Symmetric Square Research Articles

Infinite-Dimensional Lie Algebras Determined by the Space of Symmetric Squares of Hyperelliptic Curves

Построены алгебры Ли векторных полей на универсальных расслоениях симметрических квадратов гиперэллиптических кривых рода $g=1,2,…$. В каждой из этих алгебр Ли имеются коммутирующие образующие подалгебры Ли вертикальных полей, а образующие подалгебры Ли проецируемых полей задают каноническое представление подалгебры Ли с образующими $L_{2q}$, $q=-1,0,1,2,…$, алгебры Витта. В качестве приложения получены интегрируемые полиномиальные динамические системы.

Local symmetric square L-factors of representations of general linear groups

Local symmetric square L-factors of representations of general linear groups

Iwasawa theory for the symmetric square of a modular form

Abstract We construct a compatible family of global cohomology classes (an Euler system) for the symmetric square of a modular form, and apply this to bounding Selmer groups of the symmetric square Galois representation and its twists.

An improved fast plane wave expansion method for topology optimization of phononic crystals

Phononic crystals (PnCs) are artificially made materials composed of periodically arranged structures capable of manipulating acoustic/elastic wave propagation characteristics. In this paper, an improved fast plane wave expansion method (IFPWEM) is developed to obtain the band structures of PnCs. In the method presented, the continuity of the algorithm has been improved by eliminating the jump discontinuity points as well as decreasing the number of wave vectors used. Implementing these changes results in increased computational efficiency when compared to the traditional fast plane wave expansion method (FPWEM). In order to increase the band gap width produced by the PnCs, an adaptive genetic algorithm is adopted to optimize the PnCs structural topology for in-plane wave mode (xy mode). The numerical results yielded from optimization of two-dimension (2D) PnCs with a symmetric square lattice microstructure verifies that the efficiency of the IFPWEM is significantly greater than the conventional FPWEM and finite element methods.

Manifolds with the fixed point property and their squares

The Cartesian squares (powers) of manifolds with the fixed point property (f.p.p.) are considered. Examples of manifolds with the f.p.p. are constructed whose symmetric squares fail to have the f.p.p.

Nonperturbative linked-cluster expansions in long-range ordered quantum systems

We introduce a generic scheme to perform non-perturbative linked cluster expansions in long-range ordered quantum phases. Clusters are considered to be surrounded by an ordered reference state leading to effective edge-fields in the exact diagonalization on clusters which break the associated symmetry of the ordered phase. Two approaches, based either on a self-consistent solution of the order parameter or on minimal sensitivity with respect to the ground-state energy per site, are formulated to find the optimal edge-field in each NLCE order. Furthermore, we investigate the scaling behavior of the NLCE data sequences towards the infinite-order limit. We apply our scheme to gapped and gapless ordered phases of XXZ Heisenberg models on various lattices and for spins 1/2 and 1 using several types of cluster expansions ranging from a full-graph decomposition, rectangular clusters, up to more symmetric square clusters. It is found that the inclusion of edge-fields allows to regularize non-perturbative linked-cluster expansions in ordered phases yielding convergent data sequences. This includes the long-range spin-ordered ground state of the spin-1/2 and spin-1 Heisenberg model on the square and triangular lattice as well as the trimerized valence bond crystal of the spin-1 Heisenberg model on the kagome lattice.

Time varying axially symmetric vector random fields on the sphere

Abstract This paper presents a general form of the covariance matrix structure for a vector random field that is axially symmetric and mean square continuous on the sphere and provides a series representation for a longitudinally reversible one. The series representation is somehow an imitator of the covariance matrix function, and both of them have simpler forms than those proposed in the literature in terms of the associated Legendre functions and are useful for modeling and simulation. Also, a general form of the covariance matrix structure is derived for a spatio-temporal vector random field that is axially symmetric and mean square continuous over the sphere, and a series representation is given for a longitudinally reversible one.

Symmetrization of principal minors and cycle-sums

We solve the Symmetrized Principal Minor Assignment Problem, that is we show how to determine if for a given vector there is an matrix that has all principal minors equal to . We use a special isomorphism (a non-linear change of coordinates to cycle-sums) that simplifies computation and reveals hidden structure. We use the symmetries that preserve symmetrized principal minors and cycle-sums to treat three cases: symmetric, skew-symmetric and general square matrices. We describe the matrices that have such symmetrized principal minors as well as the ideal of relations among symmetrized principal minors / cycle-sums. We also connect the resulting algebraic varieties of symmetrized principal minors to tangential and secant varieties, and Eulerian polynomials.

Twisted symmetric square L-functions for $$\mathrm {GL}_n$$ GL n and invariant trilinear forms

Following the works of Bump and Ginzburg and of Takeda, we develop a theory of twisted symmetric square L-functions for \(\mathrm {GL}_n\). We characterize their pole in terms of certain trilinear period integrals, determine all irreducible summands of the discrete spectrum of \(\mathrm {GL}_n\) having nonvanishing trilinear periods, and construct nonzero local invariant trilinear forms on a certain family of induced representations.

Derivative at s = 1 of the p-adic L-function of the symmetric square of a Hilbert modular form

Let p be an odd prime and F a totally real number field. Let f be a Hilbert cuspidal eigenform of parallel weight 2, trivial Nebentypus and ordinary at p. It is possible to construct a p-adic L-function which interpolates the complex L-function associated to the symmetric square representation of f. This p-adic L-function vanishes at s=1 even if the complex L-function does not. Assuming p inert and f Steinberg at p, we give a formula for the p-adic derivative at s=1 of this p-adic L-function, generalizing unpublished work of Greenberg and Tilouine. Under some hypotheses on the conductor of f we prove a particular case of a conjecture of Greenberg on trivial zeros.

Estimates for fourier coefficients of cusp forms in weight aspect

Let f be a holomorphic Hecke eigenform of weight k for the modular group Γ = SL2(Z) and let λf (n) be the n-th normalized Fourier coefficient. In this paper, by a new estimate of the second integral moment of the symmetric square L-function related to f, the estimate $$\sum\limits_{n \leqslant x} {{\lambda _f}\left( {{n^2}} \right)} \ll {x^{\frac{1}{2}}}{k^{\frac{1}{2}}}{\left( {\log \left( {x + k} \right)} \right)^6}$$ is established, which improves the previous result.

Modular-type relations associated to the Rankin–Selberg L-function

Hafner and Stopple proved a conjecture of Zagier relating to the asymptotic behaviour of the inverse Mellin transform of the symmetric square L-function associated with the Ramanujan tau function. In this paper, we prove a similar result for any cusp form over the full modular group.

The Selberg trace formula as a Dirichlet series

Abstract We explore an idea of Conrey and Li of expressing the Selberg trace formula as a Dirichlet series. We describe two applications, including an interpretation of the Selberg eigenvalue conjecture in terms of quadratic twists of certain Dirichlet series, and a formula for an arithmetically weighted sum of the complete symmetric square L-functions associated to cuspidal Maass newforms of squarefree level N > 1 ${N>1}$ .

Lower bound for the higher moment of symmetric square $L$-functions

Let $\mathcal {S}_k(N)$ be the space of holomorphic cusp forms of weight $k$, level $N$ and let $\mathcal {B}_k(N)$ be an orthogonal basis of $\mathcal {S}_k(N)$ consisting of newforms. Let $L(s, \textup {sym}^2 f)$ be the symmetric square $L$-function of $f\in \mathcal {B}_k(N)$. In this paper, the lower bound of the higher moment of $L(1/2,\textup {sym}^2 f)$ is established, i.e., for any even positive number $r$, \[ \sum _{f\in \mathcal {B}_k(N)}\omega _f^{-1}L\bigg (\frac 12, \textup {sym}^2 f\bigg )^r \gg (\log N)^{{r(r+1)}/{2}} \] holds for $N\rightarrow \infty $.

FPGA, GPU, and CPU implementations of Jacobi algorithm for eigenanalysis

Parallel implementations of Jacobi algorithm for eigenanalysis of a matrix on most commonly used high performance computing (HPC) devices such as central processing unit (CPU), graphics processing unit (GPU), and field-programmable gate array (FPGA) are discussed in this paper. Their performances are investigated and compared. It is shown that CPU, even with multi-threaded implementation, is not a feasible option for large dense matrices. For the GPU implementation, performance impact of the global memory access patterns on the GPU board and the memory coalescing are emphasized. Three memory access methods are proposed. It is shown that one of them achieves 81.6% computational performance improvement over the traditional GPU methods, and it runs 68.5 times faster than a single-threaded CPU for a dense symmetric square matrix of size 1,024. Furthermore, FPGA implementation is presented and its performance running on chips from two major manufacturers are reported. A comparison of GPU and FPGA implementations is quantified and ranked. It is reported that FPGA design delivers the best performance for such a task while GPU is a strong competitor requiring less development effort with superior scalability. We predict that emerging big data applications will benefit from real-time and high performance computing implementations of eigenanalysis for information inference and signal analytics in the future.

L-split marker for augmented reality in aircraft assembly

In order to improve the performance of conventional square markers widely used by marker-based augmented reality systems in aircraft assembly environments, an L-split marker is proposed. Every marker consists of four separate L-shaped parts and each of them contains partial information about the marker. Geometric features of the L-shape, which are more discriminate than the symmetrical square shape adopted by conventional markers, are used to detect proposed markers from the camera images effectively. The marker is split into four separate parts in order to improve the robustness to occlusion and curvature to some extent. The registration process can be successfully completed as long as three parts are detected (up to about 80% of the area could be occluded). Moreover, when we attach the marker on nonplanar surfaces, the curvature status of the marker can be roughly analyzed with every part’s normal direction, which can be obtained since their six corners have been explicitly determined in the previous detection process. And based on the marker design, new detection and recognition algorithms are proposed and detailed. The experimental results show that the marker and the algorithms are effective.

Central value of the symmetric square L-functions related to Hecke–Maass forms

Let {ϕj(z) : j ≥ 1} be an orthonormal basis of Hecke–Maass cusp forms with Laplace eigenvalue 1/4 + tj2. For each ϕj(z), we have the automorphic L-function L(s, sym2ϕj), which is called the symmetric square L-function associated to ϕj. In this paper, we consider the average estimate of L(1/2, sym2ϕj) and prove that, for sufficiently large T, the estimate \( {\displaystyle {\sum}_{T<{t}_j\le T+M}{\left|L\left(1/2,{\mathrm{sym}}^2{\phi}_j\right)\right|}^2\ll {T}^{1+\varepsilon }M} \) holds for T1/3 + e ≤ M ≤ T1 − e.

The crystal structure and composition of pottsite, (Pb3xBi4-2x)(VO4)4·H2O (0.8 x &lt; 1.0)

Pottsite from the Linka mine (Nevada, USA) was studied by scanning electron microscopy, energy- and wavelength-dispersive analyses, and single-crystal X-ray diffraction. The formula, calculated on the basis of (V + As) = 4, shows that the Pb/Bi atomic ratio in pottsite varies between 0.86 and 1.48, and a plot of Pb vs. Bi (atoms per formula unit) suggests that the 3Pb 2+ ↔ 2Bi 3+ substitution is operative defining two compositional groups: one with low Pb/Bi atomic ratio (0.86–1.01), the other with high Pb/Bi ratio (1.22–1.48). Structure refinement shows that the crystal studied by X-ray diffraction has a low Pb/Bi ratio. Pottsite crystallizes in the tetragonal space group I 4 1 / a , a = 11.0839(5), c = 12.6516(6) A, V = 1554.29(16) A 3 , Z = 1. The crystal structure was solved by direct methods and refined to R 1 = 0.040 for 1167 unique reflections with F >4σ F . The structure contains one A site split into partially occupied A (1) and A (2) sites. The A (1) site is populated by Pb 0.500 Bi 0.380 Ca 0.030 , whereas A (2) is occupied by Pb 0.068 Ca 0.032 , with a site occupation factor (s.o.f.) of 0.07. The Bi(1) site has a s.o.f. of 0.84 and forms a distorted but symmetrical square antiprismatic coordination environment, which indicates a low degree of stereochemical activity of the lone electron pair on Bi 3+ . The symmetrically independent V(1) site has a V 0.983 As 0.017 population and is coordinated by four oxygen atoms to form a distorted VO 4 tetrahedron. The minor amount of chlorine in pottsite is restricted to the O w site (O 0.928 Cl 0.072 ) which belongs to the H 2 O molecule. The atomic arrangement in the structure of pottsite can be described as the packing of isolated V 5+ O 4 tetrahedra, linked with heavy cations via common oxygen atoms to form 2 D blocks or double-layered ( DL ) structural units that are parallel to (001). The DL units are formed by pairs of vanadate tetrahedra arranged opposite one to each other and oriented in opposite directions. Water molecules are located in between vanadate tetrahedral groups to form [(VO) 4 (H 2 O)] 3− chains which are separated from adjacent chains in the (001) plane by rows of Pb 2+ and Bi 3+ cations extending in the [120] direction. Two adjacent DL units are shifted one relative to the other by a translation of ±1/2 unit along the b axis, with an interlayer spacing of about 1.13 A, measured as the shortest distance between adjacent O–O edges of the VO 4 tetrahedra. The structure of pottsite is compared to that of related vanadate minerals based on isolated V 5+ O 4 tetrahedra with formation of DL units. The observed variations in Pb and Bi contents and the single-crystal X-ray diffraction data suggest a general formula of pottsite as (Pb 3 x Bi 4−2 x )(VO 4 ) 4 ·H 2 O, 0.8 x 3 Bi)Bi(VO 4 ) 4 ·H 2 O.

Theta distinguished representations, inflation and the symmetric square L-function

Let $\Pi_0$ be a representation of a group $H$. We say that a representation $\tau$ is $(H,\Pi_0)$-distinguished, if it is a quotient of $\Pi_0$. It is natural to ask whether this notion "inflates" to larger groups, in the sense that a representation $\mathrm{I}(\tau)$ induced from $\tau$ and $H$ to a group $G$, is $(G,\Pi)$-distinguished. We study representations distinguished by theta representations: $H=GL_n$, $\Pi_0$ is a pair of the exceptional representations of Kazhdan and Patterson, $G=GSpin_{2n+1}$ and $\Pi$ is a pair of the small representations of Bump, Friedberg and Ginzburg. We prove a Rodier-type hereditary property: a tempered representation $\tau$ is distinguished if and only if $\mathrm{I}(\tau)$ is distinguished, and the multiplicity in each model is the same. If $\tau$ is supercuspidal and distinguished, we prove that the Langlands quotient of $\mathrm{I}(\tau)$ is distinguished. As a corollary, we characterize supercuspidal distinguished representations, in terms of the pole of the local symmetric square $L$-function at $s=0$.

Numerical study of driven flows of shear thinning viscoelastic fluids in rectangular cavities

In this work, we present a numerical study of flow of shear thinning viscoelastic fluids in rectangular lid driven cavities for a wide range of aspect ratios (depth to width ratio) varying from 1/16 to 4. In particular, the effect of elasticity, inertia, model parameters and polymer concentration on flow features in rectangular driven cavity has been studied for two shear thinning viscoelastic fluids, namely, Giesekus and linear PTT. We perform numerical simulations using the symmetric square root representation of the conformation tensor to stabilize the numerical scheme against the high Weissenberg number problem. The variation in flow structures associated with merging and splitting of elongated vortices in shallow cavities and coalescence of corner eddies to yield a second primary vortex in deep cavities with respect to the variation in flow parameters is discussed. We discuss the effect of the dominant eigenvalues and the corresponding eigenvectors on the location of the primary eddy in the cavity. We also demonstrate, by performing numerical simulations for shallow and deep cavities, that where the Deborah number (based on convective time scale) characterizes the elastic behaviour of the fluid in deep cavities, Weissenberg number (based on shear rate) should be used for shallow cavities.