Minimum Resolution of the Minkowski, Schwarzschild and Kerr Differential Modules

Our recent arXiv preprints and published papers on the solution of the Riemann-Lanczos and Weyl-Lanczos problems have brought our attention on the importance of revisiting the algebraic structure of the Bianchi identities in Riemannian geometry. We also discovered in the meantime that, in our first GB book of 1978, we had already used a new way for studying the compatibility conditions (CC) of an operator that may not be necessarily formally integrable (FI) in order to construct canonical formally exact differential sequences on the jet level. The purpose of this paper is to prove that the combination of these two facts clearly shows the specific importance of the Spencer operator and the Spencer δ-cohomology, totally absent from mathematical physics today. The results obtained are unavoidable because they only depend on elementary combinatorics and diagram chasing. They also provide for the first time the purely intrinsic interpretation of the respective numbers of successive first, second, third and higher order generating CC. However, if they of course agree with the linearized Killing operator over the Minkowski metric, they largely disagree with recent publications on the respective numbers of generating CC for the linearized Killing operator over the Schwarzschild and Kerr metrics. Many similar examples are illustrating these new techniques, providing in particular a few resolutions in which the orders of the successive operators may go “up and down” surprisingly, like in the conformal situation for various dimensions.

Killing Operator for the Kerr Metric

When D: E →F is a linear differential operator of order q between the sections of vector bundles over a manifold X of dimension n, it is defined by a bundle map Φ: Jq(E) →F=F0 that may depend, explicitly or implicitly, on constant parameters a, b, c, ... . A “direct problem” is to find the generating compatibility conditions (CC) in the form of an operator D1: F0 →F1. When D is involutive, that is when the corresponding system Rq = ker (Φ) is involutive, this procedure provides successive first order involutive operators D1, ..., Dn. Though D1 οD = 0 implies ad (D) οad(D1) = 0 by taking the respective adjoint operators, then ad (D) may not generate the CC of ad (D1) and measuring such “gaps” led to introduce extension modules in differential homological algebra. They may also depend on the parameters and such a situation is well known in ordinary or partial control theory. When Rq is not involutive, a standard prolongation/projection (PP) procedure allows in general to find integers r, s such that the image of the projection at order q+r of the prolongation is involutive but it may highly depend on the parameters. However, sometimes the resulting system no longer depends on the parameters and the extension modules do not depend on the parameters because it is known that they do not depend on the differential sequence used for their definition. The purpose of this paper is to study the above problems for the Kerr (m, a), Schwarzschild (m, 0) and Minkowski (0, 0) parameters while computing the dimensions of the inclusions for the respective Killing operators. Other striking motivating examples are also presented.

Adaptive Secure Transmission for Wireless Powered Communication Networks

This paper investigates secure transmission in a wireless powered communication network(WPCN)with an energy harvesting(EH)source configured with multiple antennas.In the WPCN,the EH source harvests energy from the radio frequency(RF)signals broadcasted by a power beacon(PB),and purely relies on the harvested energy to communicate with the destination in the presence of passive eavesdroppers.It is noteworthy that the RF-EH source is equipped with a finite energy storage to accumulate the harvested energy for the future usage.Moreover,due to energy-constraint and complexitylimitation,the multi-antenna source is only configured with a single RF-chain.To enhance the security for the WPCN,we propose two adaptive transmission schemes,i.e.,energy-aware transmit antenna selection(EATAS)and energy-aware differential spatial modulation(EADSM).According to the energy status and the channel quality,the source adaptively decides whether to transmit confidential information or harvest RF energy.To evaluate the secrecy performance of the proposed schemes,we first study the evolution of the energy storage,and then derive the analytical expressions of connection outage probability(COP),secrecy outage probability(SOP)and efficient secrecy throughput(EST).Numerical results demonstrate that our proposed schemes outperform the existing schemes,i.e.,time-switching based TAS(TS-TAS)Received:May 19,2020 Revised:Sep.13,2020 Editor:Deli Qiao and accumulate-then-transmit(ATT).And,increasing the transmit power of the PB or the capacity of the source’s energy storage is helpful to improve the secrecy performance.Moreover,there exists an optimal transmission rate for each proposed scheme to achieve best secrecy performance.

Differential Correspondences and Control Theory

When a differential field K having n commuting derivations is given together with two finitely generated differential extensions L and M of K, an important problem in differential algebra is to exhibit a common differential extension N in order to define the new differential extensions L∩M and the smallest differential field (L,M ) ⊂ N containing both L and M. Such a result allows to generalize the use of complex numbers in classical algebra. Having now two finitely generated differential modules L and M over the non-commutative ring D = K [d1,...,dn] = K [d] of differential operators with coefficients in K, we may similarly look for a differential module N containing both L and M in order to define L∩M and L+M. This is exactly the situation met in linear or non-linear OD or PD control theory by selecting the inputs and the outputs among the control variables. However, in many recent books and papers, we have shown that controllability was a built-in property of a control system, not depending on the choice of inputs and outputs. The purpose of this paper is thus to revisit control theory by showing the specific importance of the two previous problems and the part plaid by N in both cases for the parametrization of the control system. An important tool will be the study of differential correspondences, a modern name for what was called Bäcklund problem during the last century, namely the elimination theory for groups of variables among systems of linear or nonlinear OD or PD equations. The main difficulty is to revisit differential homological algebra by using noncommutative localization as a way to generalize the symbolic calculus in the style of Heaviside and Mikusinski. Finally, when M is a D-module, this paper is using for the first time the fact that the system R = homK (M,K) is a D-module for the Spencer operator acting on sections, avoiding thus behaviours, trajectories and signal spaces in a purely formal way, contrary to a few recent works on this difficult subject.

A New High Rate Differential Space-Time-Frequency Modulation for MIMO-OFDM

In this paper, we propose a new differential space-time-frequency （DSTF） modulation for MIMOOFDM system with four transmit-antennas and arbitrary receive-antennas, which can improve the transmission rate since it can adopt high order quadrature amplitude modulation （QAM） modulation. Our proposed DSTF scheme embeds some full diversity full rate （FDFR） quasi-orthogonal space-time codes （QOSTBC） with QAM modulation into the frequency intervals and adopts the differential modulation in both time and frequency domains. The simulation results demonstrate that the proposed DSTF scheme can improve transmission rate greatly. Compared with the conventional differential unitary space-time modulation （DUSTM）, it can get better transmission performance in high transmission rate for MIMO-OFDM system.

THE EFFECT OF HOT DRAWN ON THE PHYSICAL AGING OF ATACTIC POLYSTYRENE

The physical aging behavior of atactic polystyrene (aPS) films achieved under different drawing conditions has been studied by optical birefringence and modulate differential scanning: calorimeter (m.d.s.c.). The results show that on annealing at specific temperature below glass transition temperatue (T-g), the enthalpy relaxation (Delta H) and T-g decrease with increasing of birefringence (Delta n). On the other baud, the effect of molecular draw ratio (MDR) is confusing: Delta H and T-g decrease with increasing of MDR in the early stage of aging, but the MDR's effect become unobvious with the development of aging. It may be concluded that the ordered domain generated by the drawing below or above glass transition temperature will affect the development of physical aging behavior. The viewpoint of cohesional entanglement is used to account for the above observations.

LDPC based differential unitary space-frequency coding for MIMO-OFDM systems

This paper proposes a novel LDPC based differential unitary space-frequency coding (DUSFC) scheme for MIMO-OFDM systems when neither the transmitter nor the receiver has access to the channel state information (CSI). The new DUSFC strategy basically consists of coding across transmit antennas and OFDM tones simultaneously as well as differential modulation in the time-domain. It can fully exploit the inherent advantages provided by the multipath fading channels, resulting in a high degree of diversity. The state-of-the-art low-density parity-check (LDPC) codes are concatenated with our DUSFC as channel coding to improve the bit error rate (BER) performance considerably. Owing to the maximum multipath diversity and large coding advantages, LDPC-DUSFC strongly outperforms the differential unitary space-time coded OFDM techniques re- cently proposed in literature. The corresponding iterative decoding algorithm without channel estimation is finally provided to offer significant performance gain. Simulation results illustrate the merits of the proposed scheme.

Novel time-frequency differential space-time modulation for multi-antenna OFDM systems

Differential space-time （DST） modulation has been proposed recently for multiple-antenna systems over Rayleigh fading channels, where neither the transmitter nor the receiver knows the fading coefficients. Among existing schemes, differential modulation is always performed in the time domain and suffers performance degradations in frequency-selective fading channels. In order to combat the fast time and frequency-selective fading, a novel time-frequency differential space-time （TF-DST） modulation scheme, which adopts differential modulation in both time and frequency domains, is proposed for multi-antenna orthogonal frequency division multiplexing （OFDM） system. A corresponding suboptimal yet low-complexity non-coherent detection approach is also proposed. Simulation results demonstrate that the proposed system is robust for time and frequency-selective Rayleigh fading channels.

Differential modulation based on space-time block codes

A differential modulation scheme using space-time block codes is put forward. Compared with other schemes, our scheme has lower computational complexity and has a simpler decoder. In the case of three or four transmitter antennas, our scheme has a higher rate a higher coding gain and a lower bit error rate for a given rate. Then we made simulations for space-time block codes as well as group codes in the case of two, three, four and five transmit antennas. The simulations prove that using two transmit antennas, one receive antenna and code rate of 4 bits/s/Hz, the differential STBC method outperform the differential group codes method by 4 dB. Useing three, four and five transmit antennas, one receive antenna, and code rate of 3 bits/s/Hz are adopted, the differential STBC method outperform the differential group codes method by 5 dB, 6. 5 dB and 7 dB, respectively. In other words, the differential modulation scheme based on space-time block code is better than the corresponding differential modulation scheme

Differential Space Frequency Mapping Schemes and Norm Criterion

This paper proposed four types of differential modulation to map the unitary code into the orthogonal frequency division multiplexing(OFDM) signal. The time-varying channel model is established and the norm of detection metric is deduced. The norm is the inherent interference of the time-varying channel, so it can be used as criterion to evaluate the performance of the mapping schemes. The simulation results agree with the analytic conclusion.

Micro capacitance detection circuit for MEMS capacitive sensor

With the development of micro-electr o-mechanical system(MEMS) technolog y,the MEMS-based capacitive sensor has been widely applied in the field of ele ctron components.However,the capacitance of the micromachined sensor is so sma ll that the detection of the smaller value change of the capacitance is a great challenge.Based on the principle of charging and discharging of the capacitor,a kind of pulse width modulated differential circuit is introduced in this paper.For subsequent amplification,a modified amplifier is presented.The different ial circuit converts the weak capacitance change to the change of the pulse widt h of the output voltage,and the linear relationship can be obtained.And the mo dified amplifier implements the processes of amplification and filtering synchro nously,and a large DC output voltage can be obtained by the lo w-pass filter.T he designed circuits have advantages as simplified circuit,high voltage stabili ty,perfect linearity and resolution.Besides,it is feasible to be integrated w ith the sensor to largely reduce the transmission error and interference.

A NOVEL LINEAR DIFFERENTIAL SPACE-TIME MODULATION

Based on the Complex Orthogonal Linear Dispersion (COLD) code,a novel linear Differ- ential Space-Time Modulation (DSTM) design is proposed in this paper.Compared with the existing nonlinear DSTM schemes based on group codes,the proposed linear DSTM scheme is easier to design, enjoys full diversity and allows for a simplified differential receiver,which can detect the transmitted symbols separately.Furthermore,compared with the existing linear DSTM based on orthogonal design, our new construction can be applied to any number of transmit antennas.Similar to other algorithms, the proposed scheme also can be demodulated with or without channel estimates at the receiver,but the performance degrades approximately by 3dB when estimates are not available.

Differential space-time modulation in rapidly fading channels

Previously proposed differential modulation schemes for time-varying channels may not achieve the full transmit diversity and the maximum Doppler diversity simultaneously. Based on an existing basis expansion model, a new differential space-time code, which wisely combines interleaver/de-interleaver with traditonal space-time transmitting technique to overcome such limitation, .is presented. Two noncoherent differential decoders, named decision-feedback differential detector （DF-DD） and Viterbi-algorithmbased multiple-symbol-detection differential detector （ MSD-DD）, are also derived. We show that our design may recover data symbols with full antenna diversity and the maximum Doppler diversity at high signal-to-noise ratio. System performance is evaluated with simulations.

Differential space-time modulation based on orthogonal design for cooperative network

Differential modulation was widely used for wireless networks in which channel estimation was diffi-cult.Based on orthogonal design,a novel distributed differential space-time coding/decoding scheme forM-PSK modulations was proposed,which had a high code rate of 2/3 and second-order diversity for thetwo-user cooperative networks.The performance of decode-and-forward (DF) protocols was evaluated.Simulations show that the differential space-time modulation scheme in this paper has better bit error rate(BER) performance or higher code rate than the schemes proposed by Tarasak and Wang when interuserchannel states are good enough.The impacts of transmission error between two users for the whole systemBER performance were also investigated.

ROTATION CONSTELLATION FOR DIFFERENTIAL UNITARY SPACE-TIME MODULATION

A new constellation which is the multiplication of the rotation matrix and the diagonal matrix ac- cording to the number of transmitters is proposed to increase the diversity product, the key property to the performance of the differential unitary space-time modulation. Analyses and the simulation results show that the proposed constellation performs better and 2dB or more coding gain can be achieved over the traditional cyclic constellation.

Novel differential unitary space-time modulation schemes for fast fading channels

Differential unitary space-time modulation （DUSTM）, which obtains full transmit diversity in slowly fiat-fading channels without channel state iufonnation, has generated significant interests recently. To combat frequency-selective fading, DUSTM has been applied to each subcarrier of an OFDM system and DUSTM-OFDM system was proposed. Both DUSTM and DUSTM-OFDM, however, are designed for slowly fading channels and suffer performance deterioration in fast fading channels. In this paper, two novel differential unitary space-time modulation schemes are proposed for fast fading channels. For fast fiat-fading channels, a subatrix interleaved DUSTM （SMI-DUSTM） scheme is proposed, in which matrix-segmentation and sub-matrix based interleaving are introduced into DUSTM system. For fast frequency-selective fading channels, a differential unitary space-frequency modulation （DUSFM） scheme is proposed, in which existing unitary space-time codes are employed across transmit antennas and OFDM subcarriers simultaneouslv and differential modulation is performed between two adjacent OFDM blocks. Compared with DUSTM and DUSTM-OFDM schemes, SMI-DUSTM and DUSFM-OFDM are more robust to fast channel fading with low decoding complexity, which is demonstrated by performance analysis and simulation resuits.

SIGNAL CONSTELLATIONS DESIGN FOR DIFFERENTIAL UNITARY SPACE-TIME IN MIMO-OFDM SYSTEM OVER FREQUENCY-SELECTIVE FADING CHANNELS

In this paper, the design of signal constellations parameters is studied for Differential Unitary Space-Time Modulation (DUSTM) based on the design criterion of maximizing the diversity product. Further, noninteger searching method for the signal constellation parameters design is proposed in order to get better codes. Experimental results show that under the different Doppler spread and data transmission rate, the proposed design performs better than the previous design using integer parameters in Multiple Input Multiple Output Orthogonal Frequency Division Multiplexing (MIMO-OFDM) system over frequency-selective fading channels.

Canonical Form Associated with an r-Jacobi Algebra

In this paper, we denote by A a commutative and unitary algebra over a commutative field K of characteristic 0 and r an integer ≥1. We define the notion of r-Jacobi algebra A and we construct the canonical form associated with the r-Jacobi algebra A.

A new principle of traveling wave differential protection on series-capacitor-compensated transmission lines

Traveling wave differential protection has the ability,in theory,to entirely eliminate the effects of distributed capacitive current,but it cannot be applied on series-capacitor-compensated lines directly.In this paper,unbalanced output of conventional forward and reverse traveling wave differential currents under normal operating conditions and external faults was analyzed.A new type of traveling wave differential current was defined by combining forward and reverse traveling wave differential currents.Expressions of the defined differential current when internal and external faults occur were deduced.On this basis,a new principle of traveling wave module differential protection on series-capacitor-compensated lines was proposed,and characteristics of module differential current under different faults were analyzed.The priniciple is immune to line distributed capacity,series capacitor positions,and presence or absence of MOV breakovers.The validity of this scheme was verified by PSACD simulations.

CONSTRUCTIONS OF THREE-TRANSMIT-ANTENNA SPACE-TIME CODES

In this paper, we give design methods for three-transmit-antenna space-time codes which have reasonable parameters. A few examples are given to show that some unitary space-time codes constructed with our methods are better than the previously best-known ones.