In this article, we modify the product structure and turn it into the so-called ε-truncated product structure Iε, for any ε∈(0,1). The new structure keeps the property of being a complete residuated ...lattice, and additionally, its semiring reduct is locally finite. We convert each fuzzy automaton A over the product structure into a fuzzy automaton Aε over Iε, and accordingly, we turn the problems of testing the existence and computing weak simulations and bisimulations between fuzzy automata A and B over the product structure into the corresponding problems for the automata Aε and Bε. Those problems concerning the automata Aε and Bε are easier to solve, due to the local finiteness of the semiring reduct of Iε, and can be solved even in cases where the corresponding problems for the automata A and B cannot be solved. We show that weak simulations and bisimulations between the automata Aε and Bε determine certain kinds of approximate weak simulations and bisimulations between the original automata A and B, which we call ε-weak simulations and bisimulations. We also prove that the existence of an ε-weak simulation or bisimulation between automata A and B witnesses the existence of a certain kind of approximate inclusion or equivalence, where the deviation measure of the fuzzy languages of those automata from language inclusion or equality does not exceed ε.
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Recent literature has linked export upgrading to income inequality, and found that regions exporting more sophisticated products tend to have lower levels of income inequality. Here, we extend the ...literature by stressing that regional income inequality is shaped not only by export product structure but also by export destination structure. We further scale down our analysis to the intra-regional level, and examine urban-rural inequality within a region. Few have related urban-rural inequality to export upgrading and a region's export product/destination structures. Hence, we contribute to the current literature by examining how regional export product/destination structures have shaped income inequality in general and urban-rural inequality in particular. Based on China's export data and income survey data, we show that export upgrading only contributes to the reduction of income inequality in China's urban areas. Urban-rural inequality tends to be more severe in regions that have more complex export product/destination structures, due to the concentration of export activities in urban areas and to some barriers that inhibits the flow of input factors (e.g. capital and labor) between rural and urban areas.
•Regional income inequality is shaped by export product structure.•Regional income inequality is shaped by export destination structure.•Urban-rural inequality is shaped by export product structure.•Urban-rural inequality is shaped by export destination structure.
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In this paper, first we introduce the notion of a phase space of a 3-Lie algebra and show that a 3-Lie algebra has a phase space if and only if it is sub-adjacent to a 3-pre-Lie algebra. Then we ...introduce the notion of a product structure on a 3-Lie algebra using the Nijenhuis condition as the integrability condition. A 3-Lie algebra enjoys a product structure if and only if it is the direct sum (as vector spaces) of two subalgebras. We find that there are four types special integrability conditions, and each of them gives rise to a special decomposition of the original 3-Lie algebra. They are also related to O-operators, Rota–Baxter operators and matched pairs of 3-Lie algebras. Parallelly, we introduce the notion of a complex structure on a 3-Lie algebra and there are also four types special integrability conditions. Finally, we add compatibility conditions between a complex structure and a product structure, between a symplectic structure and a paracomplex structure, between a symplectic structure and a complex structure, to introduce the notions of a complex product structure, a para-Kähler structure and a pseudo-Kähler structure on a 3-Lie algebra. We use 3-pre-Lie algebras to construct these structures. Furthermore, a Levi-Civita product is introduced associated to a pseudo-Riemannian 3-Lie algebra and deeply studied.
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•We consider the neural network training from a nonlinear computation point of view.•A new linear product structure initialization strategy has been developed for training neural ...networks.•Theoretical analysis shows that the LPS initialization yields a low probability of dying ReLU.
Weight initialization plays an important role in training neural networks and also affects tremendous deep learning applications. Various weight initialization strategies have already been developed for different activation functions with different neural networks. These initialization algorithms are based on minimizing the variance of the parameters between layers and might still fail when neural networks are deep, e.g., dying ReLU. To address this challenge, we study neural networks from a nonlinear computation point of view and propose a novel weight initialization strategy that is based on the linear product structure (LPS) of neural networks. The proposed strategy is derived from the polynomial approximation of activation functions by using theories of numerical algebraic geometry to guarantee to find all the local minima. We also provide a theoretical analysis that the LPS initialization has a lower probability of dying ReLU comparing to other existing initialization strategies. Finally, we test the LPS initialization algorithm on both fully connected neural networks and convolutional neural networks to show its feasibility, efficiency, and robustness on public datasets.
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Abstract
In the field of thermoplastic injection molding, there are a large number of qualified product data, mining and analyzing these data is of great significance to enterprise production. At ...present, most scholars do optimization research around given process parameters, and cannot fundamentally solve the problem of process parameter setting values. To this end, this article proposes a new method, through the analysis of the material, structure and process of historical qualified products, from the perspective of the product, find the factors that affect the process parameters, and use the BP neural network model to train the non-linear mapping relationship between the product and several main process parameters, and predict the main process parameter values of the new product. For inexperienced manufacturers of injection products, this method can greatly improve their productivity.
•An industrial network with SND agents for product optimization design is built.•Heterogeneous uncertainties are covered in performance balance design of product structures.•The random chance ...constrained programming for performance balance is conducted.•An integrated discrete cuckoo algorithm nested with random simulation is designed.•The rationality and superiority of the proposed method are verified by a case.
Optimization design of product structures is a critical point of intelligent manufacturing that is often overlooked, particularly their performance balance involving different uncertainties. It is urgent to connect product consumers, design experts, and manufacturing plants to balance the optimization of product structures. As an input of intelligent manufacturing, an industrial network based on software-defined networking (SND) for product optimization design including structure performance balance is established in this study, which breaks down the barriers between an industrial network and the performance balance optimization of product structures. Accordingly, the high-dimensional coupling clustering of product components is performed through a two-dimensional (2D) plane mapping with a design structure matrix (DSM) that aims at the entire product life cycle (PLC). A random chance-constrained programming model for the performance balance optimization of product structures is developed, where the fuzzy expert evaluation is considered to include heterogeneous uncertainties, and an integrated multi-objective discrete cuckoo algorithm nested with Monte Carlo simulation is designed to solve the model. The rationality and superiority of the proposed method are verified using a case study with a large hydraulic machine tool for sheet stamping where product modules with better comprehensive performance are generated.
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7.
The product structure of squaregraphs Hickingbotham, Robert; Jungeblut, Paul; Merker, Laura ...
Journal of graph theory,
February 2024, 2024-02-00, 20240201, Volume:
105, Issue:
2
Journal Article
Peer reviewed
Open access
A squaregraph is a plane graph in which each internal face is a 4‐cycle and each internal vertex has degree at least 4. This paper proves that every squaregraph is isomorphic to a subgraph of the ...semistrong product of an outerplanar graph and a path. We generalise this result for infinite squaregraphs, and show that this is best possible in the sense that “outerplanar graph” cannot be replaced by “forest”.
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Abstract
Our aim is to make a step toward clarification of foundations for the notion of entanglement (both physical and mathematical) by representing it in the conditional probability framework. In ...Schrödinger’s words, this is entanglement of knowledge which can be extracted via conditional measurements. In particular, quantum probabilities are interpreted as conditional ones (as, e.g., by Ballentine). We restrict considerations to perfect conditional correlations (PCC) induced by measurements (‘EPR entanglement’). Such entanglement is coupled to the pairs of observables with the projection type state update as the back action of measurement. In this way, we determine a special class of entangled states. One of our aims is to decouple the notion of entanglement from the compound systems. The rigid association of entanglement with the state of a few body systems stimulated its linking with quantum nonlocality (‘spooky action at a distance’). However, already by Schrödinger entanglement was presented as knotting of knowledge (about statistics) for one observable
A
with knowledge about another observable
B
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In this paper, a symplectic structure on a Leibniz algebra is defined to be a symmetric nondegenerate bilinear form satisfying certain compatibility condition, and a phase space of a Leibniz algebra ...is defined to be a symplectic Leibniz algebra satisfying certain conditions. We show that a Leibniz algebra has a phase space if and only if there is a compatible Leibniz-dendriform algebra, and phase spaces of Leibniz algebras are one-to-one corresponds to Manin triples of Leibniz-dendriform algebras. Product (paracomplex) structures and complex structures on Leibniz algebras are studied in terms of decompositions of Leibniz algebras. A para-Kähler structure on a Leibniz algebra is defined to be a symplectic structure and a paracomplex structure satisfying a compatibility condition. We show that a symplectic Leibniz algebra admits a para-Kähler structure if and only if the Leibniz algebra is the direct sum of two Lagrangian subalgebras as vector spaces. A complex product structure on a Leibniz algebra consists of a complex structure and a product structure satisfying a compatibility condition. A pseudo-Kähler structure on a Leibniz algebra is defined to be a symplectic structure and a complex structure satisfying a compatibility condition. Various properties and relations of complex product structures and pseudo-Kähler structures are studied. In particular, Leibniz-dendriform algebras give rise to complex product structures and pseudo-Kähler structures naturally.
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