A subalgebra of the full matrix algebra Mn(K),K a field, satisfying the identity x1,y1x2,y2⋯xq,yq=0 is called a Dq subalgebra of Mn(K). In the paper we deal with the structure, conjugation and ...isomorphism problems of maximal Dq subalgebras of Mn(K).
We show that a maximal Dq subalgebra A of Mn(K) is conjugated with a block triangular subalgebra of Mn(K) with maximal commutative diagonal blocks. By analysis of conjugations, the sizes of the obtained diagonal blocks are uniquely determined. It reduces the problem of conjugation of maximal Dq subalgebras of Mn(K) to the analogous problem in the class of commutative subalgebras of Mn(K). Further examining conjugations, in case A is contained in the upper triangular matrix algebra Un(K), we prove that A is already in a block triangular form.
We consider the isomorphism problem in a certain class of maximal Dq subalgebras of Mn(K) which contain all Dq subalgebras of Mn(K) with maximum dimension. In case K is algebraically closed, we invoke Jacobson's characterization of maximal commutative subalgebras of Mn(K) with maximum (K-)dimension to show that isomorphic subalgebras in this class are already conjugated. To illustrate it, we invoke results from 19 and find all isomorphism (equivalently conjugation) classes of Dq subalgebras of Mn(K) with maximum possible dimension, in case K is algebraically closed.
Max-plus algebra is the set ℝmax or ℝε=ℝ∪{ε} where ℝ is the set of all real number and ε = −∞ which is equipped with maximum (⊕) and plus (⊗) operations. The structure of max-plus algebra is ...semifield. Another semifield that can be learned is min-plus algebra. Min-plus algebra is the set ℝmin or ℝε′=ℝ∪{ε′} where ε′ = ∞ which is equipped with minimum (⊕ ′) and plus (⊗) operations. Max-plus algebra has been generalized into interval max-plus algebra, so that min-plus algebra can be developed into an interval min-plus algebra. Interval min-plus algebra is defined as a set I ( ℝ ) ε ′ ={x= x _ , x ¯ | x _ , x ¯ ∈ℝ , x _ ≤ x ¯ < ε ′ } which have minimum (⊕¯′) and addition (⊗¯) operations. A matrix in which its components are the element of ℝε is called matrix over max-plus algebra. Matrices over max-plus algebra has been generalized into interval matrices in which its components are the element of I(ℝ)ε. This research will discusses the interval min-plus algebraic structure and matrices over interval min-plus algebra.
We study the spin liquid candidate of the spin-1/2 J_{1}-J_{2} Heisenberg antiferromagnet on the triangular lattice by means of density matrix renormalization group (DMRG) simulations. By applying an ...external Aharonov-Bohm flux insertion in an infinitely long cylinder, we find unambiguous evidence for gapless U(1) Dirac spin liquid behavior. The flux insertion overcomes the finite size restriction for energy gaps and clearly shows gapless behavior at the expected wave vectors. Using the DMRG transfer matrix, the low-lying excitation spectrum can be extracted, which shows characteristic Dirac cone structures of both spinon-bilinear and monopole excitations. Finally, we confirm that the entanglement entropy follows the predicted universal response under the flux insertion.
Let G be a primitive strongly regular graph of order n and A its adjacency matrix. In this paper, we first associate an Euclidean Jordan algebra V to G considering the real Euclidean Jordan algebra ...spanned by the identity of order n and the natural powers of A. Next, by the analysis of the spectra of an Hadamard logarithmic series of V we establish new admissibility conditions on the parameters of the strongly regular graph G.
Variational quantum algorithms have been proposed to solve static and dynamic problems of closed many-body quantum systems. Here we investigate variational quantum simulation of three general types ...of tasks-generalized time evolution with a non-Hermitian Hamiltonian, linear algebra problems, and open quantum system dynamics. The algorithm for generalized time evolution provides a unified framework for variational quantum simulation. In particular, we show its application in solving linear systems of equations and matrix-vector multiplications by converting these algebraic problems into generalized time evolution. Meanwhile, assuming a tensor product structure of the matrices, we also propose another variational approach for these two tasks by combining variational real and imaginary time evolution. Finally, we introduce variational quantum simulation for open system dynamics. We variationally implement the stochastic Schrödinger equation, which consists of dissipative evolution and stochastic jump processes. We numerically test the algorithm with a 6-qubit 2D transverse field Ising model under dissipation.
A common challenge faced in quantum physics is finding the extremal eigenvalues and eigenvectors of a Hamiltonian matrix in a vector space so large that linear algebra operations on general vectors ...are not possible. There are numerous efficient methods developed for this task, but they generally fail when some control parameter in the Hamiltonian matrix exceeds some threshold value. In this Letter we present a new technique called eigenvector continuation that can extend the reach of these methods. The key insight is that while an eigenvector resides in a linear space with enormous dimensions, the eigenvector trajectory generated by smooth changes of the Hamiltonian matrix is well approximated by a very low-dimensional manifold. We prove this statement using analytic function theory and propose an algorithm to solve for the extremal eigenvectors. We benchmark the method using several examples from quantum many-body theory.
Using memristor crossbar arrays to accelerate computations is a promising approach to efficiently implement algorithms in deep neural networks. Early demonstrations, however, are limited to ...simulations or small‐scale problems primarily due to materials and device challenges that limit the size of the memristor crossbar arrays that can be reliably programmed to stable and analog values, which is the focus of the current work. High‐precision analog tuning and control of memristor cells across a 128 × 64 array is demonstrated, and the resulting vector matrix multiplication (VMM) computing precision is evaluated. Single‐layer neural network inference is performed in these arrays, and the performance compared to a digital approach is assessed. Memristor computing system used here reaches a VMM accuracy equivalent of 6 bits, and an 89.9% recognition accuracy is achieved for the 10k MNIST handwritten digit test set. Forecasts show that with integrated (on chip) and scaled memristors, a computational efficiency greater than 100 trillion operations per second per Watt is possible.
Large memristor arrays composed of hafnium oxide are demonstrated with suitability for computing matrix operations at higher power efficiency than digital systems. The nonmemory application of memristors is performed in an analog computing platform. Computational operations with 6 bit equivalent precision are shown and utilized to directly compute neural network inference within a memristor crossbar.
Neuromorphic photonics has recently emerged as a promising hardware accelerator, with significant potential speed and energy advantages over digital electronics for machine learning algorithms, such ...as neural networks of various types. Integrated photonic networks are particularly powerful in performing analog computing of matrix-vector multiplication (MVM) as they afford unparalleled speed and bandwidth density for data transmission. Incorporating nonvolatile phase-change materials in integrated photonic devices enables indispensable programming and in-memory computing capabilities for on-chip optical computing. Here, we demonstrate a multimode photonic computing core consisting of an array of programable mode converters based on on-waveguide metasurfaces made of phase-change materials. The programmable converters utilize the refractive index change of the phase-change material Ge
Sb
Te
during phase transition to control the waveguide spatial modes with a very high precision of up to 64 levels in modal contrast. This contrast is used to represent the matrix elements, with 6-bit resolution and both positive and negative values, to perform MVM computation in neural network algorithms. We demonstrate a prototypical optical convolutional neural network that can perform image processing and recognition tasks with high accuracy. With a broad operation bandwidth and a compact device footprint, the demonstrated multimode photonic core is promising toward large-scale photonic neural networks with ultrahigh computation throughputs.
Traditional link prediction methods are generally only calculated for the neighbor information of nodes, and the network path between nodes has not been fully utilized. Therefore, this paper proposes ...a directed network link prediction method based on path extension similarity to improve the prediction accuracy of potential edges of network nodes. Firstly, the mathematical definition of each local index is expressed in matrix form through matrix algebra; secondly, according to the algorithm principle of global and quasi-local indices, the extension form of local indices is clarified; and the path extension of each local index is carried out respectively; finally, multiple real data sets are used to analyze the benchmark indices and extended indices. The results of the AUC and Precision evaluation metrics show that the path extension similarity proposed in this paper has higher accuracy and stronger robustness than the benchmark indices.
•The utilization of reciprocity coefficient greatly improves the prediction accuracy of directed network.•The mathematical definition formula of each local similarity index is expressed in matrix form through matrix algebra.•The path extension of network node related links can improve the utilization of node information.•The proposed prediction algorithm can be applied to future wireless network.
Let
G
be a generalized matrix algebra. A linear map
ϕ
:
G
→
G
is said to be a left (right) Lie centralizer at
E
∈
G
if
ϕ
(
S
,
T
)
=
ϕ
(
S
)
,
T
(
ϕ
(
S
,
T
)
=
S
,
ϕ
(
T
)
) holds for all
S
...,
T
∈
G
with ST = E.
ϕ
is of a standard form if
ϕ
(
A
)
=
Z
A
+
γ
(
A
)
for all
A
∈
G
, where Z is in the center of
G
and γ is a linear map from
G
into its center vanishing on each commutator
S
,
T
whenever ST = E. In this paper, we give a complete characterization of
ϕ
. It is shown that, under some suitable assumptions on
G
,
ϕ
has a standard form.
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