Trivial Set Research Articles

Bifurcation for indefinite‐weighted p$p$‐Laplacian problems with slightly subcritical nonlinearity

AbstractWe study a superlinear elliptic boundary value problem involving the ‐Laplacian operator, with changing sign weights. The problem has positive solutions bifurcating from the trivial solution set at the two principal eigenvalues of the corresponding linear weighted boundary value problem.Drabek's bifurcation result applies when the nonlinearity is of power growth. We extend Drabek's bifurcation result to slightly subcritical nonlinearities. Compactness in this setting is a delicate issue obtained via Orlicz spaces.

Real-space renormalization-group treatment of quadratic chains

We have recently proposed a one-dimensional nonperiodic chain with lattice positions at 02 d, 12 d, 22 d, ... with length d a constant. The spectrum is singular-continuous, and for weak potential, the states are all extended apart from a trivial set of localized states. In this study, we obtain the exact extended-state spectrum of the quadratic chain in a nearest-neighbor tight-binding model where the quadratic modulation is in the onsite matrix elements. Then, a real-space renormalization-group method (RSRG) is used by decimation to reduce the transfer matrix for the chain into self-similar matrix products. The RSRG decimation scheme is used here to organize the calculation and facilitate numerical computation. The extended-state spectrum appears as minibands broken by numerous gaps. Previous work on quadratic chains shows that the structure factor is singular-continuous and given by a dense set of states with wavevectors with scaling exponent γ(k) = 2 as in periodic and quasi-periodic chains. The origin of extended states in this nonperiodic lattice appears to arise from a type of mechanism not yet identified in deterministic nonperiodic lattices, and is based on a hidden symmetry giving rise to an energy-dependent translational invariance of the transfer matrix.

How hysteresis produces discontinuous patterns in degenerate reaction–diffusion systems

In this paper, we study the asymptotic behaviour of the solutions to a degenerate reaction–diffusion system. This system admits a continuum of discontinuous stationary solutions due to the effect of a hysteresis process, but only one discontinuous stationary solution is compatible with a principle of preservation of locally invariant regions. Using a macroscopic mass effect which guarantees that fast particles help slow particles to displace, we establish a novel result of convergence of a non trivial set of trajectories towards a discontinuous pattern.

On the universality and isotopy-isomorphy of (r,s,t)-inverse quasigroups and loops with applications to cryptography

This paper introduced a condition called $\mathcal{R}$-condition under which $(r,s,t)$-inverse quasigroups are universal. Middle isotopic $(r,s,t)$-inverse loops, satisfying the $\mathcal{R}$-condition and possessing a trivial set of $r$-weak inverse permutations were shown to be isomorphic; isotopy-isomorphy for $(r,s,t)$-inverse loops. Isotopy-isomorphy for $(r,s,t)$-inverse loops was generally characterized. With the $\mathcal{R}$-condition, it was shown that for positive integers $r$, $s$ and $t$, if there is a $(r,s,t)$-inverse quasigroup of order $3k$ with an inverse-cycle of length $gcd(k,r+s+t)>1$, then there exists an $(r,s,t)$-inverse quasigroup of order $3k$ with an inverse-cycle of length $gcd\big(k(r+s+t), (r+s+t)^2\big)$. The procedure of application of such $(r,s,t)$-inverse quasigroups to cryptography was described and explained, while the feasibility of such $(r,s,t)$-inverse quasigroups was illustrated with sample values of $k,r,s$ and $t$.

Geometric T-Duality: Buscher Rules in General Topology

The classical Buscher rules d escribe T-duality for metrics and B-fields in a topologically trivial setting. On the other hand, topological T-duality addresses aspects of non-trivial topology while neglecting metrics and B-fields. In this article, we develop a new unifying framework for both aspects.

On interrelations between trivial and nontrivial basic sets of structurally stable diffeomorphisms of surfaces

The article is devoted to interrelations between an existence of trivial and nontrivial basic sets of A-diffeomorphisms of surfaces. We prove that if all trivial basic sets of a structurally stable diffeomorphism of surface M2 are source periodic points α1,…αk, then the non-wandering set of this diffeomorphism consists of points α1,…,αk and exactly one one-dimensional attractor Λ. We give some sufficient conditions for attractor Λ to be widely situated. Also, we prove that if a non-wandering set of a structurally stable diffeomorphism contains a nontrivial zero-dimensional basic set, then it also contains source and sink periodic points.

Probabilistic Analysis for Remaining Useful Life Prediction and Reliability Assessment

Although the importance of remaining useful life (RUL) prediction is widely recognized in industries, its implementation in real scenarios is highly restricted by the complexity of the degradation mechanism, uncertainty of machinery, and insufficiency of prior knowledge. To address such a challenge, this article proposes a model-based framework, which has the capability to integrate multiple predictive models via a probabilistic mechanism. When a new observation is fed into each predictive model, the posterior distribution of each model will be updated via Bayesian inference. Then, a grid-sampling strategy is applied to their posterior distributions for identifying the “peak” and “profile,” which are used for RUL prediction and reliability assessment, respectively. The effectiveness of this framework is validated with the experiments on a set of steel tension specimens. Theoretical interpretations and comparative studies demonstrate the superiority of the proposed framework. Besides, the proposed framework can not only reduce human workload on trivial parameter setting but also be effective with insufficient prior knowledge, making the intelligent RUL prediction easier.

Any closed 3-manifold supports A-flows with 2-dimensional expanding attractors

Abstract We prove that given any closed 3-manifold M 3, there is an A-flow f t on M 3 such that the non-wandering set NW (f t ) consists of 2-dimensional non-orientable expanding attractor and trivial basic sets.

Optimization for Medical Image Segmentation: Theory and Practice When Evaluating With Dice Score or Jaccard Index.

In many medical imaging and classical computer vision tasks, the Dice score and Jaccard index are used to evaluate the segmentation performance. Despite the existence and great empirical success of metric-sensitive losses, i.e. relaxations of these metrics such as soft Dice, soft Jaccard and Lovász-Softmax, many researchers still use per-pixel losses, such as (weighted) cross-entropy to train CNNs for segmentation. Therefore, the target metric is in many cases not directly optimized. We investigate from a theoretical perspective, the relation within the group of metric-sensitive loss functions and question the existence of an optimal weighting scheme for weighted cross-entropy to optimize the Dice score and Jaccard index at test time. We find that the Dice score and Jaccard index approximate each other relatively and absolutely, but we find no such approximation for a weighted Hamming similarity. For the Tversky loss, the approximation gets monotonically worse when deviating from the trivial weight setting where soft Tversky equals soft Dice. We verify these results empirically in an extensive validation on six medical segmentation tasks and can confirm that metric-sensitive losses are superior to cross-entropy based loss functions in case of evaluation with Dice Score or Jaccard Index. This further holds in a multi-class setting, and across different object sizes and foreground/background ratios. These results encourage a wider adoption of metric-sensitive loss functions for medical segmentation tasks where the performance measure of interest is the Dice score or Jaccard index.

On Two-Dimensional Expanding Attractors of A-Flows

We prove that given any closed $n$-manifold $M^n$, $n\geq 4$, there is an A-flow $f^t$ on $M^n$ such that the non-wandering set $NW(f^t)$ consists of 2-dimensional expanding attractor (the both, orientable and non-orientable) and trivial basic sets. For 3-manifolds, we prove that given any closed 3-manifold $M^3$, there is an A-flow $f^t$ on $M^3$ such that the non-wandering set $NW(f^t)$ consists of a non-orientable 2-dimensional expanding attractor and trivial basic sets. Moreover, there is a nonsingular A-flow $f^t$ on a 3-sphere such that the non-wandering set $NW(f^t)$ consists of an orientable 2-dimensional expanding attractor and trivial basic sets (isolated periodic trajectories).

Dynamics of a Discrete Nonlinear Prey–Predator Model

The dynamics of a discrete nonlinear prey–predator model is studied. The local dynamical results are obtained on asymptotic properties of fixed points and Neimark–Sacker bifurcations. Then global dynamics is studied by finding invariant sets. Also some achievements on attraction are shown for certain trivial invariant sets including a shadowing type result, and estimates on boundedness of orbits. Some ergodic type results are also derived. Certain issues are extended to systems with impulses by showing the influence of impulses on dynamics. Moreover, backward dynamics is investigated as well. All these results are derived analytically and numerical computations are presented to support them.

Computing from projections of random points

We study the sets that are computable from both halves of some (Martin–Löf) random sequence, which we call [Formula: see text]-bases. We show that the collection of such sets forms an ideal in the Turing degrees that is generated by its c.e. elements. It is a proper subideal of the [Formula: see text]-trivial sets. We characterize [Formula: see text]-bases as the sets computable from both halves of Chaitin’s [Formula: see text], and as the sets that obey the cost function [Formula: see text]. Generalizing these results yields a dense hierarchy of subideals in the [Formula: see text]-trivial degrees: For [Formula: see text], let [Formula: see text] be the collection of sets that are below any [Formula: see text] out of [Formula: see text] columns of some random sequence. As before, this is an ideal generated by its c.e. elements and the random sequence in the definition can always be taken to be [Formula: see text]. Furthermore, the corresponding cost function characterization reveals that [Formula: see text] is independent of the particular representation of the rational [Formula: see text], and that [Formula: see text] is properly contained in [Formula: see text] for rational numbers [Formula: see text]. These results are proved using a generalization of the Loomis–Whitney inequality, which bounds the measure of an open set in terms of the measures of its projections. The generality allows us to analyze arbitrary families of orthogonal projections. As it turns out, these do not give us new subideals of the [Formula: see text]-trivial sets; we can calculate from the family which [Formula: see text] it characterizes. We finish by studying the union of [Formula: see text] for [Formula: see text]; we prove that this ideal consists of the sets that are robustly computable from some random sequence. This class was previously studied by Hirschfeldt [D. R. Hirschfeldt, C. G. Jockusch, R. Kuyper and P. E. Schupp, Coarse reducibility and algorithmic randomness, J. Symbolic Logic 81(3) (2016) 1028–1046], who showed that it is a proper subclass of the [Formula: see text]-trivial sets. We prove that all such sets are robustly computable from [Formula: see text], and that they form a proper subideal of the sets computable from every (weakly) LR-hard random sequence. We also show that the ideal cannot be characterized by a cost function, giving the first such example of a [Formula: see text] subideal of the [Formula: see text]-trivial sets.

Performance optimization of LQR‐based PID controller for DC‐DC buck converter via iterative‐learning‐tuning of state‐weighting matrix

AbstractThis paper presents a numerically optimized linear‐quadratic‐regulator–based tuning mechanism for a ubiquitous proportional‐integral‐derivative controller to improve the output‐voltage regulation capability of a direct‐current (DC)‐DC buck converter. The linear‐quadratic‐regulator minimizes the quadratic cost of variations in the control signal and error‐dynamics of output‐voltage to provide a trivial set of optimized proportional‐integral‐derivative controller gains in the form of the state‐feedback gain vector. In order to further improve the controller's time‐domain performance and its disturbance‐rejection capability against load‐transients and input‐fluctuations, an iterative‐learning‐tuning mechanism is adopted to optimize the state‐weighting matrix of the linear‐quadratic cost function. The proposed optimization mechanism iteratively converges in the direction of the steepest gradient‐descent of another performance index that directly captures the transient response characteristics, and thus, optimally selects the weighting matrix to achieve the desired natural frequency and damping ratio of the closed‐loop system. Credible hardware‐in‐the‐loop experiments are conducted on a low‐power DC‐DC buck converter circuit to validate the aforementioned propositions.

Improvements to exact Boltzmann sampling using probabilistic divide-and-conquer and the recursive method

Abstract We demonstrate an approach for exact sampling of certain discrete combinatorial distributions, which is a hybrid of exact Boltzmann sampling and the recursive method, using probabilistic divide-and-conquer (PDC). The approach specializes to exact Boltzmann sampling in the trivial setting, and specializes to PDC deterministic second half in the first non-trivial application. A large class of examples is given for which this method broadly applies, and several examples are worked out explicitly.

Improvements to exact Boltzmann sampling using probabilistic divide-and-conquer and the recursive method

We demonstrate an approach for exact sampling of certain discrete combinatorial distributions, which is a hybrid of exact Boltzmann sampling and the recursive method, using probabilistic divide-and-conquer (PDC). The approach specializes to exact Boltzmann sampling in the trivial setting, and specializes to PDC deterministic second half in the first non-trivial application. A large class of examples is given for which this method broadly applies, and several examples are worked out explicitly.

Continuous higher randomness

We investigate the role of continuous reductions and continuous relativization in the context of higher randomness. We define a higher analogue of Turing reducibility and show that it interacts well with higher randomness, for example with respect to van Lambalgen’s theorem and the Miller–Yu/Levin theorem. We study lowness for continuous relativization of randomness, and show the equivalence of the higher analogues of the different characterizations of lowness for Martin-Löf randomness. We also characterize computing higher [Formula: see text]-trivial sets by higher random sequences. We give a separation between higher notions of randomness, in particular between higher weak 2-randomness and [Formula: see text]-randomness. To do so we investigate classes of functions computable from Kleene’s [Formula: see text] based on strong forms of the higher limit lemma.

Substructure decoupling as the identification of a set of disconnection forces

The objective of this paper is to achieve a better understanding of the substructure decoupling problem by analyzing different sets of disconnection forces. Substructure decoupling consists in the identification of a dynamic model of an unknown substructure embedded in a known structure (coupled structure). Disconnection forces can be imagined to be applied to the coupled structure in order to reproduce the dynamic behavior of the substructure to be identified. The disconnection forces are such as to cancel the effect of constraint forces exerted at the coupling DoFs of the unknown substructure by the remaining part of the structural system. Several sets of disconnection forces can be devised: the trivial set, consisting of disconnection forces acting at the coupling DoFs and opposite to the constraint forces; non trivial sets of disconnection forces acting at different DoFs but such as to cancel the effect of the constraint forces. Here, a procedure to determine disconnection forces is recalled in the framework of substructure decoupling. Different sets of disconnection forces are compared using both simulated and experimental data of a tree structure (known structure). Indications provided by disconnection forces are in agreement with the results of the decoupling procedure.

Image denoising via adaptive eigenvectors of graph Laplacian

An image denoising method via adaptive eigenvectors of graph Laplacian (EGL) is proposed. Unlike the trivial parameter setting of the used eigenvectors in the traditional EGL method, in our method, the eigenvectors are adaptively selected in the whole denoising procedure. In detail, a rough image is first built with the eigenvectors from the noisy image, where the eigenvectors are selected by using the deviation estimation of the clean image. Subsequently, a guided image is effectively restored with a weighted average of the noisy and rough images. In this operation, the average coefficient is adaptively obtained to set the deviation of the guided image to approximately that of the clean image. Finally, the denoised image is achieved by a group-sparse model with the pattern from the guided image, where the eigenvectors are chosen in the error control of the noise deviation. Moreover, a modified group orthogonal matching pursuit algorithm is developed to efficiently solve the above group sparse model. The experiments show that our method not only improves the practicality of the EGL methods with the dependence reduction of the parameter setting, but also can outperform some well-developed denoising methods, especially for noise with large deviations.

Coherent randomness tests and computing the $K$-trivial sets

We introduce Oberwolfach randomness, a notion within Demuth's framework of statistical tests with moving components; here the components' movement has to be coherent across levels. We show that a ML-random set computes all K -trivial sets if and only if it is not Oberwolfach random, and indeed that there is a K -trivial set which is not computable from any Oberwolfach random set. We show that Oberwolfach random sets satisfy effective versions of almost-everywhere theorems of analysis, such as the Lebesgue density theorem and Doob's martingale convergence theorem. We also show that random sets which are not Oberwolfach random satisfy highness properties (such as LR-hardness) which mean they are close to computing the halting problem. A consequence of these results is that a ML-random set failing the effective version of Lebesgue's density theorem for closed sets must compute all K -trivial sets. Combined with a recent result by Day and Miller, this gives a positive solution to the ML-covering problem of algorithmic randomness. On the other hand these results settle stronger variants of the covering problem in the negative: no low ML-random set computes all K -trivial sets, and not every K -trivial set is computable from both halves of a random set.

Lightweight, self-tuning data dissemination for dense nanonetworks

A nanonetwork comprises a high number of autonomous nodes with wireless connectivity, assembled at micro-to-nanoscale. In general, manufacturing and cost considerations imply that nanonetworking approaches should have minimal complexity, ideally without sacrifices in network coverage. The present paper studies a networking approach fit for static, dense topologies comprising numerous, identical, computationally-constrained nodes. These attributes are especially important in the context of recently proposed applications of nanonetworks. The presented networking approach assumes that each node is equipped with 10 bits of reclaimable storage to accommodate four integer counters, and a trivial set of integer operations on them. These modest resources are used for logging packet reception statistics. Nanonodes with good reception serve as retransmitters within the network. This classification process is based on the Misra–Gries algorithm, used for detecting frequent items into sequential streams. Evaluation via extensive simulations in various 2D and 3D topologies yields high network coverage, achieved with less resources than related approaches.