Multilinear Forms Research Articles

New inequalities for permanents and hafnians and some generalizations

We show new upper bounds for permanents and hafnians, which are particularly useful for complex matrices. Multidimensional permanents and hyperhafnians are considered as well. The permanental bounds improve on a Hadamard type inequality of Carlen et al. [An inequality of Hadamard type for permanents. Methods Appl Anal. 2006;13(1):1–17. doi: 10.4310/MAA.2006.v13.n1.a1] and Cobos et al. [Multilinear forms of Hilbert type and some other distinguished forms. Integral Equ Oper Theory. 2006;56(1):57–70. doi: 10.1007/s00020-005-1412-2]. Our proofs are based on a more general inequality, which can be applied to generalized Laplace type expansions of the matrix functions under consideration. As application, we show new bounds on the characteristic function of a random diagonal sum. A numerical comparison shows the performance of some of our permanental bounds.

Norm-Peak Multilinear Forms on the Plane with the Rotated Supremum Norm

Let 𝑛 ≥ 2. A continuous 𝑛-linear form 𝑇 on a Banach space 𝐸 is called norm-peak if there is a unique (𝑥1, … , 𝑥𝑛) ∈ 𝐸𝑛 such that ║𝑥1║ = … = ║𝑥𝑛║ = 1 and for the multilinear operator norm it holds ‖𝑇 ‖ = |𝑇 (𝑥1, … , 𝑥𝑛)|.Let 0 ≤ 𝜃 ≤ = ℝ2 with the rotated supremum norm ‖(𝑥, 𝑦)‖(∞,𝜃) = max {|𝑥 cos 𝜃 + 𝑦 sin 𝜃|, |𝑥 sin 𝜃 − 𝑦 cos 𝜃|}.In this note, we characterize all norm-peak multilinear forms on . As a corollary we characterize all norm-peak multilinear forms on = ℝ2 with the 𝓁𝑝-norm for 𝑝 = 1, ∞.

Conditions for a quadratic quantum speedup in nonlinear transforms with applications to energy contract pricing

Abstract Computing nonlinear functions over multilinear forms is a general problem with applications in risk analysis.
For instance in the domain of energy economics, accurate and timely risk management demands for
efficient simulation of millions of scenarios, largely benefiting from computational speedups. We develop a
novel hybrid quantum-classical algorithm based on polynomial approximation of nonlinear functions, computed
through Quantum Hadamard Products, and we rigorously assess the conditions for its end-to-end
speedup for different implementation variants against classical algorithms. In our setting, a quadratic quantum
speedup, up to polylogarithmic factors, can be proven only when forms are bilinear and approximating
polynomials have second degree, if efficient loading unitaries are available for the input data sets. We also
enhance the bidirectional encoding, that allows tuning the balance between circuit depth and width, proposing
an improved version that can be exploited for the calculation of inner products. Lastly, we exploit the
dynamic circuit capabilities, recently introduced on IBM Quantum devices, to reduce the average depth
of the Quantum Hadamard Product circuit. A proof of principle is implemented and validated on IBM
Quantum systems.

The norming sets of multilinear forms on a certain normed space $\mathbb{R}^n$

Let $n, m\in \mathbb{N}, n, m\geq 2$ and $E$ a Banach space. An element $(x_1, \ldots, x_n)\in E^n$ is called a~norming point of $T\in {\mathcal L}(^n E)$ if $\|x_1\|=\cdots=\|x_n\|=1$ and $|T(x_1, \ldots, x_n)|=\|T\|,$ where ${\mathcal L}(^n E)$ denotes the space of all continuous $n$-linear forms on $E.$ For $T\in {\mathcal L}(^n E),$ we define ${Norm}(T)$ as the set of all $(x_1, \ldots, x_n)\in E^n$ which are the norming points of~$T.$ Let $\mathbb{R}^n_{\|\cdot\|}=\mathbb{R}^n$ with a norm satisfying that $\{W_1, \ldots, W_n\}$ forms a basis and the set of all extreme points of $B_{\mathbb{R}^n_{\|\cdot\|}}$ is $\{\pm W_1, \ldots, \pm W_n\}$. In the paper we characterize ${Norm}(T)$ for every $T\in {\mathcal L}(^m \mathbb{R}^n_{\|\cdot\|})$ as follows: Let $ T=(T(W_{i_1}, \ldots W_{i_m}))_{\overset{1\leq i_k\leq n,}{1\leq k\leq m}}\in {\mathcal L}(^m \mathbb{R}^n_{\|\cdot\|})$, $\|T\|=1,$\ $S_T=(b_{i_1\cdots i_m})_{\overset{1\leq i_k\leq n,}{1\leq k\leq m}}\in {\mathcal L}(^m \mathbb{R}^n_{\|\cdot\|})$ such that $\displaystyle b_{i_1\cdots i_m}=T\big(W_{i_1}, \ldots W_{i_m}\big)~\mbox{if}~ |T\big(W_{i_1}, \ldots W_{i_m}\big)|=1~ \mbox{and}~ b_{i_1\cdots i_m}=1~\mbox{if}~ |T\big(W_{i_1}, \ldots W_{i_m}\big)|<1,$ and $A$ is the Cartesian product of the set $\{1, \ldots, n\}$, $M$ is the set of indices $(i_1, \ldots, i_m)\in A$ such that $|T\big(W_{i_1}, \ldots W_{i_m}\big)|<1.$ Then, \begin{gather*} {Norm}(T)=\bigcap_{(i_1, \ldots, i_m)\in M} \bigcup_{j=1}^m \Big\{\Big( \sum_{1\leq i\leq n}s_i^{(1)}W_i, \ldots, \sum_{1\leq i\leq n}s_i^{(j-1)}W_i, \sum_{1\leq i\leq n}s_i^{(j)}W_i-s_{i_j}W_{i_j}, \\ \sum_{1\leq i\leq n}s_i^{(j+1)}W_i, \ldots, \sum_{1\leq i\leq n}s_i^{(m)}W_i\Big)\colon \Big(\sum_{1\leq i\leq n}s_i^{(1)}W_i, \ldots, \sum_{1\leq i\leq n}s_i^{(m)}W_i\Big)\in {Norm}(S_T)\Big\}. \end{gather*}

Generational Time and Multilinear Form in the Climate Change Novel

Generational Time and Multilinear Form in the Climate Change Novel

Study on the behavior and mechanism of double transition points in stable fatigue crack growth of superelastic NiTi shape memory alloy

Fatigue crack growth (FCG) tests were conducted on superelastic NiTi alloys, demonstrating that the da/dN-ΔK curve in the stable crack growth stage exhibits two transition points in the double-logarithmic coordinate system, presenting a tri-linear form. Fracture surface SEM analysis indicated that the FCG mechanisms differ across the three stages on either side of the two transition points. This phenomenon is first discovered and studied in NiTi alloys. The study investigated the size and position relationships between the characteristic zones at the crack tip (phase transformation zone and cyclic plasticity zone) and the microstructure during crack growth. Based on this, a critical prediction method for the transition points was established and found to be in close agreement with the experimental results. Finally, the formation mechanism of the double transition points was explained by combining the SEM results of the fracture surfaces with every stage of FCG.

On the power of quantum entanglement in multipartite quantum XOR games

AbstractWe show that, given , there exist ‐player quantum XOR games for which the entangled bias can be arbitrarily larger than the bias of the game when the players are restricted to separable strategies. In particular, quantum entanglement can be a much more powerful resource than local operations and classical communication to play these games. This result shows a strong contrast to the bipartite case, where it was recently proved that, as a consequence of a noncommutative version of Grothendieck theorem, the entangled bias is always upper bounded by a universal constant times the one‐way classical communication bias. In this sense, our main result can be understood as a counterexample to an extension of such a noncommutative Grothendieck theorem to multilinear forms.

Spinor-helicity representations of particles of any mass in dS4 and AdS4 spacetimes

The spinor-helicity representations of massive and (partially) massless particles in four-dimensional (anti–)de Sitter (A)dS spacetime are studied within the framework of the dual pair correspondence. We show that the dual groups (also known as “little groups”) of the anti–de Sitter and de Sitter groups are, respectively, O(2N) and O*(2N). For N=1, the generator of the dual algebra so(2)≅so*(2)≅u(1) corresponds to the helicity operator, and the spinor-helicity representation describes massless particles in (A)dS4. For N=2, the dual algebra is composed of two ideals, s and mΛ. The former ideal s≅so(3) fixes the spin of the particle, while the mass is determined by the latter ideal mΛ, which is isomorphic to so(2,1), iso(2), or so(3) depending on the cosmological constant being positive, zero, or negative. In the case of a positive cosmological constant, namely dS4, the spinor-helicity representation contains all massive particles corresponding to the principal series representations and the partially massless particles corresponding to the discrete series representations leaving out only the light massive particles corresponding to the complementary series representations. The zero and negative cosmological constant cases, which had been addressed in earlier references, are also discussed briefly. Finally, we consider the multilinear form of helicity spinors invariant under (A)dS group, which can serve as the (A)dS counterpart of the scattering amplitude, and discuss technical differences and difficulties of the (A)dS cases compared to the flat spacetime case. Published by the American Physical Society 2024

Generating operators of symmetry breaking—From discrete to continuous

Based on the “generating operator” of the Rankin–Cohen brackets introduced in Kobayashi–Pevzner [arXiv:2306.16800], we present a method to construct various fundamental operators with continuous parameters such as invariant trilinear forms on infinite-dimensional representations, the Fourier and the Poisson transforms on the anti-de Sitter space, and integral symmetry breaking operators for the fusion rules, among others, out of a countable set of differential symmetry breaking operators.

Norm-peak multilinear forms on l1

Let n ∈ N, n≥ 2. A continuous n-linear form T on a Banach space E is called norm-peak if there is unique (x1, . . . , xn) ∈ En such that ∥x1∥ = ··· = ∥xn∥ = 1 and T attains its norm only at (±x1,...,±xn). In this paper, we characterize the norm-peak multilinear forms on l1.

A dual number formulation to efficiently compute higher order directional derivatives

This contribution proposes a new formulation to efficiently compute directional derivatives of order one to fourth. The formulation is based on automatic differentiation implemented with dual numbers. Directional derivatives are particular cases of symmetric multilinear forms; therefore, using their symmetric properties and their coordinate representation, we implement functions to calculate mixed partial derivatives. Moreover, with directional derivatives, we deduce concise formulas for the velocity, acceleration, jerk, and jounce/snap vectors. The utility of our formulation is proved with three examples. The first example presents a comparison against the forward mode of finite differences to compute the fourth-order directional derivative of a scalar function. To this end, we have coded the finite differences method to calculate partial derivatives until the fourth order, to any order of approximation. The second example presents efficient computations of the velocity, acceleration, jerk, and jounce/snap. Finally, the third example is related to the computation of some partial derivatives. The implemented code of the proposed formulation and the finite differences method is proportioned as additional material to this article.

Classification of anemia using Harris hawks optimization method and multivariate adaptive regression spline

Data mining methods are important for the diagnosis and prediction of diseases. Early and accurate diagnosis of patients is vital for their treatment. Various methods have been used in the literature to classify anemia. However, due to the different characteristics of patient datasets, changes in dataset sizes, different parameter numbers and features, and different numbers of patient records, algorithm performances vary according to datasets. In this study, the Harris hawks algorithm (HHA) and the multivariate adaptive regression spline (MARS) were used to classify anemia based on blood data of 1732 patients from the Kaggle database of patients with and without anemia. Six different algorithms were proposed to determine the parameters of the linear anemia approximation, namely multilinear form HHA, multilinear quadratic form HHA, multilinear exponential form HHA, first-order MARS model, second-order MARS model, and the best performing MARS model. The performance of the six proposed algorithms has been analyzed and found to be better than the previous studies in the literature.

Experimental characterization and cyclic cohesive zone modeling of mode I delamination growth in glass/epoxy composite laminates under high cycle fatigue

Large-scale fiber bridging makes a significant impact on fatigue delamination growth (FDG) in fiber-reinforced polymer composites. As bilinear cyclic cohesive zone modeling (CCZM) is not a promising modeling tool in this case, another solution is needed. The present study aims to evaluate a new easily-calibrating trilinear CCZM framework for modeling the FDG in glass/epoxy double cantilever beam (DCB) laminates with large-scale fiber bridging. First, mode I delamination growth, characterized by a significant R-curve effect, is experimentally determined under both quasi-static and high cycle fatigue regimes. Next, trilinear forms of the Turon et al. and the Kawashita-Hallett damage models are developed to predict such fatigue delamination behavior. Results show that the accuracy and efficiency of the proposed trilinear CCZM framework are enhanced by increasing the bridging length of the composite structure. Additionally, the Kawashita-Hallett trilinear model yields the most consistent and precise predictions with the least computational time.

Dynamic vibration control of non-linear buildings using multiple tuned mass dampers

In the field of civil engineering, tuned mass dampers (TMDs) serve as passive devices designed for dynamic vibration control of structures. When dealing with buildings exhibiting nonlinear behavior under dynamic loads, the effectiveness of TMDs may be affected by detuning due to the degradation of the building’s strength. Therefore, addressing the non-linear behavior requires a unique strategy involving the tuning of TMDs to specific time periods following the onset of non-linearity. The proposed approach in this study entails a pushover analysis to establish the pushover capacity curve. The regions between the origin and a target drift of 1/150 are then represented using an idealized trilinear form, with the initial segment corresponding to linearity and subsequent segments capturing non-linear behavior. The second segment spans from the onset of non-linearity to a target drift of 1/400, and the third segment covers the drift range from 1/400 to 1/150. Examining this strategy involves calculating time periods for each segment. Subsequently, three single TMD (STMD) scenarios and one multiple TMD (MTMD) scenario with 3 TMDs, each tuned to time periods corresponding to specific segments of the idealized trilinear, are compared in this study. The evaluation includes non-linear dynamic analysis of 7-story and 25-story reinforced concrete buildings equipped with these TMD scenarios. The floor maximum displacement and peak acceleration results indicate that the STMDs tuned to the time periods corresponding to the non-linear segments exhibit robustness, surpassing the performance of the STMD tuned to the fundamental period. Remarkably, the MTMD scenario demonstrates superior robustness compared to all three STMD scenarios. Further analysis under wind load on the same 25-story building confirms the effectiveness of the MTMDs and STMD tuned to the nonlinearity segment compared to the STMD tuned to the fundamental period. This research provides valuable insights into TMD design for enhanced building performance under non-linear conditions.

Multi-linear forms, graphs, and -improving measures in

Abstract The purpose of this paper is to introduce and study the following graph-theoretic paradigm. Let $$ \begin{align*}T_Kf(x)=\int K(x,y) f(y) d\mu(y),\end{align*} $$ where $f: X \to {\Bbb R}$ , X a set, finite or infinite, and K and $\mu $ denote a suitable kernel and a measure, respectively. Given a connected ordered graph G on n vertices, consider the multi-linear form $$ \begin{align*}\Lambda_G(f_1,f_2, \dots, f_n)=\int_{x^1, \dots, x^n \in X} \ \prod_{(i,j) \in {\mathcal E}(G)} K(x^i,x^j) \prod_{l=1}^n f_l(x^l) d\mu(x^l),\end{align*} $$ where ${\mathcal E}(G)$ is the edge set of G. Define $\Lambda _G(p_1, \ldots , p_n)$ as the smallest constant $C>0$ such that the inequality (0.1) $$ \begin{align} \Lambda_G(f_1, \dots, f_n) \leq C \prod_{i=1}^n {||f_i||}_{L^{p_i}(X, \mu)} \end{align} $$ holds for all nonnegative real-valued functions $f_i$ , $1\le i\le n$ , on X. The basic question is, how does the structure of G and the mapping properties of the operator $T_K$ influence the sharp exponents in (0.1). In this paper, this question is investigated mainly in the case $X={\Bbb F}_q^d$ , the d-dimensional vector space over the field with q elements, $K(x^i,x^j)$ is the indicator function of the sphere evaluated at $x^i-x^j$ , and connected graphs G with at most four vertices.

Performance-based plastic design of high-strength steel framed-tube structures fused with replaceable shear links

This study proposes high-strength steel framed-tube structures fused with replaceable shear links (HSS-FTSLs) to improve the seismic performance and post-earthquake reparability of traditional steel framed-tube structures. The conventional seismic design commonly uses the capacity design method, which primarily focuses on the ductility of the structures, resulting in localized damage concentrated on individual floors. An improved performance-based plastic design (I-PBPD) method is proposed for HSS-FTSLs based on the energy balance concept, target drifts, and preselected failure mechanisms. The proposed I-PBPD method considers the influence of post-yield stiffness of the structures. The seismic performance objectives for HSS-FTSLs are categorized into four levels, accompanied by the interstory drifts and residual interstory drifts corresponding to these objectives. The capacity curve for HSS-FTSLs is assumed as a tri-linear form to correspond with the failure modes. Three HSS-FTSL prototype structures with various heights are designed through the I-PBPD method. Nonlinear pushover and dynamic analyses were carried out to investigate their seismic performance. The results indicate that the HSS-FTSL structures designed using the I-PBPD method can achieve the expected failure modes and performance objectives under earthquakes of varying intensities, exhibiting exceptional seismic performance.

The Global and the Multilinear: Novelistic Forms for Planetary Processes

Abstract: Numerous scholars have argued that narrative multilinearity defines the contemporary novel's engagement with planetary processes ranging from globalization to ecological and migrant crises. This article seeks to develop and clarify the notion of multilinearity, adopting a narratological framework that distinguishes between three dimensions of multilinear novels—what I call their linkage, distribution, and focus. I discuss examples for each of these dimensions, examining their interaction and also investigating, from a broadly New Formalist perspective, how they speak to larger tensions, inherent in globalization, between cosmopolitan aspirations and a history of inequalities. In the final section, I turn to Hanya Yanagihara's novel To Paradise as a powerful illustration of how multilinear form is able to probe the complexity and moral murkiness of global processes, particularly when novelistic narrative resists the temptation of a closed form and instead embraces the instability of the multilinear.

The maximal subgroups of the exceptional groups F_{4}(q), E_{6}(q) and ^{2}!E_{6}(q) and related almost simple groups

This article produces a complete list of all maximal subgroups of the finite simple groups of type F_{4}, E_{6} and twisted E_{6} over all finite fields. Along the way, we determine the collection of Lie primitive almost simple subgroups of the corresponding algebraic groups. We give the stabilizers under the actions of outer automorphisms, from which one can obtain complete information about the maximal subgroups of all almost simple groups with socle one of these groups. We also provide a new maximal subgroup of ^{2}!F_{4}(8), correcting the maximal subgroups for that group from the list of Malle. This provides the first new exceptional groups of Lie type to have their maximal subgroups enumerated for three decades. The techniques are a mixture of algebraic groups, representation theory, computational algebra, and use of the trilinear form on the 27-dimensional minimal module for E_{6}. We provide a collection of supplementary Magma files that prove the author’s computational claims, yielding existence and the number of conjugacy classes of all maximal subgroups mentioned in the text.

Quantitative inverse theorem for Gowers uniformity norms and in

AbstractWe prove quantitative bounds for the inverse theorem for Gowers uniformity norms $\mathsf {U}^5$ and $\mathsf {U}^6$ in $\mathbb {F}_2^n$ . The proof starts from an earlier partial result of Gowers and the author which reduces the inverse problem to a study of algebraic properties of certain multilinear forms. The bulk of the work in this paper is a study of the relationship between the natural actions of $\operatorname {Sym}_4$ and $\operatorname {Sym}_5$ on the space of multilinear forms and the partition rank, using an algebraic version of regularity method. Along the way, we give a positive answer to a conjecture of Tidor about approximately symmetric multilinear forms in five variables, which is known to be false in the case of four variables. Finally, we discuss the possible generalization of the argument for $\mathsf {U}^k$ norms.

Centers of multilinear forms and applications

The center algebra of a general multilinear form is defined and investigated. We show that the center of a nondegenerate multilinear form is a finite dimensional commutative algebra, and center algebras can be effectively applied to direct sum decompositions of multilinear forms. As an application of the algebraic structure of centers, we show that almost all multilinear forms are absolutely indecomposable. The theory of centers can also be applied to symmetric equivalence of multilinear forms. Moreover, with a help of the results of symmetric equivalence, we are able to provide a linear algebraic proof for a well known Torelli type result which says that two complex homogeneous polynomials with the same Jacobian ideal are linearly equivalent.