Monotone Case Research Articles

Algebras of binary formulas for ℵ0-categorical weakly circularly minimal theories: monotonic case

This article concerns the notion of weak circular minimality being a variant of o-minimality for circularly ordered structures. Algebras of binary isolating formulas are studied for countably categorical weakly circularly minimal theories of convexity rank greater than 1 having both a 1-transitive non-primitive automorphism group and a non-trivial strictly monotonic function acting on the universe of a structure. On the basis of the study, the authors present a description of these algebras. It is shown that there exist both commutative and non-commutative algebras among these ones. A strict m-deterministicity of such algebras for some natural number m is also established.

Travelling waves and speed selection for the integrodifference pioneer-climax competition system

We present a general integrodifference pioneer-climax competition system and study the invasion dynamics of the system in both monotone and nonmonotone cases. For the parameter regions in which the system is monotone, we establish the existence of the spreading speed and travelling waves and show that the minimum wave speed is linearly selected if the overall dispersal of the invasive species is larger than that of the native one. For the parameter regions in which the system is nonmonotone, we estimate the spreading speed by constructing suitable auxiliary systems and prove the existence of travelling waves by the fixed point theorem and upper-lower solution method. We apply our theoretical results to an integrodiffference pioneer-climax system which is a spatial version of the model recently studied by Gilbertson and Kot [Theor. Ecol., 14(2021), 501-523].

Comparative numerical study to simulate masonry with bed joint reinforced repointing

Bed joint reinforced repointing is a retrofitting technique for unreinforced masonry structures that is commonly applied in the Netherlands to repair settlement-induced damage. Using this technique, the bed joints of masonry walls are reinforced with steel rebars that are embedded in a high strength repair mortar. Due to the increase of induced seismic events in the northern part of the Netherlands, an experimental study was carried out at Delft University of Technology to investigate the performance of this retrofitting technique for combined settlement and seismic loading. This paper aims to simulate the experimental results, with a focus on the comparison of different finite element modelling approaches for studying both un-strengthened and strengthened full-scale tested walls. To that end, three different models are investigated – comprising both macro (continuum) and simplified and detailed micro (brick-to-brick) modelling approaches. The bricks and mortar joints are modelled as one homogenous continuum in the macro model, whereas in the two brick-to-brick models these structural components are modelled separately, with the detailed model including interface elements to simulate the brick–mortar bonds. Nonlinear pushover analyses are subsequently carried out using all three modelling approaches, for both monotonic and cyclic loading cases. Based on these analyses, the detailed brick-to-brick model was found unsuitable to simulate the strengthened wall because cracks in the model mainly occur in the form of opening of the brick–mortar bond interfaces, while smeared cracking in the plane stress elements of the mortar joints is very limited. Similarly, the continuum damage model was found to be inaccurate when pre-existing damage in the experiment needed to be taken into account. The continuum damage model also showed lower axial stresses in the rebars, compared with the simplified brick-to-brick model, as the former does not allow for the direct assignment of material properties for the high strength repair mortar in the strengthened joints.

On the diagonal proximal point algorithms

We study the asymptotic behaviour of the diagonal proximal point algorithms governed by a sequence of maximally monotone operators. With a summability condition on the Brézis–Haraux functions of the sequence of operators, we obtain several ergodic and weak convergence results in the monotone and subgradient cases. This approach is of relevance to and motivated by Attouch et al. [Asymptotic behaviour of nonautonomous monotone and subgradient evolution equations. Trans Amer Math Soc. 2018;370:755–790]. Some strong convergence results are presented as well. Our results in this paper include discrete versions of the results in Attouch et al. [Asymptotic behaviour of nonautonomous monotone and subgradient evolution equations. Trans Amer Math Soc. 2018;370:755–790], and also unify and improve many existing results in Attouch et al. [Prox-penalization and splitting methods for constrained variational problems. SIAM J Optim. 2011;21:149–173], Cabot [Proximal point algorithm controlled by a slowly vanishing term: Applications to hierarchical minimization. SIAM J Optim. 2005;15:555–572], Lehdili and Moudafi [Combining the proximal algorithm and Tikhonov regularization. Optimization. 1996;37:239–252], Xu [A regularization method for the proximal point algorithm. J Glob Optim. 2006;36:115–125].

Positive solutions of the Gross–Pitaevskii equation for energy critical and supercritical nonlinearities

We consider positive and spatially decaying solutions to the following Gross–Pitaevskii equation with a harmonic potential: −Δu+|x|2u=ωu+|u|p−2uin Rd, where , p > 2 and ω > 0. For (energy-critical case) there exists a ground state u ω if and only if , where for d = 3 and for . We give a precise description on asymptotic behaviours of u ω as up to the leading order term for different values of . When (energy-supercritical case) there exists a singular solution for some . We compute the Morse index of in the class of radial functions and show that the Morse index of is infinite in the oscillatory case, is equal to 1 or 2 in the monotone case for p not large enough and is equal to 1 in the monotone case for p sufficiently large.

An analytical approach for the fracture characterization of concrete under cyclic loading

Many civil engineering infrastructures frequently encounter repetitive loading during their service life. Due to the inherent complexity observed in concrete, like quasi-brittle materials, understanding the fatigue behavior in concrete still poses a challenge. Moreover, the fracture process zone characteristics ahead of the crack tip have been observed to be different in fatigue loading than in the monotonic cases. Therefore, it is crucial to comprehend the energy dissipation associated with the fracture process zone (FPZ) due to repetitive loading. It is well known that energy dissipation due to cyclic loading provides a better understanding of the fracture behavior of concrete. Under repetitive load cycles, concrete members exhibit a two-stage energy dissipation process. Experimentally, it has been observed that the rate of energy dissipation decreases initially with an increase in crack length and subsequently increases. In this work, an attempt has been made to propose an analytical expression to predict energy dissipation as a function of crack length, and the fully developed fracture process zone length under cyclic loading condition.

A Unified Analysis of Variational Inequality Methods: Variance Reduction, Sampling, Quantization, and Coordinate Descent

In this paper, we present a unified analysis of methods for such a wide class of problems as variational inequalities, which includes minimization problems and saddle point problems. We develop our analysis on the modified Extra-Gradient method (the classic algorithm for variational inequalities) and consider the strongly monotone and monotone cases, which corresponds to strongly-convex-strongly-concave and convex-concave saddle point problems. The theoretical analysis is based on parametric assumptions about Extra-Gradient iterations. Therefore, it can serve as a strong basis for combining the already existing type methods and also for creating new algorithms. In particular, to show this we develop new robust methods, which include methods with quantization, coordinate methods, distributed randomized local methods, and others. Most of these approaches have never been considered in the generality of variational inequalities and have previously been used only for minimization problems. The robustness of the new methods is also confirmed by numerical experiments with GANs.

Iterative roots of two-dimensional mappings

AbstractAs a weak version of embedding flow, the problem of iterative roots is studied extensively in one dimension, especially in monotone case. There are few results in high dimensions because the constructive method dealing with monotone mappings is unavailable. In this paper, by introducing a kind of partial order, we define the monotonicity for two-dimensional mappings and then present some results on the existence of iterative roots for linear mappings, triangle-type mappings, and co-triangle-type mappings, respectively. Our theorems show that even the property of monotonicity for iterative roots of monotone mappings, which is a trivial result in one dimension, does not hold anymore in high dimensions. At the end of this paper, the problem of iterative roots for two well-known planar mappings, that is, Hénon mappings and coupled logistic mappings, are also discussed.

A Unified Analysis of Variational Inequality Methods: Variance Reduction, Sampling, Quantization, and Coordinate Descent

We present a unified analysis of methods for such a wide class of problems as variational inequalities, which include minimization and saddle point problems as special cases. The analysis is developed relying on the extragradient method, which is a classic technique for solving variational inequalities. We consider the monotone and strongly monotone cases, which correspond to convex-concave and strongly-convex-strongly-concave saddle point problems. The theoretical analysis is based on parametric assumptions about extragradient iterations. Therefore, it can serve as a strong basis for combining existing methods of various types and for creating new algorithms. Specifically, to show this, we develop new robust methods, including methods with quantization, coordinate methods, and distributed randomized local methods. Most of these approaches have never been considered in the generality of variational inequalities and have previously been used only for minimization problems. The robustness of the new methods is confirmed by numerical experiments with GANs.

Killed distribution dependent SDE for nonlinear Dirichlet problem

To characterize nonlinear Dirichlet problems in an open domain, we investigate killed distribution dependent SDEs. By constructing the coupling by projection and using the Zvonkin/Girsanov transforms, the well-posedness is proved for three different situations:1) monotone case with distribution dependent noise (possibly degenerate);2) singular case with non-degenerate distribution dependent noise;3) singular case with non-degenerate distribution independent noise. In the first two cases the domain is $ C^2 $ smooth such that the Lipschitz continuity in initial distributions is also derived, and in the last case the domain is arbitrary.

Partial-monotone adaptive submodular maximization

Many AI/Machine learning problems require adaptively selecting a sequence of items, each selected item might provide some feedback that is valuable for making better selections in the future, with the goal of maximizing an adaptive submodular function. Most of existing studies in this field focus on either monotone case or non-monotone case. Specifically, if the utility function is monotone and adaptive submodular, Golovin and Krause (J Artif Intell Res 42:427–486, 2011) developed \((1-1/e)\) approximation solution subject to a cardinality constraint. For the cardinality-constrained non-monotone case, Tang (Theor Comput Sci 850:249–261, 2021) showed that a random greedy policy attains an approximation ratio of 1/e. In this work, we generalize the above mentioned results by studying the partial-monotone adaptive submodular maximization problem. To this end, we introduce the notation of adaptive monotonicity ratio \(m\in [0,1]\) to measure the degree of monotonicity of a function. Our main result is to show that for the case of cardinality constraints, if the utility function has an adaptive monotonicity ratio of m and it is adaptive submodular, then a random greedy policy attains an approximation ratio of \(m(1-1/e)+(1-m)(1/e)\). Notably this result recovers the aforementioned \((1-1/e)\) and 1/e approximation ratios when \(m = 1\) and \(m = 0\), respectively. We further extend our results to consider a knapsack constraint and develop a \((m+1)/10\) approximation solution for this general case. One important implication of our results is that even for a non-monotone utility function, we still can attain an approximation ratio close to \((1-1/e)\) if this function is “close” to a monotone function. This leads to improved performance bounds for many machine learning applications whose utility functions are almost adaptive monotone.

Well-posedness and variational numerical scheme for an adaptive model in highly heterogeneous porous media

Mathematical modeling of fluid flow in a porous medium is usually described by a continuity equation and a chosen constitutive law. The latter, depending on the problem at hand, may be a nonlinear relation between the fluid's pressure gradient and velocity. The actual shape of this relation is normally chosen at the outset of the problem, even though, in practice, the fluid may experience velocities outside of its range of applicability. We propose here an adaptive model, so that the most appropriate law is locally selected depending on the computed velocity. From the analytical point of view, we show well-posedness of the problem when the law is monotone in velocity and show existence in one space dimension otherwise. From the computational point of view, we present a new approach based on regularizing via mollification the underlying dissipation, i.e., the power lost by the fluid to the porous medium through drag. The resulting regularization is shown to converge to the original problem using Γ-convergence on the dissipation in the monotone case. This approach gives rise to a variational numerical scheme which applies to very general problems and which we validate on three test cases.

Inhibitor-Induced Wavetrains and Spiral Waves in an Extended FitzHugh-Nagumo Model of Nerve Cell Dynamics.

It is well known that the FitzHugh-Nagumo model is one of the simplified forms of the four-variable Hodgkin-Huxley model that can reflect most of the significant phenomena of nerve cell action potential. However, this model cannot capture the irregular action potentials of sufficiently large periods in a one-parameter family of solutions. Motivated by this, we propose a modified FitzHugh-Nagumo reaction-diffusion system by changing its recovery kinetics. First, we investigate the parameter regime to know the existence of the wavetrains. Second, we conceive the occurrence of Eckhaus bifurcations of solutions that divide the solution region into two parts. The essential spectra at different grid points explore the occurrence of bifurcations of the waves. We find that the wavetrains of sufficiently large periods cross the stability boundary. This characteristic phenomenon is absent in the standard FitzHugh-Nagumo model. Finally, we observe a reasonable agreement between the direct PDE simulations and the solutions in the traveling wave ODEs. Furthermore, the model exhibits spiral wave for monotone and non-monotone cases that agrees with the waves observed in cellular activity.

Seismic behaviour of diaphragm-through bolted-welded joints between CFST column to steel beam

A new generation of through-diaphragm bolted-welded joints between narrow cross-section concrete filled steel tube column to H-beam were developed in this paper. In such joints, the welded connection is adopted in the upper flange of the beam, and the bolted connection is still used in bottom flange and web of the beam. The new structures of through-diaphragm bolted-welded joints were mainly used to solve the problems of reserved installation gaps and smaller joint stiffness of through-diaphragm bolted joints proposed by authors previously. To investigate the seismic behaviour of through-diaphragm bolted-welded joints, this paper presents the details of four new experiments. For comparison, both monotonic and hysteretic loading cases were considered in the experiments. The test results show that through-diaphragm bolted-welded joints have larger stiffness, which can be categorized as rigid joints, and they also have good ductility, satisfactory bearing capacity, stable and effective hysteretic behaviour and desirable energy dissipation capacity.

Morse index for the ground state in the energy supercritical Gross–Pitaevskii equation

The ground state of the energy super-critical Gross–Pitaevskii equation with a harmonic potential converges in the energy space to the singular solution in the limit of large amplitudes. The ground state can be represented by a solution curve which has either oscillatory or monotone behavior, depending on the dimension of the system and the power of the focusing nonlinearity. We address here the monotone case for the cubic nonlinearity in the spatial dimensions d≥13. By using the shooting method for the radial Schrödinger operators, we prove that the Morse index of the ground state is finite and is independent of the (large) amplitude. We also prove that it is equal to the Morse index of the limiting singular solution, which can be computed from numerical approximations. The numerical results suggest that the Morse index of the ground state is equal to one for every d≥13.

Erdős–Szekeres theorem for multidimensional arrays

The classical Erdős–Szekeres theorem dating back almost a hundred years states that any sequence of (n-1)^2 + 1 distinct real numbers contains a monotone subsequence of length n . This theorem has been generalised to higher dimensions in a variety of ways but perhaps the most natural one was proposed by Fishburn and Graham more than 25 years ago. They defined the concept of a monotone and a lex-monotone array and asked how large an array one needs in order to be able to find a monotone or a lex-monotone subarray of size n \times \cdots \times n . Fishburn and Graham obtained Ackerman-type bounds in both cases. We significantly improve these results. Regardless of the dimension we obtain at most a triple exponential bound in n in the monotone case and a quadruple exponential one in the lex-monotone case.

Sample greedy based task allocation for multiple robot systems

This paper addresses in-schedule dependent task allocation problems for multi-robot systems. One of the main issues with those problems is the inherent NP-hardness of combinatorial optimisation. To handle this issue, this paper develops a decentralised task allocation algorithm by leveraging the submodularity concept and a sampling process of task sets. Our theoretical analysis reveals that the proposed algorithm can provide an approximation guarantee of 1/2 of the optimal solution for the monotone submodular case and 1/4 for the non-monotone submodular case, both with polynomial time complexity. To examine the performance of the proposed algorithm and validate the theoretical analysis, we introduce two task allocation scenarios and perform numerical simulations. The simulation results confirm that the proposed algorithm achieves a solution quality which is comparable to state-of-the-art algorithms in the monotone case and much better quality in the non-monotone case with significantly lower computational complexity.

Fréchet fuzzy metric

The aim of this note is to introduce a fuzzy counterpart of the Fréchet distance between curves. We consider both monotonic and non-monotonic case.

Effect of 0.5% CNT reinforcement of a glass fiber composite on strength and cyclic damage induced by transverse and out-of-plane compressive loads

The impact of the addition of 0.5 wt% multiwall carbon nanotubes (MWCNT) on the quasi-static transverse compressive resistance and damage evolution of unidirectional glass fiber reinforced composite (FRC) fabricated by resin transfer molding is investigated. The degradation of macroscopic mechanical properties such as elastic modulus and compressive strength vs. plastic strains are correlated with crack density and microcrack propagation behavior including analysis of failure phenomena such as interface debonding, matrix cracking and interface cracking. The experimental campaign includes compression tests with monotonic and cyclic loads on cubic specimens in the out-of-plane (0°) and transverse (90°) directions for masterbatch MWCNT-based and CNT-free glass FRC. In the monotonic case, the elastic modulus and maximum compressive strength increased for the glass FRC with 0.5 wt% CNTs, particularly in the transverse (0°) direction by 11% and 13%, respectively. Under cyclic loading, the elastic modulus degradation stagnates at a lower plastic deformation for composite with 0.5 wt% CNTs. In addition, CNTs reduces by 20% the cumulative plastic deformation in the 0° direction and almost 30% in the 90° direction. At the microscopic scale, image analysis showed that the CNTs improved the properties of the interface by delaying decohesion failure and reducing by around 30% and 40% the computed crack density in the 0° and 90° loading direction, respectively. In addition, image analysis in the out-of-plane direction (0°) shows that fiber-matrix decohesion is predominant before crack propagation at the interlaminar level and prior to final failure. By contrast, at 90° loadings, no decohesion is observed and crack propagation is almost purely interlaminar. Finally, the addition of 0.5% MWCNTs to the glass FRC increases the mechanical resistance of the composite material in the transverse direction demonstrated by the delaying of cracks during the failure analysis of different cycles as well as the decrease of the elastic modulus degradation and increase of compressive strength.

Plasticity evolution of an aluminum-magnesium alloy under abrupt strain path changes

During the forming and manufacturing of engineering materials, plasticity behavior could be evolving significantly due to complex deformation history. Therefore, this study aims to characterize the plasticity evolution of an aluminum-magnesium alloy under simple monotonic and non-monotonic loading with abrupt strain path changes. Instead of focusing only on one single stress state in the first-step loading for most of the studies in the literature, the current non-monotonic strain path testing program investigates three stress states – uniaxial, plane-strain, and biaxial tension – in the first-step loading and combines them with a second-step uniaxial loading along and orthogonal to the initial loading direction. This combination generates non-monotonic stress–strain data in a quite large and distributed spectrum in terms of the Schmitt parameter. It is found that the aluminum-magnesium alloy shows a unique phenomenon with a lower yield strength at reloading compared to monotonic cases coupled with a steady increase of stress overshooting the monotonic one at large strains. This increase of stress as well as the strain hardening rate lasts till the uniform strain and is therefore referred to as permanent hardening. The comprehensive non-monotonic behavior delivered by the new experimental program in this study could further assist the development of material models and an in-depth understanding of the underlying mechanisms.