Generalized Macaulay-Northcott Modules and Tor-Groups

Let (S,≤) be a strictly totally ordered monoid which is also artinian, and R a right noetherian ring. Assume that M is a finitely generated right R-module and N is a left Rmodule. Denote by [[MS,≤]] and [NS,≤] the module of generalized power series over M, and the generalized Macaulay-Northcott module over N, respectively. Then we show that there exists an isomorphism of Abelian groups:Tori[[ RS,≤]]([[MS,≤]],[NS,≤])≌ s∈S ToriR (M,N).

Design of Protograph LDPC-Coded MIMO-VLC Systems with Generalized Spatial Modulation

This paper investigates the bit-interleaved coded generalized spatial modulation(BICGSM) with iterative decoding(BICGSM-ID) for multiple-input multiple-output(MIMO) visible light communications(VLC). In the BICGSM-ID scheme, the information bits conveyed by the signal-domain(SiD) symbols and the spatial-domain(SpD) light emitting diode(LED)-index patterns are coded by a protograph low-density parity-check(P-LDPC) code. Specifically, we propose a signal-domain symbol expanding and re-allocating(SSER) method for constructing a type of novel generalized spatial modulation(GSM) constellations, referred to as SSERGSM constellations, so as to boost the performance of the BICGSM-ID MIMO-VLC systems.Moreover, by applying a modified PEXIT(MPEXIT) algorithm, we further design a family of rate-compatible P-LDPC codes, referred to as enhanced accumulate-repeat-accumulate(EARA) codes,which possess both excellent decoding thresholds and linear-minimum-distance-growth property. Both analysis and simulation results illustrate that the proposed SSERGSM constellations and P-LDPC codes can remarkably improve the convergence and decoding performance of MIMO-VLC systems. Therefore, the proposed P-LDPC-coded SSERGSM-mapped BICGSMID configuration is envisioned as a promising transmission solution to satisfy the high-throughput requirement of MIMO-VLC applications.

Construction of semisimple category over generalized Yetter-Drinfeld modules

Let H be a commutative, noetherian, semisimple and cosemisimple Hopf algebra with a bijective antipode over a field k. Then the semisimplicity of YD（H） is considered, where YD （H） means the disjoint union of the category of generalized Yetter-Drinfeld modules nYD^H（ α, β） for any α, β E Aut Hopf（H）. First, the fact that YD（H） is closed under Mor is proved. Secondly, based on the properties of finitely generated projective modules and semisimplicity of H, YD（H） satisfies the exact condition. Thus each object in YD（H） can be decomposed into simple ones since H is noetherian and cosemisimple. Finally, it is proved that YD （H） is a sernisimple category.

RESEARCH ON THE GENERALIZED MODULAR DESIGN METHOD WITH FLEXIBLE MODULES OF PRODUCT STRUCTURES

A new generalized modular design (GMD) method is proposed based on designpractice of frame structure of hydraulic press machines. By building a series of flexible modules(FMs), design knowledge and structure features are integrated into parametric models. Then,parametric design and variational analysis methods for GMD are presented according to user defineddesign objectives and customized product characteristics. A FM-centered GMD system is developed andsuccessfully used in the rapid design of relevant products.

On generalized extending modules

A module Mis called generalized extending if for any submodule N of M, there is a direct summand K of M such that N≤K and KIN is singular. Any extending module and any singular module are generalized extending. Any homomorphic image of a generalized extending module is generalized extending. Any direct sum of a singular （uniform） module and a semi-simple module is generalized extending. A ring R is a right Co-H-ring if and only if all right R modules are generalized extending modules.

PS-modules over Rings of Generalized Power Series

Let R be a commutative ring and (S, ≤) a strictly totally ordered monoid which satisfies the condition that 0 ≤ s for every s ∈ S. In this paper we show that if RM is a PS-module, then the module [[MS≤]] of generalized power series over M is a PS [[RS,≤]]-module.

STABILITY OF GENERALIZED DERIVATIONS ON HILBERT C*-MODULES ASSOCIATED WITH A PEXIDERIZED CAUCHY-JENSEN TYPE FUNCTIONAL EQUATION

In this article, we introduce the notion of generalized derivations on Hilbert C＊-modules. We use a fixed-point method to prove the generalized Hyers-Ulam-Rassias stability associated to the Pexiderized Cauchy-Jensen type functional equationrf（x＋y/r）＋sg（x-y/s）=2h（x）for r, s ∈ R ／ {0} on Hilbert C＊-modules, where f, g, and h are mappings from a Hilbert C＊-module M to M.

A Note on Generalized Long Modules

Let HLB be the category of generalized Long modules, that is, H-modules and B-comodules over Hopf algebras B and H. We describe a new Turaev braided group category over generalized Long module HLB （S（π）） where the opposite group S（π） of the semidirect product of the opposite group πopof a group π by π. As an application, we show that this is a Turaev braided group-category HLBfor a quasitriangular Turaev group-coalgebra H and a coquasitriangular Turaev group-algebra B.

CONTROL STRATEGY FOR ELECTRO-HYDRAULIC POSITION SERVO SYSTEM WITH GENERALIZED PULSE CODE MODULATION

A hybrid control strategy has been designed and developed for the electro-hydraulic posi-tion servo control system with generalized Pulse code modulation （GPCM）, which is suitable for the area where the work condition is poor and a large flow rate is required. It is difficult to control the GPCM system because the system is discrete. With consideration of the stability and speediness of the GPCM position servo control system, a control strategy is developed through the theoretical and ex-perimental analyses. The control strategy integrates the merits of Bang-Bang control, PID control and fuzzy control. With this hybrid control strategy, the electro hydraulic control system has good per-formances, and the servo control is carried out with GPCM through on-off valves.

A FIXED POINT APPROACH TO THE STABILITY OF A GENERAL MIXED ADDITIVE-CUBIC EQUATION ON BANACH MODULES

Using a fixed-point method, we establish the generalized Hyers-Ulam stability of a general mixed additive-cubic equation： f（kx ＋ y） ＋ f（kx - y） = kf（x ＋ y） ＋ kf（x - y） ＋ 2f（kx） - 2kf（x） in Banach modules over a unital Banach algebra.

A Unified Bit-to-Symbol Mapping for Generalized Constellation Modulation

Gray mapping is a well-known way to improve the performance of regular constellation modulation,but it is challenging to be applied directly for irregular alternative.To address this issue,in this paper,a unified bit-to-symbol mapping method is designed for generalized constellation modulation(i.e.,regular and irregular shaping).The objective of the proposed approach is to minimize the average bit error probability by reducing the hamming distance(HD)of symbols with larger values of pairwise error probability.Simulation results show that the conventional constellation modulation(i.e.,phase shift keying and quadrature amplitude modulation(QAM)with the proposed mapping rule yield the same performance as that of classical gray mapping.Moreover,the recently developed golden angle modulation(GAM)with the proposed mapping method is capable of providing around1 d B gain over the conventional mapping counterpart and offers comparable performance to QAM with Gray mapping.

Adaptive Mapping Generalized Space Shift Keying Modulation

Generalized Space Shift Keying （GSSK） modulation is a low-complexity spatial nmltiplexing technique for nmltiple-antenna wireless systems. However, effective transmit antenna combinations have to be preselected, and there exist redundant antenna combinations which are not used in GSSK. In this paper, a novel adaptive mapping scheme for GSSK modulation, named as Adaptive Mapping Generalized Space Shift Keying （AMGSSK）, is presented. Compared with GSSK, the antenna combinations are updated adaptively according to the Channel State Inforrmtion （CSI） in the proposed AMGSSK system, and the perfonrance of average Symbol Error Rate （SER） is reduced considerably. In the proposed scheme, two algorithrrs for selecting the optimum antenna combinations are described. The SER perfonmnce of AMGSSK is analyzed theoretically, and validated by Monte Carlo sinmlation. It is shown that the proposed AMGSSK scheme has good perfonmnce in SER and spectral efficiency.

A Note on Finitely Generated Modules over a <i>PID</i>

In this note, we provide an effective proof of the fundamental structure theorem of finitely generated modules over a principal ideal domain, from which we find the minimality of decomposition for a finitely generated module over a principal ideal domain.

Semi-Supervised Learning with Generative Adversarial Networks on Digital Signal Modulation Classification

Deep Learning(DL)is such a powerful tool that we have seen tremendous success in areas such as Computer Vision,Speech Recognition,and Natural Language Processing.Since Automated Modulation Classification(AMC)is an important part in Cognitive Radio Networks,we try to explore its potential in solving signal modulation recognition problem.It cannot be overlooked that DL model is a complex model,thus making them prone to over-fitting.DL model requires many training data to combat with over-fitting,but adding high quality labels to training data manually is not always cheap and accessible,especially in real-time system,which may counter unprecedented data in dataset.Semi-supervised Learning is a way to exploit unlabeled data effectively to reduce over-fitting in DL.In this paper,we extend Generative Adversarial Networks(GANs)to the semi-supervised learning will show it is a method can be used to create a more dataefficient classifier.

Geometrical Langlands Ramifications and Differential Operators Classification by Coherent D-Modules in Field Theory

Spaces of equivalence modulo a relation of congruence are constructed on field solutions to establish a theory of the universe that includes the theory QFT （Quantum Field theory）, the SUSY （Super-symmetry theory） and HST （heterotic string theory） using the sheaves correspondence of differential operators of the field equations and sheaves of coherent D - Modules [1]. The above mentioned correspondence use a Zuckerman functor that is a factor of the universal functor of derived sheaves of Harish-Chandra to the Langlands geometrical program in mirror symmetry [2, 3]. The obtained development includes complexes of D - modules of infinite dimension, generalizing for this way, the BRST-cohomology in this context. With it, the class of the integrable systems is extended in mathematical physics and the possibility of obtaining a general theory of integral transforms for the space - time （integral operator cohomology [4]）, and with it the measurement of many of their observables [5]. Also tends a bridge to complete a classification of the differential operators for the different field equations using on the base of Verma modules that are classification spaces of SO（l, n ＋ 1）, where elements of the Lie algebra al（1, n ＋ 1）, are differential operators, of the equations in mathematical physics [1]. The cosmological problem that exists is to reduce the number of field equations that are resoluble under the same gauge field （Verma modules） and to extend the gauge solutions to other fields using the topological groups symmetries that define their interactions. This extension can be given by a global Langlands correspondence between the Hecke sheaves category on an adequate moduli stack and the holomorphic L G - bundles category with a special connection （Deligne connection）. The corresponding D - modules may be viewed as sheaves of conformal blocks （or co-invariants） （images under a version of the Penrose transform [1, 6]） naturally arising in the framework of conformal field theory.

Some Studies on Quasi AP-injective Modules

Let R be a ring. R is called right AP-injective if, for any a E R, there exists a left ideal of R such that lr（a） = Ra＋Xa. We extend this notion to modules. A right R-module M with S = End（MR） is called quasi AP-injective if, for any s∈S, there exists a left ideal Xs of S such that ls（Ker（s）） = Ss＋Xs. In this paper, we give some characterizations and properties of quasi AP-injective modules which generalize results of Page and Zhou.

Three-step self-calibrating generalized phase-shifting interferometry

An accurate and fast three-step self-calibrating generalized phase-shifting interferomertry(SGPSI) is proposed. In this approach, two new phase-shifting signals are constructed by the difference interferograms normalization and noise suppressing, then the unknown phase shift between the two difference phase-shifting signals is estimated quickly through searching the minimum coefficient of variation of the modulation amplitude, a limited number of pixels are selected to participate in the search process to further save time, and finally the phase is reconstructed through the searched phase shift. Through the reconstruction of phase map by the simulation and experiment, and the comparison with several mature algorithms, the good performance of the proposed algorithm is proved, and it eliminates the limitation of requiring more than three phase-shifting interferograms for high-precision SGPSI. We expect this method to be widely used in the future.

A General Algorithm for Biorthogonal Functions and Performance Analysis of Biorthogonal Scramble Modulation System

Applying the theorems of Mobius inverse and Dirichlet inverse, a general algorithm to obtain biorthogonal functions based on generalized Fourier series analysis is introduced. In the algorithm, the orthogonal function can be not only Fourier or Legendre series, but also can be any one of all orthogonal function systems. These kinds of biorthogonal function sets are used as scramble signals to construct biorthogonal scramble modulation (BOSM) wireless transmission systems. In a BOSM system, the transmitted signal has significant security performance. Several different BOSM and orthogonal systems are compared on aspects of BER performance and spectrum efficiency, simulation results show that the BOSM systems based on Chebyshev polynomial and Legendre polynomial are better than BOSM system based on Fourier series, also better than orthogonal MCM and OFDM systems.

Some Equivalences and Dualities via Static Modules

For a ring A, an extension ring B, a fixed right A-module M, the endomorphism ring D formed by M, the endomorphism ring E formed by , and the endomorphism ring F formed by HomA (B,M), we present equivalences and dualities between subcategories of B-modules which are finitely cogenerated injective as A-modules and E-modules and F-modules which are finitely generated projective as D-modules.

Gelfand-Kirillov Dimension and Reducibility of Scalar Generalized Verma Modules for Classical Lie Algebras

Let g be a classical complex simple Lie algebra and q be a parabolic subalgebra.Let M be a generalized Verma module induced from a one dimensional representation of q.Such M is called a scalar generalized Verma module.In this paper,we will determine the reducibility of scalar generalized Verma modules associated to maximal parabolic subalgebras by computing explicitly the Gelfand-Kirillov dimension of the corresponding highest weight modules.