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Although the idea of information granulation has been shown to be a research craze in short-term time series forecasting, it is still urgent to develop a granular framework so that information granulation can characterize the trend distribution of data to the significant extent under a common concept of time. This paper puts forward a novel granulation algorithm involving two-stage partitioning scheme so that information granule established there exhibits well-articulated semantics at the time level, while at the same time, it gives full consideration to the varying patterns of data. On this basis, a new association rule based on this type of information granules is presented. Unlike most fuzzy association rules, the proposed association rules can extract and derive the correlations between two collections of trend features corresponding to the past and future, which is in accord with humans reasoning. Keeping in mind that the prediction process should center on essential rules while freeing from the interference of irrelevant rules, which contributes to a reliable prediction result; thus, a rule selection algorithm is involved so as one makes sure the accuracy and interpretability of the forecasting results. The design of short-term forecasting model based on fuzzy inference system is implemented, where the concept of granulation eliminates the commonly used alternative, i.e., the recursive iterations of one-step prediction. Experimental results have verified the effectiveness of the proposed model.

Traditional intelligent fault diagnosis models are required to be trained and tested under an identical probability distribution. However, the shift in data distributions is inevitable due to changes in environmental and operational conditions, which results in diagnostic performance degradation. Currently, transfer learning has been successfully applied to learn a discriminative diagnosis model in the presence of a shift. But conventional transfer learning approaches encounter obstacles without adequately considering the feature interactivity and transferable ability at different layers. In this study, a joint adaptive transfer learning framework based on multi-layer feature fusion for reliable cross-domain diagnosis is presented to address these issues. Firstly, the multilinear map is employed to implement a novel multi-layer feature fusion. This fusion is key to realizing a substantial improvement of feature representation capability and effectively embedding joint distribution of multi-layer features. Furthermore, a novel joint adaptive transfer learning (JATL) framework is devised to facilitate reliable cross-domain adaption by making utmost use of cross-domain-invariant features with a small amount of data. Experiments with different transfer scenarios on two benchmark datasets have been conducted, and experimental results demonstrate the superiority of the proposed approach.

We define the β-index of a nonzero multilinear polynomial, which gives a kind of classification of a nonzero multilinear polynomial. By using the β-index we give a complete description of the images of multilinear polynomials in non-commutative variables evaluated on upper triangular matrix algebras, which solves the Fagundes-Mello Conjecture proposed by Fagundes and Mello in 2019. The Fagundes-Mello Conjecture is an important variation of the old and famous Lvov-Kaplansky Conjecture.

In Kahrobaei et al. [Multilinear cryptography using nilpotent groups, Proceedings of Elementary Theory of Groups and Group Rings, and Related Topics conference. Conference held at Fairfield University and at the Graduate Center, CUNY, New York, NY, USA, November 1-2, 2018, De Gruyter, 2020, pp. 127-133] we generalized the definition of a multilinear map to arbitrary groups and introduced two multiparty key-exchange protocols using nilpotent groups. In this paper we have a closer look at the protocols and will address some incorrect cryptanalysis which has been proposed in Roman'kov [Discrete logarithm for nilpotent groups and cryptanalysis of polylinear cryptographic system, Prikl. Diskretn. Mat. Suppl. (12), (2019), pp. 154-160].

We introduce the theory of random tensors, which naturally extends the method of random averaging operators in our earlier work (Deng et al. in: Invariant Gibbs measures and global strong solutions for 4009 nonlinear Schrodinger equations in dimension two, arXiv:1910.08492 ), to study the propagation of randomness under nonlinear dispersive equations. By applying this theory we establish almost-sure local well-posedness for semilinear Schrodinger equations in the full subcritical range relative to the probabilistic scaling (Theorem 1.1). The solution we construct has an explicit expansion in terms of multilinear Gaussians with adapted random tensor coefficients. As a byproduct we also obtain new results concerning regular data and long-time solutions, in particular Theorem 1.6, which provides long-time control for random homogeneous data, demonstrating the highly nontrivial fact that the first energy cascade happens at a much later time than in the deterministic setting. In the random setting, the probabilistic scaling is the natural scaling for dispersive equations, and is different from the natural scaling for parabolic equations. Our theory of random tensors can be viewed as the dispersive counterpart of the existing parabolic theories (regularity structures, para-controlled calculus and renormalization group techniques).

In this article, we investigate the multilinear distorted multiplier estimate (Coifman-Meyer type theorem) associated with the Schr\"{o}dinger operator $H=-\Delta + V$ in the framework of the corresponding distorted Fourier transform. Our result is the "distorted" analog of the multilinear Coifman-Meyer multiplier operator theorem in \cite{CM1}, which extends the bilinear estimates of Germain, Hani and Walsh's in \cite{PZS} to the multilinear case for all dimensions. As applications, we give the estimate of Leibniz's law of integer order derivations for the multilinear distorted multiplier for the first time and we obtain small data scattering for a kind of generalized mass-critical NLS with good potential in low dimensions $d=1,2$., Comment: 24pages. arXiv admin note: text overlap with arXiv:1303.4354 by other authors

Multilinear model predictive control (MLMPC) can regulate a nonlinear process with wide operating regions based on a set of linear models. Although online computational cost is reduced compare to nonlinear MPC (NMPC), it is difficult to obtain a reliable full nonlinear model or set of linear models in practice. In this paper, we propose a combination of MLMPC with differential dynamic programming (DDP), so that the system can be controlled offset-free in the absence of a full nonlinear model. DDP is a ‘trajectory-centric’ optimization technique that solves nonlinear optimal control problems. The trajectory can be optimized even if the full model for the system is unknown, because DDP uses only the gradients around the visited trajectory, which is easily obtained by input excitations. Moreover, the gradient information can provide linear models in the subsequent MLMPC step. In the proposed scheme, a novel model selection based on gap metric and weighting method are employed for MLMPC. We prove the offset tracking property of DDP assisted MLMPC. A continuous stirred tank reactor (CSTR) process is studied to demonstrate the effectiveness of the proposed algorithms. Simulation studies show that CDDP designed by the proposed algorithm improves the trajectory over iterations, and the resulting MLMPC achieves offset-free tracking property regardless of an initial point and a set-point in the operating region.

We present concentration inequalities on the multislice which are based on (modified) log-Sobolev inequalities. This includes bounds for convex functions and multilinear polynomials. As an application, we show concentration results for the triangle count in the G(n, M) Erdős–Rényi model resembling known bounds in the G(n, p) case. Moreover, we give a proof of Talagrand’s convex distance inequality for the multislice. Interpreting the multislice in a sampling without replacement context, we furthermore present concentration results for n out of N sampling without replacement. Based on a bounded difference inequality involving the finite-sampling correction factor $$1 - (n / N)$$ 1 - ( n / N ) , we present an easy proof of Serfling’s inequality with a slightly worse factor in the exponent, as well as a sub-Gaussian right tail for the Kolmogorov distance between the empirical measure and the true distribution of the sample.

In this paper, we provide a constructive proof of $\mathbf {H}^{1}(\mathbb {R}^{n})$ factorization in terms of multilinear Calderon–Zygmund operators in Morrey spaces. As a direct application, we obtain a characterization of functions in $\text {BMO}(\mathbb {R}^{n})$ via commutators of multilinear Calderon–Zygmund operators. Furthermore, we prove a Morrey compactness characterization of [b, T]l, the commutator in the l-th entry.

In this article, we introduce local completely positive k-linear maps between locally C∗-algebras and obtain Stinespring type representation by adopting the notion of ‘invariance’ defined by J. Heo...

Intelligent mechanical fault diagnosis is a crucial measure to ensure the safe operation of equipment. To address the issue of model collapse in domain adversarial training and the problem posed by different training samples having different transferability not considered in transfer tasks, this article proposes a weighted entropy minimization based deep conditional adversarial diagnosis approach of rotating machines under variable working conditions. First, the features of vibration signals in the source domain and target domain are extracted by a weight-sharing one-dimensional deep convolution neural network. The feature vectors and category prediction vectors are then fused by multilinear mapping to carry out adversarial training in domain adaptation. The entropy of the output of the domain discrimination model provides the index by which to measure the transferability of training samples. The transferability weights of samples are applied to the entropy minimization loss to eliminate the influence of these samples that are hard to transfer in adversarial domain adaptation. Experimental datasets under variable working conditions support the value of our approach.

Let $F$ be an infinite field and let $f$ be a nonzero multilinear polynomial with coefficients in $F$. We prove that for every positive integer $d$ there exists a positive integer $s$ such that $f(M_{s}(F))$, the image of $f$ in $M_{s}(F)$, contains all trace zero $d \times d$ matrices. In particular, the image of $f$ in the algebra of all finitary matrices contains all trace zero finitary matrices., 11 pages, 0 figures, submited to Linear Algebra and Its Applications

Computer vision plays a crucial role in Advanced Assistance Systems. Most computer vision systems are based on Deep Convolutional Neural Networks (deep CNN) architectures. However, the high computational resource to run a CNN algorithm is demanding. Therefore, the methods to speed up computation have become a relevant research issue. Even though several works on architecture reduction found in the literaturehave not yet been achievedsatisfactory results for embedded real-time system applications. This paper presents an alternative approach based on the Multilinear Feature Space (MFS) method resorting to transfer learning from large CNN architectures. The proposed method uses CNNs to generate feature maps, although it does not work as complexity reduction approach. After the training process, the generated features maps are used to create vector feature space. We use this new vector space to make projections of any new sample to classify them. Our method, named AMFC, uses the transfer learning from pre-trained CNN to reduce the classification time of new sample image, with minimal accuracy loss. Our method uses the VGG-16 model as the base CNN architecture for experiments; however, the method works with any similar CNN model. Using the well-known Vehicle Image Database and the German Traffic Sign Recognition Benchmark, we compared the classification time of the original VGG-16 model with the AMFCmethod, and our method is, on average, 17 times faster. The fast classification time reduces the computational and memory demands in embedded applications requiring a large CNN architecture.

Given a discrete-time system, if all roots of corresponding characteristic polynomial belong to the open unit disc, then the corresponding system is called (Schur) stable. The stability domain of nth-order monic polynomials is defined as the set of all stable n-dimensional vectors in the coefficient space. The main hindrance of the stability and stabilization problems is the nonconvexity of the stability domain. Therefore, inner convex approximations of this domain are very important. In this paper, by using known geometrical and topological properties of the set of stable polynomials (the edge theorem, nonconvexity, openness, polytopic property and the structure of the boundary), we define a new multilinear map from the n-dimensional open cube $$(-1,1)^n=(-1,1)\times \cdots \times (-1,1)$$ onto the stability region and box coefficients. The main advantage of our map is that the stability of polytopes edges is equivalent to the stability of the convex combinations of third- and fourth-order factor polynomials. Inner approximations of the stability region by polytopes and applications to stabilizability problems are given.

In this paper, the authors study the boundedness properties of a class of multilinear strongly singular integral operator with generalized kernels on product of weighted Lebesgue spaces and product of variable exponent Lebesgue spaces, respectively. Moreover, the types $ L^{\infty}\times \dots \times L^{\infty}\rightarrow BMO $ and $ BMO\times \dots \times BMO\rightarrow BMO $ endpoint estimates are also obtained.

In this paper, the weighted boundedness for some multilinear operators generated by the pseudo-differential operators and the weighted Lipschitz functions are obtained.

In this paper, we establish the optimal multilinear restriction estimate for $n-1$ hypersurfaces with some curvature, where $n$ is the dimension of the underlying space. The result is sharp up to the end point and the role of curvature is made precise in terms of the shape operator.

In this paper, we introduce weighted Morrey-Herz spaces $ M\dot K^{\alpha, \lambda}_{q, p(\cdot)}(w~^{p(\cdot)}) $ with variable exponent $ p(\cdot) $. Then we prove the boundedness of multilinear Calderón-Zygmund singular operators on weighted Lebesgue spaces and weighted Morrey-Herz spaces with variable exponents.

The paper deals with approximations of periodic functions that play a significant role in harmonic analysis. The approach revisits the trigonometric polynomials, seen as combinations of functions, and proposes to extend the class of models of the combined functions to a wider class of functions. The key here is to use structured functions, that have low complexity, with suitable functional representation and adapted parametrizations for the approximation. Such representation enables to approximate multivariate functions with few eventually random samples. The new parametrization is determined automatically with a greedy procedure, and a low rank format is used for the approximation associated with each new parametrization. A supervised learning algorithm is used for the approximation of a function of multiple random variables in tree-based tensor format, here the particular Tensor Train format. Adaptive strategies using statistical error estimates are proposed for the selection of the underlying tensor bases and the ranks for the Tensor-Train format. The method is applied for the estimation of the wall pressure for a flow over a cylinder for a range of low to medium Reynolds numbers for which we observe two flow regimes: a laminar flow with periodic vortex shedding and a laminar boundary layer with a turbulent wake (sub-critic regime). The automatic re-parametrization enables here to take into account the specific periodic feature of the pressure.

In this paper, the boundedness of certain multilinear operator related to the multiplier operator from Lebesgue spaces to Orlicz spaces is obtained.

The multilinear polytope of a hypergraph is the convex hull of a set of binary points satisfying a collection of multilinear equations. We introduce the running intersection inequalities, a new class of facet-defining inequalities for the multilinear polytope. Accordingly, we define a new polyhedral relaxation of the multilinear polytope, referred to as the running intersection relaxation, and identify conditions under which this relaxation is tight. Namely, we show that for kite-free beta-acyclic hypergraphs, a class that lies between gamma-acyclic and beta-acyclic hypergraphs, the running intersection relaxation coincides with the multilinear polytope and it admits a polynomial size extended formulation.

Nowadays, multi-dimensional data (tensor data) have shown their capability of preserving multilinear structures. Due to the measuring error or other non-human factors, these data often suffer signal corruptions or missing values, or even both. To address these issues simultaneously, this paper studies the Robust Tensor Completion (RTC) problem, a mixed problem of the known Low-Rank Tensor Completion (LRTC) and Robust Principal Component Analysis (RPCA). Based on Tensor-Train rank (TT rank), the proposed model is able to capture the latent structure information of tensor data by recovering the low-rank component and separating the sparse component from the partial observations. To make TT rank more effective, an auto-weighted mechanism is utilized to balance the importance of different matricizations from the same tensor. We also propose a more flexible tensor augmentation approach called Tree Ket Augmentation (Tree-KA) to obtain a higher-order tensor from a lower one with it a new general explanation. Alternating direction method of multipliers (ADMM) is employed to solve the resulting model and extensive numerical experiments have verified the effectiveness of the proposed model compared with other state-of-the-art methods.