Volume form on moduli spaces of d-differentials

Given $d\in \mathbb{N}$, $g\in \mathbb{N} \cup\{0\}$, and an integral vector $\kappa=(k_1,\dots,k_n)$ such that $k_i>-d$ and $k_1+\dots+k_n=d(2g-2)$, let $\Omega^d\mathcal{M}_{g,n}(\kappa)$ denote the moduli space of meromorphic $d$-differentials on Riemann surfaces of genus $g$ whose zeros and poles have orders prescribed by $\kappa$. We show that $\Omega^d\mathcal{M}_{g,n}(\kappa)$ carries a canonical volume form that is parallel with respect to its affine complex manifold structure, and that the total volume of $\mathbb{P}\Omega^d\mathcal{M}_{g,n}(\kappa)=\Omega^d\mathcal{M}_{g,n}/\mathbb{C}^*$ with respect to the measure induced by this volume form is finite.Comment: Streamlined, minor corrections added, definition of the volume form independent of the choice of a d-th root of unit

Translation surfaces and the curve graph in genus two

Let $S$ be a (topological) compact closed surface of genus two. We associate to each translation surface $(X,\omega) \in \mathcal{H}(2)\sqcup\mathcal{H}(1,1)$ a subgraph $\hat{\mathcal{C}}_{\rm cyl}$ of the curve graph of $S$. The vertices of this subgraph are free homotopy classes of curves which can be represented either by a simple closed geodesic, or by a concatenation of two parallel saddle connections (satisfying some additional properties) on $X$. The subgraph $\hat{\mathcal{C}}_{\rm cyl}$ is by definition $\mathrm{GL}^+(2,\mathbb{R})$-invariant. Hence, it may be seen as the image of the corresponding Teichm\"uller disk in the curve graph. We will show that $\hat{\mathcal{C}}_{\rm cyl}$ is always connected and has infinite diameter. The group ${\rm Aff}^+(X,\omega)$ of affine automorphisms of $(X,\omega)$ preserves naturally $\hat{\mathcal{C}}_{\rm cyl}$, we show that ${\rm Aff}^+(X,\omega)$ is precisely the stabilizer of $\hat{\mathcal{C}}_{\rm cyl}$ in ${\rm Mod}(S)$. We also prove that $\hat{\mathcal{C}}_{\rm cyl}$ is Gromov-hyperbolic if $(X,\omega)$ is completely periodic in the sense of Calta. It turns out that the quotient of $\hat{\mathcal{C}}_{\rm cyl}$ by ${\rm Aff}^+(X,\omega)$ is closely related to McMullen's prototypes in the case $(X,\omega)$ is a Veech surface in $\mathcal{H}(2)$. We finally show that this quotient graph has finitely many vertices if and only if $(X,\omega)$ is a Veech surface for $(X,\omega)$ in both strata $\mathcal{H}(2)$ and $\mathcal{H}(1,1)$.Comment: 47 pages, 17 figures. Minor changes, some proofs improved. Comments welcome

The Lagrange multipliers and existence of competitive equilibrium in an intertemporal model with endogenous leisure

This paper proves the existence of competitive equilibrium in a single sector dynamic economy with elastic labor supply. The method of proof relies on some recent results (see Le Van and Saglam [2004]) concerning the existence of Lagrange multipliers in infinite dimensional spaces and their representation as a summable sequence.Optimal growth model, Lagrange multipliers, competitive equilibrium, elastic labor supply.

Growth and convergence in a model with renewable and non-renewable resources: existence, transitional dynamics, and empirical evidence

This paper studies an optimal endogenous growth model using physical capital, labor and two kinds of natural resources in the final goods sector and employing labor to accumulate knowledge. Based on results in calculus of variations, a direct proof of existence of optimal solution is provided. Analytical solutions for the planner case and the balanced growth paths are found for a specific CRRA utility and Cobb-Douglas production function. Transitional dynamics to the steady state from the theoretical model are used to derive three convergence equations of output intensity growth rate, exhaustible resource growth rate and renewable growth rate, which are tested based on data on production and energy consumption in 27 OECD countries.Optimal growth, existence of equilibrium, transitional dynamics, energy, renewable resource, non-renewable resource.

Phase-ambiguity resolution for QPSK modulation systems. Part 2: A method to resolve offset QPSK

Part 2 presents a new method to resolve the phase-ambiguity for Offset QPSK modulation systems. When an Offset Quaternary Phase-Shift-Keyed (OQPSK) communications link is utilized, the phase ambiguity of the reference carrier must be resolved. At the transmitter, two different unique words are separately modulated onto the quadrature carriers. At the receiver, the recovered carrier may have one of four possible phases, 0, 90, 180, or 270 degrees, referenced to the nominally correct phase. The IF portion of the channel may cause a phase-sense reversal, i.e., a reversal in the direction of phase rotation for a specified bit pattern. Hence, eight possible phase relationships (the so-called eight ambiguous phase conditions) between input and output of the demodulator must be resolved. Using the In-phase (I)/Quadrature (Q) channel reversal correcting property of an OQPSK Costas loop with integrated symbol synchronization, four ambiguous phase conditions are eliminated. Thus, only four possible ambiguous phase conditions remain. The errors caused by the remaining ambiguous phase conditions can be corrected by monitoring and detecting the polarity of the two unique words. The correction of the unique word polarities results in the complete phase-ambiguity resolution for the OQPSK system

Wavelet-Based Kernel Construction for Heart Disease Classification

© 2019 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERINGHeart disease classification plays an important role in clinical diagnoses. The performance improvement of an Electrocardiogram classifier is therefore of great relevance, but it is a challenging task too. This paper proposes a novel classification algorithm using the kernel method. A kernel is constructed based on wavelet coefficients of heartbeat signals for a classifier with high performance. In particular, a wavelet packet decomposition algorithm is applied to heartbeat signals to obtain the Approximation and Detail coefficients, which are used to calculate the parameters of the kernel. A principal component analysis algorithm with the wavelet-based kernel is employed to choose the main features of the heartbeat signals for the input of the classifier. In addition, a neural network with three hidden layers in the classifier is utilized for classifying five types of heart disease. The electrocardiogram signals in nine patients obtained from the MIT-BIH database are used to test the proposed classifier. In order to evaluate the performance of the classifier, a multi-class confusion matrix is applied to produce the performance indexes, including the Accuracy, Recall, Precision, and F1 score. The experimental results show that the proposed method gives good results for the classification of the five mentioned types of heart disease.Peer reviewedFinal Published versio