity functional as well as the second moment measure of the boundary length. It extends results obtained for the one-dimensional case to the two-dimensional case, with tools borrowed from crossings theory, in particular Rice methods, and from integral and stochastic geometry.

+ Birch's theorem with shifts. (arXiv:1410.7789v1 [math.NT])

Authors: Sam Chow

Let $f_1, ..., f_R$ be rational forms of degree $d \ge 2$ in $n > \sigma + R(R+1)(d-1)2^{d-1}$ variables, where $\sigma$ is the dimension of the affine variety cut out by the condition $\mathrm{rank}(\nabla f_k)_{k=1}^R < R$. Assume that $\mathbf{f} = \mathbf{0}$ has a nonsingular real solution, and that the forms $(1,...,1) \cdot \nabla f_k$ are linearly independent. Let $\boldsymbol{\tau} \in \mathbb{R}^R$, let $\mu$ be an irrational real number, and let $\eta$ be a positive real number. We consider the values taken by $\mathbf{f}(x_1 + \mu, ..., x_n + \mu)$ for integers $x_1, ..., x_n$. We show that these values are dense in $\mathbb{R}^R$, and prove an asymptotic formula for the number of integer solutions $\mathbf{x} \in [-P,P]^n$ to the system of inequalities $|f_k(x_1 + \mu, ..., x_n + \mu) - \tau_k| < \eta$ ($1 \le k\le R$).

+ A systolic inequality for geodesic flows on the two-sphere. (arXiv:1410.7790v1 [math.DG])

Authors: Alberto Abbondandolo, Barney Bramham, Umberto L. Hryniewicz, Pedro A. S. Salomão

Given a smooth Riemannian two-sphere $(S^2,g)$, consider $l_{min}(S^2,g)$ defined as the minimum of all lengths of non-constant closed geodesics. Our main result asserts that if $g$ is $\delta$-pinched for some $\delta>(4+\sqrt 7)/8=0.8307...$ then the systolic inequality $l_{min}(S^2,g)^2 \leq \pi \ {Area}(S^2,g)$ holds, with equality if and only if $(S^2,g)$ is Zoll. The proof is based on Toponogov's comparison theorem and on a theorem relating the Calabi invariant to the action of fixed points for certain area-preserving annulus maps admitting a generating function.

+ H\"older stability for Serrin's overdetermined problem. (arXiv:1410.7791v1 [math.AP])

Authors: Giulio Ciraolo, Rolando Magnanini, Vincenzo Vespri

In a bounded domain $\Omega$, we consider a positive solution of the problem $\Delta u+f(u)=0$ in $\Omega$, $u=0$ on $\partial\Omega$, where $f:\mathbb{R}\to\mathbb{R}$ is a locally Lipschitz continuous function. Under sufficient conditions on $\Omega$, we show that $\partial\Omega$ is contained in a spherical annulus of radii $r_i<r_e$, where $r_e-r_i\leq C\,[u_\nu]_{\partial\Omega}^\alpha$ for some constants $C>0$ and $\alpha\in (0,1]$. Here, $[u_\nu]_{\partial\Omega}$ is the Lipschitz seminorm on $\partial\Omega$ of the normal derivative of $u$. This result improves to H\"older stability the logarithmic estimate obtained in [1] for Serrin's overdetermined problem. It also extends to a large class of semilinear equations the H\"older estimate obtained in [6] for the case of torsional rigidity ($f\equiv 1$) by means of integral identities. The proof hinges on ideas contained in [1] and uses Carleson-type estimates and improved Harnack inequalities in cones.

+ Hyperplanes in the space of convergent sequences and preduals of $\ell_1$. (arXiv:1410.7801v1 [math.FA])

Authors: E. Casini, E. Miglierina, Ł. Piasecki

The main aim of the present paper is to investigate various structural properties of hyperplanes of $c$, the Banach space of the convergent sequences. In particular, we give an explicit formula for the projection constants and we prove that an hyperplane of $c$ is isometric to the whole space if and only if it is $1$-complemented. Moreover, we obtain the classification of those hyperplanes for which their duals are isometric to $\ell_{1}$ and we give a complete description of the preduals of $\ell_{1}$ under the assumption that the standard basis of $\ell_{1}$ is weak$^{*}$-convergent.

+ Homological Projective Duality for Determinantal Varieties. (arXiv:1410.7803v1 [math.AG])

Authors: Marcello Bernardara, Michele Bolognesi, Daniele Faenzi

In this paper we prove Homological Projective Duality for crepant categorical resolutions of several classes of linear determinantal varieties. By this we mean varieties that are cut out by the minors of a given rank of a n x m matrix of linear forms on a given projective space. As applications, we obtain pairs of derived-equivalent Calabi-Yau manifolds, and address a question by A. Bondal asking whether the derived category of any smooth projective variety can be fully faithfully embedded in the derived category of a smooth Fano variety. Moreover we discuss the relation between rationality and categorical representability in codimension two for determinantal varieties.

+ Glick's conjecture on the point of collapse of axis-aligned polygons under the pentagram maps. (arXiv:1410.7806v1 [math.MG])

Authors: Zijian Yao

The pentagram map has been studied in a series of papers by Schwartz and others. Schwartz showed that an axis-aligned polygon collapses to a point under a predictable number of iterations of the pentagram map. Glick gave a different proof using cluster algebras, and conjectured that the point of collapse is always the center of mass of the axis-aligned polygon. In this paper, we answer Glick's conjecture positively, and generalize the statement to higher and lower dimensional pentagram maps. For the latter map, we define a new system -- the mirror pentagram map -- and prove a closely related result. In addition, the mirror pentagram map provides a geometric description for the lower dimensional pentagram map, defined algebraically by Gekhtman, Shapiro, Tabachnikov and Vainshtein.

Mathematics News -- ScienceDaily

+ Same votes, different voting districts would alter election results in NC: Math study bolsters call for non-partisan redistricting reform
Researchers have developed a mathematical model that shows how changes in congressional voting districts affect election outcomes. Focusing on the last election, they show the outcome of the 2012 US House of Representatives elections in North Carolina would have been very different had the state's congressional districts been drawn with only the legal requirements of redistricting in mind. The researchers hope the study will bolster calls for redistricting reform in 2016.
+ Lack of A level maths leading to fewer female economists in England
A study has found there are far fewer women studying economics than men, with women accounting for just 27 per cent of economics students, despite them making up 57 per cent of the undergraduate population in UK universities. The findings suggest less than half as many girls (1.2 per cent) as boys (3..8 percent) apply to study economics at university, while only 10 per cent of females enroll at university with an A level in maths, compared to 19 per cent of males.
+ An effective, cost-saving way to detect natural gas pipeline leaks
Major leaks from oil and gas pipelines have led to home evacuations, explosions, millions of dollars in lawsuit payouts and valuable natural resources escaping into the air, ground and water. Scientists say they have now developed a new software-based method that finds leaks even when they're small, which could help prevent serious incidents -- and save money for customers and industry.
+ Technology helps even the odds for blind students
Technology to help a blind student see math clearly and pursue a degree has been uncovered by researchers. Despite losing her vision three years ago due to complications from the flu, one study entered university last fall with the specific goal of pursuing a dual degree in mathematics and business. Technology is helping her make this a reality.
+ 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.
+ Doing math with your body
You do math in your head most of the time, but you can also teach your body how to do it. Researchers investigated how our brain processes and understands numbers and number size. They show that movements and sensory perception help us understand numbers.
+ 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.