Symmetric Basis Research Articles

The SHAI property for the operators on Lp

A Banach space X has the SHAI (surjective homomorphisms are injective) property provided that for every Banach space Y, every continuous surjective algebra homomorphism from the bounded linear operators on X onto the bounded linear operators on Y is injective. The main result gives a sufficient condition for X to have the SHAI property. The condition is satisfied for Lp(0,1) for 1<p<∞, spaces with symmetric bases that have finite cotype, and the Schatten p-spaces for 1<p<∞.

Deformation pattern classification and energy absorption optimization of the eccentric body centered cubic lattice structures

Lattice structures have shown excellent mechanical properties, such as high energy absorption capacity. In this work, the extended body centered cubic lattice was proposed by offsetting the position of the body center. The pattern recognition method of nonlinear discriminant function was creatively used to classify the proposed eccentric body centered cubic (EBCC) lattice structures according to their deformation patterns. The results indicated that the lattice structures belong to X-Pattern possessed the best energy absorption capability. Further, the relationship between the energy absorption properties and the configuration was captured using a modified symmetric radius basis function (RBF) surrogate model showing higher computational efficiency and approximately equal accuracy compared to the original RBF model. The nondominated sorting genetic algorithm II (NSGA-II) was carried out to obtain the Pareto optimal set, rendering the EBCC structure with balanced properties of high specific energy absorption and low peak crushing force. This work presents a first effort in optimizing the energy absorption of lattice structures through pattern recognition, thus expanding the approaches for designing new lattice structures.

Wild dynamics and asymptotically separated sets

Let X X be a separable infinite dimensional (real or complex) Banach space. Hájek and Smith, in 2010, constructed a linear bounded operator T T on X X such that A T ≔ { x ∈ X : ‖ T n x ‖ → ∞ } A_T≔\{x\in X:~ \|T^nx\|\to \infty \} is not dense and has nonempty interior, whenever X X admits a symmetric basis. Augé, in 2012, extended the previous construction to each separable Banach space, introducing the notion of wild operators. This work is divided in two parts. In the first part we define and explore the notion of asymptotically separated sets on Banach spaces, giving several examples whenever the ambient space has either finite or infinite dimension. We show how these sets can be used to construct operators with non-intuitive dynamics. Specifically, operators T : X → X T:X\rightarrow X for which the set A T A_T and the set of recurrent points form a partition of the space. Secondly, we investigate geometric and spectral properties of operators with wild dynamic and we provide a wild operator whose spectrum is equal to the closed unit disk. We end this paper giving some comments about the norm closure of the set of wild operators.

Bilocal Bell Inequalities Violated by the Quantum Elegant Joint Measurement.

Network Bell experiments give rise to a form of quantum nonlocality that conceptually goes beyond Bell's theorem. We investigate here the simplest network, known as the bilocality scenario. We depart from the typical use of the Bell state measurement in the network central node and instead introduce a family of symmetric isoentangled measurement bases that generalize the so-called "elegant joint measurement." This leads us to report noise-tolerant quantum correlations that elude bilocal variable models. Inspired by these quantum correlations, we introduce network Bell inequalities for the bilocality scenario and show that they admit noise-tolerant quantum violations. In contrast to many previous studies of network Bell inequalities, neither our inequalities nor their quantum violations are based on standard Bell inequalities and standard quantum nonlocality. Moreover, we pave the way for an experimental realization by presenting a simple two-qubit quantum circuit for the implementation of the elegant joint measurement and our generalization.

Low-rank matrix completion in a general non-orthogonal basis

This paper considers theoretical analysis of recovering a low rank matrix given a few expansion coefficients with respect to any basis. The current approach generalizes the existing analysis for the low-rank matrix completion problem with sampling under entry sensing or with respect to a symmetric orthonormal basis. The analysis is based on dual certificates using a dual basis approach. We introduce a condition on the basis called the correlation condition. This condition can be computed in time O(n3) and holds for many cases of deterministic basis. If the correlation condition holds and the underlying low rank matrix obeys the coherence condition with parameter ν, under additional mild assumptions, our main result shows that the true matrix can be recovered with very high probability from O(nrνlog2⁡n) uniformly random expansion coefficients.

A dichotomy for subsymmetric basic sequences with applications to Garling spaces

Our aim in this article is to contribute to the study of the structure of subsymmetric basic sequences in Banach spaces (even, more generally, in quasi-Banach spaces). For that we introduce the notion of positionings and develop new tools which lead to a dichotomy theorem that holds for general spaces with subsymmetric bases. As an illustration of how to use this dichotomy theorem we obtain the classification of all subsymmetric sequences in certain types of spaces. To be more specific, we show that Garling sequence spaces have a unique symmetric basic sequence but no symmetric basis and that these spaces have a continuum of subsymmetric basic sequences.

Tribaryons in a constituent quark model

We calculate the matrix elements of the color-spin interaction for all possible multi-quark states of tribaryons in flavor SU(3) broken case. For that purpose, we construct the flavor$\otimes$color$\otimes$spin wave functions of the tribaryons, which are taken to be antisymmetric to satisfy the Pauli exclusion principle. Furthermore, we analyze the diquark structure of the tribaryon configurations using the symmetric and antisymmetric basis set of flavor, color and spin states.

Symmetric method of approximate particular solutions for solving certain partial differential equations

In this study, a symmetric method of approximate particular solutions (MAPS) is proposed for solving certain partial differential equations (PDEs). Inspired by the unsymmetric MAPS and symmetric radial basis function collocation method (RBFCM), the symmetric MAPS is developed by using the bi-particular solutions of the multiquadrics (MQ). Similar to the unsymmetric MAPS, the right-hand-side of the governing equation is mainly approximated by the MQ in the proposed method. In addition, the system matrix of the prescribed method is symmetric. Numerical examples are solved by the unsymmetric & symmetric RBFCM and MAPS for different problems with different types of governing equations and boundary conditions. Numerical results with different shape parameters are analyzed to show that the symmetric methods are more stable. In addition, the accuracy improvement of the symmetric MAPS is studied. Finally, the stability performance of the symmetric MAPS is further studied for convection-diffusion problems at high Péclet numbers.

A meshless Radial Basis Method (RBM) for solving the detailed population balance equation

The Radial Basis Method (RBM) is introduced for solving the Population Balance Equation (PBE). In this method, the continuous number density function is approximated by a series of symmetric multiquadric basis functions at selected global collocation points along the internal coordinates. Therefore, the RBM is considered as a meshless method. The RBM is convenient and suitable for complex problems involving nucleation, growth, breakage and aggregation phenomena. The method provides naturally a continuously differentiable, or “smooth”, solution and avoids solving ill-conditioned problems. The numerical solutions are validated analytically and numerically using different case-studies for uniform stirred-tank. For these examples, 20 collocation points were found to be enough to achieve accurate predictions. The RBM ability to predict multimodal distribution functions and moving steep fronts is moreover highlighted. The proposed method proves to be very efficient for solving the detailed PBE in terms of accuracy, stability and CPU time requirements.

On quasisymmetric power sums

In the 1995 paper entitled “Noncommutative symmetric functions”, Gelfand et al. defined two noncommutative symmetric function analogues for the power sum basis of the symmetric functions. This paper explores the combinatorial properties of their duals, two distinct quasisymmetric power sum bases. In contrast to the symmetric power sums, the quasisymmetric power sums have a more complex combinatorial description. This paper offers a first detailed exploration of these two relatively unstudied quasisymmetric bases, in which we show that they refine the classical symmetric power sum basis, we give transition matrices to other well-understood bases, and we provide explicit formulas for products of quasisymmetric power sums.

A New Class of Symmetric Beta Type Distributions Constructed by Means of Symmetric Bernstein Type Basis Functions

The main aim of this paper is to define and investigate a new class of symmetric beta type distributions with the help of the symmetric Bernstein-type basis functions. We give symmetry property of these distributions and the Bernstein-type basis functions. Using the Bernstein-type basis functions and binomial series, we give some series and integral representations including moment generating function for these distributions. Using generating functions and their functional equations, we also give many new identities related to the moments, the polygamma function, the digamma function, the harmonic numbers, the Stirling numbers, generalized harmonic numbers, the Lah numbers, the Bernstein-type basis functions, the array polynomials, and the Apostol–Bernoulli polynomials. Moreover, some numerical values of the expected values for the logarithm of random variable are given.

Imaging the Three-Dimensional Ionospheric Structure with a Blob Basis Functional Ionospheric Tomography Model.

A new ionospheric tomography model is presented in this work. In the new model, the traditional voxel basis function is replaced by the blob basis function. Due to the overlapping nature of their rotational symmetric basis functions, the new model introduces certain weighting from nearby tomographic spherical blobs. To confirm the feasibility of the new tomography model, a numerical simulation scheme is devised, and the simulation demonstrates that the reconstructed quality of the blob basis tomographic model is higher than that of the voxel basis tomographic model. Meanwhile, the variable blob radius is adopted in order to improve the efficiency of the new model. Finally, the new ionospheric tomography model is applied to reconstruct the temporal-spatial distribution of ionospheric electron density using actual global navigation satellite system observations. The comparisons between the tomographic profiles and those obtained from ionosonde data further demonstrate the reliability and the superiority of the new ionospheric tomography model.

LLT polynomials, elementary symmetric functions and melting lollipops

We conjecture an explicit positive combinatorial formula for the expansion of unicellular LLT polynomials in the elementary symmetric basis. This is an analogue of the Shareshian–Wachs conjecture previously studied by Panova and the author in 2018. We show that the conjecture for unicellular LLT polynomials implies a similar formula for vertical-strip LLT polynomials. We prove positivity in the elementary symmetric basis for the class of graphs called “melting lollipops” previously considered by Huh, Nam and Yoo. This is done by proving a curious relationship between a generalization of charge and orientations of unit-interval graphs. We also provide short bijective proofs of Lee’s three-term recurrences for unicellular LLT polynomials, and we show that these recurrences are enough to generate all unicellular LLT polynomials associated with abelian area sequences.

Upwind skewed radial basis functions (USRBF) for solution of highly convective problems over meshfree nodes

An upwind skewed radial basis function (USRBF)-based solution scheme is presented for stabilized solutions of convection-dominated problems over meshfree nodes. The conventional, radially symmetric radial basis functions (RBFs) are multiplied with an upwinding factor which skews the RBFs toward the upwind direction. The upwinding factor is a function of flow direction, intensity of convection, size of local support domain, and nodal distribution. The use of USRBFs modifies the weight values such that the necessary artificial diffusion is added only along the flow direction, whereas the crosswind diffusion is avoided. Subsequently, these skewed radial basis functions are employed in finite difference mode (RBF-FD) for derivative approximation. The performance and accuracy of the proposed scheme is studied by solving convection–diffusion problems over uniform and random distribution of meshfree nodes with various convection intensities. The upwinding effectively suppresses non-physical perturbation in numerical solution of convection-dominated problems. The results show that significant improvement in accuracy can be achieved by using the proposed USRBF-based solution scheme, particularly at higher convection intensities.

Compatible observable decompositions for coarse-grained representations of real molecular systems.

Coarse-grained (CG) observable expressions, such as pressure or potential energy, are generally different than their fine-grained (FG, e.g., atomistic) counterparts. Recently, we analyzed this so-called "representability problem" in Wagner et al. [J. Chem. Phys. 145, 044108 (2016)]. While the issue of representability was clearly and mathematically stated in that work, it was not made clear how to actually determine CG observable expressions from the underlying FG systems that can only be simulated numerically. In this work, we propose minimization targets for the CG observables of such systems. These CG observables are compatible with each other and with structural observables. Also, these CG observables are systematically improvable since they are variationally minimized. Our methods are local and data efficient because we decompose the observable contributions. Hence, our approaches are called the multiscale compatible observable decomposition (MS-CODE) and the relative entropy compatible observable decomposition (RE-CODE), which reflect two main approaches to the "bottom-up" coarse-graining of real FG systems. The parameterization of these CG observable expressions requires the introduction of new, symmetric basis sets and one-body terms. We apply MS-CODE and RE-CODE to 1-site and 2-site CG models of methanol for the case of pressure, as well as to 1-site methanol and acetonitrile models for potential energy.

Momentum space approach to crossing symmetric CFT correlators. Part II. General spacetime dimension

Our previous work [1] constructed, in three-dimensional momentum space, a manifestly crossing symmetric basis for scalar conformal four-point functions, based on the factorization property proposed by Polyakov. This work extends this construction to general dimensional conformal field theory. To facilitate the treatment of symmetric traceless tensors, we exploit techniques of spherical harmonics in general dimensions.

On invertible algebras

An algebra A over a field K is said to be invertible if it has a basis B consisting only of units; if B−1 is again a basis, A is invertible-2, or I2. The question of when an invertible algebra is necessarily I2 arises naturally. The study of these algebras was recently initiated by López-Permouth, Moore, Szabo, Pilewski [13], [14]. In this paper, we prove several positive results on this problem, answering also some questions and generalizing a few results from these papers. We show that every field is an I2 algebra over any subfield, and that any subalgebra of the rational functions field K(x) which strictly contains K[x], with K an algebraically closed field, has a symmetric basis B=B−1. Using this, we expand the class of examples of algebras known to be invertible or I2 with several classes, such as semiprimary rings over fields K≠F2 satisfying some additional mild condition. We also show that every commutative affine invertible algebra is almost I2 in the sense that it becomes I2 after localization at a single element.

On using signal magnitude in diffusion magnetic resonance measurements of restricted motion

Tissue microstructure has significance as a biomarker, however its accurate inference with diffusion magnetic resonance (MR) is still an open problem. With few exceptions, diffusion weighted (DW) MR models either process diffusion MR data using signal magnitude, whereby microstructural information is forcefully confined to symmetry due to Fourier transform properties, or directly use symmetric basis expansions.Herein, information loss from magnitude utilization is demonstrated by numerically simulating particles undergoing diffusion near a fully reflective infinite wall and an orthogonal corner.Simulation results show that the loss of the Hermitian property when using signal magnitude impedes DW–MR from accurately inferring microstructural information in both of the geometries.

Constructing 2-Arc-Transitive Covers of Hypercubes

We introduce the notion of a symmetric basis of a vector space equipped with a quadratic form, and provide a sufficient and necessary condition for the existence to such a basis. Symmetric bases are then used to study Cayley graphs of certain extraspecial 2-groups of order 2^{2r+1} (r\geq 1), which are further shown to be normal Cayley graphs and 2-arc-transitive covers of 2r-dimensional hypercubes.

Exponential shapelets: basis functions for data analysis of isolated features

Abstract We introduce one- and two-dimensional ‘exponential shapelets’: orthonormal basis functions that efficiently model isolated features in data. They are built from eigenfunctions of the quantum mechanical hydrogen atom, and inherit mathematics with elegant properties under Fourier transform, and hence (de)convolution. For a wide variety of data, exponential shapelets compress information better than Gauss–Hermite/Gauss–Laguerre (‘shapelet’) decomposition, and generalize previous attempts that were limited to 1D or circularly symmetric basis functions. We discuss example applications in astronomy, fundamental physics, and space geodesy.