The generalized Kupershmidt deformation for constructing new discrete integrable systems

It is known that the KdV6 equation can be represented as the Kupershmidt deformation of the KdV equation. We propose a generalized Kupershmidt deformation for constructing new discrete integrable systems starting from the bi-Hamiltonian structure of a discrete integrable system. We consider the Toda, Kac-van Moerbeke, and Ablowitz-Ladik hierarchies and obtain Lax representations for these new deformed systems. The generalized Kupershmidt deformation provides a new way to construct discrete integrable systems.     

Envelopes and Covers by Modules of Finite FP-Injective Dimensions

Let R be a ring, n a fixed non-negative integer and ? the class of all left R-modules of FP-injective dimensions at most n. It is proved that all left R-modules over a left coherent ring R have ?-preenvelopes and ?-covers. Left (right) ?-resolutions and the left derived functors of Hom are used to study the FP-injective dimensions of modules and rings.     

Full automorphism group of the generalized symplectic graph

Let Fq be a finite field of odd characteristic, m, ν the integers with 1≤m≤ν and Ka 2ν× 2ν nonsingular alternate matrix over Fq. In this paper, the generalized symplectic graph GSp2ν (q, m) relative to K over Fq is introduced. It is the graph with m-dimensional totally isotropic subspaces of the 2ν-dimensional symplectic space F(2ν)q as its vertices and two vertices P and Q are adjacent if and only if the rank of PKQT is 1 and the dimension of P ∩ Q is m-1. It is proved that the full automorphism group of the graph GSp2ν(q, m) is the projective semilinear symplectic group PΣp(2ν, q).     

Lagrangian conditions for approximate solutions on nonconvex set-valued optimization problems

The purpose of this paper is to consider the set-valued optimization problem in Asplund spaces without convexity assumption. By a scalarization function introduced by Tammer and Weidner (J Optim Theory Appl 67:297–320, 1990), we obtain the Lagrangian condition for approximate solutions on set-valued optimization problems in terms of the Mordukhovich coderivative.     

Two new triangles of q-integers via q-Eulerian polynomials of type A and B

The classical Eulerian polynomials can be expanded in the basis t ^k?1(1+t) ^n+1?2k (1≤k≤?(n+1)/2?) with positive integral coefficients. This formula implies both the symmetry and the unimodality of the Eulerian polynomials. In this paper, we prove a q-analogue of this expansion for Carlitz’s q-Eulerian polynomials as well as a similar formula for Chow–Gessel’s q-Eulerian polynomials of type B. We shall give some applications of these two formulas, which involve two new sequences of polynomials in the variable q with positive integral coefficients. It is an open problem to give a combinatorial interpretation for these polynomials.     

DLSTM-Based Successive Cancellation Flipping Decoder for Short Polar Codes

Polar code has been adopted as the control channel coding scheme for the fifth generation (5G), and the performance of short polar codes is receiving intensive attention. The successive cancellation flipping (SC flipping) algorithm suffers a significant performance loss in short block lengths. To address this issue, we propose a double long short-term memory (DLSTM) neural network to locate the first error bit. To enhance the prediction accuracy of the DLSTM network, all frozen bits are clipped in the output layer. Then, Gaussian approximation is applied to measure the channel reliability and rank the flipping set to choose the least reliable position for multi-bit flipping. To be robust under different codewords, padding and masking strategies aid the network architecture to be compatible with multiple block lengths. Numerical results indicate that the error-correction performance of the proposed algorithm is competitive with that of the CA-SCL algorithm. It has better performance than the machine learning-based multi-bit flipping SC (ML-MSCF) decoder and the dynamic SC flipping (DSCF) decoder for short polar codes.     

Tapered Bragg reflection waveguide edge emitting lasers with near circular twin-beam emission

An edge emitting laser with two symmetrical near-circular spots located far field （FF） is demonstrated using tapered double-sided Bragg reflection waveguides （BRWs）. The BRWs consist of six pairs of top p-type and bottom n-type A10.Ga0.9 As/A10.3 Ga0.7As Bragg reflectors with a period thickness of 850 nm. The device has a 4° tapered angle configuration and exhibits two stable circular beams with a separation angle of 52°. Typical FF angles of 5.87° and 7.8° in the lateral and vertical directions, respectively, are achieved. The lateral FF angle in the ridged section is independent of the injection current （〉0.8 A） beeause of narrow ridge （-10 μm） confinement. By contrast, the FF angle in the tapered section shows an increase rate of 1.2 1.66°/A. The periodic modulation of the lasing wavelength is observed to be sensitive to self-heating effects.     

Near-infrared downconversion in Eu2＋ and Pr3＋ co-doped KSrPO4 phosphor

A novel near-infrared (NIR) downconversion (DC) phosphor KSrPO4 :Eu2+ , Pr3+ is synthesized by the conventional high temperature solid-state reaction. The Eu2+ acts as an efficient sensitizer for Pr3+ in the KSrPO4 host. With broad-band near-ultraviolet light excitation induced by the 4f→5d transition of Eu2+ , the characteristic NIR emission of Pr3+ , peaking at 974 nm and 1019 nm due to 3P0 → 1G4 and 1G4 → 3H4 transitions, is generated as a result of the energy transfer from Eu2+ to Pr3+ . The luminescence spectra in both the visible and the NIR regions and the decay lifetime curves of Eu2+ prove the energy transfer from Eu2+ to Pr3+ . This Eu2+ and Pr3+ co-doped KSrPO4 phosphor may be a promising candidate to modify the spectral mismatch behavior of crystalline solar cells and sunlight.     

A meshless model for transient heat conduction analyses of 3D axisymmetric functionally graded solids

A meshless numerical model is developed for analyzing transient heat conductions in three-dimensional （3D） axisymmetric continuously nonhomogeneous functionally graded materials （FGMs）. Axial symmetry of geometry and boundary conditions reduces the original 3D initial-boundary value problem into a two-dimensional （2D） problem. Local weak forms are derived for small polygonal sub-domains which surround nodal points distributed over the cross section. In order to simplify the treatment of the essential boundary conditions, spatial variations of the temperature and heat flux at discrete time instants are interpolated by the natural neighbor interpolation. Moreover, the using of three-node triangular finite element method （FEM） shape functions as test functions reduces the orders of integrands involved in domain integrals. The semi-discrete heat conduction equation is solved numerically with the traditional two-point difference technique in the time domain. Two numerical examples are investigated and excellent results are obtained, demonstrating the potential application of the proposed approach.