On injective tensor powers of ℓ1 Causey, R.M.; Galego, E.M.; Samuel, C.
Journal of mathematical analysis and applications,
02/2021, Letnik:
494, Številka:
1
Journal Article
Recenzirano
Odprti dostop
In this paper we prove that the 3-fold injective tensor product ℓ1⊗ˆεℓ1⊗ˆεℓ1 is not isomorphic to any subspace of ℓ1⊗ˆεℓ1. This result provides a new solution to a problem of Diestel on the ...projective tensor products of c0. Moreover, this result implies that for any infinite countable compact metric space K, the 3-fold projective tensor product C(K)⊗ˆπC(K)⊗ˆπC(K) is not isomorphic to any quotient of C(K)⊗ˆπC(K).
This is Part II of the two-part comprehensive survey devoted to a computing framework most commonly known under the names Hyperdimensional Computing and Vector Symbolic Architectures (HDC/VSA). Both ...names refer to a family of computational models that use high-dimensional distributed representations and rely on the algebraic properties of their key operations to incorporate the advantages of structured symbolic representations and vector distributed representations. Holographic Reduced Representations 321, 326 is an influential HDC/VSA model that is well known in the machine learning domain and often used to refer to the whole family. However, for the sake of consistency, we use HDC/VSA to refer to the field.Part I of this survey 222 covered foundational aspects of the field, such as the historical context leading to the development of HDC/VSA, key elements of any HDC/VSA model, known HDC/VSA models, and the transformation of input data of various types into high-dimensional vectors suitable for HDC/VSA. This second part surveys existing applications, the role of HDC/VSA in cognitive computing and architectures, as well as directions for future work. Most of the applications lie within the Machine Learning/Artificial Intelligence domain; however, we also cover other applications to provide a complete picture. The survey is written to be useful for both newcomers and practitioners.
This paper studies chaotic image encryption technology and an application of matrix semi-tensor product theory, and a Boolean network encryption algorithm for a synchronous update process is ...proposed. A 2D-LASM chaotic system is used to generate a random key stream. First, a Boolean network is coded, and a Boolean matrix is generated. If necessary, the Boolean network matrix is diffused in one round so that the Boolean matrix can be saved in the form of an image. Then, three random position scramblings are used to scramble the plaintext image. Finally, using a matrix semi-tensor product technique to generate an encrypted image in a second round of diffusion, a new Boolean network can be generated by encoding the encrypted image. In secure communications, users can choose to implement an image encryption transmission or a Boolean network encryption transmission according to their own needs. Compared with other algorithms, this algorithm exhibits good security characteristics.
In this paper, a chaotic image encryption algorithm based on the matrix semi-tensor product (STP) with a compound secret key is designed. First, a new scrambling method is designed. The pixels of the ...initial plaintext image are randomly divided into four blocks. The pixels in each block are then subjected to different numbers of rounds of Arnold transformation, and the four blocks are combined to generate a scrambled image. Then, a compound secret key is designed. A set of pseudosecret keys is given and filtered through a synchronously updating Boolean network to generate the real secret key. This secret key is used as the initial value of the mixed linear-nonlinear coupled map lattice (MLNCML) system to generate a chaotic sequence. Finally, the STP operation is applied to the chaotic sequences and the scrambled image to generate an encrypted image. Compared with other encryption algorithms, the algorithm proposed in this paper is more secure and effective, and it is also suitable for color image encryption.
•A 3D model encryption algorithm (3DME-SC) is proposed.•A 2D chaotic system (2D-LAIC), which has good dynamic behavior, is proposed.•Simulation experiments show that 3DME-SC exhibits good security ...characteristics and effectiveness.
With the birth of the metaverse, 3D models have received extensive attention, and the security of information transmission continues to be an important issue. In this paper, we propose a 3D model encryption method based on a 2D chaotic system constructed via the coupling of the logistic map and infinite collapse (2D-LAIC) and on semi-tensor product (STP) theory. In terms of Lyapunov exponents, NIST test results, bifurcation diagrams, etc., 2D-LAIC exhibits better dynamical behavior than classical chaotic systems. 2D-LAIC can generate an unpredictable keystream, which is highly suitable for cryptography. Therefore, we propose a new 3D model encryption algorithm based on 2D-LAIC, named 3DME-SC. For a 3D model of the floating-point data type, XOR and STP processing are applied to the integer part and fractional part, respectively, of the model to obtain a 3D ciphertext model. The keystream required for XOR and STP processing is generated by 2D-LAIC. The results of a detailed security analysis and a comparative experimental analysis show that 3DME-SC exhibits good performance and effectiveness.
(Code: https://github.com/Gao5211996/3D-model-encryption)
ON -ALGEBRAS WHICH DETECT NUCLEARITY POP, FLORIN
Bulletin of the Australian Mathematical Society,
02/2023, Letnik:
107, Številka:
1
Journal Article
Recenzirano
Abstract
A
$C^{*}$
-algebra
A
is said to detect nuclearity if, whenever a
$C^{*}$
-algebra
B
satisfies
$A\otimes _{\mathrm{min}} B = A\otimes _{\mathrm{max}} B,$
it follows that
B
is nuclear. In this ...note, we survey the main results associated with this topic and present the background and tools necessary for proving the main results. In particular, we show that the
$C^{*}$
-algebra
$A = C^{*}(\mathbb {F}_{\infty })\otimes _{\mathrm{min}} B(\ell ^{2})/K(\ell ^{2})$
detects nuclearity. This result is known to experts, but has never appeared in the literature.
This study discusses the robust stability problem of Boolean networks (BNs) with data loss and disturbances, where data loss is appropriately described by random Bernoulli distribution sequences. ...Firstly, a BN with data loss and disturbances is converted into an algebraic form via the semi-tensor product (STP) technique. Accordingly, the original system is constructed as a probabilistic augmented system, based on which the problem of stability with probability one for the original system becomes a set stability with probability one for the augmented system. Subsequently, certain criteria are proposed for the robust stability of the systems. Moreover, an algorithm is developed to verify the robust set stability of the augmented system based on truth matrices. Finally, the validity of the obtained results is demonstrated by an illustrative example.
This paper studies the minimum observability of probabilistic Boolean networks (PBNs), the main objective of which is to add the fewest measurements such that an unobservable PBN becomes observable. ...First of all, the algebraic form of a PBN is established with the help of semi-tensor product (STP) of matrices. By combining the algebraic forms of two identical PBNs into a parallel system, a method to search the states that need to be H-distinguishable is proposed based on the robust set reachability technique. Secondly, a necessary and sufficient condition is given to find the minimum measurements such that a given set can be H-distinguishable. Moreover, by comparing the numbers of measurements for all the feasible H-distinguishable state sets, the least measurements that make the system observable are gained. Finally, an example is given to verify the validity of the obtained results.
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