Symmetric Square Research Articles

Higher power moments of symmetric square L-function

Let f(z) be a Maass cusp form for the full modular group SL2(Z). And let λsym2f(n) be the n-th coefficient of symmetric square L-function associated with f(z). For A≥2 fixed we define M(A) as the infimum of all numbers M such that∫1T|L(1/2+it,sym2f)|Adt≪fTM+ϵ, where L(s,sym2f) is the automorphic L-function attached to f. In this paper, we will establish the upper bounds of M(A) and get the zero density estimates for L(s,sym2f).

The quantitative distribution of Hecke eigenvalues of Maass forms

Let f be a normalized Hecke–Maass cusp form of weight zero for the group \(SL_2({\mathbb {Z}})\). This article presents several quantitative results about the distribution of Hecke eigenvalues of f. Applications to the \(\Omega _{\pm }\)-results for the Hecke eigenvalues of f and its symmetric square sym\(^2(f)\) are also given.

Nanopore-Patterned CuSe Drives the Realization of the PbSe-CuSe Lateral Heterostructure.

Monolayer PbSe has been predicted to be a two-dimensional (2D) topological crystalline insulator (TCI) with crystalline symmetry-protected Dirac-cone-like edge states. Recently, few-layered epitaxial PbSe has been grown on the SrTiO3 substrate successfully, but the corresponding signature of the TCI was only observed for films not thinner than seven monolayers, largely due to interfacial strain. Here, we demonstrate a two-step method based on molecular beam epitaxy for the growth of the PbSe-CuSe lateral heterostructure on the Cu(111) substrate, in which we observe a nanopore-patterned CuSe layer that acts as the template for lateral epitaxial growth of PbSe. This further results in a PbSe-CuSe lateral heterostructure with an atomically sharp interface. Scanning tunneling microscopy and spectroscopy measurements reveal a fourfold symmetric square lattice of such PbSe with a quasi-particle band gap of 1.8 eV, a value highly comparable with the theoretical value of freestanding PbSe. The weak monolayer-substrate interaction is further supported by both density functional theory (DFT) and projected crystal orbital Hamilton population, with the former predicting the monolayer's anti-bond state to reside below the Fermi level. Our work demonstrates a practical strategy to fabricate a high-quality in-plane heterostructure, involving a monolayer TCI, which is viable for further exploration of the topology-derived quantum physics and phenomena in the monolayer limit.

Analytic periods via twisted symmetric squares

We study the symmetric square of Picard-Fuchs operators of genus one curves and the thereby induced generalized Clausen identities. This allows the computation of analytic expressions for the periods of all one-parameter K3 manifolds in terms of elliptic integrals. The resulting expressions are globally valid throughout the moduli space and allow the explicit inversion of the mirror map and the exact computation of distances, useful for checks of the Swampland Distance Conjecture. We comment on the generalization to multi-parameter models and provide a two-parameter example.

Sparse Approximation of the Precision Matrices for the Wide-Swath Altimeters

The upcoming technology of wide-swath altimetry from space will deliver a large volume of data on the ocean surface at unprecedentedly high spatial resolution. These data are contaminated by errors caused by the uncertainties in the geometry and orientation of the on-board interferometer and environmental conditions, such as sea surface roughness and atmospheric state. Being highly correlated along and across the swath, these errors present a certain challenge for accurate processing in operational data assimilation centers. In particular, the error covariance matrix R of the Surface Water and Ocean Topography (SWOT) mission may contain trillions of elements for a transoceanic swath segment at kilometer resolution, and this makes its handling a computationally prohibitive task. Analysis presented here shows, however, that the SWOT precision matrix R−1 and its symmetric square root can be efficiently approximated by a sparse block-diagonal matrix within an accuracy of a few per cent. A series of observational system experiments with simulated data shows that such approximation comes at the expense of a relatively minor reduction in the assimilation accuracy, and, therefore, could be useful in operational systems targeted at the retrieval of submesoscale variability of the ocean surface.

Magneto-Nanofluid Flow via Mixed Convection Inside E-Shaped Square Chamber

Nanofluids play a crucial role in the augmentation of heat transfer in several energy systems. They exhibit better thermal conductivity and physical strength compared to normal fluids. Here, we conduct an evaluative investigation of the magnetized flow of water–copper nanofluid and its heat transport inside a symmetrical E-shaped square chamber via mixed convective impact with a heated corner. The chamber was constructed symmetrically with an inclined magnetic field strength, and the upper surface of the chamber was isolated and set to move at a fixed velocity. The heated corner was set at a fixed hot temperature in both the left and lower directions. The right side was maintained at a fixed cold temperature, while the remaining portions of the left and lower parts were isolated. The investigation was implemented computationally, solving each of the energy and Navier–Stokes models via the application of a symmetrical finite volume method. The following topics have been addressed in this study: the consequences of the magnetic field, the volumetric fraction of nanoparticles, the heat generation–absorption parameters, and the effects of heat-source length and Richardson number on the fluid comportment and heat transport. The outputs of this symmetric study enabled us to arrive at the following derivation: the magnetic field reduces the fluid circulation inside the E-shaped square chamber. The augmentation of the Richardson number leads to an increase in the heat transfer. Moreover, the decrease in heat generation coefficient lowers the nanofluid temperature and weakens the flow fields.

Construction and Functionalization of Highly Strained N -Doped Zigzag Hydrocarbon Belts

Construction and Functionalization of Highly Strained <i>N</i> -Doped Zigzag Hydrocarbon Belts

Spin-Ruijsenaars, q-Deformed Haldane–Shastry and Macdonald Polynomials

We study the q-analogue of the Haldane–Shastry model, a partially isotropic (xxz-like) long-range spin chain that by construction enjoys quantum-affine (really: quantum-loop) symmetries at finite system size. We derive the pairwise form of the Hamiltonian, found by one of us building on work of D. Uglov, via ‘freezing’ from the affine Hecke algebra. To this end we first obtain explicit expressions for the spin-Macdonald operators of the (trigonometric) spin-Ruijsenaars model. Through freezing these give rise to the higher Hamiltonians of the spin chain, including another Hamiltonian of the opposite ‘chirality’. The sum of the two chiral Hamiltonians has a real spectrum also when |mathsf {q}|=1, so in particular when q is a root of unity. For generic mathsf {q} the eigenspaces are known to be labelled by ‘motifs’. We clarify the relation between these patterns and the corresponding degeneracies (multiplicities) in the crystal limit textsf {q}rightarrow infty . For each motif we obtain an explicit expression for the exact eigenvector, valid for generic q, that has (‘pseudo’ or ‘l-’) highest weight in the sense that, in terms of the operators from the monodromy matrix, it is an eigenvector of A and D and annihilated by C. It has a simple component featuring the ‘symmetric square’ of the q-Vandermonde polynomial times a Macdonald polynomial—or more precisely its quantum spherical zonal special case. All other components of the eigenvector are obtained from this through the action of the Hecke algebra, followed by ‘evaluation’ of the variables to roots of unity. We prove that our vectors have highest weight upon evaluation. Our description of the exact spectrum is complete. The entire model, including the quantum-loop action, can be reformulated in terms of polynomials. Our main tools are the Y-operators from the affine Hecke algebra. From a more mathematical perspective the key step in our diagonalisation is as follows. We show that on a subspace of suitable polynomials the first M ‘classical’ (i.e. no difference part) Y-operators in N variables reduce, upon evaluation as above, to Y-operators in M variables with parameters at the quantum zonal spherical point.

Almost all optimally coloured complete graphs contain a rainbow Hamilton path

A subgraph H of an edge-coloured graph is called rainbow if all of the edges of H have different colours. In 1989, Andersen conjectured that every proper edge-colouring of Kn admits a rainbow path of length n−2. We show that almost all optimal edge-colourings of Kn admit both (i) a rainbow Hamilton path and (ii) a rainbow cycle using all of the colours. This result demonstrates that Andersen's Conjecture holds for almost all optimal edge-colourings of Kn and answers a recent question of Ferber, Jain, and Sudakov. Our result also has applications to the existence of transversals in random symmetric Latin squares.

Higher moments of the Fourier coefficients of symmetric square L-functions on certain sequence

Higher moments of the Fourier coefficients of symmetric square L-functions on certain sequence

Investigation of rectangular apertures and their application on speckle imaging

BackgroundAn aperture in the form of four squares arranged symmetrically along the cartesian coordinates with equal distances from the center investigated. Three models are suggested in the computation of the Point Spread Function PSF using the FFT technique. In the 1st model, circular annulus is placed in the center, while in the 2nd model a square annulus is shown, and in the 3rd model, two symmetric squares in the models 1, 2 are replaced by two symmetric rectangles while the center remains of square annulus. In all the models, central obstruction is made seeking to improve the PSF.ResultsAn analytical formula for the PSF for the aperture described in the 1st model is obtained. In addition, the autocorrelation corresponding to these apertures are computed and compared with the known autocorrelation corresponding to the whole square aperture. An application on speckle imaging is given using these apertures combined with the diffuser. All images for the design of the apertures and the speckle images are made using the MATLAB code.ConclusionsThe resolution computed from the FWHM showed an improvement for the suggested square apertures as compared with the uniform square aperture where the total width is kept constant. In addition, the strength of the legs in the PSF for the suggested apertures is much higher than that corresponding the uniform aperture which makes it useful for imaging of extended objects.

Optimal Target Shape for LiDAR Pose Estimation

Targets are essential in problems such as object tracking in cluttered or textureless environments, camera (and multi-sensor) calibration tasks, and simultaneous localization and mapping (SLAM). Target shapes for these tasks typically are symmetric (square, rectangular, or circular) and work well for structured, dense sensor data such as pixel arrays (i.e., image). However, symmetric shapes lead to pose ambiguity when using sparse sensor data such as LiDAR point clouds and suffer from the quantization uncertainty of the LiDAR. This paper introduces the concept of optimizing target shape to remove pose ambiguity for LiDAR point clouds. A target is designed to induce large gradients at edge points under rotation and translation relative to the LiDAR to ameliorate the quantization uncertainty associated with point cloud sparseness. Moreover, given a target shape, we present a means that leverages the target's geometry to estimate the target's vertices while globally estimating the pose. Both the simulation and the experimental results (verified by a motion capture system) confirm that by using the optimal shape and the global solver, we achieve centimeter error in translation and a few degrees in rotation even when a partially illuminated target is placed 30 meters away. All the implementations and datasets are available at https://github.com/UMich-BipedLab/optimal_shape_global_pose_estimation.

Discrete mean square of the coefficients of symmetric square L-functions on certain sequence of positive numbers

Discrete mean square of the coefficients of symmetric square L-functions on certain sequence of positive numbers

On the spectra and eigenspaces of the universal adjacency matrices of arbitrary lifts of graphs

The universal adjacency matrix U of a graph Γ, with adjacency matrix A, is a linear combination of A, the diagonal matrix D of vertex degrees, the identity matrix I, and the all-1 matrix J with real coefficients, that is, , with and . Thus, in particular cases, U may be the adjacency matrix, the Laplacian, the signless Laplacian, and the Seidel matrix. In this paper, we develop a method for determining the universal spectra and bases of all the corresponding eigenspaces of arbitrary lifts of graphs (regular or not). As an example, the method is applied to give an efficient algorithm to determine the characteristic polynomial of the Laplacian matrix of the symmetric squares of odd cycles, together with closed formulas for some of their eigenvalues.

The post-pandemic city: speculation through simulation

Although we can explain the way centripetal and centrifugal forces determine the form and function of contemporary cities, our abilities to predict their futures are severely limited. The pandemic has led to changes in locational and travel behaviour as well as regulated lockdowns with respect to where people work, live and social distance from one another. This makes it impossible to predict a ‘new normal’ reflecting ways we are able to control and manage the pandemic. As we have little data pertaining to this future, to engage in informed discussion, we develop a hypothetical city organised around theories of spatial interaction, urban hierarchy, density, and heterogeneity of movement. We propose a symmetric square grid of locations, simulate the interactions using gravitational models, and then lock it down. We release the lockdown in the transition to a new normal, assuming different parameter values controlling the effects of distance, illustrating the difficulty of generating highly decentralised city forms. We apply the model to London, locking down the metropolis, and exploring seven functional forms that provide us with a sample of different city shapes and densities. Our approach provides a framework for speculating about the future using what we call ‘computable thought experiments’.

3D-Stacked Multistage Inertial Microfluidic Chip for High-Throughput Enrichment of Circulating Tumor Cells.

Whether for cancer diagnosis or single-cell analysis, it remains a major challenge to isolate the target sample cells from a large background cell for high-efficiency downstream detection and analysis in an integrated chip. Therefore, in this paper, we propose a 3D-stacked multistage inertial microfluidic sorting chip for high-throughput enrichment of circulating tumor cells (CTCs) and convenient downstream analysis. In this chip, the first stage is a spiral channel with a trapezoidal cross-section, which has better separation performance than a spiral channel with a rectangular cross-section. The second and third stages adopt symmetrical square serpentine channels with different rectangular cross-section widths for further separation and enrichment of sample cells reducing the outlet flow rate for easier downstream detection and analysis. The multistage channel can separate 5 μm and 15 μm particles with a separation efficiency of 92.37% and purity of 98.10% at a high inlet flow rate of 1.3 mL/min. Meanwhile, it can separate tumor cells (SW480, A549, and Caki-1) from massive red blood cells (RBCs) with a separation efficiency of >80%, separation purity of >90%, and a concentration fold of ~20. The proposed work is aimed at providing a high-throughput sample processing system that can be easily integrated with flowing sample detection methods for rapid CTC analysis.

HoloCast+: Hybrid Digital-Analog Transmission for Graceful Point Cloud Delivery With Graph Fourier Transform

Point cloud is an emerging data format useful for various applications such has holographic display, autonomous vehicle, and augmented reality. Conventionally, communications of point cloud data have relied on digital compression and digital modulation for three-dimensional (3D) data streaming. However, such digital-based delivery schemes have fundamental issues called cliff and leveling effects, where the 3D reconstruction quality is a step function in terms of wireless channel quality. We propose a novel scheme of point cloud delivery, called HoloCast+, to overcome cliff and leveling effects. Specifically, our method utilizes hybrid digital-analog coding, integrating digital compression and analog coding based on graph Fourier transform (GFT), to gracefully improve 3D reconstruction quality with the improvement of channel quality. We demonstrate that HoloCast+ offers better 3D reconstruction quality in terms of the symmetric mean square error (sMSE) by up to 18.3 dB and 10.5 dB, respectively, compared to conventional digital-based and analog-based delivery methods in wireless fading environments.

Enumeration of Latin squares with conjugate symmetry

AbstractA Latin square has six conjugate Latin squares obtained by uniformly permuting its (row, column, symbol) triples. We say that a Latin square has conjugate symmetry if at least two of its six conjugates are equal. We enumerate Latin squares with conjugate symmetry and classify them according to several common notions of equivalence. We also do similar enumerations under additional hypotheses, such as assuming the Latin square is reduced, diagonal, idempotent or unipotent. Our data corrected an error in earlier literature and suggested several patterns that we then found proofs for, including (1) the number of isomorphism classes of semisymmetric idempotent Latin squares of order equals the number of isomorphism classes of semisymmetric unipotent Latin squares of order , and (2) suppose and are totally symmetric Latin squares of order . If and are paratopic then and are isomorphic.

Sato–Tate equidistribution for families of Hecke–Maass forms on SL(n, ℝ)∕SO(n)

We establish the Sato-Tate equidistribution of Hecke eigenvalues on average for families of Hecke--Maass cusp forms on SL(n,R)/SO(n). For each of the principal, symmetric square and exterior square L-functions we verify that the families are essentially cuspidal and deduce the level distribution with restricted support of the low-lying zeros. We also deduce average estimates toward Ramanujan.

A novel approach based on combining deep learning models with statistical methods for COVID-19 time series forecasting.

The COVID-19 pandemic has disrupted the economy and businesses and impacted all facets of people’s lives. It is critical to forecast the number of infected cases to make accurate decisions on the necessary measures to control the outbreak. While deep learning models have proved to be effective in this context, time series augmentation can improve their performance. In this paper, we use time series augmentation techniques to create new time series that take into account the characteristics of the original series, which we then use to generate enough samples to fit deep learning models properly. The proposed method is applied in the context of COVID-19 time series forecasting using three deep learning techniques, (1) the long short-term memory, (2) gated recurrent units, and (3) convolutional neural network. In terms of symmetric mean absolute percentage error and root mean square error measures, the proposed method significantly improves the performance of long short-term memory and convolutional neural networks. Also, the improvement is average for the gated recurrent units. Finally, we present a summary of the top augmentation model as well as a visual representation of the actual and forecasted data for each country.