Quasimodularity and large genus limits of Siegel-Veech constants

Quasimodular forms were first studied in the context of counting torus coverings. Here we show that a weighted version of these coverings with Siegel-Veech weights also provides quasimodular forms. We apply this to prove conjectures of Eskin and Zorich on the large genus limits of Masur-Veech volumes and of Siegel-Veech constants. In Part I we connect the geometric definition of Siegel-Veech constants both with a combinatorial counting problem and with intersection numbers on Hurwitz spaces. We introduce modified Siegel-Veech weights whose generating functions will later be shown to be quasimodular. Parts II and III are devoted to the study of the quasimodularity of the generating functions arising from weighted counting of torus coverings. The starting point is the theorem of Bloch and Okounkov saying that q-brackets of shifted symmetric functions are quasimodular forms. In Part II we give an expression for their growth polynomials in terms of Gaussian integrals and use this to obtain a closed formula for the generating series of cumulants that is the basis for studying large genus asymptotics. In Part III we show that the even hook-length moments of partitions are shifted symmetric polynomials and prove a formula for the q-bracket of the product of such a hook-length moment with an arbitrary shifted symmetric polynomial. This formula proves quasimodularity also for the (-2)-nd hook-length moments by extrapolation, and implies the quasimodularity of the Siegel-Veech weighted counting functions. Finally, in Part IV these results are used to give explicit generating functions for the volumes and Siegel-Veech constants in the case of the principal stratum of abelian differentials. To apply these exact formulas to the Eskin-Zorich conjectures we provide a general framework for computing the asymptotics of rapidly divergent power series.Comment: 107 pages, final version, to appear in J. of the AM

Resonant Interactions in Rotating Homogeneous Three-dimensional Turbulence

Direct numerical simulations of three-dimensional (3D) homogeneous turbulence under rapid rigid rotation are conducted to examine the predictions of resonant wave theory for both small Rossby number and large Reynolds number. The simulation results reveal that there is a clear inverse energy cascade to the large scales, as predicted by 2D Navier-Stokes equations for resonant interactions of slow modes. As the rotation rate increases, the vertically-averaged horizontal velocity field from 3D Navier-Stokes converges to the velocity field from 2D Navier-Stokes, as measured by the energy in their difference field. Likewise, the vertically-averaged vertical velocity from 3D Navier-Stokes converges to a solution of the 2D passive scalar equation. The energy flux directly into small wave numbers in the $k_z=0$ plane from non-resonant interactions decreases, while fast-mode energy concentrates closer to that plane. The simulations are consistent with an increasingly dominant role of resonant triads for more rapid rotation

Hybrid power semiconductor

The voltage rating of a bipolar transistor may be greatly extended while at the same time reducing its switching time by operating it in conjunction with FETs in a hybrid circuit. One FET is used to drive the bipolar transistor while the other FET is connected in series with the transistor and an inductive load. Both FETs are turned on or off by a single drive signal of load power, the second FET upon ceasing conductions, rendering one power electrode of the bipolar transistor open. Means are provided to dissipate currents which flow after the bipolar transistor is rendered nonconducting

Network support for integrated design

A framework of network support for utilization of integrated design over the Internet has been developed. The techniques presented also applicable for Intranet/Extranet. The integrated design system was initially developed for local application in a single site. With the network support, geographically dispersed designers can collaborate a design task through out the total design process, quickly respond to clients’ requests and enhance the design argilty. In this paper, after a brief introduction of the integrated design system, the network support framework is presented, followed by description of two key techniques involved: Java Saverlet approach for remotely executing a large program and online CAD collaboration

Positivity Of Equivariant Gromov–Witten Invariants

We show that the equivariant Gromov–Witten invariants of a projective homogeneous space G/P exhibit Graham-positivity: when expressed as polynomials in the positive roots, they have nonnegative coefficients

Rare-earth ions doped transparent oxyfluoride glass-ceramics

In recent years, rare-earth ions doped transparent oxyfluoride glass-ceramics have attracted great attentions for their low phonon energy environments of fluoride nanocrystals and high chemical and mechanical stabilities of oxide glassy matrix. In this chapter, firstly, the crystallization behaviors of the transparent glass ceramics containing CaF2 nanocrystals are presented to demonstrate the controllable microstructure evolution of nano-composites. Secondly, the optical properties of the newly developed transparent glass-ceramics containing β-YF3 nanocrystals are systematically reviewed. The rare-earth ions are inclined to partition into the YF3 nanocrystals after crystallization. Through variation of the rare-earth doping and control of the microstructures, the glass-ceramics could exhibit high-stimulated emission cross-section, broadband near infrared emission, high efficient ultraviolet upconversion emission and bright white light emission, indicating their potential multifunctional applications in solid state laser, upconversion, optical amplifier, three-dimensional display, and so on

Equivariant Quantum Schubert Polynomials

We establish an equivariant quantum Giambelli formula for partial flag varieties. The answer is given in terms of a specialization of universal double Schubert polynomials. Along the way, we give new proofs of the presentation of the equivariant quantum cohomology ring, as well as Graham-positivity of the structure constants in equivariant quantum Schubert calculus. (C) 2013 Elsevier Inc. All rights reserved

Machine learning phases in statistical physics

Conventionally, the study of phases in statistical mechan- ics is performed with the help of random sampling tools. Among the most powerful are Monte Carlo simulations consisting of a stochastic importance sampling over state space and evaluation of estimators for physical quantities. The ability of modern machine learning techniques to classify, identify, or in- terpret massive data sets provides a complementary paradigm to the above approach to analyze the exponentially large number of states in statistical physics. In this report, it is demonstrated by application on Ising-type models that deep learning has potential wide applications in solving many-body statis- tical physics problems. In application of supervised learning, we showed that the feed-forward neural network can identify phases and phase transitions in the ferromagnetic Ising model and the convolutional neural network (CNN) is extremely powerful in classifying T = 0 and T = ∞ phases in the Ising gauge model; In application of unsupervised learning, we illustrated that a deep auto-encoder constructed by stacked restricted Boltzmann machines (RBM) is closely related to the renormalization group (RG) method well understood in modern physics and our reconstruction of Ising spin configurations in the ferromagnetic Ising model is similar to the hand-written digits reconstruction.Statistic

Different thermodynamic pathways to the solvation free energy of a spherical cavity in a hard sphere fluid

This paper determines the excess free energy associated with the formation of a spherical cavity in a hard sphere fluid. The solvation free energy can be calculated by integration of the structural changes induced by inserting the cavity using a number of different exact thermodynamic pathways. We consider three such pathways, including a new density route derived here. Structural information about the nonuniform hard sphere fluid in the presence of a general external field is given by the recently developed hydrostatic linear response (HLR) integral equation. Use of the HLR results in the different pathways gives a generally accurate determination of the solvation free energy for cavities over a wide range of sizes, from zero to infinity. Results for a related method, the Gaussian Field Model, are also discussed.Comment: 10 pages, 4 figures; submitted to J. Chem. Phy