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Articles 871 - 900 of 1644
Full-Text Articles in Electrical and Computer Engineering
Orbital Angular Momentum Generation And Detection By Geometric-Phase Based Metasurfaces, Menglin L.N. Chen, Li (Lijun) Jun Jiang, Wei E.I. Sha
Orbital Angular Momentum Generation And Detection By Geometric-Phase Based Metasurfaces, Menglin L.N. Chen, Li (Lijun) Jun Jiang, Wei E.I. Sha
Electrical and Computer Engineering Faculty Research & Creative Works
We present a comprehensive review on the geometric phase based metasurfaces for orbital angular momentum (OAM) generation and detection. Thesemetasurfacesmanipulate the electromagnetic (EM) wave by introducing abrupt phase change, which is strongly dependent on the polarization state of incident EMwave and can be interpreted by geometric phase. Hence, the conventional bulk devices that based on the accumulated phase change along the optical path can be avoided.
Space-Angle Signal Processing Using A Modulated Scatter Array, Nasr Alkhafaji
Space-Angle Signal Processing Using A Modulated Scatter Array, Nasr Alkhafaji
Electrical and Computer Engineering PhD Day
A UHF signal processing technique is described in the Fourier Space-Angle domain that uses an array of scattering elements with reflection coefficients modulated at baseband. The illumination arrival angle distributes a UHF phase across the array, and the baseband modulator phase at each of the array elements determines the radiation angle of the desired scattered product. Undesired products are re-radiated in different directions. Example 1, 2 and 4 element arrays operate at 432 MHz array with 700 Hz modulation. 2 element arrays cancel one sideband, and the 4 element array cancels both the undesired sideband and 2nd order products. Reflection …
From A Locally Competitive Algorithm To Sensory Relevance Models, Walter Woods
From A Locally Competitive Algorithm To Sensory Relevance Models, Walter Woods
Electrical and Computer Engineering PhD Day
This poster addresses the development of a new Machine Learning (ML) mechanism, the Sensory Relevance Model (SRM), as a means of splitting information processing tasks into two sub-tasks with more intuitive properties. Specifically, SRMs are a front-end for other ML techniques, re-mapping the input data to a similar space with significantly more sparsity, achieved through the transformation and suppression of inputs irrelevant to the task. Prior work has attempted to reveal this information for Neural Networks (NNs) either as a post-processing step via saliency maps or through a simple masking of the input achieved with a dot product (so-called ``attention'' …
Coaxial Magnetic Gearbox With A Flux Concentration Halbach Rotor And Consequent Pole Rotor Typology, Ho Yin Wong
Coaxial Magnetic Gearbox With A Flux Concentration Halbach Rotor And Consequent Pole Rotor Typology, Ho Yin Wong
Electrical and Computer Engineering PhD Day
This paper presents the performance comparison between a flux concentration and triple permanent magnet consequent pole magnetic gear rotor typology with the same 7.5:1 gear ratio. The triple permanent magnet consequent pole magnetic gear is unique in that it can create torque change through two field heterodyning pole combinations. The performance with respect to volumetric and mass torque density is considered. It is shown that the flux concentration typology can operate with a higher mass and volumetric higher torque density, even when using a low flux concentration ratio rotor.
Magnetically Geared Lead Screw, Mojtaba Bahrami Kouhshahi
Magnetically Geared Lead Screw, Mojtaba Bahrami Kouhshahi
Electrical and Computer Engineering PhD Day
New magnetically geared lead screw (MGLS) is proposed in this work, which relies only on magnetic loading. Therefore it can provide speed and force conversion like its mechanical counterparts, but unlike the mechanical gears, a MGLS contact-less operation enables it to operate without lubrication and it can operate with low noise, and higher efficiency. Its reliance on magnetic loading also provides overload protection. MGLS combines the concepts of two previously proposed actuators named linear magnetic gear (LMG) and magnetic lead screw (MLS). However, unlike the LMG and MLS the translator on a MGLS does not contain magnets and therefore for …
Joint Pilot Allocation And Robust Transmission Design For Ultra-Dense User-Centric Tdd C-Ran With Imperfect Csi, Cunhua Pan, Hani Mehrpouyan, Yuanwei Liu, Maged Elkashlan, Arumugam Nallanathan
Joint Pilot Allocation And Robust Transmission Design For Ultra-Dense User-Centric Tdd C-Ran With Imperfect Csi, Cunhua Pan, Hani Mehrpouyan, Yuanwei Liu, Maged Elkashlan, Arumugam Nallanathan
Electrical and Computer Engineering Faculty Publications and Presentations
This paper considers the unavailability of complete channel state information (CSI) in ultra-dense cloud radio access networks (C-RANs). The user-centric cluster is adopted to reduce the computational complexity, while the incomplete CSI is considered to reduce the heavy channel training overhead, where only large-scale inter-cluster CSI is available. Channel estimation for intra-cluster CSI is also considered, where we formulate a joint pilot allocation and user equipment (UE) selection problem to maximize the number of admitted UEs with fixed number of pilots. A novel pilot allocation algorithm is proposed by considering the multi-UE pilot interference. Then, we consider robust beam-vector optimization …
Crumpling Effects And Specific Absorption Rates Of Flexible Amc Integrated Antennas, Sima Noghanian, Ala Alemaryeen
Crumpling Effects And Specific Absorption Rates Of Flexible Amc Integrated Antennas, Sima Noghanian, Ala Alemaryeen
Electrical Engineering Faculty Publications
This study presents the design of a wearable textile-based monopole antenna integrated with a flexible artificial magnetic conductor (AMC). The proposed design covers the industrial, scientific, and medical 2.45 GHz band. An AMC reflector is utilised to isolate the human body from undesired electromagnetic radiation and to improve the antenna radiation characteristics. A monopole antenna was backed with an AMC reflector and tested in free space and on human body models. The inclusion of an AMC reflector increases the front-to-back ratio and antenna gain in free space. On the human body models, the AMC antenna provides stable performance and a …
Review: Electric Field Driven Pumping In Microfluidic Device, Mohammad R. Hossan, Diganta Dutta, Nazmul Islam, Prashanta Dutta
Review: Electric Field Driven Pumping In Microfluidic Device, Mohammad R. Hossan, Diganta Dutta, Nazmul Islam, Prashanta Dutta
Electrical and Computer Engineering Faculty Publications
Pumping of fluids with precise control is one of the key components in a microfluidic device. The electric field has been used as one of the most popular and efficient nonmechanical pumping mechanism to transport fluids in microchannels from the very early stage of microfluidic technology development. This review presents fundamental physics and theories of the different microscale phenomena that arise due to the application of an electric field in fluids, which can be applied for pumping of fluids in microdevices. Specific mechanisms considered in this report are electroosmosis, AC electroosmosis, AC electrothermal, induced charge electroosmosis, traveling wave dielectrophoresis, and …
Drafting In Self-Timed Circuits, Christopher Cowan
Drafting In Self-Timed Circuits, Christopher Cowan
Electrical and Computer Engineering PhD Day
Intervals between data items propagating in self-timed circuits are controlled by handshake signals rather than by a clock. The sequence of handshakes can be abstracted as the movement of “tokens”. In many self-timed designs, a trailing token will catch up with a leading token, even when it trails by thousands of gate delays. Simulations in SPICE of a simple GasP circular FIFO reveal this effect. Contrary to earlier work, we find the cause of drafting to be charge stored on an isolated node between two series transistors. This mechanism occurs in many decision gates that implement a logical AND. The …
Nonlinear Control Framework For Gimbal And Multirotor In Target Tracking, Jae Hun Lee
Nonlinear Control Framework For Gimbal And Multirotor In Target Tracking, Jae Hun Lee
Theses and Dissertations
This thesis presents some existing gimbal and UAV control algorithms as well as novel algorithms developed as the extensions of the existing ones. The existing image-based visual servoing algorithms for both gimbal and UAV require the depth information to the object of interest. The depth information is not measurable when only a monocular camera is used for tracking. This thesis is the result of contemplation to the question: how can the necessity for a depth measurement be removed? A novel gimbal algorithm using adaptive control is developed and presented with simulation and hardware results. Although the estimated depth using the …
A Flexible Fpga-Assisted Framework For Remote Attestation Of Internet Connected Embedded Devices, Jared Russell Patten
A Flexible Fpga-Assisted Framework For Remote Attestation Of Internet Connected Embedded Devices, Jared Russell Patten
Theses and Dissertations
Embedded devices permeate our every day lives. They exist in our vehicles, traffic lights, medical equipment, and infrastructure controls. In many cases, improper functionality of these devices can present a physical danger to their users, data or financial loss, etc. Improper functionality can be a result of software or hardware bugs, but now more than ever, is often the result of malicious compromise and tampering, or as it is known colloquially "hacking". We are beginning to witness a proliferation of cyber-crime, and as more devices are built with internet connectivity (in the so called "Internet of Things"), security should be …
A New Wiretap Channel Model And Its Strong Secrecy Capacity, Mohamed Nafea, Aylin Yener
A New Wiretap Channel Model And Its Strong Secrecy Capacity, Mohamed Nafea, Aylin Yener
Electrical and Computer Engineering Faculty Research & Creative Works
In this paper, a new wiretap channel model is proposed, where the legitimate transmitter and receiver communicate over a discrete memoryless channel. The wiretapper has perfect access to a fixed-length subset of the transmitted code word symbols of her choosing. Additionally, she observes the remainder of the transmitted symbols through a discrete memoryless channel. This new model subsumes the classical wiretap channel and wiretap channel II with noisy main channel as its special cases and is termed as the generalized wiretap channel for that reason. The strong secrecy capacity of the proposed channel model is identified. Achievability is established by …
Deep Learning Nuclei Detection In Digitized Histology Images By Superpixels, Sudhir Sornapudi, R. Joe Stanley, William V. Stoecker, Haidar Almubarak, Rodney Long, Sameer Antani, George Thoma, Rosemary Zuna, Shelliane R. Frazier
Deep Learning Nuclei Detection In Digitized Histology Images By Superpixels, Sudhir Sornapudi, R. Joe Stanley, William V. Stoecker, Haidar Almubarak, Rodney Long, Sameer Antani, George Thoma, Rosemary Zuna, Shelliane R. Frazier
Electrical and Computer Engineering Faculty Research & Creative Works
Background: Advances in image analysis and computational techniques have facilitated automatic detection of critical features in histopathology images. Detection of nuclei is critical for squamous epithelium cervical intraepithelial neoplasia (CIN) classification into normal, CIN1, CIN2, and CIN3 grades.
Methods: In this study, a deep learning (DL)-based nuclei segmentation approach is investigated based on gathering localized information through the generation of superpixels using a simple linear iterative clustering algorithm and training with a convolutional neural network.
Results: The proposed approach was evaluated on a dataset of 133 digitized histology images and achieved an overall nuclei detection (object-based) accuracy of 95.97%, with …
Standalone Wireless Humming Music Synthesizer (Humsynth), Anna Shabrova, Chad Carlson
Standalone Wireless Humming Music Synthesizer (Humsynth), Anna Shabrova, Chad Carlson
Electrical Engineering
The HumSynth is a voice controlled music synthesizer. Unlike the commonly found voice modulators on the market, this product uses a vocal input in place of the traditional keyboard interface. This allows the instrument to serve as an intuitive learning tool for those not proficient in music theory. By using their voice, the consumer is able to get a feel for different scales and keys additionally bypassing the physical learning curve of learning a traditional instrument. By the date of submission this project performs basic functions in recreating the pitch input but does not meet the full specs. The enclosure …
Effects Of Position Uncertainty On Passive Multistatic Sar Resolution Measures, Duy K. Nguyen
Effects Of Position Uncertainty On Passive Multistatic Sar Resolution Measures, Duy K. Nguyen
Theses and Dissertations
Given numerous exciting potentials including substantial cost savings, passive multistatic synthetic aperture radar (PMSAR) has been gaining more attention in the radar community. However, PMSAR research efforts such as emitters of opportunity selection thus far have not considered radar position uncertainty. Seeking to fill this knowledge gap, we run Monte Carlo trials to simulate transmitter position uncertainty and examine the effects on the multistatic resolution of radar sets of different geometries, center frequencies and bandwidths. By monitoring the 3dB contour of the multistatic point spread function, we find that the multistatic resolution worsens on average and the data spread grows …
Influencing Effective Electrical Distribution Modernization Through Advanced Metering, Jared R. Erickson
Influencing Effective Electrical Distribution Modernization Through Advanced Metering, Jared R. Erickson
Theses and Dissertations
In 2013, the Department of Defense (DoD) required its services to implement advanced metering for the purpose of reducing energy usage (Department of Defense, 2013). Additionally, the DoD has aimed to improve its ability to assure a continuous energy supply to all of its installations. This study investigated processes for applying advanced meters on Air Force bases to increase energy assurance. This study also identified strategies for using advanced meters to influence infrastructure funding. This was accomplished through the use of extensive advanced meter data. The data was analyzed for outages and a procedure was created to locate outages in …
Evolutionary Learning Of Boosted Features For Visual Inspection Automation, Meng Zhang
Evolutionary Learning Of Boosted Features For Visual Inspection Automation, Meng Zhang
Theses and Dissertations
Feature extraction is one of the major challenges in object recognition. Features that are extracted from one type of objects cannot always be used directly for a different type of objects, therefore limiting the performance of feature extraction. Having an automatic feature learning algorithm could be a big advantage for an object recognition algorithm. This research first introduces several improvements on a fully automatic feature construction method called Evolution COnstructed Feature (ECO-Feature). These improvements are developed to construct more robust features and make the training process more efficient than the original version. The main weakness of the original ECO-Feature algorithm …
Towards Tools For Achieving Third-Party Ip Assurance, Sean Talbot Jensen
Towards Tools For Achieving Third-Party Ip Assurance, Sean Talbot Jensen
Theses and Dissertations
Intellectual Property (IP) is used to speed up the design process and save money. The use of IP and complex CAD tools reduce visibility into the design and what is actually happening during synthesis and implementation. All of the complexity makes it easier for an attacker to insert malicious logic or tamper with the design in ways that are difficult to detect. Not very much work has been done towards the creation of tools to facilitate the safe use of 3rd-party IP. This work presents Physical and Functional Assurance, two approaches that aim to accomplish this task through physically and …
Bandgap Profiling In Cigs Solar Cells Via Valence Electron Energy-Loss Spectroscopy, Julia I. Deitz, Shankar Karki, Sylvain X. Marsillac, Tyler J. Grassman
Bandgap Profiling In Cigs Solar Cells Via Valence Electron Energy-Loss Spectroscopy, Julia I. Deitz, Shankar Karki, Sylvain X. Marsillac, Tyler J. Grassman
Electrical & Computer Engineering Faculty Publications
A robust, reproducible method for the extraction of relative bandgap trends from scanning transmission electron microscopy (STEM) based electron energy-loss spectroscopy (EELS) is described. The effectiveness of the approach is demonstrated by profiling the bandgap through a CuIn1-xGaxSe2 solar cell that possesses intentional Ga/(In + Ga) composition variation. The EELS-determined bandgap profile is compared to the nominal profile calculated from compositional data collected via STEM-based energy dispersive X-ray spectroscopy. The EELS based profile is found to closely track the calculated bandgap trends, with only a small, fixed offset difference. This method, which is particularly advantageous …
Compiler-Assisted Software Fault Tolerance For Microcontrollers, Matthew Kendall Bohman
Compiler-Assisted Software Fault Tolerance For Microcontrollers, Matthew Kendall Bohman
Theses and Dissertations
Commercial off-the-shelf (COTS) microcontrollers can be useful for non-critical processing on spaceborne platforms. Many of these microprocessors are inexpensive and consume little power. However, the software running on these processors is vulnerable to radiation upsets, which can cause unpredictable program execution or corrupt data. Space missions cannot allow these errors to interrupt functionality or destroy gathered data. As a result, several techniques have been developed to reduce the effect of these upsets. Some proposed techniques involve altering the processor hardware, which is impossible for a COTS device. Alternately, the software running on the microcontroller can be modified to detect or …
Techniques For Low-Latency In Software-Defined Radio-Based Networks, Daniel D. Hart
Techniques For Low-Latency In Software-Defined Radio-Based Networks, Daniel D. Hart
Theses and Dissertations
Decreased budgets have pushed the United States Air Force towards using existing systems in new ways. The use of unmanned aerial vehicle swarms is one example of reuse of existing systems. One problem with the increased utilization of these swarms is the congestion of the electromagnetic spectrum. Software-defined or cognitive radios have been proposed as a basis for a potential robust communications solution. The present research aims to develop and test a genetic algorithm-based cognitive engine to begin looking at real-time engines that could be used in future swarms. Here, latency is the optimization objective of primary importance. In testing …
Navigation Using Vector And Tensor Measurements Of The Earth's Magnetic Anomaly Field, Lauren A. Mount
Navigation Using Vector And Tensor Measurements Of The Earth's Magnetic Anomaly Field, Lauren A. Mount
Theses and Dissertations
This research explores the viability of using a navigation system that relies on measurements of the magnetic anomaly field as an alternative to GPS navigation. Previous research has been conducted on developing a navigation system using the intensity of the Earth's magnetic anomaly field as an alternative signal. This research focuses on using vector and tensor measurements, as opposed to scalar measurements of the anomaly field, as a means of obtaining accurate position and orientation solutions. This paper presents two navigation systems. The first uses an Extended Kalman Filter (EKF) with vector measurements of the magnetic anomaly field to aid …
Efficient Dynamic Centrality Metrics For Election Advertising - A Case Study, Soheil Eshghi, Leandros Tassiulas
Efficient Dynamic Centrality Metrics For Election Advertising - A Case Study, Soheil Eshghi, Leandros Tassiulas
Yale Day of Data
In prior work [1], we have shown how advertising channels should be chosen by a budget-constrained electoral campaign. In this poster, we apply the resulting proposed algorithm to the MIT Social Evolution [2] data-set (N=84), which captured political discussions, inclinations, and voting behaviors around the 2008 US Presidential Election within a student dorm. We compare the resulting centrality metrics developed from our algorithm (which have a direct mapping to optimal channel choice decisions) against more traditional static centralities, and show how employing them leads to more votes.
[1] Eshghi, S., Preciado, V.M., Sarkar, S., Venkatesh, S.S., Zhao, Q., D'Souza, R. …
Synthesis And Electrochemical Properties Of Nickel-Rich Cathode Material Lini0.6Co0.2Mn0.2O2 With High Initial Coulombic Efficiency, Xiao-Yun Guan, Chao-Yu Hong, Jian-Ping Zhu, Wei-Li Wang, Yi-Xiao Li, Yong Yang
Synthesis And Electrochemical Properties Of Nickel-Rich Cathode Material Lini0.6Co0.2Mn0.2O2 With High Initial Coulombic Efficiency, Xiao-Yun Guan, Chao-Yu Hong, Jian-Ping Zhu, Wei-Li Wang, Yi-Xiao Li, Yong Yang
Journal of Electrochemistry
Nickel-rich cathode materials LiNi0.6Co0.2Mn0.2O2(NCM622)were synthesized by a co-precipitation-solid state sintering method at different temperatures. The structure, morphology and electrochemical performance of the as-prepared materials were investigated by X-ray powder diffraction (XRD) /Rietveld refinement, scanning electron microscope (SEM) and electrochemical experiments. It is found that NCM622 calcined at 800 ℃ showed the lowest degree of cation disorder (~1.97%) with a high initial Coulombic effiency of 92.2% and the capacity retention of 81.4% after 100 cycles.
Preparation And Properties Of Nanocrystalline Nickel By Pulse Electrodeposition On Glassy Carbon, Ya-Li Zhang, Ming-Xian Zhang, Teng-Da Guo, Yan-Jie Zhao, Sheng-Xue Yu
Preparation And Properties Of Nanocrystalline Nickel By Pulse Electrodeposition On Glassy Carbon, Ya-Li Zhang, Ming-Xian Zhang, Teng-Da Guo, Yan-Jie Zhao, Sheng-Xue Yu
Journal of Electrochemistry
Nanocrystalline nickel exhibits electrocatalytic activity for methanol. In this work, nanocrystalline nickel coating has been obtained by using the double pulse method on glassy carbon materials surface. The influence of pulse conditions on the grain size of nanocrystalline nicke was studied. The concentrations and compositions of the plating bath were 300 g·L-1 NiSO4·6H2O,45 g·L-1 NiCl2·6H2O,40 g·L-1 H3BO3,5 g·L-1 C7H5NO3S,and 0.05 g·L-1 C12H25NaO4S. The optimal pulse process parameters including the average pulse …
Preparation And Electrocatalytic Activity Of Nitrogen-Doping Tungsten Carbide Catalyst, Pian-Pian Yang, Li-Zhen Huang, Ying-Ying Li, Mei-Qin Shi, Chun-An Ma
Preparation And Electrocatalytic Activity Of Nitrogen-Doping Tungsten Carbide Catalyst, Pian-Pian Yang, Li-Zhen Huang, Ying-Ying Li, Mei-Qin Shi, Chun-An Ma
Journal of Electrochemistry
Tungsten carbide (WC) is a promising electrocatalyst, however, its electrocatalytic activity is far inferior to Pt and Pt-group metal. In this work, nitrogen-doping tungsten carbide (WN|WC) catalysts with a nanoplate morphology were prepared via the tungsten nitride (WN) as the precursor and sodium tungstate as the tungsten source. The SEM and TEM results indicated that carbon atoms entered into the WN lattice to form the hexagonal close packed WC phase. In this way, an atomic scale heterostructure involving the closely linked interfaces between WN and WC was created . The XRD data confirmed that the cubic crystal structure of WN …
Intrinsic Kinetic Properties Of Ternary Material For Lithium Ion Batteries Assessed By Single Particle Microelectrode, Yi-Min Wei
Journal of Electrochemistry
Electrochemical performances such as capacity, rate, cycle and thermal stability of the nickel (Ni), cobalt (Co) and manganese (Mn) ternary cathode material, LiNixCoyMnzO2 (x + y + z = 1), are significantly influenced by the proportion of Ni, Co, and Mn elements. To obtain higher specific capacity, LiNi0.6Co0.2Mn 0.2O2 (NCM622) and LiNi0.8Co0.1Mn0.1O2 (NCM811) with high amounts of Ni element were employed for the lithium ion batteries. By now, many studies have been focusing on the thermal and cycling stabilities of …
Applications Of Electrospinning In Lithium-Air Batteries, Yue Fu, Jin Wang, Hai-Yang Yu, Jing-Hua Tian, Rui-Zhi Yang
Applications Of Electrospinning In Lithium-Air Batteries, Yue Fu, Jin Wang, Hai-Yang Yu, Jing-Hua Tian, Rui-Zhi Yang
Journal of Electrochemistry
Lithium-air batteries have received increasing attention as the “ultimate batteries” due to their extremely high theoretical capacity and environment friendly. However, there are still many obstacles hindering the wide applications of the lithium-air batteries owing to the limitations of large charge-discharge over-potentials, poor Columbic efficiency and cycling performance. Exploring of high-efficiency, low-cost and eco-friendly catalysts is one of the vital issues for lithium-air batteries. Meanwhile, electrospinning techniques have been under substantial developments due to its simple fabrication process, high efficiency and production. This paper reviews the recent progresses in fabrications of various novel catalysts for lithium-air batteries with electrospinning techniques, …
The 3d Stress-Tensor Bootstrap, Anatoly Dymarsky, Filip Kos, Petr Kravchuk, David Poland, David Simmons-Duffin
The 3d Stress-Tensor Bootstrap, Anatoly Dymarsky, Filip Kos, Petr Kravchuk, David Poland, David Simmons-Duffin
Physics and Astronomy Faculty Publications
We study the conformal bootstrap for 4-point functions of stress tensors in parity-preserving 3d CFTs. To set up the bootstrap equations, we analyze the constraints of conformal symmetry, permutation symmetry, and conservation on the stress-tensor 4-point function and identify a non-redundant set of crossing equations. Studying these equations numerically using semidefinite optimization, we compute bounds on the central charge as a function of the independent coefficient in the stress-tensor 3-point function. With no additional assumptions, these bounds numerically reproduce the conformal collider bounds and give a general lower bound on the central charge. We also study the effect of gaps …
Three Dimensional View Of Arbitrary Q Syk Models, Sumit R. Das, Aminik Ghosh, Antal Jevicki, Kenta Suzuki
Three Dimensional View Of Arbitrary Q Syk Models, Sumit R. Das, Aminik Ghosh, Antal Jevicki, Kenta Suzuki
Physics and Astronomy Faculty Publications
In [15] it was shown that the spectrum and bilocal propagator of SYK model with four fermion interactions can be realized as a three dimensional model in AdS2 ×S1/Z2 with nontrivial boundary conditions in the additional dimension. In this paper we show that a similar picture holds for generalizations of the SYK model with q-fermion interactions. The 3D realization is now given on a space whose metric is conformal to AdS2 × S1/Z2 and is subject to a non-trivial potential in addition to a delta function at the center of …