Symmetric Basis Research Articles

Characterization of symmetric distributions based on concomitants of ordered variables from FGMs family of bivariate distributions

Several characterization results of a symmetric distribution based on concomitants of order statistics as well as k-records from Farlie-Gumbel-Morgenstern (FGM) family of bivariate distributions are established. These include characterizations of a symmetric distribution on the basis of equality in distribution, moments, R?nyi and Tsallis entropies of concomitants of upper and lower order statistics, also in terms of the same properties of concomitants of upper and lower k-records.

Symmetric Radial Basis Function Method for Simulation of Elliptic Partial Differential Equations

In this paper, the symmetric radial basis function method is utilized for the numerical solution of two- and three-dimensional elliptic PDEs. Numerical results are obtained by using a set of uniform or random points. Numerical tests are accomplished to demonstrate the efficacy and accuracy of the method on both regular and irregular domains. Furthermore, the proposed method is tested for the solution of elliptic PDE in the case of various frequencies.

Music and choreography interaction in the stage dances of musical theater productions of the 17th – the first half of the 18th century

Music and choreography interaction in the stage dances of musical theater productions of the 17th – the first half of the 18th century

Approximation of norms on Banach spaces

Relatively recently it was proved that if Γ is an arbitrary set, then any equivalent norm on c0(Γ) can be approximated uniformly on bounded sets by polyhedral norms and C∞ smooth norms, with arbitrary precision. We extend this result to more classes of spaces having uncountable symmetric bases, such as preduals of the ‘discrete’ Lorentz spaces d(w,1,Γ), and certain symmetric Nakano spaces and Orlicz spaces. We also show that, given an arbitrary ordinal number α, there exists a scattered compact space K having Cantor–Bendixson height at least α, such that every equivalent norm on C(K) can be approximated as above.

Momentum space approach to crossing symmetric CFT correlators

We construct a crossing symmetric basis for conformal four-point functions in momentum space by requiring consistent factorization. Just as scattering amplitudes factorize when the intermediate particle is on-shell, non-analytic parts of conformal correlators enjoy a similar factorization in momentum space. Based on this property, Polyakov, in his pioneering 1974 work, introduced a basis for conformal correlators which manifestly satisfies the crossing symmetry. He then initiated the bootstrap program by requiring its consistency with the operator product expansion. This approach is complementary to the ordinary bootstrap program, which is based on the conformal block and requires the crossing symmetry as a consistency condition of the theory. Even though Polyakov’s original bootstrap approach has been revisited recently, the crossing symmetric basis has not been constructed explicitly in momentum space. In this paper we complete the construction of the crossing symmetric basis for scalar four-point functions with an intermediate operator with a general spin, by using new analytic expressions for three-point functions involving one tensor. Our new basis manifests the analytic properties of conformal correlators. Also the connected and disconnected correlators are manifestly separated, so that it will be useful for the study of large N CFTs in particular.

Garling sequence spaces

The aim of this paper is to introduce and investigate a new class of separable Banach spaces modeled after an example of Garling from 1968. For each $1\leqslant p<\infty$ and each nonincreasing weight $\textbf{w}\in c_0\setminus\ell_1$ we exhibit an $\ell_p$-saturated, complementably homogeneous, and uniformly subprojective Banach space $g(\textbf{w},p)$. We also show that $g(\textbf{w},p)$ admits a unique subsymmetric basis despite the fact that for a wide class of weights it does not admit a symmetric basis. This provides the first known examples of Banach spaces where those two properties coexist.

Optimality of the rearrangement inequality with applications to Lorentz-type sequence spaces

We characterize the sequences (wi)8i=1 of non-negative numbers for which ∞Σi=1aiwi is of the same order as supnnΣi=1aiw1+n-i when (ai)8i=1 runs over all non-increasing sequences of non-negative numbers. As a by-product of our work we settle a problem raised in [1] and prove that Garling sequences spaces have no symmetric basis. © ELEMEN , Zagreb.

Optimality of the rearrangement inequality with applications to Lorentz-type sequence spaces

We characterize the sequences $(w_i)_{i=1}^\infty$ of non-negative numbers for which \[ \sum_{i=1}^\infty a_i w_i \quad \text{ is of the same order as } \quad \sup_n \sum_{i=1}^n a_i w_{1+n-i} \] when $(a_i)_{i=1}^\infty$ runs over all non-increasing sequences of non-negative numbers. As a by-product of our work we settle a problem raised in [F. Albiac, Jose L. Ansorena and B. Wallis; arXiv:1703.07772[math.FA]] and prove that Garling sequences spaces have no symmetric basis.

Adaptive multiscale methods for elliptic PDEs

AbstractThis paper deals with the numerical solution of elliptic partial differential equations with an adaptive multiscale algorithm, which uses symmetric collocation and radial basis functions. (© 2017 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

A note on subsymmetric renormings of Banach spaces

We revisit the concept of a subsymmetric norm and construct a subsymmetric renorming of a Banach space with a subsymmetric basis. As a by-product of our work we introduce the concept of a lower symmetric basis and investigate its connection with subsymmetric bases and subsymmetric renormings.

Second rank direction cosine spherical tensor operators and the nuclear electric quadrupole hyperfine structure Hamiltonian of rotating molecules

Transformations of vector or tensor properties from a space-fixed to a molecule-fixed axis system are often required in the study of rotating molecules. Spherical components λμ,ν of a first rank irreducible tensor can be obtained from the direction cosines between the two axis systems, and a second rank tensor with spherical components λμ,ν(2) can be built from the direct product λ × λ.It is shown that the treatment of the interaction between molecular rotation and the electric quadrupole of a nucleus is greatly simplified, if the coefficients in the axis-system transformation of the gradient of the electric field of the outer charges at the coupled nucleus are arranged as spherical components λμ,ν(2). Then the reduced matrix elements of the field gradient operators in a symmetric top eigenfunction basis, including their dependence on the molecule-fixed z-angular momentum component k, can be determined from the knowledge of those of λ(2). The hyperfine structure Hamiltonian Hq is expressed as the sum of terms characterized each by a value of the molecule-fixed index ν, whose matrix elements obey the rule Δk = ν. Some of these terms may vanish because of molecular symmetry, and the specific cases of linear and symmetric top molecules, orthorhombic molecules, and molecules with symmetry lower than orthorhombic are considered.Each ν-term consists of a contraction of the rotational tensor λ(2) and the nuclear quadrupole tensor in the space-fixed frame, and its matrix elements in the rotation-nuclear spin coupled representation can be determined by the standard spherical tensor methods.

Optimal model for 4-band biorthogonal wavelets bases for fast calculation

A class of 4-band symmetric biorthogonal wavelet bases has been constructed, in which any wavelet system the high-pass filters can be determined by exchanging position and changing the sign of the two low-pass filters. Thus, the least restrictive conditions are needed for forming a wavelet so that the free degrees can be reversed for application requirement. Some concrete examples with high vanishing moments are also given, the properties of the transformation matrix are studied and the optimal model is constructed. These wavelets can process the boundary conveniently, and they lead to highly efficient computations in applications.

Invariant convex subcones of the Tits cone of a linear Coxeter group

We investigate the faces and the face lattices of arbitrary Coxeter group invariant convex subcones of the Tits cone of a linear Coxeter system as introduced by E.B. Vinberg. Particular examples are given by certain Weyl group invariant polyhedral cones, which underlie the theory of normal reductive linear algebraic monoids as developed by M.S. Putcha and L.E. Renner. We determine the faces and the face lattice of the Tits cone and the imaginary cone, generalizing some of the results obtained for linear Coxeter systems with symmetric root bases by M. Dyer, and for linear Coxeter systems with free root bases by E. Looijenga, P. Slodowy, and the author.

The Delta Conjecture at 𝑞=1

We use a weight-preserving, sign-reversing involution to find a combinatorial expansion of Δ e k e n \Delta _{e_k} e_n at q = 1 q=1 in terms of the elementary symmetric function basis. We then use a weight-preserving bijection to prove the Delta Conjecture of Haglund, Remmel, and Wilson at q = 1 q=1 . The method of proof provides a variety of structures which can compute the inner product of Δ e k e n | q = 1 \Delta _{e_k} e_n|_{q=1} with any symmetric function.

Systematic study of multi-quasiparticleK-isomeric bands in tungsten isotopes by the extended projected shell model

Background: The interplay between collective and single-particle degrees of freedom is an important structure aspect to study. The nuclei in theA ≈ 180 mass region are often denoted as good examples to study such problems because these nuclei are known to exhibit many rotational bands based on multi-quasiparticle K isomers. Purpose: A large set of high-quality experimental data on high-K isomeric states in the A ≈ 180 mass region has accumulated. A systematic description of them is a theoretical challenge as it requires a method going beyond the usual mean field with multi-quasiparticle configurations built in the shell-model basis. The K-isomer data provide an ideal testing ground for theoretical models. Method: The recently extended projected shell model (PSM) by the Pfaffian method is employed with a sufficiently large configuration space including up to 10 quasiparticles. The restoration of rotational symmetry which is broken in the deformed mean field is obtained by means of angular-momentum projection. With axial symmetry in the basis deformation, each multi-quasiparticle state, classified by a K quantum number, represents the major component of a rotational K band. Shell-model diagonalization in such a projected basis defines the K mixing, which is the key ingredient of the present method. Results: Quasiparticle structure and rotational properties of high-K isomers in even-even neutron-rich 174−186W isotopes are described. The rotational evolution of the yrast and near-yrast bands is discussed with successive band crossings. Multi-quasiparticle K isomers and associated rotational bands in each W isotope are studied with detailed quasiparticle configurations given. Electromagnetic transition properties are also studied and the calculated B(E2), B(M1), and g-factors are compared with experiment if data exist. Conclusions: Many nuclei of the A ≈ 180 mass region exhibit properties of an axially symmetric shape and K is approximately a good quantum number. For such nuclei, the extended PSM assuming an axially symmetric basis but including K mixing through diagonalization of the two-body Hamiltonian is an appropriate method to study multi-quasiparticle K isomers and K violations in these states. For special examples where one finds highly K-forbidden transitions the present model needs to be further improved.

Synthesis, Characterization and Biological Evaluation of New Series of Schiff Bases Derived from Hexamethylenediamine as Potential Antibacterial and Antifungal Agents

&lt;div&gt;&lt;p&gt;&lt;em&gt;We report herein synthesis, characterization and antimicrobial activity of four Shiff bases derived from hexamethylenediamine. Substitution of a nitro group on each aromatic ring in ortho, meta or para positions of N,N'-bis(phenylmethyl)hexane-1,6-diimineallowed to have a homogeneous series of positional isomers. These four symmetric diimine Schiff bases were characterized by conventional spectrometry methods (NMR, IR, MS), then tested against Gram-positive and Gram-negative bacterial strains. Among them, compounds &lt;strong&gt;1b&lt;/strong&gt;, &lt;strong&gt;1c&lt;/strong&gt;, &lt;strong&gt;1d&lt;/strong&gt; were found to be active against bacterial strain Staphylococcus aureus CIP with MIC value of 375&lt;/em&gt;&lt;em&gt;μ&lt;/em&gt;&lt;em&gt;g/ ml, 187.5&lt;/em&gt;&lt;em&gt;μ&lt;/em&gt;&lt;em&gt;g /ml and 375&lt;/em&gt;&lt;em&gt;μ&lt;/em&gt;&lt;em&gt;g /ml respectively. Candida Albicans fungal strain showed resistance to all synthesized Schiff base compounds, butin the other hand, Candida glabrata has been sensitive to all compounds with MIC of 1500&lt;/em&gt;&lt;em&gt;μ&lt;/em&gt;&lt;em&gt;g/ ml and one more time except &lt;strong&gt;1a&lt;/strong&gt;.&lt;/em&gt;&lt;/p&gt;&lt;/div&gt;

Symmetric minimally entangled typical thermal states for canonical and grand-canonical ensembles

Based on the density matrix renormalization group (DMRG), strongly correlated quantum many-body systems at finite temperatures can be simulated by sampling over a certain class of pure matrix product states (MPS) called minimally entangled typical thermal states (METTS). When a system features symmetries, these can be utilized to substantially reduce MPS computation costs. It is conceptually straightforward to simulate canonical ensembles using symmetric METTS. In practice, it is important to alternate between different symmetric collapse bases to decrease autocorrelations in the Markov chain of METTS. To this purpose, we introduce symmetric Fourier and Haar-random block bases that are efficiently mixing. We also show how grand-canonical ensembles can be simulated efficiently with symmetric METTS. We demonstrate these approaches for spin-1/2 XXZ chains and discuss how the choice of the collapse bases influences autocorrelations as well as the distribution of measurement values and, hence, convergence speeds.

Synthesis, Characterization and Antimicrobial Evaluation of Symmetric Α-Diimine Schiff Bases Derived from Cis and Trans Racemic Mixture of Cyclohexanediamine

&lt;div&gt;&lt;pre&gt;&lt;em&gt;From&lt;strong&gt; &lt;/strong&gt;N,N'-bis(phenylmethylene)cyclohexane-1,2-diamine, substitution of a nitro group on each aromatic ring and its systematic displacement in the positions ortho,&lt;strong&gt; &lt;/strong&gt;meta and para positions allowed to synthesize a homogeneous series of positional isomers. These four symmetric α-diimine Schiff bases derived from cis and trans racemic mixture of cyclohexanediamine have been characterized by conventional spectroscopic methods (NMR, IR and MS). Antimicrobial screening showed that, unlike N, N'-bis (phenylmethylene) cyclohexane-1,2-diamine, the bacterial strain Staphylococcus aureus CIP is sensitive to the other three compounds with MIC values of 93.75&lt;/em&gt;&lt;em&gt;μ&lt;/em&gt;&lt;em&gt;g/ml, 187.5&lt;/em&gt;&lt;em&gt;μ&lt;/em&gt;&lt;em&gt;g/ml and 375&lt;/em&gt;&lt;em&gt;μ&lt;/em&gt;&lt;em&gt;g/ml. The Candida albicans fungal strain shows resistance to all synthesized compounds, but Candida glabrata is sensitive to the non-substituted N, N'-bis(phenylmethylene)cyclohexane-1,2-diamine and ortho substituted compound with a MIC value of 1500&lt;/em&gt;&lt;em&gt;μ&lt;/em&gt;&lt;em&gt;g/ml. &lt;/em&gt;&lt;/pre&gt;&lt;/div&gt;

Nonadiabatic laser-induced alignment of molecules: Reconstructing ⟨𝖼𝗈𝗌𝟤 θ⟩ directly from ⟨𝖼𝗈𝗌𝟤 θ2D⟩ by Fourier analysis.

We present an efficient, noise-robust method based on Fourier analysis for reconstructing the three-dimensional measure of the alignment degree, ⟨cos2θ⟩, directly from its two-dimensional counterpart, ⟨cos2θ2D⟩. The method applies to nonadiabatic alignment of linear molecules induced by a linearly polarized, nonresonant laser pulse. Our theoretical analysis shows that the Fourier transform of the time-dependent ⟨cos2θ2D⟩ trace over one molecular rotational period contains additional frequency components compared to the Fourier transform of ⟨cos2θ⟩. These additional frequency components can be identified and removed from the Fourier spectrum of ⟨cos2θ2D⟩. By rescaling of the remaining frequency components, the Fourier spectrum of ⟨cos2θ⟩ is obtained and, finally, ⟨cos2θ⟩ is reconstructed through inverse Fourier transformation. The method allows the reconstruction of the ⟨cos2θ⟩ trace from a measured ⟨cos2θ2D⟩ trace, which is the typical observable of many experiments, and thereby provides direct comparison to calculated ⟨cos2θ⟩ traces, which is the commonly used alignment metric in theoretical descriptions. We illustrate our method by applying it to the measurement of nonadiabatic alignment of I2 molecules. In addition, we present an efficient algorithm for calculating the matrix elements of cos2θ2D and any other observable in the symmetric top basis. These matrix elements are required in the rescaling step, and they allow for highly efficient numerical calculation of ⟨cos2θ2D⟩ and ⟨cos2θ⟩ in general.

Duality preserving Gray maps for codes over rings

Given a finite ring [Formula: see text] which is a free left module over a subring [Formula: see text] of [Formula: see text], two types of [Formula: see text]-bases, pseudo-self-dual bases (similar to trace orthogonal bases) and symmetric bases, are defined which in turn are used to define duality preserving maps from codes over [Formula: see text] to codes over [Formula: see text]. Both types of bases are generalizations of similar concepts for fields. Many illustrative examples are given to shed light on the advantages to such mappings as well as their abundance.