Some new classes of Kadison-Singer lattices in Hilbert spaces

Let N be a maximal and discrete nest on a separable Hilbert space H,E the projection from H onto the subspace[C]spanned by a particular separating vector for N′and Q the projection from K=H⊕H onto the closed subspace{(,):∈H}.Let L be the closed lattice in the strong operator topology generated by the projections(E 00 0),{(E 00 0):E∈N}and Q.We show that L is a Kadison-Singer lattice with trivial commutant,i.e.,L′=CI.Furthermore,we similarly construct some Kadison-Singer lattices in the matrix algebras M2n(C)and M2n.1(C).

On Discrete Hopf Fibrations, Grand Unification Groups, the Barnes-Wall, Leech Lattices, and Quasicrystals

A discrete Hopf fibration of S15 over S8 with S7 (unit octonions) as fibers leads to a 16D Polytope P16 with 4320 vertices obtained from the convex hull of the 16D Barnes-Wall lattice Λ16. It is argued (conjectured) how a subsequent 2-1 mapping (projection) of P16 onto a 8D-hyperplane might furnish the 2160 vertices of the uniform 241 polytope in 8-dimensions, and such that one can capture the chain sequence of polytopes 241,231,221,211in D=8,7,6,5dimensions, leading, respectively, to the sequence of Coxeter groups E8,E7,E6,SO(10)which are putative GUT group candidates. An embedding of the E8⊕E8and E8⊕E8⊕E8lattice into the Barnes-Wall Λ16 and Leech Λ24 lattices, respectively, is explicitly shown. From the 16D lattice E8⊕E8one can generate two separate families of Elser-Sloane 4D quasicrystals (QC’s) with H4 (icosahedral) symmetry via the “cut-and-project” method from 8D to 4D in each separate E8 lattice. Therefore, one obtains in this fashion the Cartesian product of two Elser-Sloane QC’s Q×Qspanning an 8D space. Similarly, from the 24D lattice E8⊕E8⊕E8one can generate the Cartesian product of three Elser-Sloane 4D quasicrystals (QC’s) Q×Q×Qwith H4 symmetry and spanning a 12D space.

Band structures of strained kagome lattices

Materials with kagome lattices have attracted significant research attention due to their nontrivial features in energy bands.We theoretically investigate the evolution of electronic band structures of kagome lattices in response to uniaxial strain using both a tight-binding model and an antidot model based on a periodic muffin-tin potential.It is found that the Dirac points move with applied strain.Furthermore,the flat band of unstrained kagome lattices is found to develop into a highly anisotropic shape under a stretching strain along y direction,forming a partially flat band with a region dispersionless along ky direction while dispersive along kx direction.Our results shed light on the possibility of engineering the electronic band structures of kagome materials by mechanical strain.

Rule acquisition of three-way semi-concept lattices in formal decision context

Three-way concept analysis is an important tool for information processing,and rule acquisition is one of the research hotspots of three-way concept analysis.However,compared with three-way concept lattices,three-way semi-concept lattices have three-way operators with weaker constraints,which can generate more concepts.In this article,the problem of rule acquisition for three-way semi-concept lattices is discussed in general.The authors construct the finer relation of three-way semi-concept lattices,and propose a method of rule acquisition for three-way semi-concept lattices.The authors also discuss the set of decision rules and the relationships of decision rules among object-induced three-way semi-concept lattices,object-induced three-way concept lattices,classical concept lattices and semi-concept lattices.Finally,examples are provided to illustrate the validity of our conclusions.

Three-Dimensional Printing Conformal Cooling with Structural Lattices for Plastic Injection Molding

The design of three-dimensional printing based conformal cooling channels(CCCs)in injection molding holds great significance.Compared to CCCs,conformal cooling(CC)cavity solutions show promise in delivering enhanced cooling performance for plastic products,although they have been underexplored.In this research,CC cavity is designed within the mold geometry,reinforced by body-centered cubic(BCC)lattice structures to enhance mechanical strength.Three distinct BCC lattice variations have been integrated into the CC cavity:the BCC structure,BCC with cubes,and BCC with pillars.The thermal performances of the BCC lattice-added CC cavity are assessed numerically after experimental validation.To provide feasible solutions from viewpoints of thermal performances,various BCC lattice structure thicknesses are analyzed in the range of 0.8–1.2mm.Thermal simulation outcomes reveal that thicker lattice structures enhance mechanical strength but simultaneously lead to an increase in cooling time.Upon examining all the proposed CC cavity solutions supported by BCC,the cooling times range from 2.2 to 4 s,resulting in a reduction of 38.6%to 66.1%when compared to conventional straightdrilled channels.In contrast to CCCs,CC cavities have the potential to decrease the maximum temperature nonuniformity from 8.5 to 6 K.Nevertheless,the presence of lattice structures in CC cavity solutions results in an elevated pressure drop,reaching 2.8MPa,whereas the results for CCCs remain below2.1MPa.

Statistical segmentation model on lattices

To reduce the difficulty of implementation and shorten the runtime of the traditional Kim-Fisher model, an entirely discrete Kim-Fisher-like model on lattices is proposed. The discrete model is directly built on the lattices, and the greedy algorithm is used in the implementation to continually decrease the energy function. First, regarding the gray values in images as discrete-valued random variables makes it possible to make a much simpler estimation of conditional entropy. Secondly, a uniform method within the level set framework for two-phase and multiphase segmentations without extension is presented. Finally, a more accurate approximation to the curve length on lattices with multi-labels is proposed. The experimental results show that, compared with the continuous Kim-Fisher model, the proposed model can obtain comparative results, while the implementation is much simpler and the runtime is dramatically reduced.

The Classification of Positive Definite Unimodular Lattices Over Z[(1+21^(1/2))/2]

In this paper, the author applies adjacent lattice method and Siegel mass formula to determine the classes of positive definite unimodular lattices of rank 4 over Z , and obtains that the class number of unit genus gen( I 4 ) is nine and the class number of even unimodular lattices is three, and also gives the representative lattices of each class.

The Classification of Positive Definite Unimodular Lattices Over Z[(1+211/2)/2]

In this paper, the author applies adjacent lattice method and Siegel mass formula to determine the classes of positive definite unimodular lattices of rank 4 over Z , and obtains that the class number of unit genus gen( I 4 ) is nine and the class number of even unimodular lattices is three, and also gives the representative lattices of each class.

一些Kadison-Singer代数的例子

设H是一个复的Hilbert空间。L={(0),L,M,K,H}是一个五角子空间格,满足M是L和x0的闭线性扩张,其中非零元x0属于K┴但不属于K+L。本文证明了Alg L是一个半单的Kadison-Singer代数。同时还给出两个不是套代数的CSL代数的例子,一个相似于某个Kadison-Singer代数,另一个却与任何一个Kadison-Singer代数都不相似。

GENERALIZATIONS AND CARTESIAN CLOSED SUBCATEGORIES OF SEMICONTINUOUS LATTICES

In this article, the authors mainly study how to obtain new semicontinuous lattices from the given semicontinuous lattices and discuss the conditions under which the image of a semicontinuous projection operator is also semicontinuous. Moreover, the authors investigate the relation between semicontinuous lattices and completely distributive lattices. Finally, it is proved that the strongly semicontinuous lattice category is a Cartesian closed category.

Stereolithographic 3D Printing-Based Hierarchically Cellular Lattices for High-Performance Quasi-Solid Supercapacitor

3D printing-based supercapacitors have been extensively explored,yet the rigid rheological requirement for corresponding ink preparation significantly limits the manufacturing of true 3D architecture in achieving superior energy storage.We proposed the stereolithographic technique to fabricate the metallic composite lattices with octet-truss arrangement by using electroless plating and engineering the 3D hierarchically porous graphene onto the scaffolds to build the hierarchically cellular lattices in quasi-solid supercapacitor application.The supercapacitor device that is composed of composite lattices span several pore size orders from nm to mm holds promising behavior on the areal capacitance(57.75 mF cm-2),rate capability(70% retention,2-40 mA cm-2),and long lifespan(96% after 5000 cycles),as well as superior energy density of 0.008 mWh cm-2,which are comparable to the state-of-the-art carbon-based supercapacitor.By synergistically combining this facile stereolithographic 3D printing technology with the hierarchically porous graphene architecture,we provide a novel route of manufacturing energy storage device as well as new insight into building other high-performance functional electronics.

Digital image encryption with chaotic map lattices

This paper proposes a secure approach for encryption and decryption of digital images with chaotic map lattices. In the proposed encryption process, eight different types of operations are used to encrypt the pixels of an image and one of them will be used for particular pixels decided by the outcome of the chaotic map lattices. To make the cipher more robust against any attacks, the secret key is modified after encrypting each block of sixteen pixels of the image. The experimental results and security analysis show that the proposed image encryption scheme achieves high security and efficiency.

Cascades with coupled map lattices in preferential attachment community networks

In this paper, cascading failure is studied by coupled map lattice （CML） methods in preferential attachment community networks. It is found that external perturbation R is increasing with modularity Q growing by simulation. In particular, the large modularity Q can hold off the cascading failure dynamic process in community networks. Furthermore, different attack strategies also greatly affect the cascading failure dynamic process. It is particularly significant to control cascading failure process in real community networks.

Tolerance of edge cascades with coupled map lattices methods

This paper studies the cascading failure on random networks and scale-free networks by introducing the tolerance parameter of edge based on the coupled map lattices methods. The whole work focuses on investigating some indices including the number of failed edges, dynamic edge tolerance capacity and the perturbation of edge. In general, it assumes that the perturbation is attributed to the normal distribution in adopted simulations. By investigating the effectiveness of edge tolerance in scale-free and random networks, it finds that the larger tolerance parameter λ can more efficiently delay the cascading failure process for scale-free networks than random networks. These results indicate that the cascading failure process can be effectively controlled by increasing the tolerance parameter λ. Moreover, the simulations also show that, larger variance of perturbation can easily trigger the cascading failures than the smaller one. This study may be useful for evaluating efficiency of whole traffic systems, and for alleviating cascading failure in such systems.

Vortex solitons in the semi-infinite gap of optically induced periodic lattices

Vortex solitons with a ring vortex core residing in a single lattice site in the semi-infinite gap of square optical lattices are reported. These solitons are no longer bound states of the Bloch-wave unit （Bloch-wave distribution in one lattice site） at the band edge of the periodic lattice, and consequently they do not bifurcate from the corresponding band edge. For saturable nonlinearity, one family of such solitons is found, and its existing curve forms a closed loop, which is very surprising. For Kerr nonlinearity, two families of such vortex solitons are found.

New Identity Based Proxy Re-Encryption Scheme from Lattices

Identity-Based Proxy Re-Encryption (IB-PRE) allows a semi-trusted proxy to convert the ciphertext encrypted under Alice’s identity into Bob’s ciphertext of the same message without leaking plaintext. Lattice-based cryptography enjoys potential resistance to quantum analysis and low computational complexity. A multi-hop and unidirectional IB-PRE from lattices is presented. We split the functions of decryption and ciphertext transformation separately, and design the double private keys mechanism, where two keys are generated for each user, one key is used to decrypt the ciphertext by Pre-Image Sampling technique, and the other is used to generate the re-encryption key by Bonsai Trees technique. The generation of the re-encryption key is non-interactive and collusion resistant. Moreover, its IND-sID-CPA security over the decisional Learning With Errors (LWE) assumption under the standard model is proved. Compared with some previous IBPRE schemes from Bilinear Pairings, the format of transformed ciphertext in our scheme remains unchanged, furthermore, it can also resist quantum analysis. Compared with some existing IB-PRE schemes from lattices with similar properties, the space of the message in our scheme is a vector of length l and the encryption process remains a lower encryption blowup factor. At last, a proof-of-concept implementation is provided.

ON GENERALIZED BUFFON NEEDLE PROBLEM FOR LATTICES

Two new concepts, the generalized support function and restricted chord function, both referring to a convex set, were introduced in [1]. General formulae to yield the kinematic measure of a segment of fixed length in a convex set were established based on these concepts. In this article , using the partial intersection method, we consider the generalized Buffon problem for three kinds of lattices. We determine the probability of intersection of a body test needle of length l, l a.

一类生成M_(mn)(C)的Kadison-Singer格

研究了矩阵代数Mmn(C)中不变子空间投影格生成的Von Neumann代数″;证明了是生成Mmn(C)的Kadison-Singer格;并通过格的关系图给出了的Hasses关系图.

Partial and complete periodic synchronization in coupled discontinuous map lattices

The partial and complete periodic synchronization in coupled discontinuous map lattices consisting of both discon- tinuous and non-invertible maps are discussed. We classify three typical types of periodic synchronization states, which give rise to different spatiotemporal patterns including static partial periodic synchronization, dynamically periodic syn- chronization, and complete periodic synchronization patterns. A special prelude dynamics of partial and complete periodic synchronization motion, which is shown by five separated concave curves in the time series plots of the order parameters, is observed. The detailed analysis shows that the special prelude dynamics is induced by the competition between two synchronized clusters, and the analytical expression for the corresponding order parameter is obtained.

Cryptanalysis and improvement of a digital image encryption method with chaotic map lattices

A digital image encryption scheme using chaotic map lattices has been proposed recently. In this paper, two fatal flaws of the cryptosystem are pointed out. According to these two drawbacks, cryptanalysts could recover the plaintext by applying the chosen plaintext attack. Therefore, the proposed cryptosystem is not secure enough to be used in the image transmission system. Experimental results show the feasibility of the attack. As a result, we make some improvements to the encryption scheme, which can completely resist our chosen plaintext attack.