Multilinear Forms Research Articles

The Orlicz inequality for multilinear forms

The Orlicz (ℓ2,ℓ1)-mixed inequality states that(∑j1=1n(∑j2=1n|A(ej1,ej2)|)2)12≤2‖A‖ for all bilinear forms A:Kn×Kn→K and all positive integers n, where Kn denotes Rn or Cn endowed with the supremum norm. In this paper we extend this inequality to multilinear forms, with Kn endowed with ℓp norms for all p∈[1,∞].

Summability of the coefficients of a multilinear form

Let T be an m -linear form defined on a product of \ell_p -spaces and let \Lambda\subset\mathbb{N}^m . We investigate the best exponent s such that the sequence of coefficients of T belongs to \ell_s(\Lambda) . The cases \Lambda=\mathbb{N}^m and \Lambda is the diagonal of \mathbb{N}^m are already known. We study the intermediate cases using notions like combinatorial dimension of sets, multiple summing maps and random polynomials.

Fast evaluation of finite element weak forms using python tensor contraction packages

In finite element calculations, the integral forms are usually evaluated using nested loops over elements, and over quadrature points. Many such forms (e.g. linear or multi-linear) can be expressed in a compact way, without the explicit loops, using a single tensor contraction expression by employing the Einstein summation convention. To automate this process and leverage existing high performance codes, we first introduce a notation allowing trivial differentiation of multi-linear finite element forms. Based on that we propose and describe a new transpiler from Einstein summation based expressions, augmented to allow defining multi-linear finite element weak forms, to regular tensor contraction expressions. The resulting expressions are compatible with a number of Python scientific computing packages, that implement, optimize and in some cases parallelize the general tensor contractions. We assess the performance of those packages, as well as the influence of operand memory layouts and tensor contraction paths optimizations on the elapsed time and memory requirements of the finite element form evaluations. We also compare the efficiency of the transpiled weak form implementations to the C-based functions available in the finite element package SfePy.

Geometry of multilinear forms on l_1

We characterize extreme, exposed and smooth points in the Banach space L(nE) of continuous n-linear forms on E, and in its subspace Ls(nE) of symmetric n-linear forms on E when E = l1 and E = l1m for n,m ∈ N with n,m ≥ 2 .

Geometry of multilinear forms on ℝm with a certain norm

Geometry of multilinear forms on ℝm with a certain norm

A Kalman Filter for Multilinear Forms and Its Connection with Tensorial Adaptive Filters.

The Kalman filter represents a very popular signal processing tool, with a wide range of applications within many fields. Following a Bayesian framework, the Kalman filter recursively provides an optimal estimate of a set of unknown variables based on a set of noisy observations. Therefore, it fits system identification problems very well. Nevertheless, such scenarios become more challenging (in terms of the convergence and accuracy of the solution) when the parameter space becomes larger. In this context, the identification of linearly separable systems can be efficiently addressed by exploiting tensor-based decomposition techniques. Such multilinear forms can be modeled as rank-1 tensors, while the final solution is obtained by solving and combining low-dimension system identification problems related to the individual components of the tensor. Recently, the identification of multilinear forms was addressed based on the Wiener filter and most well-known adaptive algorithms. In this work, we propose a tensorial Kalman filter tailored to the identification of multilinear forms. Furthermore, we also show the connection between the proposed algorithm and other tensor-based adaptive filters. Simulation results support the theoretical findings and show the appealing performance features of the proposed Kalman filter for multilinear forms.

Abundant interaction between lump and k-kink, periodic and other analytical solutions for the (3+1)-D Burger system by bilinear analysis

In this paper, we study the (3+1)-dimensional Burger system which is considered in soliton theory and generated by considering the Hirota bilinear operators. The bilinear frame to the Burger system by using the multi-dimensional Bell polynomials is constructed. Also, based on the binary Bäcklund transformations, the generalized Bell polynomials are written. We retrieve some novel exact analytical solutions, containing interaction between lump and two kink wave solutions, interaction between lump and periodic wave solutions, interaction between stripe and periodic solutions, breather wave solutions, cross-kink wave solutions, interaction between kink and periodic wave solutions, multi-wave solutions, and finally solitary wave solutions for the (3+1)-dimensional Burger system by Maple symbolic computations. The required conditions of the analyticity and positivity of the solutions can be easily achieved by taking special choices of the involved parameters. The main ingredients for this scheme are to recover the Hirota trilinear forms and their generalized equivalences.

Nonlinear dynamics of (2 + 1)‐dimensional Bogoyavlenskii–Schieff equation arising in plasma physics

In this literature, the dynamic characteristics of the Bogoyavlenskii–Schieff equation in (2 + 1)‐dimension that arises in plasma physics are studied. Several characteristics of multi‐soliton solutions, complex rogue wave, M‐lump solutions, fusion solutions, and interaction phenomena between M‐lump and soliton solutions also with a fusion solution are discussed. A logarithmic variable transform is used to convert the studied nonlinear equation to a Hirota trilinear form. For all solutions, three‐dimensional figures are presented to more understand its dynamic behaviors. All findings are recent, and no experts have reported them.

Wronskian and lump wave solutions to an extended second KP equation

The letter aims to generalize the second KP equation to a new one which still possesses a trilinear form. We construct Wronskian determinant solutions, based on its associated trilinear equation, instead of a Hirota bilinear form. Multi-soliton solutions are generated from the presented Wronskian formulation with higher-order dispersion relations. It is then shown that the extended second KP equation possesses resonant N-wave solutions. Moreover, generic one-lump and two-lump waves are built via the improved long wave limit procedure.

Mixed Rational Lump-Solitary Wave Solutions to an Extended (2+1)-Dimensional KdV Equation

Based on the bilinear method, rational lump and mixed lump-solitary wave solutions to an extended (2+1)-dimensional KdV equation are constructed through the different assumptions of the auxiliary function in the trilinear form. It is found that the rational lump decays algebraically in all directions in the space plane and its amplitude possesses one maximum and two minima. One kind of the mixed solution describes the interaction between one lump and one line solitary wave, which exhibits fission and fusion phenomena under the different parameters. The other kind of the mixed solution shows one lump interacting with two paralleled line solitary waves, in which the evolution of the lump gives rise to a two-dimensional rogue wave. This shows that these three interesting phenomena exist in the corresponding physical model.

A quantitative stability theorem for convolution on the Heisenberg group

Although the convolution operators on Euclidean space and the Heisenberg group satisfy the same $L^p$ bounds with the same optimal constants, the former has maximizers while the latter does not. However, as work of Christ has shown, it is still possible to characterize near-maximizers. Specifically, any near-maximizing triple of the trilinear form for convolution on the Heisenberg group must be close to a particular type of triple of ordered Gaussians after adjusting by symmetry. In this paper, we use the expansion method to prove a quantitative version of this characterization.

Attenuation and Basin Amplification Revealed by the Dense Ground Motions of the 12 July 2020 Ms 5.1 Tangshan, China, Earthquake

AbstractOn 12 July 2020, an Ms 5.1 moderate earthquake occurred on the north segment of the Tangshan fault in North China, which was the seismogenic fault of the 1976 Ms 7.8 Tangshan earthquake and numerous small-to-moderate earthquakes in recent decades in the Tangshan seismic zone. The Ms 5.1 event was well-recorded by dense ground-motion observation stations, including the national strong-motion stations and seismic intensity stations. This many ground-motion recordings, obtained for such a moderate event in North China for the first time, provided a rare opportunity to investigate the attenuation and site effects on ground motion. The distance decay in the Tangshan seismic zone was first explored using the spectral amplitudes from the vertical component. The strong anelastic attenuation and weak geometrical spreading effects were clearly found. The hinged trilinear form may be more effective at describing the geometrical spreading. No geometrical spreading decay was visible at medium distances (60–100 km). Anomalous areas with extraordinary high amplitudes occurred in the spatial distribution of peak ground accelerations and peak ground velocities that we attribute to significant basin amplification effects, which was confirmed by the wideband and high amplifications on the standard spectral ratio and the later-arriving, long-period surface waves observed in waveforms in the Ninghe–Baodi area and south of Beijing. The basin-induced surface waves in the 2–5 s period were most prominent in the Ninghe–Baodi area. We further inferred that basin effects may be responsible for the high-intensity anomaly areas observed in the 1976 Ms 7.8 Tangshan earthquake.

Imposing evenness, preventing combination: charting the international dynamics of socio-technical imaginaries of innovation in American foreign policy

The Socio-Technical Imaginaries (STI) approach in Science and Technology Studies (STS) has illuminated the central role of social imaginaries in shaping the politics of technology. Its emphasis on the multilinear forms of socio-technical development is a useful corrective to universalist explanations of technological change. However, STI lacks a clear account of how inter-societal interaction shapes the imaginaries of any given political community. Synthesizing STI with the theory of Uneven and Combined Development (UCD) can correct this shortcoming. UCD offers an ontology of universals and accompanying methodology of incorporated comparison, enabling STI to integrate inter-societal causality into its theoretical framework. A combined UCD and STI framework is examined in this paper through a focus on imaginaries of technological innovation in contemporary American foreign policy. Responding to the ‘whip of external necessity’, US foreign policy seeks to upend technological diffusion and impose global regulatory evenness on national forms of technological innovation.

Early-age tensile constitutive relationships for steel and polypropylene fiber reinforced concrete

Early-age fracture tests are conducted on concrete beams with discrete steel, macro polypropylene and blends of macro and micro polypropylene fibers. The fracture behavior with aging of the concrete matrix containing 0.66% volume fraction of fibers is investigated at 1, 3, 7, and 28 days of age. The age-dependent multi-linear cohesive stress crack-separation (σ-w) relationship and the modulus of elasticity of the fiber reinforced composite material are obtained from the measured fractured responses. The generic multi-linear form of the σ-w relationship captures the age dependent variations in the stress with crack opening associated with initial softening and the hardening produced by fibers. The σ-w relationships determined from the fracture responses indicate that all fibers types significantly increase the tensile strength of concrete at 1 day. By 28 days, the tensile strength of the fiber reinforced concrete is comparable to the plain concrete. The steel fibers significantly increase the Young’s modulus of the fiber reinforced concrete in the first 3 days. At any age, steel fibers provide the largest σ as a function of w, when compared to the polypropylene fibers. The differences between σ-w responses of the polypropylene and steel fiber reinforced concretes increase with age and there is a steady increase in the fracture energy with age. The improvement in the post-cracking response of steel fibers with age is due to increases in the resistances to crack propagation and pullout of fibers. Blends of micro and macro polypropylene fibers exhibit higher resistance to crack propagation and to pullout of fibers when compared with the macro polypropylene fibers alone.

АНАЛИЗ ВЛИЯНИЯ КОНСЕРВАТИВНОГО СГЛАЖИВАНИЯ КОНЕЧНО-ЭЛЕМЕНТНОГО РЕШЕНИЯ НА ПОДАВЛЕНИЕ МОД НУЛЕВОЙ ЭНЕРГИИ

The problem of high-speed penetration of a non-deformable cylinder into a steel plate is considered. The defining system of equations is formulated in Lagrange variables in a three-dimensional formulation. The equation of motion is derived from virtual work capacities balance. Kinematic relations are recorded in the metric of the current state. The relations of the flow theory with kinematic and isotropic hardening are used as equations of state. The contact interaction of the cylinder and the plate is modeled by non-penetration conditions. The numerical solution of the problem under given boundary and initial conditions is based on the moment scheme of the finite element method and “cross” type explicit time integration scheme. To discretize the defining system of equations with respect to spatial variables, 8-node isoparametric finite elements with multilinear form functions are used. To suppress the high-frequency oscillations of the numerical solution, the procedure of nodal displacement velocities conservative smoothing is used. The smoothing algorithm is based on the momentum conservation law, focused on finite element grids consisting of blocks that are mutually unambiguously mapped to a unit cube. To analyze the nodal displacement velocities monotonicity, the numerical solution splitting in the directions of the finite element grid lines is used. As the results of computer modeling have shown, the finite elements of the plate are exposed large deformations and rotation angles as a rigid whole during local intense dynamic loading. The conservative smoothing procedure influence on the numerical solution stability is analyzed. It is shown that in the problem under consideration, without applying the conservative smoothing procedure, zero-energy modes develop in the contact zone in the finite-element grid of the plate (an hourglass-type instability) and the collision process cannot be modeled before the cylinder rebounds.

Extended group finite element method

Interpolation methods for nonlinear finite element discretizations are commonly used to eliminate the computational costs associated with the repeated assembly of the nonlinear systems. While the group finite element formulation interpolates nonlinear terms onto the finite element approximation space, we propose the use of a separate approximation space that is tailored to the nonlinearity. In many cases, this allows for the exact reformulation of the discrete nonlinear problem into a quadratic problem with algebraic constraints. Furthermore, the substitution of the nonlinear terms often shifts general nonlinear forms into trilinear forms, which can easily be described by third-order tensors. The numerical benefits as well as the advantages in comparison to the original group finite element method are studied using a wide variety of academic benchmark problems.

Averages of simplex Hilbert transforms

We study a multilinear singular integral obtained by taking averages of simplex Hilbert transforms. This multilinear form is also closely related to Calder\'on commutators and the twisted paraproduct. We prove $L^p$ bounds in dimensions two and three and give a conditional result valid in all dimensions.

Versatile mixed methods for the incompressible Navier–Stokes equations

In the spirit of the “Principle of Equipresence” introduced by Truesdell & Toupin, The Classical Field Theories (1960), we use the full version of the viscous stress tensor ν∇u+∇uT−23∇⋅uI which was originally derived for compressible flows, instead of the classical incompressible stress tensor ν∇u. (Note that, here ν is the dynamic viscosity coefficient, and u is the velocity field.) In our approach, the divergence-free constraint for the viscous stress term is not enforced ahead of discretization. Instead, our formulation allows the scheme itself to “choose” a consistent way to interpret the divergence-free constraint: i.e., the divergence-free constraint is interpreted (or enforced) in a consistent fashion in both the mass conservation equation and the stress tensor term (in the momentum equation). Furthermore, our approach preserves the original symmetrical properties of the stress tensor, e.g. its rotational invariance, and it remains physically correct in the context of compressible flows. As a result, our approach facilitates versatility and code reuse. In this paper, we introduce our approach and establish some important mathematical properties for the resulting class of finite element schemes. More precisely, for general mixed methods, which are not necessarily pointwise divergence-free, we establish the existence of a new norm induced by the full, viscous bilinear form. Thereafter, we prove the coercivity of the viscous bilinear form and the semi-coercivity of a convective trilinear form. In addition, we demonstrate L2-stability of the discrete velocity fields for the general class of methods and (by deduction) the H(div)-conforming methods. Finally, we run some numerical experiments to illustrate the behavior of the versatile mixed methods, and we make careful comparisons with a conventional H(div)-conforming scheme.

Extrapolation for multilinear Muckenhoupt classes and applications

In this paper we solve a long standing problem about the multivariable Rubio de Francia extrapolation theorem for the multilinear Muckenhoupt classes Ap→, which were extensively studied by Lerner et al. and which are the natural ones for the class of multilinear Calderón-Zygmund operators as well as for some bilinear rough singular integral operators. Furthermore, we go beyond the classes Ap→ and extrapolate within the classes Ap→,r→ which appear naturally associated to the weighted norm inequalities for multilinear sparse forms which control fundamental operators such as the bilinear Hilbert transform. We give several applications which can be easily obtained using extrapolation. First, weighted norm inequalities (scalar and vector-valued) for the bilinear rough singular integral operators are established. Second, for the bilinear Hilbert transform one can extrapolate from the recent result of Culiuc et al. who considered the Banach range, extend the estimates to the quasi-Banach range, and furthermore, prove trivially vector-valued inequalities. We also extend recent results of Carando et al. on Marcinkiewicz-Zygmund estimates for multilinear Calderón-Zygmund operators. Finally, our last application gives new weighted estimates (scalar and vector-valued) for the commutators of multilinear Calderón-Zygmund operators, bilinear rough singular integral operators, and for the bilinear Hilbert transform with BMO functions using ideas from Bényi et al.

New Periodic Wave, Cross-Kink Wave, Breather, and the Interaction Phenomenon for the (2 + 1)-Dimensional Sharmo–Tasso–Olver Equation

In this paper, with the aid of symbolic computation, several kinds of exact solutions including periodic waves, cross-kink waves, and breather are proposed by using a trilinear form for the (2 + 1)-dimensional Sharmo–Tasso–Olver equation. Then, by combing the different forms, the interactions between a lump and one-kink soliton and between a lump and periodic waves are generated. Moreover, the dynamic characteristics of interaction solutions are analyzed graphically by selecting suitable parameters with the help of Maple.