Manifolds with nonnegative isotropic curvature

We prove that if (M-n, g), n >= 4, is a compact, orientable, locally irreducible Riemannian manifold with nonnegative isotropic curvature,then one of the following possibilities hold: (i) M admits a metric with positive isotropic curvature. (ii) (M, g) is isometric to a locally symmetric space. (iii) (M, g) is Kahler and biholomorphic to CPn/2. (iv) (M, g) is quaternionic-Kahler. This is implied by the following result: Let (M-2n, g) be a compact, locally irreducible Kahler manifold with nonnegative isotropic curvature. Then either M is biholomorphic to CPn or isometric to a compact Hermitian symmetric space. This answers a question of Micallef and Wang in the affirmative. The proof is based on the recent work of Brendle and Schoen on the Ricci flow.

New algorithms of the Q-learning type

We propose two algorithms for Q-learning that use the two-timescale stochastic approximation methodology. The first of these updates Q-values of all feasible state–action pairs at each instant while the second updates Q-values of states with actions chosen according to the ‘current’ randomized policy updates. A proof of convergence of the algorithms is shown. Finally, numerical experiments using the proposed algorithms on an application of routing in communication networks are presented on a few different settings.

Quench dynamics and defect production in the Kitaev and extended Kitaev models

We study quench dynamics and defect production in the Kitaev and the extended Kitaev models. For the Kitaev model in one dimension, we show that in the limit of slow quench rate, the defect density n∼1/√τ, where 1/τ is the quench rate. We also compute the defect correlation function by providing an exact calculation of all independent nonzero spin correlation functions of the model. In two dimensions, where the quench dynamics takes the system across a critical line, we elaborate on the results of earlier work [K. Sengupta, D. Sen, and S. Mondal, Phys. Rev. Lett. 100, 077204 (2008)] to discuss the unconventional scaling of the defect density with the quench rate. In this context, we outline a general proof that for a d-dimensional quantum model, where the quench takes the system through a d−m dimensional gapless (critical) surface characterized by correlation length exponent ν and dynamical critical exponent z, the defect density n∼1/τmν/(zν+1). We also discuss the variation of the shape and spatial extent of the defect correlation function with both the rate of quench and the model parameters and compute the entropy generated during such a quenching process. Finally, we study the defect scaling law, entropy generation and defect correlation function of the two-dimensional extended Kitaev model.

Uniqueness of Walkup's 9-vertex 3-dimensional Klein bottle

Via a computer search, Altshuler and Steinberg found that there are 1296+1 combinatorial 3-manifolds on nine vertices, of which only one is non-sphere. This exceptional 3-manifold View the MathML source triangulates the twisted S2-bundle over S1. It was first constructed by Walkup. In this paper, we present a computer-free proof of the uniqueness of this non-sphere combinatorial 3-manifold. As opposed to the computer-generated proof, ours does not require wading through all the 9-vertex 3-spheres. As a preliminary result, we also show that any 9-vertex combinatorial 3-manifold is equivalent by proper bistellar moves to a 9-vertex neighbourly 3-manifold.

Ultra high throughput sequencing excludes MDH1 as candidate gene for RP28-linked retinitis pigmentosa

Purpose: Mutations in IDH3B, an enzyme participating in the Krebs cycle, have recently been found to cause autosomal recessive retinitis pigmentosa (arRP). The MDH1 gene maps within the RP28 arRP linkage interval and encodes cytoplasmic malate dehydrogenase, an enzyme functionally related to IDH3B. As a proof of concept for candidate gene screening to be routinely performed by ultra high throughput sequencing (UHTs), we analyzed MDH1 in a patient from each of the two families described so far to show linkage between arRP and RP28. Methods: With genomic long-range PCR, we amplified all introns and exons of the MDH1 gene (23.4 kb). PCR products were then sequenced by short-read UHTs with no further processing. Computer-based mapping of the reads and mutation detection were performed by three independent software packages. Results: Despite the intrinsic complexity of human genome sequences, reads were easily mapped and analyzed, and all algorithms used provided the same results. The two patients were homozygous for all DNA variants identified in the region, which confirms previous linkage and homozygosity mapping results, but had different haplotypes, indicating genetic or allelic heterogeneity. None of the DNA changes detected could be associated with the disease. Conclusions: The MDH1 gene is not the cause of RP28-linked arRP. Our experimental strategy shows that long-range genomic PCR followed by UHTs provides an excellent system to perform a thorough screening of candidate genes for hereditary retinal degeneration.

Asymmetric cross-coupled differential pair configuration to realize neuron activation function and its derivative

In this brief, we present a new circuit technique to generate the sigmoid neuron activation function (NAF) and its derivative (DNAF). The circuit makes use of transistor asymmetry in cross-coupled differential pair to obtain the derivative. The asymmetry is introduced through external control signal, as and when required. This results in the efficient utilization of the hard-ware by realizing NAF and DNAF using the same building blocks. The operation of the circuit is presented in the subthreshold region for ultra low-power applications. The proposed circuit has been experimentally prototyped and characterized as a proof of concept on the 1.5-mum AMI technology.

On the performance analysis of a class of neuron circuits

A new performance metric, Peak-Error Ratio (PER) has been presented to benchmark the performance of a class of neuron circuits to realize neuron activation function (NAF) and its derivative (DNAF). Neuron circuits, biased in subthreshold region, based on the asymmetric cross-coupled differential pair configuration and conventional configuration of applying small external offset voltage at the input have been compared on the basis of PER. It is shown that the technique of using transistor asymmetry in a cross-coupled differential pair performs on-par with that of applying external offset voltage. The neuron circuits have been experimentally prototyped and characterized as a proof of concept on the 1.5 mu m AMI technology.

Solving MDPs using two-timescale simulated annealing with multiplicative weights

We develop extensions of the Simulated Annealing with Multiplicative Weights (SAMW) algorithm that proposed a method of solution of Finite-Horizon Markov Decision Processes (FH-MDPs). The extensions developed are in three directions: a) Use of the dynamic programming principle in the policy update step of SAMW b) A two-timescale actor-critic algorithm that uses simulated transitions alone, and c) Extending the algorithm to the infinite-horizon discounted-reward scenario. In particular, a) reduces the storage required from exponential to linear in the number of actions per stage-state pair. On the faster timescale, a 'critic' recursion performs policy evaluation while on the slower timescale an 'actor' recursion performs policy improvement using SAMW. We give a proof outlining convergence w.p. 1 and show experimental results on two settings: semiconductor fabrication and flow control in communication networks.

Optimal threshold policies for admission control in communication networks via discrete parameter stochastic approximation

The problem of admission control of packets in communication networks is studied in the continuous time queueing framework under different classes of service and delayed information feedback. We develop and use a variant of a simulation based two timescale simultaneous perturbation stochastic approximation (SPSA) algorithm for finding an optimal feedback policy within the class of threshold type policies. Even though SPSA has originally been designed for continuous parameter optimization, its variant for the discrete parameter case is seen to work well. We give a proof of the hypothesis needed to show convergence of the algorithm on our setting along with a sketch of the convergence analysis. Extensive numerical experiments with the algorithm are illustrated for different parameter specifications. In particular, we study the effect of feedback delays on the system performance.

On the performance analysis of a class of neuron circuits

A new performance metric, Peak-Error Ratio (PER) has been presented to benchmark the performance of a class of neuron circuits to realize neuron activation function (NAF) and its derivative (DNAF). Neuron circuits, biased in subthreshold region, based on the asymmetric cross-coupled differential pair configuration and conventional configuration of applying small external offset voltage at the input have been compared on the basis of PER. It is shown that the technique of using transistor asymmetry in a cross-coupled differential pair performs on-par with that of applying external offset voltage. The neuron circuits have been experimentally prototyped and characterized as a proof of concept on the 1.5 mu m AMI technology.

Ultra high throughput sequencing excludes MDH1 as candidate gene for RP28-linked retinitis pigmentosa

Purpose: Mutations in IDH3B, an enzyme participating in the Krebs cycle, have recently been found to cause autosomal recessive retinitis pigmentosa (arRP). The MDH1 gene maps within the RP28 arRP linkage interval and encodes cytoplasmic malate dehydrogenase, an enzyme functionally related to IDH3B. As a proof of concept for candidate gene screening to be routinely performed by ultra high throughput sequencing (UHTs), we analyzed MDH1 in a patient from each of the two families described so far to show linkage between arRP and RP28. Methods: With genomic long-range PCR, we amplified all introns and exons of the MDH1 gene (23.4 kb). PCR products were then sequenced by short-read UHTs with no further processing. Computer-based mapping of the reads and mutation detection were performed by three independent software packages. Results: Despite the intrinsic complexity of human genome sequences, reads were easily mapped and analyzed, and all algorithms used provided the same results. The two patients were homozygous for all DNA variants identified in the region, which confirms previous linkage and homozygosity mapping results, but had different haplotypes, indicating genetic or allelic heterogeneity. None of the DNA changes detected could be associated with the disease.

A generalized mathematical theory and experimental verification of proportional notches

A theory and generalized synthesis procedure is advocated for the design of weir notches and orifice-notches having a base in any given shape, to a depth a, such that the discharge through it is proportional to any singular monotonically-increasing function of the depth of flow measured above a certain datum. The problem is reduced to finding an exact solution of a Volterra integral equation in Abel form. The maximization of the depth of the datum below the crest of the notch is investigated. Proof is given that for a weir notch made out of one continuous curve, and for a flow proportional to the mth power of the head, it is impossible to bring the datum lower than (2m − 1)a below the crest of the notch. A new concept of an orifice-notch, having discontinuity in the curve and a division of flow into two distinct portions, is presented. The division of flow is shown to have a beneficial effect in reducing the datum below (2m − 1)a from the crest of the weir and still maintaining the proportionality of the flow. Experimental proof with one such orifice-notch is found to have a constant coefficient of discharge of 0.625. The importance of this analysis in the design of grit chambers is emphasized.

Syntheses of the trans,meso- and racemic C11 acids, degradation products of diterpene acids

Syntheses of the isomers of the C11 acid, 1(a),3(a)- dimethylcyclohexane-1 (e),2(e),3(e)-tricarboxylic acid (A) and 1(a),3(e)-dimethylcyclohexane-1(e),2(e),3(a)-tricarboxylic acid (B), the latter by two different routes, are reported. Two of the four possible isomers of the precursor triester, trimethyl 1-methylcyclohexane-1,2,3-tricarboxylate, on individual methylation followed by hydrolysis, gave the trans,meso-acid (A), identified by comparison with an authentic sample, and the cis,trans-form (B) whose structure and configuration were proved by comparison with a specimen obtained by the unambiguous and highly stereoselective second synthesis. This demonstrated that methylation of the triester isomers occurs stereospecifically and exclusively at C-3. In the second sequence, it has been possible to assign definite conformations to four key intermediates and the final product, directly from n.m.r. spectra, from changes in these spectra accompanying specific steps, and from chemical evidence. Comparison of the n.m.r. spectra of the isomeric triesters (A) and (B) has provided unequivocal proof of the accepted trans,meso configuration for the abietic acid degradation product (A).

Design of incentive compatible protocols for wireless networks: A game theoretic approach

In this thesis work, we design rigorous and efficient protocols/mechanisms for different types of wireless networks using a mechanism design [1] and game theoretic approach [2]. Our work can broadly be viewed in two parts. In the first part, we concentrate on ad hoc wireless networks [3] and [4]. In particular, we consider broadcast in these networks where each node is owned by independent and selfish users. Being selfish, these nodes do not forward the broadcast packets. All existing protocols for broadcast assume that nodes forward the transit packets. So, there is need for developing new broadcast protocols to overcome node selfishness. In our paper [5], we develop a strategy proof pricing mechanism which we call immediate predecessor node pricing mechanism (IPNPM) and an efficient new broadcast protocol based on IPNPM. We show the efficacy of our proposed broadcast protocol using simulation results.

An efficient pricing based protocol for broadcasting in wireless ad hoc networks

In many applications of wireless ad hoc networks, wireless nodes are owned by rational and intelligent users. In this paper, we call nodes selfish if they are owned by independent users and their only objective is to maximize their individual goals. In such situations, it may not be possible to use the existing protocols for wireless ad hoc networks as these protocols assume that nodes follow the prescribed protocol without deviation. Stimulating cooperation among these nodes is an interesting and challenging problem. Providing incentives and pricing the transactions are well known approaches to stimulate cooperation. In this paper, we present a game theoretic framework for truthful broadcast protocol and strategy proof pricing mechanism called Immediate Predecessor Node Pricing Mechanism (IPNPM). The phrase strategy proof here means that truth revelation of cost is a weakly dominant-strategy (in game theoretic terms) for each node. In order to steer our mechanism-design approach towards practical implementation, we compute the payments to nodes using a distributed algorithm. We also propose a new protocol for broadcast in wireless ad hoc network with selfish nodes based on IPNPM. The features of the proposed broadcast protocol are reliability and a significantly reduced number of packet forwards compared to the number of network nodes, which in turn leads to less system-wide power consumption to broadcast a single packet. Our simulation results show the efficacy of the proposed broadcast protocol.