Multilinear Research Articles

Robust Multilinear Target-Based Decision Analysis Considering High-dimensional Interactions

The Multilinear Target-based Preference Functions (MTPFs) support multi-attribute decision problems characterized by attribute interactions and targets. However, existing research falls short in flexibly modeling high-dimensional interactions and lacks robustness in decision-making recommendations when faced with uncertain parameters and targets. The paper proposes a robust multilinear target-based decision analysis framework considering high-dimensional interactions and uncertainty in parameters and targets. First, the necessity of high-dimensional interactions and the limitations of available MTPFs in modeling high-dimensional interactions are demonstrated. Second, the MTPFs based on the 2-interactive fuzzy measure and the Nonmodularity index are proposed to model the high-dimensional interactions, while simultaneously reducing the computational challenges of parameter identification. Third, new descriptive measures are proposed based on the Stochastic Multicriteria Acceptability Analysis to evaluate the robustness of decision recommendations subject to uncertain targets and parameters. The validation and advantages of the framework are illustrated with simulation studies and an application in customer competitive evaluation of smart thermometer patches.

A Note on Multilinear Mappings in Modules

The linear compactness of certain linearly topologized modules of multilinear mappings is established, as well as the continuity of multilinear mappings in a known class of linearly topologized modules.

Structure and arithmetic of multivariate Ore extensions

We give the basic structure of the multivariable Ore extensions S = A [ t ¯ ; σ , δ ¯ ] introduced in the work of Martínez-Peñas and Kschischang. The Pseudo multilinear transformations (PMT’s) are introduced and correspond to modules over S. These maps are strongly connected to the evaluation of polynomials in S. A general product formula is obtained. PMT’s help to put some structure on the set of roots of a polynomial f ( t ) ∈ S .

Bounds of Different Integral Operators in Tensorial Hilbert and Variable Exponent Function Spaces

In dynamical systems, Hilbert spaces provide a useful framework for analyzing and solving problems because they are able to handle infinitely dimensional spaces. Many dynamical systems are described by linear operators acting on a Hilbert space. Understanding the spectrum, eigenvalues, and eigenvectors of these operators is crucial. Functional analysis typically involves the use of tensors to represent multilinear mappings between Hilbert spaces, which can result in inequality in tensor Hilbert spaces. In this paper, we study two types of function spaces and use convex and harmonic convex mappings to establish various operator inequalities and their bounds. In the first part of the article, we develop the operator Hermite–Hadamard and upper and lower bounds for weighted discrete Jensen-type inequalities in Hilbert spaces using some relational properties and arithmetic operations from the tensor analysis. Furthermore, we use the Riemann–Liouville fractional integral and develop several new identities which are used in operator Milne-type inequalities to develop several new bounds using different types of generalized mappings, including differentiable, quasi-convex, and convex mappings. Furthermore, some examples and consequences for logarithm and exponential functions are also provided. Furthermore, we provide an interesting example of a physics dynamical model for harmonic mean. Lastly, we develop Hermite–Hadamard inequality in variable exponent function spaces, specifically in mixed norm function space (lq(·)(Lp(·))). Moreover, it was developed using classical Lebesgue space (Lp) space, in which the exponent is constant. This inequality not only refines Jensen and triangular inequality in the norm sense, but we also impose specific conditions on exponent functions to show whether this inequality holds true or not.

Quantitative weighted estimates for the multilinear pseudo-differential operators in function spaces

Abstract In this paper, the weighted estimates for multilinear pseudo-differential operators were systematically studied in rearrangement invariant Banach and quasi-Banach spaces. These spaces contain the Lebesgue space, the classical Lorentz space and Marcinkiewicz space as typical examples. More precisely, the weighted boundedness and weighted modular estimates, including the weak endpoint case, were established for multilinear pseudo-differential operators and their commutators. As applications, we show that the above results also hold for the multilinear Fourier multipliers, multilinear square functions, and a class of multilinear Calderón–Zygmund operators.

A local magnitude scale (ML) for Northern Algeria

This study presents a local magnitude scale (ML) based on the original Richter definition and designed for use within the Algerian Digital Seismic Network (ADSN). The magnitude scale is derived from the analysis of 17,377 zero-peak maximum amplitude traces extracted from the vertical component, simulated as Wood-Anderson seismograms. These traces are taken from a dataset of 1901 earthquakes recorded between January 1, 2010, and June 1, 2022, at a minimum of five stations in the ADSN network. To better account for the attenuation of direct and refracted waves in northern Algeria, amplitude decay analysis reveals the presence of two transition distances at 90 and 190 km, resulting in three segments. A distance correction term, −log10(A0), is introduced and described by the following trilinear function:−logA0=0.6747∗log10R+0.0002∗R+1.6306R≤901.7736∗log10R+0.0002∗R−0.516990<R≤1902.4580∗log10R+0.0002∗R−2.0765R>190R represents the hypocentral distance in kilometers. The derived distance correction formula provides a well-constrained ML relationship for northern Algeria that is valid over a distance range of 5 to 600 km. Compared to other local magnitude relationships, the methodology proposed in this study consistently gives ML values slightly higher than those calculated by the Southern California relationship over all distances, with an average difference of 0.2 units. We computed corrections for 72 stations by minimizing the ML residuals. These corrections range from −0.50 to 0.54, highlighting the influence of local site effects on the amplitude of the seismic signal. The magnitude residuals using our magnitude relationship and incorporating the station corrections, show that the standard deviation has improved significantly, from 0.34 to 0.24. An ML relationship specific to the northern Algerian region provides a valuable tool for seismic monitoring, hazard assessment, and earthquake research in the region.

Bumpless transfer control for switched positive time-delay systems with linear-type dissipative inequalities

In this paper, a family of amplitude limited controllers and a hysteresis switching paradigm, for switched positive systems with time-varying delays, are simultaneously determined to achieve strict (q,r) -α-dissipativity. The concept of strictly (q,r)-α-dissipative, induced by linear-type dissipative inequalities, is introduced. The main feature of the presented method lies in two aspects. The first one is that the magnitude of controller discontinuities, at the occurrence of switching, is here restricted. The other one is that the high switching frequency is refrained commonly the appearance of in state-dependent switching schemes. Moreover, dissipativity analysis as well as feedback dissipativity are taken into account. Novel criteria are derived for the considered systems to be dissipativity by resorting to multi-linear copositive storage functionals. Eventually, the validity and superiority of the presented control strategy are demonstrated through two illustrative examples.

Simpler characterizations of total orderization invariant maps

Given a finite subset [Formula: see text] of a distributive lattice, its total orderization [Formula: see text] is a natural transformation of [Formula: see text] into a totally ordered set. Recently, the author showed that multivariate maps on distributive lattices which remain invariant under total orderizations generalize various maps on vector lattices, including bounded orthosymmetric multilinear maps and finite sums of bounded orthogonally additive polynomials. Therefore, a study of total orderization invariant maps on distributive lattices provides new perspectives for maps widely researched in vector lattice theory. However, the unwieldy notation of total orderizations can make calculations extremely long and difficult. In this paper we resolve this complication by providing considerably simpler characterizations of total orderization maps. Utilizing these easier representations, we then prove that a lattice multi-homomorphism on a distributive lattice is total orderization invariant if and only if it is symmetric, and we show that the diagonal of a symmetric lattice multi-homomorphism is a lattice homomorphism, extending known results for orthosymmetric vector lattice homomorphisms.

KGR: A Kernel-Mapping Based Group Recommender System Using Trust Relations

A massive amount of information explosion over the internet has caused a possible difficulty of information overload. To overcome this, Recommender systems are systematic tools that are rapidly being employed in several domains such as movies, travel, E-commerce, and music. In the existing research, several methods have been proposed for single-user modeling, however, the massive rise of social connections potentially increases the significance of group recommender systems (GRS). A GRS is one that jointly recommends a list of items to a collection of individuals based on their interests. Moreover, the single-user model poses several challenges to recommender systems such as data sparsity, cold start, and long tail problems. On the contrary hand, another hotspot for group-based recommendation is the modeling of user preferences and interests based on the groups to which they belong using effective aggregation strategies. To address such issues, a novel “KGR” group recommender system based on user-trust relations is proposed in this study using kernel mapping techniques. In the proposed model, user-trust networks or relations are exploited to generate trust-based groups of users which is one of the important behavioral and social aspects. More precisely, in KGR the group kernels and group residual matrices are exploited as well as seeking a multi-linear mapping between encoded vectors of group-item interactions and probability density function indicating how groups will rate the items. Moreover, to emphasize the relevance of individual preferences of users in a group to which they belong, a hybrid approach is also suggested in which group kernels and individual user kernels are merged as additive and multiplicative models. Furthermore, the proposed KGR is validated on two different trust-based datasets including Film Trust and CiaoDVD. In addition, KGR outperforms with an RMSE value of 0.3306 and 0.3013 on FilmTrust and CiaoDVD datasets which are lower than the 1.8176 and 1.1092 observed with the original KMR.

A parareal exponential integrator finite element method for semilinear parabolic equations

AbstractIn this article, we present a parareal exponential finite element method, with the help of variational formulation and parareal framework, for solving semilinear parabolic equations in rectangular domains. The model equation is first discretized in space using the finite element method with continuous piecewise multilinear rectangular basis functions, producing the semi‐discrete system. We then discretize the temporal direction using the explicit exponential Runge–Kutta approach accompanied by the parareal framework, resulting in the fully‐discrete numerical scheme. To further improve computational speed, we design a fast solver for our method based on tensor product spectral decomposition and fast Fourier transform. Under certain regularity assumption, we successfully derive optimal error estimates for the proposed parallel‐based method with respect to ‐norm. Extensive numerical experiments in two and three dimensions are also carried out to validate the theoretical results and demonstrate the performance of our method.

Jordan norms for multilinear maps on $C^{\ast}$-algebras and Grothendieck's inequalities

There exists a generalization of the concept completely bounded norm, for multilinear maps on $C^{\ast }$-algebras. We will use the word Jordan norm, for this norm and denote it by $\lVert \cdot \rVert _J$. The Jordan norm $\lVert\Phi\rVert_J$ of a multilinear map is obtained via factorizations of $\Phi$ in the form $$\Phi (a_1, \dots , a_n) = T_0 \sigma _1(a_1)T_1 \cdots T_{(n-1)}\sigma _n(a_n)T_n ,$$ where the maps $\sigma _i$ are Jordan homomorphisms. We show that any bounded bilinear form on a pair of $C^{\ast }$-algebras is Jordan bounded and satisfies $\lVert B\rVert _J \leq 2\lVert B\rVert $.

Multi-Server Multi-Function Distributed Computation.

The work here studies the communication cost for a multi-server multi-task distributed computation framework, as well as for a broad class of functions and data statistics. Considering the framework where a user seeks the computation of multiple complex (conceivably non-linear) tasks from a set of distributed servers, we establish the communication cost upper bounds for a variety of data statistics, function classes, and data placements across the servers. To do so, we proceed to apply, for the first time here, Körner's characteristic graph approach-which is known to capture the structural properties of data and functions-to the promising framework of multi-server multi-task distributed computing. Going beyond the general expressions, and in order to offer clearer insight, we also consider the well-known scenario of cyclic dataset placement and linearly separable functions over the binary field, in which case, our approach exhibits considerable gains over the state of the art. Similar gains are identified for the case of multi-linear functions.

Completely continuous multilinear mappings on 𝐿₁

A useful result of H. Rosenthal and J. Bourgain states that, given a Banach space X X , an operator T : L 1 [ 0 , 1 ] → X T:L_1[0,1]\to X is completely continuous if and only if its composition with the natural inclusion i ∞ : L ∞ [ 0 , 1 ] → L 1 [ 0 , 1 ] i_\infty :L_\infty [0,1] \to L_1[0,1] is compact. We extend this result to multilinear mappings on products of L 1 [ 0 , 1 ] L_1[0,1] spaces, and consider also the composition with the natural inclusion i : C [ 0 , 1 ] → L 1 [ 0 , 1 ] i:C[0,1]\to L_1[0,1] . We show that a multilinear mapping on a product of L 1 [ 0 , 1 ] L_1[0,1] spaces is completely continuous if and only if its associated polymeasure has a relatively norm compact range.

Grothendieck-type characterization of representable multilinear mappings

Let μ be a finite measure, T : L 1 ( μ ) → X be an operator into a Banach space X, and I : L ∞ ( μ ) → L 1 ( μ ) be the natural inclusion. A well-known theorem due to Grothendieck states that T is representable if and only T ∘ I is nuclear. In a 2022 paper we gave a multilinear version of this theorem replacing nuclearity by the weaker property that we call the left integral ℓ 1 -factorization property. Here we prove the rather surprising result that the multilinear version is also true for nuclearity. As an application, we show that the composition P ∘ T of an integral operator T with a polynomial P whose Aron–Berner extension to the bidual takes values in the original range space is nuclear. This extends (and strengthens) to the polynomial setting a result on composition of operators also due to Grothendieck.

Reduced-complexity multiple parameters estimation via toeplitz matrix triple iteration reconstruction with bistatic MIMO radar

In this advanced exploration, we focus on multiple parameters estimation in bistatic Multiple-Input Multiple-Output (MIMO) radar systems, a crucial technique for target localization and imaging. Our research innovatively addresses the joint estimation of the Direction of Departure (DOD), Direction of Arrival (DOA), and Doppler frequency for incoherent targets. We propose a novel approach that significantly reduces computational complexity by utilizing the Temporal-Spatial Nested Sampling Model (TSNSM). Our methodology begins with a multi-linear mapping mechanism to efficiently eliminate unnecessary virtual Degrees of Freedom (DOFs) and reorganize the remaining ones. We then employ the Toeplitz matrix triple iteration reconstruction method, surpassing the traditional Temporal-Spatial Smoothing Window (TSSW) approach, to mitigate the single snapshot effect and reduce computational demands. We further refine the high-dimensional ESPRIT algorithm for joint estimation of DOD, DOA, and Doppler frequency, eliminating the need for additional parameter pairing. Moreover, we meticulously derive the Cramér-Rao Bound (CRB) for the TSNSM. This signal model allows for a second expansion of DOFs in time and space domains, achieving high precision in target angle and Doppler frequency estimation with low computational complexity. Our adaptable algorithm is validated through simulations and is suitable for sparse array MIMO radars with various structures, ensuring higher precision in parameter estimation with less complexity burden.

Switching positive system event triggering T-S fuzzy control

This paper presents a saturation control problem with a positive switching T-S fuzzy system based on event-triggering mechanism under continuous time. Firstly, a 1-norm event triggering mechanism is designed. Then, the multilinear co-positive Lyapunov function is applied to the closed-loop positive switching system with a T-S fuzzy control strategy under continuous time. Next, we consider constructing a cone invariant set to transform the actuator saturation problem into an interval form. The designed controller with a linear programming method guarantees both the positivity and stability of the closed-loop system. Finally, an example of a communication data network proves the effectiveness of the design. The main contents are (i) The event-triggering strategy under continuous time is used in the T-S switching positive system. (ii) A controller with input saturation is introduced into controllable subsystems. (iii) When determining the attractor domain of the system state, a cone domain is considered as a potential restriction.

DANNMCTG: Domain-Adversarial Training of Neural Network for multicenter antenatal cardiotocography signal classification

Intelligent classification of cardiotocography (CTG) based on machine learning (ML), a useful tool to improve the accuracy of fetal abnormality detection, can assist obstetricians with clinical decisions. With the advancement of information technologies and medical devices, there are development opportunities for multicenter clinical research and obtaining more digital CTG signals. However, most of the existing clinical multicenter CTG datasets are partially annotated and have discrepancies which do not satisfy the ML condition of independent identical distribution. Therefore, this paper focuses on an unsupervised domain adaptation (UDA) algorithm to realize cross-domain intelligent classification of multimodal CTG signals. We propose a method dubbed domain adversarial training of neural network for multicenter CTG (DANNMCTG), which mainly consists of a label classifier, a feature extractor and a domain discriminator. To match different distribution of fetal heart rate (FHR), uterine contraction (UC) and fetal movement (FetMov) signals, we condition the domain alignment on label predictions by defining the multi-linear map. For analysis, two datasets from the hospital central station and home monitoring devices were considered as the source and target domains. The results showed that the accuracy value, F1 value and area under the curve (AUC) value of the DANNMCTG were 71.25%, 76.08% and 0.7705, respectively. This method significantly improved the performance of the deep learning models without exploiting any information in the target domain, and outperformed the state-of-the-art UDA algorithms for CTG classification. In summary, the DANNMCTG can effectively mitigate the influence of domain shift for multicenter intelligent prenatal fetal monitoring.

Neural Network Approximators for Marginal MAP in Probabilistic Circuits

Probabilistic circuits (PCs) such as sum-product networks efficiently represent large multi-variate probability distributions. They are preferred in practice over other probabilistic representations, such as Bayesian and Markov networks, because PCs can solve marginal inference (MAR) tasks in time that scales linearly in the size of the network. Unfortunately, the most probable explanation (MPE) task and its generalization, the marginal maximum-a-posteriori (MMAP) inference task remain NP-hard in these models. Inspired by the recent work on using neural networks for generating near-optimal solutions to optimization problems such as integer linear programming, we propose an approach that uses neural networks to approximate MMAP inference in PCs. The key idea in our approach is to approximate the cost of an assignment to the query variables using a continuous multilinear function and then use the latter as a loss function. The two main benefits of our new method are that it is self-supervised, and after the neural network is learned, it requires only linear time to output a solution. We evaluate our new approach on several benchmark datasets and show that it outperforms three competing linear time approximations: max-product inference, max-marginal inference, and sequential estimation, which are used in practice to solve MMAP tasks in PCs.

A numerical range approach to Birkhoff–James orthogonality with applications

The main aim of this paper is to provide characterizations of Birkhoff–James orthogonality (BJ-orthogonality in short) in a number of families of Banach spaces in terms of the elements of significant subsets of the unit ball of their dual spaces, which makes the characterizations more applicable. The tool to do so is a fine study of the abstract numerical range and its relation with the BJ-orthogonality. Among other results, we provide a characterization of BJ-orthogonality for spaces of vector-valued bounded functions in terms of the domain set and the dual of the target space, which is applied to get results for spaces of vector-valued continuous functions, uniform algebras, Lipschitz maps, injective tensor products, bounded linear operators with respect to the operator norm and to the numerical radius, multilinear maps, and polynomials. Next, we study possible extensions of the well-known Bhatia–Šemrl theorem on BJ-orthogonality of matrices, showing results in spaces of vector-valued continuous functions, compact linear operators on reflexive spaces, and finite Blaschke products. Finally, we find applications of our results to the study of spear vectors and spear operators. We show that no smooth point of a Banach space can be BJ-orthogonal to a spear vector of Z. As a consequence, if X is a Banach space containing strongly exposed points and Y is a smooth Banach space with dimension at least two, then there are no spear operators from X to Y. Particularizing this result to the identity operator, we show that a smooth Banach space containing strongly exposed points has numerical index strictly smaller than one. These latter results partially solve some open problems.

Positive maps from the walled Brauer algebra

We present positive maps and matrix inequalities for variables from the positive cone. These inequalities contain partial transpose and reshuffling operations, and can be understood as positive multilinear maps that are in one-to-one correspondence with elements from the walled Brauer algebra. Using our formalism, these maps can be obtained in a systematic and clear way by manipulating partially transposed permutation operators under a partial trace. Additionally, these maps are reasonably easy in construction by combining an algorithmic approach with graphical calculus.