Symmetric Basis Research Articles

Secondary carbon skeletons constructed in porous carbon for electric double layer capacitors

In this work, a secondary skeleton was constructed by filling the larger spaces of porous carbon derived from passion fruit peels with glucose to created more carbon surface in same volume. The combined use of KOH activation and simultaneous etching of micropores on the carbon skeleton and secondary skeleton resulted in the preparation of porous carbon (PGC1-0.5) containing a large number of micropores. PGC1-0.5 have less total pore volume (0.987 cm3 g-1), but more micropore volume (0.821 cm3 g-1), different with 1.171 cm3 g-1, 0.668 cm3 g-1 of PGC1-0, which without glucose. Symmetric supercapacitor bases on PPGC1-0.5 have an obvious improvement in specific capacity (275.6 F g-1 (175 F cm-3), 0.1 A g-1) in 6M KOH electrolyte compared to PGC1-0 (215.2 F g-1 (121.5 F cm-3), 0.1 A g-1). Additionally, the device exhibits excellent cycling performance and retained 103% of its specific capacity after 10,000 cycles and notable reduction in transfer internal resistance in comparison to the sample without glucose. This study shows that filling space with glucose to reduce macropores is an effective method for adjusting the pore size distribution of porous carbon.

2-Rotund norms for unconditional and symmetric sequence spaces

A reflexive Banach space with an unconditional basis admits an equivalent 1-unconditional 2R norm and embeds into a reflexive space with a 1-symmetric 2R norm. Partial results on 1-symmetric 2R renormings of spaces with a symmetric basis are obtained.

Time Resolved Quantum Tomography in Molecular Spectroscopy by the Maximal Entropy Approach.

Attosecond science offers unprecedented precision in probing the initial moments of chemical reactions, revealing the dynamics of molecular electrons that shape reaction pathways. A fundamental question emerges: what role, if any, do quantum coherences between molecular electron states play in photochemical reactions? Answering this question necessitates quantum tomography─the determination of the electronic density matrix from experimental data, where the off-diagonal elements represent these coherences. The Maximal Entropy (MaxEnt) based Quantum State Tomography (QST) approach offers unique advantages in studying molecular dynamics, particularly with partial tomographic data. Here, we explore the application of MaxEnt-based QST on photoexcited ammonia, necessitating the operator form of observables specific to the performed measurements. We present two methodologies for constructing these operators: one leveraging Molecular Angular Distribution Moments (MADMs) which accurately capture the orientation-dependent vibronic dynamics of molecules and another utilizing Angular Momentum Coherence Operators to construct measurement operators for the full rovibronic density matrix in the symmetric top basis. A key revelation of our study is the direct link between Lagrange multipliers in the MaxEnt formalism and the unique set of MADMs. Additionally, we visualize the electron density within the molecular frame, demonstrating charge migration across the molecule. Furthermore, we achieve a groundbreaking milestone by constructing, for the first time, the entanglement entropy of the electronic subsystem─a metric that was previously inaccessible. The entropy vividly reveals and quantifies the effects of coupling between the excited electron and nuclear degrees of freedom. Consequently, our findings open new avenues for research in ultrafast molecular spectroscopy within the broader domain of quantum information science, offering profound implications for the study of molecular systems under excitation using quantum tomographic schemes.

Powersum Bases in Quasisymmetric Functions and Quasisymmetric Functions in Non-Commuting Variables

We introduce a new powersum basis for the Hopf algebra of quasisymmetric functions that refines the powersum symmetric basis. Unlike the quasisymmetric powersums of types 1 and 2, our basis is defined combinatorially: its expansion in quasisymmetric monomial functions is given by fillings of matrices. This basis has a shuffle product, a deconcatenate coproduct, and has a change of basis rule to the quasisymmetric fundamental basis by using tuples of ribbons. We lift our quasisymmetric powersum P basis to the Hopf algebra of quasisymmetric functions in non-commuting variables by introducing fillings with disjoint sets. This new basis has a shifted shuffle product and a standard deconcatenate coproduct, and certain basis elements agree with the fundamental basis of the Malvenuto-Reutenauer Hopf algebra of permutations. Finally we discuss how to generalize these bases and their properties by using total orders on indices.

Analytic Invariants of Semidirect Products of Symmetric Groups on Banach Spaces

We consider algebras of polynomials and analytic functions that are invariant with respect to semidirect products of groups of bounded operators on Banach spaces with symmetric bases. In particular, we consider algebras of so-called block-symmetric and double-symmetric analytic functions on Banach spaces ℓp(Cn) and the homomorphisms of these algebras. In addition, we describe an algebraic basis in the algebra of double-symmetric polynomials and discuss a structure of the spectrum of the algebra of double-symmetric analytic functions on ℓp(Cn).

Counterexamples in isometric theory of symmetric and greedy bases

We continue the study initiated in Albiac and Wojtaszczyk (2006) of properties related to greedy bases in the case when the constants involved are sharp, i.e., in the case when they are equal to 1. Our main goal here is to provide an example of a Banach space with a basis that satisfies Property (A) but fails to be 1-suppression unconditional, thus settling Problem 4.4 from Albiac and Ansorena (2017). In particular, our construction demonstrates that bases with Property (A) need not be 1-greedy even with the additional assumption that they are unconditional and symmetric. We also exhibit a finite-dimensional counterpart of this example, and show that, at least in the finite-dimensional setting, Property (A) does not pass to the dual. As a by-product of our arguments, we prove that a symmetric basis is unconditional if and only if it is total, thus generalizing the well-known result that symmetric Schauder bases are unconditional.

Duals of Tirilman spaces have unique subsymmetric basic sequences

AbstractThe Tirilman spaces , , were introduced by Casazza and Shura as variations of the spaces constructed by Tzafriri. We prove that all subsymmetric basic sequences in the dual space are equivalent to its canonical subsymmetic but not symmetric basis.

Weak weight-semi-greedy Markushevich bases

The main purpose of this paper is to study weight-semi-greedy Markushevich bases, and in particular, find conditions under which such bases are weight-almost greedy. In this context, we prove that, for a large class of weights, the two notions are equivalent. We also show that all weight semi-greedy bases are truncation quasi-greedy and weight-superdemocratic. In all of the above cases, we also bring to the context of weights the weak greedy and Chebyshev greedy algorithms—which are frequently studied in the literature on greedy approximation. In the course of our work, a new property arises naturally and its relation with squeeze symmetric and bidemocratic bases is given. In addition, we study some parameters involving the weak thresholding and Chebyshevian greedy algorithms. Finally, we give examples of conditional bases with some of the weighted greedy-type conditions we study.

Explicit consistency conditions for fully symmetric cubature on the tetrahedron

A novel fully symmetric basis is derived for the S4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${S}_4$$\\end{document}-invariant polynomial space, by using symmetric polynomials and invariant theory. This new basis enables deriving explicitly the consistency conditions for non-overdeterminedness of moment equations in the case of fully symmetric cubature rules on the tetrahedron. Solving the corresponding linear integer programming problem, optimal and quasi-optimal rule structures are derived. Explicit formulas to calculate the estimated lower bounds in the number of integration points are also given. Additionally, the new basis is of practical interest in calculating specific cubature rules, since it allows decomposing the moment equations into a series of successively independent smaller subsystems, which can be exploited in designing more efficient solution methods. Solving the moment equations analytically we obtain several interesting new results.

Mutually Unbiased Measurements, Hadamard Matrices, and Superdense Coding

Mutually unbiased bases (MUBs) are highly symmetric bases on complex Hilbert spaces, and the corresponding rank-1 projective measurements are ubiquitous in quantum information theory. In this work, we study a recently introduced generalisation of MUBs called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">mutually unbiased measurements</i> (MUMs). These measurements inherit the essential property of complementarity from MUBs, but the Hilbert space dimension is no longer required to match the number of outcomes. This operational complementarity property renders MUMs highly useful for device-independent quantum information processing. It has been shown that MUMs are strictly more general than MUBs. In this work we provide a complete proof of the characterisation of MUMs that are direct sums of MUBs. We then construct new examples of MUMs that are not direct sums of MUBs. A crucial technical tool for this construction is a correspondence with quaternionic Hadamard matrices, which allows us to map known examples of such matrices to MUMs that are not direct sums of MUBs. Furthermore, we show that–in stark contrast with MUBs–the number of MUMs for a fixed outcome number is unbounded. Next, we focus on the use of MUMs in quantum communication. We demonstrate how any pair of MUMs with <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> outcomes defines a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> -dimensional superdense coding protocol. Using MUMs that are not direct sums of MUBs, we disprove a recent conjecture due to Nayak and Yuen on the rigidity of superdense coding, for infinitely many dimensions. The superdense coding protocols arising in the refutation reveal how shared entanglement may be used in a manner heretofore unknown.

Quantum key distribution rates from semidefinite programming

Computing the key rate in quantum key distribution (QKD) protocols is a long standing challenge. Analytical methods are limited to a handful of protocols with highly symmetric measurement bases. Numerical methods can handle arbitrary measurement bases, but either use the min-entropy, which gives a loose lower bound to the von Neumann entropy, or rely on cumbersome dedicated algorithms. Based on a recently discovered semidefinite programming (SDP) hierarchy converging to the conditional von Neumann entropy, used for computing the asymptotic key rates in the device independent case, we introduce an SDP hierarchy that converges to the asymptotic secret key rate in the case of characterised devices. The resulting algorithm is efficient, easy to implement and easy to use. We illustrate its performance by recovering known bounds on the key rate and extending high-dimensional QKD protocols to previously intractable cases. We also use it to reanalyse experimental data to demonstrate how higher key rates can be achieved when the full statistics are taken into account.

Spectral methods for solving elliptic PDEs on unknown manifolds

In this paper, we propose a mesh-free numerical method for solving elliptic PDEs on unknown manifolds, identified with randomly sampled point cloud data. The PDE solver is formulated as a spectral method where the test function space is the span of the leading eigenfunctions of the Laplacian operator, which are approximated from the point cloud data. While the framework is flexible for any test functional space, we will consider the eigensolutions of a weighted Laplacian obtained from a symmetric Radial Basis Function (RBF) method induced by a weak approximation of a weighted Laplacian on an appropriate Hilbert space. In this paper, we consider a test function space that encodes the geometry of the data yet does not require us to identify and use the sampling density of the point cloud. To attain a more accurate approximation of the expansion coefficients, we adopt a second-order tangent space estimation method to improve the RBF interpolation accuracy in estimating the tangential derivatives. This spectral framework allows us to efficiently solve the PDE many times subjected to different parameters, which reduces the computational cost in the related inverse problem applications. In a well-posed elliptic PDE setting with randomly sampled point cloud data, we provide a theoretical analysis to demonstrate the convergence of the proposed solver as the sample size increases. We also report some numerical studies that show the convergence of the spectral solver on simple manifolds and unknown, rough surfaces. Our numerical results suggest that the proposed method is more accurate than a graph Laplacian-based solver on smooth manifolds. On rough manifolds, these two approaches are comparable. Due to the flexibility of the framework, we empirically found improved accuracies in both smoothed and unsmoothed Stanford bunny domains by blending the graph Laplacian eigensolutions and RBF interpolator.

New characterizations of the unit vector basis of or

AbstractMotivated by Altshuler’s famous characterization of the unit vector basis of $c_0$ or $\ell _p$ among symmetric bases (Altshuler [1976, Israel Journal of Mathematics, 24, 39–44]), we obtain similar characterizations among democratic bases and among bidemocratic bases. We also prove a separate characterization of the unit vector basis of $\ell _1$ .

A novel generalized symmetric spectral Galerkin numerical approach for solving fractional differential equations with singular kernel

&lt;abstract&gt;&lt;p&gt;Polynomial based numerical techniques usually provide the best choice for approximating the solution of fractional differential equations (FDEs). The choice of the basis at which the solution is expanded might affect the results significantly. However, there is no general approach to determine which basis will perform better with a particular problem. The aim of this paper is to develop a novel generalized symmetric orthogonal basis which has not been discussed in the context of numerical analysis before to establish a general numerical treatment for the FDEs with a singular kernel. The operational matrix with four free parameters was derived for the left-sided Caputo fractional operator in order to transform the FDEs into the corresponding algebraic system with the aid of spectral Galerkin method. Several families of the existing polynomials can be obtained as a special case from the new basis beside other new families generated according to the value of the free parameters. Consequently, the operational matrix in terms of these families was derived as a special case from the generalized one up to a coefficient diagonal matrix. Furthermore, different properties relevant to the new generalized basis were derived and the error associated with function approximation by the new basis was performed based on the generalized Taylor's formula.&lt;/p&gt;&lt;/abstract&gt;

Chirality As a Symmetric Basis of Self-Organization of Biomacromolecules

Chirality As a Symmetric Basis of Self-Organization of Biomacromolecules

Spectral properties of symmetric quantum states and symmetric entanglement witnesses

We introduce and explore two questions concerning spectra of operators that are of interest in the theory of entanglement in symmetric (i.e., bosonic) quantum systems. First, we investigate the inverse eigenvalue problem for symmetric entanglement witnesses—that is, we investigate what their possible spectra are. Second, we investigate the problem of characterizing which separable symmetric quantum states remain separable after conjugation by an arbitrary unitary acting on symmetric space—that is, which states are separable in every orthonormal symmetric basis. Both of these questions have been investigated thoroughly in the non-symmetric setting, and we contrast the answers that we find with their non-symmetric counterparts.

Correction: Yalcin, F.; Simsek, Y. A New Class of Symmetric Beta Type Distributions Constructed by Means of Symmetric Bernstein Type Basis Functions. Symmetry 2020, 12, 779

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Critical coupling for two-dimensional ϕ4 theory in discretized light-cone quantization

We solve for the critical coupling in the symmetric phase of two-dimensional ${\ensuremath{\phi}}^{4}$ field theory using discretized light-cone quantization. We adopt periodic boundary conditions, neglect the zero mode, and obtain a critical coupling consistent with the critical coupling reported using conformal truncation in light-front quantization. We find a 17% difference from the critical coupling reported with light-front quantization in a symmetric polynomial basis.

Lipschitz symmetric functions on Banach spaces with symmetric bases

We investigate Lipschitz symmetric functions on a Banach space $X$ with a symmetric basis. We consider power symmetric polynomials on $\ell_1$ and show that they are Lipschitz on the unbounded subset consisting of vectors $x\in \ell_1$ such that $|x_n|\le 1.$ Using functions $\max$ and $\min$ and tropical polynomials of several variables, we constructed a large family of Lipschitz symmetric functions on the Banach space $c_0$ which can be described as a semiring of compositions of tropical polynomials over $c_0$.

Approximations of Symmetric Functions on Banach Spaces with Symmetric Bases

This paper is devoted to studying approximations of symmetric continuous functions by symmetric analytic functions on a Banach space X with a symmetric basis. We obtain some positive results for the case when X admits a separating polynomial using a symmetrization operator. However, even in this case, there is a counter-example because the symmetrization operator is well defined only on a narrow, proper subspace of the space of analytic functions on X. For X=c0, we introduce ε-slice G-analytic functions that have a behavior similar to G-analytic functions at points x∈c0 such that all coordinates of x are greater than ε, and we prove a theorem on approximations of uniformly continuous functions on c0 by ε-slice G-analytic functions.