Symmetric Pairs Research Articles

Symplectic symmetric pairs of contragredient Lie superalgebras

Let L be a contragredient Lie superalgebra. A symmetric pair of L is a pair $$({\mathfrak {g}},{\mathfrak {g}}^\sigma )$$ , where $${\mathfrak {g}}$$ is a real form of L, and $$\sigma $$ is a $${\mathfrak {g}}$$ -involution with invariant subalgebra $${\mathfrak {g}}^\sigma $$ . We show that a symmetric pair carries invariant symplectic forms if and only if $${\mathfrak {g}}^\sigma $$ has a 1-dimensional center. Furthermore, the symplectic form is pseudo-Kahler if and only if the center of $${\mathfrak {g}}^\sigma $$ is compact. As an application, we classify the symplectic symmetric pairs, as well as the pseudo-Kahler symmetric pairs.

Damping of Vacuum Rabi Oscillations in a Two-Qubit Structure in a High-Q Cavity

The article studies the attenuation of vacuum Rabi oscillations for a system of two superconducting solid-state qubits placed in a high-Q microwave cavity. Two different cases are considered in which: 1) the first qubit is excited at the initial time and 2) the initial state comprises an entangled symmetric and antisymmetric pair of states. The dependence of the attenuation on various parameters, primarily on the coupling constant between qubits and the field and on the distance between qubits, is studied in detail. It is shown that, for some parameters, the relaxation time of the excited state of a qubit in such a system is significantly longer than that of a single qubit in the cavity.

Mirror symmetry of nonabelian Landau–Ginzburg orbifolds with loop type potentials

We study the bigraded vector space structure of Landau–Ginzburg orbifolds with the permutation symmetries of variables. We calculate the formula of the Poincaré polynomial for a certain loop polynomial orbifoldized by a semidirect product of a diagonal symmetry group and cyclic permutation group and check that the generalization of Berglund–Hübsch transpose gives mirror symmetric pairs for these cases.

Theory of electrolytes including steric, attractive, and hydration interactions.

We present a continuum theory of electrolytes composed of a waterlike solvent and univalent ions. First, we start with a density functional F for the coarse-grained solvent, cation, and anion densities, including the Debye-Hückel free energy, the Coulombic interaction, and the direct interactions among these three components. These densities fluctuate obeying the distribution ∝exp(-F/kBT). Eliminating the solvent density deviation in F, we obtain the effective non-Coulombic interactions among the ions, which consist of the direct ones and the solvent-mediated ones. We then derive general expressions for the ion correlation, the apparent partial volume, and the activity and osmotic coefficients up to linear order in the average salt density ns. Second, we perform numerical analysis using the Mansoori-Carnahan-Starling-Leland model [J. Chem. Phys. 54, 1523 (1971)] for three-component hardspheres. The effective interactions sensitively depend on the cation and anion sizes due to competition between the steric and hydration effects, which are repulsive between small-large ion pairs and attractive between symmetric pairs. These agree with previous experiments and Collins' rule [Biophys. J. 72, 65 (1997)]. We also give simple approximate expressions for the ionic interaction coefficients valid for any ion sizes.

Color and cross diagonal symmetric pair co-occurrence matrix

Objectives: The main goal of this study is to derive an efficient feature vector to capture both the color and texture information. This makes the proposed descriptor as a multipurpose descriptor. Methodology: This study derived color information by combining the individual histograms of H, S and V plane. A texture descriptor is derived by considering the relationship: i) between pairs of symmetric cross pixels and ii) between pairs of symmetric diagonal pixels of the 3 x 3 neighborhood. The relationship is derived using XOR and AND logical functions. Findings: This study derived a unique six bit code and constructed a co-occurrence matrix. The GLCM features are derived on this. To reduce the complexity, this study also derived another descriptor by indexing the six bit code using the rotational invariant property. Novelty: The proposed color and texture descriptors give better results, and a significant improvement is achieved in this study by concatenating these two descriptors for CBIR. Keywords: CBIR; firstorder; symmetric; derivatives; histogram; indexed; GLCM

Braid group actions for quantum symmetric pairs of type AIII/AIV

In the present paper we construct braid group actions on quantum symmetric pair coideal subalgebras of type AIII/AIV. This completes the proof of a conjecture by Kolb and Pellegrini in the case where the underlying Lie algebra is sln. The braid group actions are defined on the generators of the coideal subalgebras and the defining relations and braid relations are verified by explicit calculations.

Regular functions on spherical nilpotent orbits in complex symmetric pairs: Exceptional cases

Given an exceptional simple complex algebraic group G and a symmetric pair ( G , K ) , we study the spherical nilpotent K -orbit closures in the isotropy representation of K . We show that they are all normal except in one case in type G 2 , and we compute the K -module structure of the ring of regular functions on their normalizations.

A memristive chaotic oscillator with controllable amplitude and frequency

Abstract Broadband frequency property of chaos stands in the way of application. Internal parameter of a memristor leaves a new opportunity for easy amplitude/frequency control. Based on a variable-boostable chaotic system, such an amplitude-frequency controllable chaotic system is derived, which exhibits relatively robust chaotic oscillation compared with other multistable memristive systems. Furthermore, the controllable chaotic system gives a symmetric pair of oscillations, which finally merges to be a single symmetric one. Circuit experiment based on STM32 agrees well with the systematic analysis and numerical simulation.

On the symmetric Gelfand pair (ℋn ×ℋn−1,diag(ℋn−1))

We show that the [Formula: see text]-conjugacy classes of [Formula: see text] where [Formula: see text] is the hyperoctahedral group on [Formula: see text] elements, are indexed by marked bipartitions of [Formula: see text] This will lead us to prove that [Formula: see text] is a symmetric Gelfand pair and that the induced representation [Formula: see text] is multiplicity free.

Independent Control of the Chirality and Polarity for the Magnetic Vortex in Symmetric Nanodot Pairs

We studied the vortex states during the magnetization process for nanomagnetic dot pairs in different geometries, including a series of dual regular polygons with 4–16 sides and irregular shape dot pairs. All geometries demonstrated independent control of the vortex chirality and polarity and could be accomplished by adjusting the in-plane magnetic field direction. To achieve chirality and polarity control, both shape anisotropy and coupling interaction play a vital role. For the regular polygons, the effect of shape anisotropy wanes as the number of side increases, and the coupled interaction is enhanced relatively. According to the results, and combined with those for dual-circle and dual-rectangular magnetic disks, we state the principle for the geometry of the disk to achieve independent control of the chirality and polarity.

Alignment of Cell Lineage Trees Elucidates Genetic Programs for the Development and Evolution of Cell Types

SummaryA full understanding of the developmental process requires fine-scale characterization of cell divisions and cell types, which are naturally organized as the developmental cell lineage tree (CLT). Technological breakthroughs facilitated determination of more CLTs, but complete comprehension of the data remains difficult without quantitative comparison among CLTs. We hereby quantified phenotypic similarity between CLTs using a novel computational method that exhaustively searches for optimal correspondence between individual cells meanwhile retaining their topological relationships. The revealed CLT similarities allowed us to infer functional similarity at the transcriptome level, identify cell fate transformations, predict functional relationships between mutants, and find evolutionary correspondence between cell types of different species. By allowing quantitative comparison between CLTs, our work is expected to greatly enhance the interpretability of relevant data and help answer the myriad of questions surrounding the developmental process.

A SERRE PRESENTATION FOR THE ıQUANTUM GROUPS

Let (U, ) be a quasi-split quantum symmetric pair of arbitrary Kac–Moody type, where “quasi-split” means the corresponding Satake diagram contains no black node. We give a presentation of the group with explicit relations. The verification of new relations is reduced to some new q-binomial identities. Consequently, is shown to admit a bar involution under suitable conditions on the parameters.

A criterion for discrete branching laws for Klein four symmetric pairs and its application to E6(−14)

Let [Formula: see text] be a noncompact connected simple Lie group, and [Formula: see text] a Klein four-symmetric pair. In this paper, we show a necessary condition for the discrete decomposability of unitarizable simple [Formula: see text]-modules for Klein for symmetric pairs. Precisely, if certain conditions hold for [Formula: see text], there does not exist a unitarizable simple [Formula: see text]-module that is discretely decomposable as a [Formula: see text]-module. As an application, for [Formula: see text], we obtain a complete classification of Klein four symmetric pairs [Formula: see text], with [Formula: see text] noncompact, such that there exists at least one nontrivial unitarizable simple [Formula: see text]-module that is discretely decomposable as a [Formula: see text]-module and is also discretely decomposable as a [Formula: see text]-module for some nonidentity element [Formula: see text].

Analytical and FE calculations for determining the nominal tooth root stress for external, cylindrical gears with symmetric and asymmetric profile

The present work deals with the geometry dependence of the nominal tooth root stress of external toothed, cylindrical gears. The profile geometries required to perform the calculations are derived by our own program in MATLAB. Finite element simulations are executed in Abaqus. When designing the models, the geometric constraints of each tooth crown were optimized, keeping in mind the accuracy of the simulation. In addition to the analysis of the significant tooth stress value of symmetrical element pairs, special emphasis is placed on the development of the position of the critical cross-section. The numerical results obtained are also compared with the most significant standardized methods used in practice. The effect of the asymmetric design of the tooth profile on the nominal tooth root stress is reviewed in our investigations. The purpose of the numerical simulations carried out here is to determine the effect of the coast side angle on the dominant tooth root stress. In the evaluation of the results, the location of the critical cross-section, in addition to the magnitude of the stress, is also considered.

Synthetic Strategies and Computational Inhibition Activity Study for Triazinyl-Substituted Benzenesulfonamide Conjugates with Polar and Hydrophobic Amino Acids as Inhibitors of Carbonic Anhydrases.

Various sulfonamide derivatives are intensively studied as anticancer agents owing to their inhibitory activity against human tumor-associated carbonic anhydrase isoforms. In this work, different synthetic procedures for the series of 1,3,5-triazinyl-aminobenzenesulfonamide conjugates with amino acids, possessing polar uncharged, negatively charged, and hydrophobic side chain, were studied and optimized with respect to the yield/purity of the synthesis/product as well as the time of synthetic reaction. These procedures were compared to each other via characteristic HPLC-ESI-DAD/QTOF/MS analytical product profiles, and their benefits as well as limitations were discussed. For new sulfonamide derivatives, incorporating s-triazine with a symmetric pair of polar and some less-polar proteinogenic amino acids, inhibition constants (KIs) against four human carboanhydrases (hCAs), namely cytosolic hCA I, II, transmembrane hCA IV, and the tumor-associated, membrane-bound hCA IX isoforms, were computationally predicted applying various methods of the advanced statistical analysis. Quantitative structure-activity relationship (QSAR) analysis indicated an impressive KI ratio (hCA II/hCA IX) 139.1 and hCA IX inhibition constant very similar to acetazolamide (KI = 29.6 nM) for the sulfonamide derivative disubstituted with Gln. The derivatives disubstituted with Ser, Thr, and Ala showed even lower KIs (8.7, 13.1, and 8.4 nM, respectively).

Regular functions on spherical nilpotent orbits in complex symmetric pairs: Classical Hermitian cases

Given a classical semisimple complex algebraic group G and a symmetric pair (G,K) of Hermitian type, we study the closures of the spherical nilpotent K-orbits in the isotropy representation of K. We show that all such orbit closures are normal and describe the K-module structure of their ring of regular functions.

A Generalization of Steinberg Theory and an Exotic Moment Map

Abstract For a reductive group $G$, Steinberg established a map from the Weyl group to the set of nilpotent $G$-orbits by using moment maps on double flag varieties. In particular, in the case of the general linear group, it provides a geometric interpretation of the Robinson–Schensted correspondence between permutations and pairs of standard tableaux of the same shape. We extend Steinberg’s approach to the case of a symmetric pair $(G, K)$ to obtain two different maps, namely a generalized Steinberg map and an exotic moment map. Although the framework is general, in this paper we focus on the pair $(G,K) = (\textrm{GL}_{2n}({\mathbb{C}}), \textrm{GL}_n({\mathbb{C}}) \times \textrm{GL}_n({\mathbb{C}}))$. Then the generalized Steinberg map is a map from partial permutations to the pairs of nilpotent orbits in $ \mathfrak{gl}_n({\mathbb{C}}) $. It involves a generalization of the classical Robinson–Schensted correspondence to the case of partial permutations. The other map, the exotic moment map, establishes a combinatorial map from the set of partial permutations to that of signed Young diagrams, that is, the set of nilpotent $ K$-orbits in the Cartan space $(\textrm{Lie}(G)/\textrm{Lie}(K))^* $. We explain the geometric background of the theory and combinatorial algorithms, which produce the above-mentioned maps.

Optimal Nonparametric Prediction Intervals of Electricity Load

A novel machine learning based mixed integer programming model is developed for the optimal nonparametric prediction intervals (PIs) of electricity load, which minimizes interval width subject to target hit probability constraint. Binary variables are employed to analytically formulate PI coverage states and further impose calibration constraint. Considering the quantile interpretation on lower and upper bounds of PIs, an innovative binary variable reduction strategy is proposed to significantly accelerate model training process. Compared with traditional central PIs, the resultant optimal nonparametric PIs are featured with adaptive symmetric and asymmetric quantile proportion pairs for interval shortening. Numerical experiments based on actual substation data validate the remarkable superiority of the presented method with respect to both PI quality and computational efficiency.

The deceptive nature of associative word pairs: the effects of associative direction on judgments of learning.

The accuracy of judgments of learning (JOLs) in forecasting later recall of cue-target pairs is sensitive to associative direction. JOLs are generally well calibrated for forward associative pairs (e.g., credit-card), but recall accuracy is often overestimated for backward pairs (e.g., card-credit). The present study further examines the effect of associative direction on JOL accuracy by comparing forward and backward pairs to unrelated pairs and symmetrical associates (e.g., salt-pepper)-a novel comparison. The correspondence between initial JOLs and recall accuracy was examined when study was either self-paced with concurrent JOLs (Experiment 1), when study/JOL duration was equated across pair types (Experiment 2), when JOLs were made immediately following study (Experiment 3), and when JOLs were made after a delay (Experiment 4). Across experiments, JOLs accurately estimated correct recall for forward pairs, but overestimated recall for symmetrical, backward, and unrelated pairs-an overestimation that was particularly robust for backward pairs. Calibration plots depicting JOL ratings against their corresponding recall accuracy indicated overestimations occurred for all pair types, though overestimations only occurred at high JOL ratings for symmetrical and forward pairs, a qualitative difference that was not captured in standard analyses of mean JOL and recall rates.

RF Frequency Synthesizer Based on Self-Mode-Locked Multimode Lasers

Low-noise local oscillators are very important in modern communication systems. To realize a stable solution with a limited size, symmetric multimode multi-section semiconductor laser pairs are reported in this article for building a chip-level RF synthesizer. The symmetric laser structure utilizes a shared distributed Bragg reflector (DBR) and counterpropagating laser output with the help of an external feedback circuit to achieve self-mode-locking. The achieved phase noise performance is −92 dBc/Hz at a 10 kHz offset frequency for intermodal operation at 26.4 GHz using 5 locked modes, which is a 57 dB improvement compared to the free-running case. Moreover, an RF synthesizer covering frequencies of 11–13 GHz is realized based on the achieved mode-locked system using current tuning. The related phase noise of the laser pair beat-note output is −100 dBc/Hz at a 10 kHz offset of carrier frequencies of 11–13 GHz, which is 58 dB better than the phase noise of the beat-note output without mode-locking over 11–13 GHz.