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An Algebraic Circuit for a multivariate polynomial P is a computational model for constructing the polynomial P using only additions and multiplications. It is a syntactic model of computation, as opposed to the Boolean Circuit model, and hence lower bounds for this model are widely expected to be easier to prove than lower bounds for Boolean circuits. Despite this, we do not have superpolynomial lower bounds against general algebraic circuits of depth 3 (except over constant-sized finite fields) and depth 4 (over any field other than F2), while constant-depth Boolean circuit lower bounds have been known since the early 1980s. In this paper, we prove the first superpolynomial lower bounds against algebraic circuits of all constant depths over all fields of characteristic 0. We also observe that our super-polynomial lower bound for constant-depth circuits implies the first deterministic sub-exponential time algorithm for solving the Polynomial Identity Testing (PIT) problem for all small-depth circuits using the known connection between algebraic hardness and randomness. [ABSTRACT FROM AUTHOR]

An introduction is presented to the article focusing on topics related to iterated matrix multiplication (IMM) polynomials and algebraic complexity and circuits.

A new report from the University of Burgos in Spain introduces and investigates a class of noncommutative spacetimes called "T-Minkowski." These spacetimes have unique characteristics in their quantum Poincaré group of isometries. The research focuses on the identification of all the quantum Poincaré groups allowed by certain assumptions, as well as the associated quantum homogeneous spacetimes, differential calculi, and braiding constructions. This paper is the first in a series and provides a foundation for further exploration of these topics. [Extracted from the article]

This paper presents two novel methods based on geometric algebra (GA) to estimate two-dimensional (2D) direction-of-arrival (DOA) of non-circular (NC) signals for uniform rectangular array (URA). Traditional methods treat the received NC signals as a long vector which will inevitably lose orthogonality inside each electromagnetic vector sensor (EMVS) and thus miss some information of second-order statistical properties. Furthermore, the computational complexity will also increase. By contrast, the GA-based estimating signal parameter via rotational invariance techniques (ESPRIT) and propagation method (PM) algorithms are proposed to estimation DOA of NC signals. Taking advantage of GA, the relationship among multidimensional signals can be maintained. First, the six components of the EMVS are represented as a multivector in GA space. Then, we construct the GA-based extended covariance matrix to utilize the signal information more completely. The DOA parameters can be estimated through the ESPRIT and PM principle. The proposed GA-based estimation of signal parameter via rotational invariance techniques for NC signals processing (GANC-ESPRIT) can estimate DOA with high estimation accuracy. The proposed GA-based propagation method for NC signals estimation (GANC-PM) uses linear transformation to calculate angle parameters. Our model has much lower memory requirements and less computational burden compared with long vector models. Simulation results demonstrate the robustness and superiority of the proposed GANC-ESPRIT algorithm in terms of angular resolution. Complexity analysis shows that the proposed GANC-PM algorithm performances better with less computations. [ABSTRACT FROM AUTHOR]

The paper focuses on the Generalized Mutual Exclusion Constraints (GMECs) problem in the Network of Timed Event Graphs (NTEGs). A new formal method for state feedback control using Min-Plus algebra formalisms is proposed. For this purpose, the dynamic behavior of the NTEGs and GMECs are respectively described by linear equations and weighted inequalities in Min-Plus algebra. Based on these mathematical tools, control laws are computed to satisfy the constraints that are imposed on the NTEGs in case sufficient conditions are verified. The established controllers are represented by monitor places which supervise the system to ensure all GMECs. Finally, a case study of an AGV network is carried out to illustrate theoretical results and demonstrate the efficiency of developed approaches. [ABSTRACT FROM AUTHOR]

A study conducted at Linkoping University in Sweden explores the noncommutative Riemannian geometry of Kronecker algebras in mathematics. The researchers discuss the framework of derivation-based differential calculi, metrics, and bimodule connections compatible with the algebra's structure. They also apply these concepts to the noncommutative torus and examine the geometric aspects of generalized Kronecker algebras. The study concludes by constructing non-trivial Levi-Civita connections for various choices of Lie algebras of derivations. This research has been peer-reviewed and published in the Journal of Geometry and Physics. [Extracted from the article]