The Invariant Measures of some Infinite Interval Exchange Maps

We classify the locally finite ergodic invariant measures of certain infinite interval exchange transformations (IETs). These transformations naturally arise from return maps of the straight-line flow on certain translation surfaces, and the study of the invariant measures for these IETs is equivalent to the study of invariant measures for the straight-line flow in some direction on these translation surfaces. For the surfaces and directions for which our methods apply, we can characterize the locally finite ergodic invariant measures of the straight-line flow in a set of directions of Hausdorff dimension larger than 1/2. We promote this characterization to a classification in some cases. For instance, when the surfaces admit a cocompact action by a nilpotent group, we prove each ergodic invariant measure for the straight-line flow is a Maharam measure, and we describe precisely which Maharam measures arise. When the surfaces under consideration are finite area, the straight-line flows in the directions we understand are uniquely ergodic. Our methods apply to translation surfaces admitting multi-twists in a pair of cylinder decompositions in non-parallel directions.Comment: 107 pages, 11 figures. Minor improvement

Overlap properties of geometric expanders

The {\em overlap number} of a finite $(d+1)$-uniform hypergraph $H$ is defined as the largest constant $c(H)\in (0,1]$ such that no matter how we map the vertices of $H$ into $\R^d$, there is a point covered by at least a $c(H)$-fraction of the simplices induced by the images of its hyperedges. In~\cite{Gro2}, motivated by the search for an analogue of the notion of graph expansion for higher dimensional simplicial complexes, it was asked whether or not there exists a sequence $\{H_n\}_{n=1}^\infty$ of arbitrarily large $(d+1)$-uniform hypergraphs with bounded degree, for which $\inf_{n\ge 1} c(H_n)>0$. Using both random methods and explicit constructions, we answer this question positively by constructing infinite families of $(d+1)$-uniform hypergraphs with bounded degree such that their overlap numbers are bounded from below by a positive constant $c=c(d)$. We also show that, for every $d$, the best value of the constant $c=c(d)$ that can be achieved by such a construction is asymptotically equal to the limit of the overlap numbers of the complete $(d+1)$-uniform hypergraphs with $n$ vertices, as $n\rightarrow\infty$. For the proof of the latter statement, we establish the following geometric partitioning result of independent interest. For any $d$ and any $\epsilon>0$, there exists $K=K(\epsilon,d)\ge d+1$ satisfying the following condition. For any $k\ge K$, for any point $q \in \mathbb{R}^d$ and for any finite Borel measure $\mu$ on $\mathbb{R}^d$ with respect to which every hyperplane has measure $0$, there is a partition $\mathbb{R}^d=A_1 \cup \ldots \cup A_{k}$ into $k$ measurable parts of equal measure such that all but at most an $\epsilon$-fraction of the $(d+1)$-tuples $A_{i_1},\ldots,A_{i_{d+1}}$ have the property that either all simplices with one vertex in each $A_{i_j}$ contain $q$ or none of these simplices contain $q$

Directional Force Measurement Using Specialized Single-Mode Polarization-Maintaining Fibers

Two different types of specialist single-mode polarization-maintaining side-hole(s) fibers have been specifically chosen in this paper for the direct measurement of transverse force, and their performance characteristics have been recorded and cross compared. To achieve this, side-hole fibers have been used which were investigated both theoretically and experimentally for their respective pressure sensitivities as a function of rotation angles and magnitudes of the applied external force. The experimental results obtained have shown good agreement with theoretical predictions for situations where an external force applied was within a certain range. It was thus concluded that the pressure measurement sensitivities of these specialist fibers are strongly dependent upon the direction of the force applied (with reference to the fast or slow axis of the fibers). Therefore, devices based on these fibers can be used effectively as sensors for the measurement of pressure, force, and mass of an object through an appropriate device configuration, enabling measurements over a wide range and in real time

Weighted L2L^2-cohomology of Coxeter groups

Given a Coxeter system $(W,S)$ and a positive real multiparameter \bq, we study the "weighted $L^2$-cohomology groups," of a certain simplicial complex $\Sigma$ associated to $(W,S)$. These cohomology groups are Hilbert spaces, as well as modules over the Hecke algebra associated to $(W,S)$ and the multiparameter $q$. They have a "von Neumann dimension" with respect to the associated "Hecke - von Neumann algebra," $N_q$. The dimension of the $i^th$ cohomology group is denoted $b^i_q(\Sigma)$. It is a nonnegative real number which varies continuously with $q$. When $q$ is integral, the $b^i_q(\Sigma)$ are the usual $L^2$-Betti numbers of buildings of type $(W,S)$ and thickness $q$. For a certain range of $q$, we calculate these cohomology groups as modules over $N_q$ and obtain explicit formulas for the $b^i_q(\Sigma)$. The range of $q$ for which our calculations are valid depends on the region of convergence of the growth series of $W$. Within this range, we also prove a Decomposition Theorem for $N_q$, analogous to a theorem of L. Solomon on the decomposition of the group algebra of a finite Coxeter group.Comment: minor change

Sensing with coupled-core optical fiber Bragg gratings

[EN] Sensitive bending and vibration sensors based on a coupled-core optical fiber with Bragg gratings are proposed and demonstrated. The interrogation of such sensors is cost effective without comprising the sensors performance.This work was supported by the Spanish Ministry of Science and Innovation under projects No. PGC2018-101997-B100 and RTI2018-0944669-BC31 and the Universitat Politècnica de València with the scholarship PAID-01-18.Flores-Bravo, JA.; Madrigal-Madrigal, J.; Zubia, J.; Margulis, W.; Sales Maicas, S.; Villatoro, J. (2021). Sensing with coupled-core optical fiber Bragg gratings. Optica Publishing Group. 1-2. https://doi.org/10.1364/FIO.2021.FM2C.21

Food Sovereignty and Agricultural Land Use Planning: The Need to Integrate Public Priorities Across Jurisdictions

Recent calls for national food policies that promote greater food sovereignty represent an emerging concern of public policy. Such a shift in food policy toward greater citizen control over domestic food supplies would have significant implications for all aspects of the agri-food system. One area of concern is the conservation and use of agricultural land because, in the end, every act of producing and consuming food has direct or indirect impacts on the land base. Yet no research has considered the potential interactions and implications between food sovereignty and agricultural land use planning. This gap in research presents an opportunity to critically examine the effects of the changing roles and values on agricultural land use planning within and across jurisdictions. We believe that a better understanding of the dominant policy regimes within the agri-food system, including global competitiveness, farmland preservation, and food sovereignty, can lead to land use planning practices that are most beneficial for integrating not only multiple interests across jurisdictions, but also multiple perspectives

Numerical Study of Length Spectra and Low-lying Eigenvalue Spectra of Compact Hyperbolic 3-manifolds

In this paper, we numerically investigate the length spectra and the low-lying eigenvalue spectra of the Laplace-Beltrami operator for a large number of small compact(closed) hyperbolic (CH) 3-manifolds. The first non-zero eigenvalues have been successfully computed using the periodic orbit sum method, which are compared with various geometric quantities such as volume, diameter and length of the shortest periodic geodesic of the manifolds. The deviation of low-lying eigenvalue spectra of manifolds converging to a cusped hyperbolic manifold from the asymptotic distribution has been measured by $\zeta-$ function and spectral distance.Comment: 19 pages, 18 EPS figures and 2 GIF figures (fig.10) Description of cusped manifolds in section 2 is correcte

Automatic estimation of harmonic tension by distributed representation of chords

The buildup and release of a sense of tension is one of the most essential aspects of the process of listening to music. A veridical computational model of perceived musical tension would be an important ingredient for many music informatics applications. The present paper presents a new approach to modelling harmonic tension based on a distributed representation of chords. The starting hypothesis is that harmonic tension as perceived by human listeners is related, among other things, to the expectedness of harmonic units (chords) in their local harmonic context. We train a word2vec-type neural network to learn a vector space that captures contextual similarity and expectedness, and define a quantitative measure of harmonic tension on top of this. To assess the veridicality of the model, we compare its outputs on a number of well-defined chord classes and cadential contexts to results from pertinent empirical studies in music psychology. Statistical analysis shows that the model's predictions conform very well with empirical evidence obtained from human listeners.Comment: 12 pages, 4 figures. To appear in Proceedings of the 13th International Symposium on Computer Music Multidisciplinary Research (CMMR), Porto, Portuga

Optimal network topologies: Expanders, Cages, Ramanujan graphs, Entangled networks and all that

We report on some recent developments in the search for optimal network topologies. First we review some basic concepts on spectral graph theory, including adjacency and Laplacian matrices, and paying special attention to the topological implications of having large spectral gaps. We also introduce related concepts as ``expanders'', Ramanujan, and Cage graphs. Afterwards, we discuss two different dynamical feautures of networks: synchronizability and flow of random walkers and so that they are optimized if the corresponding Laplacian matrix have a large spectral gap. From this, we show, by developing a numerical optimization algorithm that maximum synchronizability and fast random walk spreading are obtained for a particular type of extremely homogeneous regular networks, with long loops and poor modular structure, that we call entangled networks. These turn out to be related to Ramanujan and Cage graphs. We argue also that these graphs are very good finite-size approximations to Bethe lattices, and provide almost or almost optimal solutions to many other problems as, for instance, searchability in the presence of congestion or performance of neural networks. Finally, we study how these results are modified when studying dynamical processes controlled by a normalized (weighted and directed) dynamics; much more heterogeneous graphs are optimal in this case. Finally, a critical discussion of the limitations and possible extensions of this work is presented.Comment: 17 pages. 11 figures. Small corrections and a new reference. Accepted for pub. in JSTA

Regular graphs of large girth and arbitrary degree

For every integer d > 9, we construct infinite families {G_n}_n of d+1-regular graphs which have a large girth > log_d |G_n|, and for d large enough > 1,33 log_d |G_n|. These are Cayley graphs on PGL_2(q) for a special set of d+1 generators whose choice is related to the arithmetic of integral quaternions. These graphs are inspired by the Ramanujan graphs of Lubotzky-Philips-Sarnak and Margulis, with which they coincide when d is prime. When d is not equal to the power of an odd prime, this improves the previous construction of Imrich in 1984 where he obtained infinite families {I_n}_n of d+1-regular graphs, realized as Cayley graphs on SL_2(q), and which are displaying a girth > 0,48 log_d |I_n|. And when d is equal to a power of 2, this improves a construction by Morgenstern in 1994 where certain families {M_n}_n of 2^k+1-regular graphs were shown to have a girth > 2/3 log_d |M_n|.Comment: (15 pages) Accepted at Combinatorica. Title changed following referee's suggestion. Revised version after reviewing proces