Monotone Continuous Function Research Articles

Certain Inequalities Related to the Generalized Numeric Range and Numeric Radius That Are Associated with Convex Functions

In this paper, we delve into the intricate connections between the numerical ranges of specific operators and their transformations using a convex function. Furthermore, we derive inequalities related to the numerical radius. These relationships and inequalities are built upon well-established principles of convexity, which are applicable to non-negative real numbers and operator inequalities. To be more precise, our investigation yields the following outcome: consider the operators A and B, both of which are positive and have spectra within the interval m,M, denoted as σA and σB. In addition, let us introduce two monotone continuous functions, namely, g and h, defined on the interval m,M. Let f be a positive, increasing, convex function possessing a supermultiplicative property, which means that for all real numbers t and s, we have fts≤ftfs. Under these specified conditions, we establish the following inequality: for all 0≤ν≤1, this outcome highlights the intricate relationship between the numerical range of the expression gνAXh1−ν when transformed by the convex function f and the norm of X. Importantly, this inequality holds true for a broad range of values of ν. Furthermore, we provide supportive examples to validate these results.

On application of Gaussian kernels and Laplace functions combined with Kolmogorov's theorem for approximation of functions of several variables

A special class of approximations of measurable functions of several variables on the unit coordinate cube is investigated. The class is constructed on the base of Kolmogorov's theorem (in version by Sprecher–Golubkov) stating that a continuous function $f$ of several variables can be represented as a finite superposition of continuous single-variable functions — so called outer functions (which depend of $f$) and inner one $\Psi$ (which is independent of $f$ and is monotone). In the case of continuous functions $f$ the class under study is obtained with outer functions approximated by linear combinations of quadratic exponentials (also known as Gaussian functions or Gaussian kernels) and with the inner function $\Psi$ approximated by Laplace functions. As is known, a measurable function $f$ can be approximated by a continuous one (up to a set of small measure) with the help of classical Luzin's theorem. The effectiveness of such approach is based on assertions that, firstly, the Mexican hat mother wavelet on any fixed bounded interval can be approximated as accurately as desired by a linear combination of two Gaussian functions, and, secondly, that a continuous monotone function on such an interval can be approximated as accurately as desired by a linear combination of translations and dilations of the Laplace integral (in other words, Laplace functions). It is proved that the class of approximations under study is dense everywhere in the class of continuous multivariable functions on the coordinate cube. For the case of continuous and piecewise continuous functions of two variables, numerical results are presented that confirm the effectiveness of approximations of the studied class.

Supremum Distribution of Weighted Sum of Random Processes from Orlicz Spaces of Exponential Type with Continuous Deviation

The paper studies distribution of sum of random processes from Orlicz spaces of exponential type weighted by continuous functions, in particular, processes from spaces Subϕ (Ω), SSubϕ (Ω) and class V (ϕ, ψ) are considered. Such spaces and classes of random variables and corresponding stochastic processes provide generalizations of Gaussian and sub-Gaussian random variables and processes and are important for various applications, for example, in queuing theory and financial mathematics. We derive the estimates for the distribution of supremum of weighted sum of such processes deviated by a continuous monotone function using the entropy method. As examples, weighted sum of Wiener and weighted sum of fractional Brownian motion processes with different Hurst indices from classes V (ϕ, ψ) are considered. Corresponding estimates of the probability of exceeding by trajectories of such weighted sums a positive level determined by a linear function are obtained. In the insurance risk theory, such aproblem arises during estimating a ruin probability of the corresponding risk process with a constant premium income, and in the communications theory, it appears for the buffer overflow probability for a single server with a constant service rate.

On Properties of Continuous Monotone Functions

On Properties of Continuous Monotone Functions

On monotone approximation of piecewise continuous monotone functions with the help of translations and dilations of the Laplace integral

For piecewise continuous monotone functions defined on a bounded interval $[-b;b]$, a monotone smooth approximation $Q(x)$ of any prescribed accuracy in the metric of the space $\mathbf{C}(\Pi)$ with as small as desired measure of the difference $[-b;b]\setminus\Pi$, $\Pi\subset[-b;b]$, is constructed using translations and dilations of the Laplace function (integral). In fact, this extends to the case of piecewise continuous monotone functions the result (obtained by the author formerly) on arbitrarily exact in the metric of the space $\mathbf{C}[-b;b]$ monotone approximation of continuous monotone functions with the help of translations and dilations of the Laplace integral. Besides, we suggest a new way of approximation in the form of linear combination of translations and dilations of the Laplace integral. Finally, we give and discuss concrete numerical examples of using approximation ways under study for a piecewise constant (stepwise) monotone function and for a piecewise continuous monotone function. Here, we also compare the results obtained for two discussed ways of approximation.

Properties and Algorithm of Lattice Pseudo-Submodular Functions

In this paper, we propose a concept of a lattice pseudo-submodular (LPS) function and consider maximizing a monotone continuous real LPS function [Formula: see text] under a convex polytope constraint. The concept of LPS function was proposed to describe the properties of some discrete functions or nonconvex continuous functions. It is a generalization of the lattice submodular function. For the real LPS maximization problem, we design the monotone Pseudo Frank-Wolfe (PFW) algorithm by taking advantage of the second derivative bound. The PFW algorithm iterates by constantly optimize linear gradient function [Formula: see text], and finally outputs the solution. We theoretically prove that PFK algorithm has an approximation ratio of [Formula: see text] (where [Formula: see text]), and it needs at least [Formula: see text] rounds (where [Formula: see text] is a parameter given in advance). The PFW algorithm is also useful for multilinear extension of discrete lattice pseudo-submodular maximization problems.

Online risk-averse submodular maximization

We present a polynomial-time online algorithm for maximizing the conditional value at risk (CVaR) of a monotone stochastic submodular function. Given T i.i.d. samples from an underlying distribution arriving online, our algorithm produces a sequence of solutions that converges to a (1-1/e)-approximate solution with a convergence rate of O(T^{-1/4}) for monotone continuous DR-submodular functions. Compared with previous offline algorithms, which require Omega (T) space, our online algorithm only requires O(sqrt{T}) space. We extend our online algorithm to portfolio optimization for monotone submodular set functions under a matroid constraint. Experiments conducted on real-world datasets demonstrate that our algorithm can rapidly achieve CVaRs that are comparable to those obtained by existing offline algorithms.

On Uniform Monotone Approximation of Continuous Monotone Functions with the Help of Translations and Dilations of the Laplace Integral

On Uniform Monotone Approximation of Continuous Monotone Functions with the Help of Translations and Dilations of the Laplace Integral

Continuous Nowhere Differentiable Function with Fractal Properties Defined in Terms of Q2-Representation

We construct a continuous nowhere monotone and nondifferentiable function depending on a single parameter q0 ϵ (0; 1). For functions from this continual class, we describe their structural, variational, fractal, and integrodifferential properties.

Wirtinger integral inequalities for pseudo-integrals and pseudo-additive measure

The main purpose of this paper is to show Wirtinger type inequalities for the pseudo-integral. We are concerned with pseudo-integrals based on the following three canonical cases: in the first case, the real semiring with pseudo-operation is generated by a strictly monotone continuous function g; in the second case, the pseudo-operations include a pseudo-multiplication and a power arithmetic addition; in the last case, ⊕-measures are interval-valued. Examples are given to illustrate these equalities.

Hyperspaces through regular and meager subcontinua

Given a metric continuum X, let C(X) and F1(X) be the hyperspaces of subcontinua and the one-point sets of X, respectively. Let D(X) be the collection of all regular subsets in X belonging to C(X) and let M(X) be all the subcontinua of X with empty interior. In the first part of this paper we are interested in the problem to classify continua that satisfies one of the following equalities: F1(X)=M(X), D(X)=D(X)−F1(X), D(X)=C(X)−M(X) and C(X)=D(X)∪M(X). We show that for hereditarily locally connected continua these equalities and the property of not contain a continuum called dendrite D1 are equivalent each other. In the second part, we consider a continuum X for which there exists a continuous surjective and monotone function f:X→[0,1]. We show that the conditions that X is a λ-type continuum and every t in [0,1] is a cohesion point are equivalent to the equalities D(X)={A∈C(X):f(A)∈D([0,1])} and M(X)={A∈C(X):f(A)∈M([0,1])}. Throughout this paper we pose open problems.

Big and little Lipschitz one sets

Given a continuous function f:{mathbb {R}}rightarrow {mathbb {R}}, we denote the so-called “big Lip” and “little lip” functions by and respectively. We are interested in the following question. Given a set E subset {mathbb {R}}, is it possible to find a continuous function f such that or ? For monotone continuous functions we provide a rather straightforward answer. For arbitrary continuous functions the answer is much more difficult to find. We introduce the concept of uniform density type (UDT) and show that if E is G_delta and UDT then there exists a continuous function f satisfying , that is, E is a set. In the other direction we show that every set is G_delta and weakly dense. We also show that the converse of this statement is not true, namely that there exist weakly dense G_{{delta }} sets which are not . We say that a set Esubset mathbb {R} is if there is a continuous function f such that . We introduce the concept of strongly one-sided density and show that every set is a strongly one-sided dense F_sigma set.

The continuous weak order

The set of permutations on a finite set can be given the lattice structure known as the weak Bruhat order. This lattice structure is generalized to the set of words on a fixed alphabet Σ={x,y,z,…}, where each letter has a fixed number of occurrences. These lattices are known as multinomial lattices and, when card(Σ)=2, as lattices of lattice paths. By interpreting the letters x,y,z,… as axes, these words can be interpreted as discrete increasing paths on a grid of a d-dimensional cube, with d=card(Σ).We show how to extend this ordering to images of continuous monotone functions from the unit interval to a d-dimensional cube and prove that this ordering is a lattice, denoted by L(Id). This construction relies on a few algebraic properties of the quantale of join-continuous functions from the unit interval of the reals to itself: it is cyclic ⋆-autonomous and it satisfies the mix rule.We investigate structural properties of these lattices, which are self-dual and not distributive. We characterize join-irreducible elements and show that these lattices are generated under infinite joins from their join-irreducible elements, they have no completely join-irreducible elements nor compact elements. We study then embeddings of the d-dimensional multinomial lattices into L(Id). We show that these embeddings arise functorially from subdivisions of the unit interval and observe that L(Id) is the Dedekind-MacNeille completion of the colimit of these embeddings. Yet, if we restrict to embeddings that take rational values and if d>2, then every element of L(Id) is only a join of meets of elements from the colimit of these embeddings.

A new family of (A, N)-implications: Construction and properties

This paper deals with a new family of (A, N)-implications that derived from fuzzy negation and continuous monotone function, and some properties, like the left neutrally, the exchange principle, the ordering property and the laws of contraposition, of the proposed family of (A, N)-implications are studied. Moreover, the T-conditionality and the Modus tollens for the new family of (A, N)-implications are also studied and some results are obtained.

Stochastic Conditional Gradient++: (Non)Convex Minimization and Continuous Submodular Maximization

In this paper, we consider the general nonoblivious stochastic optimization where the underlying stochasticity may change during the optimization procedure and depends on the point at which the function is evaluated. We develop Stochastic Frank--Wolfe++ (SFW++), an efficient variant of the conditional gradient method for minimizing a smooth nonconvex function subject to a convex body constraint. We show that SFW++ converges to an $\epsilon$-first order stationary point by using $O(1/\epsilon^3)$ stochastic gradients. Once further structures are present, SFW++'s theoretical guarantees, in terms of the convergence rate and quality of its solution, improve. In particular, for minimizing a convex function, SFW++ achieves an $\epsilon$-approximate optimum while using $O(1/\epsilon^2)$ stochastic gradients. It is known that this rate is optimal in terms of stochastic gradient evaluations. Similarly, for maximizing a monotone continuous DR-submodular function, a slightly different form of SFW++, called Stochastic Continuous Greedy++ (SCG++), achieves a tight $[(1-1/e){\text{OPT}} -\epsilon]$ solution while using $O(1/\epsilon^2)$ stochastic gradients. Through an information theoretic argument, we also prove that SCG++'s convergence rate is optimal. Finally, for maximizing a nonmonotone continuous DR-submodular function, we can achieve a $[(1/e){\text{OPT}} -\epsilon]$ solution by using $O(1/\epsilon^2)$ stochastic gradients. We should highlight that our results and our novel variance reduction technique trivially extend to the standard and easier oblivious stochastic optimization settings for (non)convex and continuous submodular settings.

Psychophysics of the hoverfly: categorical or continuous color discrimination?

There is increasing interest in flies as potentially important pollinators. Flies are known to have a complex visual system, including 4 spectral classes of photoreceptors that contribute to the perception of color. Our current understanding of how color signals are perceived by flies is based on data for the blowfly Lucilia sp., which after being conditioned to rewarded monochromatic light stimuli, showed evidence of a categorical color visual system. The resulting opponent fly color space has 4 distinct categories, and has been used to interpret how some fly pollinators may perceive flower colors. However, formal proof that flower flies (Syrphidae) only use a simple, categorical color process remains outstanding. In free-flying experiments, we tested the hoverfly Eristalis tenax, a Batesian mimic of the honeybee, that receives its nutrition by visiting flowers. Using a range of broadband similar–dissimilar color stimuli previously used to test color perception in pollinating hymenopteran species, we evaluated if there are steep changes in behavioral choices with continuously increasing color differences as might be expected by categorical color processing. Our data revealed that color choices by the hoverfly are mediated by a continuous monotonic function. Thus, these flies did not use a categorical processing, but showed evidence of a color discrimination function similar to that observed in several bee species. We therefore empirically provide data for the minimum color distance that can be discriminated by hoverflies in fly color space, enabling an improved understanding of plant–pollinator interactions with a non-model insect species.

Properties of fuzzy transform obtained from Lp minimization and a connection with Zadeh’s extension principle

In 2006 Perfilieva proved that the components Fk of the fuzzy transform are actually minimizers of certain weighted L2 type distances, obtained from a fuzzy partition. Inspired by this property, we consider and analyze the properties of a fuzzy transform when the components Fk are obtained as minimizers with respect to weighted Lp type distances, where p ≥ 1. In this way we can generate new approximation operators based on the fuzzy transform method. For these operators we obtain good rates of uniform convergence and what is more, they satisfy all the properties required for an effective approximation operator, such as additivity and homogeneity. If we consider the extended fuzzy transform and inverse fuzzy transform, respectively, in this setting of Lp type distances, then we get some nice supplementary properties like monotonicity of the operator and preservation of monotone functions. Finally, we consider fuzzy-valued versions of the L1 and L2 type fuzzy transforms, proposed recently as quantile and expectile smoothing tools, and we prove that the L1 type fuzzy-valued transform of a continuous monotonic function f coincides with the Zadeh’s fuzzy extension fEP evaluated at suitable fuzzy numbers, obtained from the basic functions of the used fuzzy partition.

Risk-Sensitive Submodular Optimization

The conditional value at risk (CVaR) is a popular risk measure which enables risk-averse decision making under uncertainty. We consider maximizing the CVaR of a continuous submodular function, an extension of submodular set functions to a continuous domain. One example application is allocating a continuous amount of energy to each sensor in a network, with the goal of detecting intrusion or contamination. Previous work allows maximization of the CVaR of a linear or concave function. Continuous submodularity represents a natural set of nonconcave functions with diminishing returns, to which existing techniques do not apply. We give a (1 - 1/e)-approximation algorithm for maximizing the CVaR of a monotone continuous submodular function. This also yields an algorithm for submodular set functions which produces a distribution over feasible sets with guaranteed CVaR. Experimental results in two sensor placement domains confirm that our algorithm substantially outperforms competitive baselines.

Second-order expansions for maxima of dynamic bivariate normal copulas

In this paper, we establish the second-order distributional expansions of normalized maxima of n independent observations, where the ith observation follows from a normal copula with its correlation coefficient being a monotone continuous function. These expansions can be used to deduce the convergence rates of distributions of normalized maxima to their limits.

Fetal pulmonary venous blood flow velocities in a normal population and new calculated reference values.

The aim of this study is to establish new Doppler reference values in a population with normal pulmonary veins during different phases of the fetal cardiac cycle for maximum blood flow velocities during ventricular systole (S), early ventricular diastole (D), atrial contraction (a), intensity-weighted mean velocity (Vmean) and the indices derived from these parameters [(S - a)/D] and [(S - a)/Vmean]. The pulsed-wave color advanced dynamic flow Doppler technique was used in a prospective cross-sectional study of 365 low-risk pregnancies between 18 and 41 completed gestational weeks. Reference ranges were calculated for the individual measuring parameters based on a growth function from a four-parameter class of monotonic continuous functions according to the smallest square principle. During the observation period of 18 to 41 gestational weeks, a continuous increase was evident in maximum flow velocities for ventricular systole (S), early ventricular diastole (D), atrial contraction (a) and intensity-weighted mean velocity (Vmean) of 14.7 to 431.1, 12.5 to 426.2, 5.4 to 411.9 and 8.7 to 421.6 cm/s, respectively. Venous indices, however, showed a decrease with increasing gestational age: (S - a)/D from 0.9 to 0.7 and (S - a)/Vmean from 1 to 0.9. The newly established normal values for the fetal pulmonary vein can be used in standardized Doppler sonography measurement to assess the left atrial pressure in the fetal heart. © 2016 John Wiley & Sons, Ltd.