We study Cheeger and \(p\)-eigenvalue partition problems depending on a given evaluation function \(\Phi\) for \(p\in1,\infty)\). We prove existence and regularity of minima, relations among the ...problems, convergence, and stability with respect to \(p\) and to \(\Phi\).
Summary
This work presents a generalized substructuring‐based topology optimization method for the design hierarchical lattice structures to maximize the first eigenvalue. In this method, the ...macrostructure is assumed to be composed of substructures with a common artificial lattice geometry pattern. And two different yet connected scales are considered through a lattice geometry feature parameter. The feature parameter, which can control the material distribution of the substructure, determines the relative density of corresponding substructure. Each substructure is condensed into a super‐element to obtain the associated density‐related matrices. A surrogate model using cubic spline interpolation has been particularly built to map the density to stiffness and mass matrices of condensed super‐elements. The derivatives of super‐element matrices to the associated densities can be evaluated efficiently and accurately. Here, an augmented penalized density for this surrogate model is introduced. And the conventional optimality criteria method is selected as updating method of the density design variables. Numerical examples under two lattice patterns of substructures are shown to validate the correctness and superiority of this substructure‐based topology optimization method.
In this note, we address an inconsistency in our recent article (Hakimi-Moghaddam and Ferrante, 2021). In particular, we provide an example showing that Lemma III.1 of that article is not correct. We ...show how the statement of this lemma can be modified in order to resolve this inconsistency. We also describe how this modification impact the other results of the article. Notwithstanding the fact that Lemma III.1 plays a central role in the proof of the main result of (Hakimi-Moghaddam and Ferrante, 2021), we show that such a result is indeed correct as it may be derived by using the modified (and correct) version of the lemma. An illustrative example is provided to support the results.
In this note, the problem of second-order leader-following consensus by a novel distributed event-triggered sampling scheme in which agents exchange information via a limited communication medium is ...studied. Event-based distributed sampling rules are designed, where each agent decides when to measure its own state value and requests its neighbor agents broadcast their state values across the network when a locally-computed measurement error exceeds a state-dependent threshold. For the case of fixed topology, a necessary and sufficient condition is established. For the case of switching topology, a sufficient condition is obtained under the assumption that the time-varying directed graph is uniformly jointly connected. It is shown that the inter-event intervals are lower bounded by a strictly positive constant, which excludes the Zeno-behavior before the consensus is achieved. Numerical simulation examples are provided to demonstrate the correctness of theoretical results.
Nonlinear frequency division multiplexing (NFDM) system is an optional candidate to overcome the fiber nonlinearity limit. A full-spectrum modulated NFDM system, modulating data on combined ...continuous spectrum (CS) and discrete spectrum (DS) together, was proposed in recent years to improve the data rates and spectral efficiency (SE) by exploiting all the degrees of freedom offered in the nonlinear spectrum. However, the selection of discrete eigenvalues greatly affects the performance of both CS and DS. Designing appropriate eigenvalues of DS is an important issue to ensure the high SE and excellent performance of the system. In this paper, we discussed the selcection principle of eigenvalues and analyzed it from multiple perspectives, 11 eigenvalues with 64-quadrature amplitude modulation (64QAM) are selected for DS. Besides optimizing the eigenvalues at the transmitter, the linear minimum mean-square estimate (LMMSE) method was used at the receiver to furtherly improve the performance of DS. Through the numerical simulation, a 113 Gb/s (SE of 2.8 bits/s/Hz) full-spectrum modulated NFDM system was set up and transmitted 1120 km distance, where the Q-factors of both CS and DS are above the hard-decision forward error correction (HD-FEC) threshold. The results provide a way to design an efficiently full-spectrum modulated NFDM system.
For α∈(0,π), let Uα denote the infinite planar sector of opening 2α,
Uα={(x1,x2)∈R2:|arg(x1+ix2)|<α},and Tαγ be the Laplacian in L2(Uα), Tαγu=−Δu, with the Robin boundary condition ∂νu=γu, where ∂ν ...stands for the outer normal derivative and γ>0. The essential spectrum of Tαγ does not depend on the angle α and equals −γ2,+∞), and the discrete spectrum is non‐empty if and only if α<π2. In this case we show that the discrete spectrum is always finite and that each individual eigenvalue is a continous strictly increasing function of the angle α. In particular, there is just one discrete eigenvalue for α≥π6. As α approaches 0, the number of discrete eigenvalues becomes arbitrary large and is minorated by κ/α with a suitable κ>0, and the nth eigenvalue En(Tαγ) of Tαγ behaves as
En(Tαγ)=−γ2(2n−1)2α2+O(1)and admits a full asymptotic expansion in powers of α2. The eigenfunctions are exponentially localized near the origin. The results are also applied to δ‐interactions on star graphs.
We introduce a Laplacian and a signless Laplacian for the distance matrix of a connected graph, called the distance Laplacian and distance signless Laplacian, respectively. We show the equivalence ...between the distance signless Laplacian, distance Laplacian and the distance spectra for the class of transmission regular graphs. There is also an equivalence between the Laplacian spectrum and the distance Laplacian spectrum of any connected graph of diameter 2. Similarities between n, as a distance Laplacian eigenvalue, and the algebraic connectivity are established.
Consider a Gaussian vector z = (x′, y′)′, consisting of two sub-vectors x and y with dimensions p and q, respectively. With n independent observations of z, we study the correlation between x and y, ...from the perspective of the canonical correlation analysis. We investigate the high-dimensional case: both p and q are proportional to the sample size n. Denote by Σ
uv
the population cross-covariance matrix of random vectors u and v, and denote by Suv
the sample counterpart. The canonical correlation coefficients between x and y are known as the square roots of the nonzero eigenvalues of the canonical correlation matrix
Σ
x
x
−
1
Σ
x
y
Σ
y
y
−
1
Σ
y
x
. In this paper, we focus on the case that Σ
xy
is of finite rank k, that is, there are k nonzero canonical correlation coefficients, whose squares are denoted by r₁ ≥ ··· ≥ rk
> 0. We study the sample counterparts of ri, i = 1, ..., k, that is, the largest k eigenvalues of the sample canonical correlation matrix
S
x
x
−
1
S
x
y
S
y
y
−
1
S
y
x
, denoted by λ₁ ≥ ··· ≥ λk
. We show that there exists a threshold rc
∈ (0, 1), such that for each i ∈ {1, ..., k}, when ri
≤ rc, λi
converges almost surely to the right edge of the limiting spectral distribution of the sample canonical correlation matrix, denoted by d
+. When ri
> rc, λi
possesses an almost sure limit in (d
+, 1, from which we can recover ri
’s in turn, thus provide an estimate of the latter in the high-dimensional scenario. We also obtain the limiting distribution of λi
’s under appropriate normalization. Specifically, λi
possesses Gaussian type fluctuation if ri
> rc
, and follows Tracy–Widom distribution if ri
< rc
. Some applications of our results are also discussed.
The solutions of a forced gyroscopic system of ODEs may undergo large oscillations whenever some eigenvalues of the corresponding quadratic eigenvalue problem (QEP) (λ2M+λG+K)v=0,0≠v∈ℂn, are close to ...the frequency of the external force (both M,K are symmetric, M is positive definite, K is definite and G is skew-symmetric). This is the phenomenon of the so-called resonance. One way to avoid resonance is to modify some (or all) of the coefficient matrices, M, G, and K∈Rn×n in such a way that the new system has no eigenvalues close to these frequencies. This is known as the frequency isolation problem. In this paper we present frequency isolation algorithms for tridiagonal systems in which only the gyroscopic term G is modified. To derive these algorithms, the real gyroscopic QEP is first transformed into a complex hyperbolic one, which allows to translate many of the ideas in Moro and Egaña (2016) for undamped systems into the full quadratic framework. Some numerical experiments are presented.
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