Singular Blocks Research Articles

Literature Study of Learning-Based Video Compression

Developments in telecommunications technology today, such as cellular with the fifth generation (5G), the development of IoT prototypes, and the migration of analog TV to digital TV starting in 2022. The development of various research using machine learning. The problem with video format information is that the video file size is quite large, so the transmission process requires a large bandwidth. In addition, sharing services such as Video on Demand (VoD) and Video Broadcasting are sensitive to delay. In comparison, the transmission media has limited capacity, such as terrestrial TV, Ethernet/Fast Ethernet, and wireless cellular data such as 2G, 3G HSPA, 4G, etc. Based on reports from Cisco, the development of internet users has increased by 10% per year, with 80% of total traffic using video. Developments in various video compression standards, such as the most recent H.264 and H.265, produce high-quality, low-bitrate video. Much research has been carried out with various proposed compression methods based on machine learning. Either uses singular block learning based or end-to-end. This research focuses on the literature study of video compression with machine learning.

A unique universal block in QCA technology

The limitations of Complementary Metal Oxide Semiconductor (CMOS) such as high-power dissipation, short channel effects, and high lithography has prompted scientists to look forward to alternatives. Many nanotechnologies were emerged and still in the competition such as Fin Field-Effect Transistor (FinFET), Carbon Nanotube Field-Effect Transistor (CNFET), Quantum-dot Cellular Automata (QCA). QCA is a new nanoelectronics technology used a new method for calculations and it is a promising technology in terms of power dissipation. In QCA, circuit complexity is depending on cell count, area and latency. For commercial purposes a universal gate is necessary, this gate can be used alone to design any Boolean function, where in CMOS technology, the NAND or NOT gates have the ability to be a universal gate. The dominant gates in QCA are the majority gate with inverter where QCA circuits cannot be designed using only a universal gate. Because the structure of the gate is important in QCA, this paper introduces a new low complexity singular block that can be used as a universal block. The proposed block can be used to form AND, OR, NOT, Exclusive-OR (XOR) and Multiplexer (MUX). So, this block not just used to design any circuit but also it can be used to form other important functions XOR and MUX to build large circuits in an optimum form.

Lightweight Encryption Assisted Man-in-The-Middle Attack-Resilient Steganography Model for Secure Satellite Imagery Services: LEMARS

The role of communication technologies has become increasingly vital in various fields such as industrial communication, surveillance and monitoring, healthcare, and data communication, leading to a surge in demand for these technologies in recent years. In this research paper, the LEMARS model is proposed as a novel and robust lightweight encryption assisted Man-in-The-Middle (MITM) attack-resilient steganography model for secure satellite imagery services. The proposed solution combines lightweight encryption and attack-resilient embedding to achieve higher attack-resilience with low computational cost and optimal reconstruction quality. The use of Feistel architecture-driven substitution and permutation-based block-cipher encryption and heuristic-driven pixel adjustment method for MITM-Adaptive B-Channel Embedding results in higher imperceptibility and superior attack-resilience under uncertain channels. The strategic amalgamation of FSPN-BCE, L2D-IWT, and HD-OMABE enabled attack-resilient steganography for secure satellite communication. The performance of LEMARS has been evaluated using Peak Signal-to-Noise Ratio (PSNR), entropy, Non-Parametric Change Rate (NPCR), Unified Average Change Intensity (UACI), and regular and singular block pattern changes. LEMARS achieved a PSNR value of 58.02 dB, an entropy value of 6.15 dB, an NPCR value of 99.84%, a UACI value of 33.70, and exhibited minimal block pattern changes. These results indicate that the proposed model achieved higher attack-resilience with low computational cost and optimal reconstruction quality. The research aimed to improve encryption and embedding methods to achieve an MITM-attack resilient embedding solution for secure satellite imagery services. The proposed LEMARS model exhibited superiority over other state-of-the-art methods, affirming the robustness of the proposed LEMARS model for real-time multimedia data security purposes, including remote sensing, satellite imagery services, telemedicine, and EHR. In conclusion, the LEMARS model offers an optimal solution for multimedia data security, combining lightweight encryption and attack-resilient embedding to achieve higher attack-resilience with low computational cost and optimal reconstruction quality.

Cohomology in singular blocks of parabolic category

AbstractWe determine the dimensions of $\textrm{Ext}$ -groups between simple modules and dual generalized Verma modules in singular blocks of parabolic versions of category $\mathcal{O}$ for complex semisimple Lie algebras and affine Kac-Moody algebras.

On certain Hochschild cohomology groups for the small quantum group

We apply the sheaf cohomology BGG method developed by the authors and Lachowska-Qi to the computation of Hochschild cohomology groups of various blocks of the small quantum group. All our computations of the center of the corresponding block agree with the conjectures of Lachowska-Qi. In the case of the nontrivial singular block for g=sl3, we obtain the H•(u,C)=C[N]-module structure of HH•(uλ).

Explicit calculations in an infinitesimal singular block of SLn

AbstractLet $G= SL_{n+1}$ be defined over an algebraically closed field of characteristic $p > 2$. For each $n \geq 1$, there exists a singular block in the category of $G_1$-modules, which contains precisely $n+1$ irreducible modules. We are interested in the ‘lift’ of this block to the category of $G_1T$-modules. Imposing only mild assumptions on $p$, we will perform a number of calculations in this setting, including a complete determination of the Loewy series for the baby Verma modules and all possible extensions between the irreducible modules. In the case where $p$ is extremely large, we will also explicitly compute the Loewy series for the indecomposable projective modules.

Projective and Whittaker functors on category [formula omitted

We show that the Whittaker functor on a regular block of the BGG-category O of a semisimple complex Lie algebra can be obtained by composing a translation to the wall functor with Soergel and Miličić's equivalence between the category of Whittaker modules and a singular block of O. We show that the Whittaker functor is a quotient functor that commutes with all projective functors and endomorphisms between them.

Convergence of a generalized PMHSS method for a class of singular block two-by-two linear systems

In this work, we introduce a generalization of the preconditioned modified Hermitian and skew-Hermitian splitting (PMHSS) iteration method, named as generalized PMHSS (GPMHSS) iteration method, to solve a class of singular block two-by-two system of linear equations. Theoretical analyses show that the GPMHSS iteration method converges unconditionally to the minimum norm least squares solution for any initial guess no matter the system is consistent or inconsistent. Besides, with the preconditioner derived from the GPMHSS iteration method, the preconditioned generalized minimal residual (GMRES) method also determines the minimum norm least squares solution of the consistent singular block two-by-two linear systems at breakdown. Numerical experiments are presented to show the effectiveness and the robustness of the GPMHSS iteration method and the corresponding preconditioner.

Categorification via blocks of modular representations for

AbstractBernstein, Frenkel, and Khovanov have constructed a categorification of tensor products of the standard representation of $\mathfrak {sl}_2$ , where they use singular blocks of category $\mathcal {O}$ for $\mathfrak {sl}_n$ and translation functors. Here we construct a positive characteristic analogue using blocks of representations of $\mathfrak {s}\mathfrak {l}_n$ over a field $\mathbf {k}$ of characteristic p with zero Frobenius character, and singular Harish-Chandra character. We show that the aforementioned categorification admits a Koszul graded lift, which is equivalent to a geometric categorification constructed by Cautis, Kamnitzer, and Licata using coherent sheaves on cotangent bundles to Grassmanians. In particular, the latter admits an abelian refinement. With respect to this abelian refinement, the stratified Mukai flop induces a perverse equivalence on the derived categories for complementary Grassmanians. This is part of a larger project to give a combinatorial approach to Lusztig’s conjectures for representations of Lie algebras in positive characteristic.

BGG complexes in singular blocks of category [formula omitted

Using translation from the regular block, we construct and analyze properties of BGG complexes in singular blocks of BGG category O. We provide criteria, in terms of the Kazhdan-Lusztig-Vogan polynomials, for such complexes to be exact. In the Koszul dual picture, exactness of BGG complexes is expressed as a certain condition on a generalized Verma flag of an indecomposable projective object in the corresponding block of parabolic category O.In the second part of the paper, we construct BGG complexes in a more general setting of balanced quasi-hereditary algebras and show how our results for singular blocks can be used to construct BGG resolutions of simple modules in S-subcategories in O.

Regularised primal–dual interior‐point method for dynamic optimal power flow with block‐angular structures

To implement privacy protection and high-efficiency distributed computing of the large-scale dynamic optimal power flow (DOPF) of the multi-area interconnected power system, the regularised term (RT) and primal-dual interior-point method (PDIPM), denoted by R-PDIPM, is proposed to distribute and parallel such DOPF solutions. First, the DOPF is transformed into a prime block-angular problem through the replication of the coupling node variables, and the multi-area decoupling of the power grid is realised. However, it is easier to develop a singular (ill-conditioned) block matrix of Newton's system under the distributed computing of PDIPM. Second, though introducing the RT into the Lagrangian function of PDIPM based on regular technology, the matrix of Newton's system becomes quasi-definite and strongly factorisable. The robustness of distributed computing of PDIPM is enhanced, and the convergence speed is also improved. Case studies on the 3012 and 3074 node systems over 2-4 time intervals are presented. The results show that the R-PDIPM is more robust in efficiently solving the problem than PDIPM in distributed computing, which is suited for distributed calculation of DOPF in a large-scale multi-area power system.

A NOTE ON BLOCK PRECONDITIONER FOR GENERALIZED SADDLE POINT MATRICES WITH HIGHLY SINGULAR (1, 1) BLOCK

In this paper, we present a block triangular preconditioner for generalized saddle point matrices whose coefficient matrices have singular (1, 1) blocks. Theoretical analysis shows that all the eigenvalues of the preconditioned matrix are strongly clustered when choosing an optimal parameter. Numerical experiments are given to demonstrate the efficiency of the presented preconditioner.

A block form of a singular pencil of operators and a method of obtaining it

A block form of a singular operator pencil $\lambda A+B$, where $\lambda$ is a complex parameter, and the linear operators $A$, $B$ act in finite-dimensional spaces, is described. An operator pencil $\lambda A+B$ is called regular if $n = m = rk(\lambda A+B)$, where $rk(\lambda A+B)$ is the rank of the pencil and $m$, $n$ are the dimensions of spaces (the operators map an $n$-dimensional space into an $m$-dimensional one); otherwise, if $n \ne m$ or $n = m$ and $rk(\lambda A+B)<n$, the pencil is called singular (irregular). The block form (structure) consists of a singular block, which is a purely singular pencil, i.e., it is impossible to separate out a regular block in this pencil, and a regular block. In these blocks, zero blocks and blocks, which are invertible operators, are separated out. A method of obtaining the block form of a singular operator pencil is described in detail for two special cases, when $rk(\lambda A+B) = m < n$ and $rk(\lambda A+B) = n < m$, and for the general case, when $rk(\lambda A+B) < n, m$. Methods for the construction of projectors onto subspaces from the direct decompositions, relative to which the pencil has the required block form, are given. Using these projectors, we can find the form of the blocks and, accordingly, the block form of the pencil. Examples of finding the block form for the various types of singular pencils are presented. To obtain the block form, in particular, the results regarding the reduction of a singular pencil of matrices to the canonical quasidiagonal form, which is called the Weierstrass-Kronecker canonical form, are used. Also, methods of linear algebra are used. The obtained block form of the pencil and the corresponding projectors can be used to solve various problems. In particular, it can be used to reduce a singular semilinear differential-operator equation to the equivalent system of purely differential and purely algebraic equations. This greatly simplifies the analysis and solution of differential-operator equations.

The center of small quantum groups II: singular blocks

We generalize to the case of singular blocks the result in \cite{BeLa} that describes the center of the principal block of a small quantum group in terms of sheaf cohomology over the Springer resolution. Then using the method developed in \cite{LQ1}, we present a linear algebro-geometric approach to compute the dimensions of the singular blocks and of the entire center of the small quantum group associated with a complex semisimple Lie algebra. A conjectural formula for the dimension of the center of the small quantum group at an $l$th root of unity is formulated in type A.

Modified block preconditioner for generalized saddle point matrices with highly singular(1,1) blocks

ABSTRACTIn this paper, based on the preconditioners presented by Zhang [A new preconditioner for generalized saddle matrices with highly singular(1,1) blocks. Int J Comput Maths. 2014;91(9):2091-2101], we consider a modified block preconditioner for generalized saddle point matrices whose coefficient matrices have singular (1,1) blocks. Moreover, theoretical analysis gives the eigenvalue distribution, forms of the eigenvectors and the minimal polynomial. Finally, numerical examples show the eigenvalue distribution with the presented preconditioner and confirm our analysis.

Cohomology in Singular Blocks for a Quantum Group at a Root of Unity

Let Uζ be a Lusztig quantum enveloping algebra associated to a complex semisimple Lie algebra mathfrak {g} and a root of unity ζ. When L, L′ are irreducible Uζ-modules having regular highest weights, the dimension of text {Ext}^{n}_{U_{zeta }}(L,L^{prime }) can be calculated in terms of the coefficients of appropriate Kazhdan-Lusztig polynomials associated to the affine Weyl group of Uζ. This paper shows for L, L′ irreducible modules in a singular block that dim text {Ext}^{n}_{U_{zeta }}(L,L^{prime }) is explicitly determined using the coefficients of parabolic Kazhdan-Lusztig polynomials. This also computes the corresponding cohomology for q-Schur algebras and many generalized q-Schur algebras. The result depends on a certain parity vanishing property which we obtain from the Kazhdan-Lusztig correspondence and a Koszul grading of Shan-Varagnolo-Vasserot for the corresponding affine Lie algebra.

Precoding Design and Power Allocation in Two-User MU-MIMO Wireless Ad Hoc Networks

In this paper, we consider the precoding design and power allocation problem for multi-user multiple-input multiple-output (MU-MIMO) wireless ad hoc networks. In the first timeslot, the source node (SN) transmits energy and information to a relay node (RN) simultaneously within the simultaneous wireless information and power transfer (SWIPT) framework. Then, in the second timeslot, based on the decoder and the forwarding (DF) protocol, after reassembling the received signal and its own signal, the RN forwards the information to the main user (U1) and simultaneously sends its own information to the secondary user (U2). In this paper, when the transmission rate of the U1 is restricted, the precoding, beamforming, and power splitting (PS) transmission ratio are jointly considered to maximize the transmission rate of U2. To maximize the system rate, we design an optimal beamforming matrix and solve the optimization problem by semi-definite relaxation (SDR), considering the high complexity of implementing the optimal solution. Two sub-optimal precoding programs are also discussed: singular value decomposition and block diagonalization. Finally, the performance of the optimization and sub-optimization schemes are compared using a simulation.

Singular blocks of restricted [formula omitted

We compute generators and relations for the basic algebra of a non-semisimple singular block of the restricted enveloping algebra of sl3 over an algebraically closed field of characteristic p>3. Working directly with the basic algebra we compute its centre and the internal degree zero part of its first Hochschild cohomology, and show its Verma modules are Koszul.

Convergence of block triangular preconditioner for generalized saddle point matrices with highly singular (1,1) blocks

In this paper, based on the preconditioners presented by Cao [J. Comput. Appl. Math. 2013;238:109–115], we present a block triangular preconditioner for generalized saddle point matrices whose coefficient matrices have singular (1,1) blocks. Moreover, theoretical analysis gives the eigenvalue distribution, forms of the eigenvectors and its minimal polynomial. Finally, numerical examples confirm our analysis when optimal parameters are chosen.

Singular localization of $\mathfrak{g}$-modules and applications to representation theory

We prove a singular version of Beilinson–Bernstein localization for a complex semi-simple Lie algebra following ideas from the positive characteristic case settled by [BMR06]. We apply this theory to translation functors, singular blocks in the Bernstein–Gelfand–Gelfand category O and Whittaker modules.