Real-valued Functions Research Articles

On global extrema of power Takagi functions

By construction, power Takagi functions Sp are similar to Takagi's continuous nowhere differentiable function described in 1903. These real-valued functions Sp(x) have one real parameter p>0 . They are defined on the real axis R by the series Sp(x)=∑∞n=0(S0(2nx)/2n)p , where S0(x) is the distance from real number x to the nearest integer number. We show that for every p>0 , the functions Sp are everywhere continuous, but nowhere differentiable on R . Next, we derive functional equations for Takagi power functions. With these, it is possible, in particular, to calculate the values Sp(x) at rational points x . In addition, for all values of the parameter p from the interval (0;1) , we find the global extrema of the functions Sp , as well as the points where they are reached. It turns out that the global maximum of Sp equals to 2p/(3p(2p−1)) and is reached only at points q+1/3 and q+2/3 , where q is an arbitrary integer. The global minimum of the functions Sp equals to 0 and is reached only at integer points. Using the results on global extremes, we obtain two-sided estimates for the functions Sp and find the points at which these estimates are reached.

( X , Y , φ ) -Stable semigroups, periodic solutions, and applications

ABSTRACT Motivated by the smoothing properties of heat semigroups on unbounded domains and the conditional stability of hyperbolic semigroups, we develop a unified approach toward the problems on the existence of periodic solutions to the evolution equation . Our method is based on the analysis of -stability of the semigroup generated by A, i.e. , t>0, for certain couple of Banach spaces and real-valued function φ satisfying . Our theory covers both cases corresponding to smoothing properties and the conditional stability of hyperbolic semigroups as well as some other important situations relating to the polynomial or exponential stability of semigroups. As illustrations for our theory, we give applications to the existence and uniqueness of periodic solutions to Navier–Stokes and damped wave equations.

Molecular Assays Simulator to Unravel Predictors Hacking in Goal-Directed Molecular Generations.

Generative models are being increasingly used in drug discovery, very often coupled with absorption, distribution, metabolism, and excretion (ADME) bioassays or quantitative structure-activity relationship (QSAR) models to optimize a given set of properties. The molecules proposed by these algorithms are often revealed to be false positives; that is, they are predicted to be active and turn out to be inactive after synthesis and testing, mostly due to overoptimization of the predicted scores, which leads to an actual decrease or stagnation of the real scores. This behavior is also known as the "hacking" of the predictive models by the generative model during the optimization step. This issue is reminiscent of adversarial examples in machine learning and it can be seen as enunciated by Goodhart's law: "when a measure becomes a target, it ceases to be a good measure." This issue is even more apparent in a multiparameter optimization (MPO) case, where the models need to extrapolate outside the training set distribution because there are no known molecules satisfying all the objectives simultaneously in the initial training set. Experimental evaluation of this problem is a hard and expensive task since it requires synthesis and testing of the generated molecules. Thus, efforts have been made to develop in silico "oracles"─real-valued functions used as proxies for molecular properties─to help with the evaluation of these generative-model-based pipelines. However, these oracles have had a limited value so far because they are often too easy to model in comparison with biological assays and are usually limited to mono-objective cases. In this work, we introduce a simulator of multitarget assays using a smartly initialized neural network (NN) that returns continuous values for any input molecule. We use this oracle to replicate a real-world prospective lead optimization (LO) scenario. First, we trained predictive models on an initial small sample of molecules aimed at predicting their oracle values. Afterward, we generated new optimized molecules using the open-source GuacaMol package coupled with the previously built predictive models. Finally, we selected compounds matching the candidate drug target profile (CDTP) according to the predicted values and evaluated them by computing the true oracle values. We observed that even when the predictive models had excellent estimated performance metrics, the final selection still contained multiple false positives according to the NN-based oracle. Then, we evaluated the optimization behavior in mono- and bi-objective scenarios using either a logistic regression or a random forest predictive model. We also propose and evaluate several methods to help mitigate the hacking issue.

Functions with Uniform Sublevel Sets on Cones

Extended real-valued functions on a real vector space with uniform sublevel sets are important in optimization theory. Weidner studied these functions in [1]. In the present paper, we study the class of these functions, which coincides with the class of Gerstewitz functionals, on cones. These cone are not necessarily embeddable in vector spaces. Almost any Weidner's results are not true on cones without extra conditions. We show that the mentioned conditions are necessary, by nontrivial examples. Specially for element k from the cone $\mathcal{P}$, we define $k$-directional closed subsets of the cone and prove some properties of them. For a subset $A$ of the cone $\mathcal{P}$, we characterize domain of the $\varphi_{A,k}$ (function with uniform sublevel set) and show that this function is $k$-transitive. One of the important conditions for satisfying the results, is that $k$ has the symmetric element in the cone. Also, we prove that, under some conditions, the class of Gerstewitz functionals coincides with the class of $k$-translative functions on $\mathcal{P}$.

Nonlinear disjointness/supplement preservers of nonnegative continuous functions

Let F(X),F(Y) be sufficiently large sets of nonnegative continuous real-valued functions defined on completely regular spaces X,Y, respectively. Let Φ:F(X)→F(Y) be a surjective map satisfying thatf∨g>0⟺Φ(f)∨Φ(g)>0,∀f,g∈F(X). In many cases, we show that there is a homeomorphism τ:Y→X such thatΦ(f)(y)≠0⟺f(τ(y))≠0,∀f∈F(X),∀y∈Y.Assume X,Y are locally compact Hausdorff (resp. separable and metrizable) and Φ:C0(X)+→C0(Y)+ (resp. Φ:Cb(X)+→Cb(Y)+) is a surjective map. We show that Φ preserves the norms of infima, i.e.,‖Φ(f)∧Φ(g)‖=‖f∧g‖,∀f,g∈C0(X)+ (resp.Cb(X)+), if and only if there is a homeomorphism τ:Y→X such thatΦ(f)(y)=f(τ(y)),∀f∈C0(X)+ (resp.Cb(X)+), ∀y∈Y.

One Axiom to Rule Them All: A Minimalist Axiomatization of Quantiles

We offer a minimalist axiomatization of quantiles among all real-valued mappings on a general set of distributions through only one axiom. This axiom is called ordinality: Quantiles are the only mappings that commute with all increasing and continuous transforms. Other convenient properties of quantiles—monotonicity, semicontinuity, comonotonic additivity, elicitability, and locality in particular—follow from this axiom. Furthermore, on the set of convexly supported distributions, the median is the only mapping that commutates with all monotone and continuous transforms. On a general set of distributions, the median interval is pinned down as the unique minimal interval-valued mapping that commutes with all monotone and continuous transforms. Finally, our main result, put in a decision-theoretic setting, leads to a minimalist axiomatization of quantile preferences. In banking and insurance, quantiles are known as the standard regulatory risk measure Value-at-Risk (VaR), and thus an axiomatization of VaR is obtained with only one axiom among law-based risk measures.

Real-boards of ideals in C(X) and their applications to characterize some spaces

In this paper, using the concept of the real-boards of ideals in C(X), the ring of all real-valued continuous functions on a completely regular Hausdorff space X, some characterizations of nearly pseudocompact and nearly realcompact spaces are given. For an ideal I⊆C(X), the real-board of I is the largest subspace of βX, the Stone-Čech compactification of X, on which every member of I can be extended continuously. Using this concept, the subrings CKI(X) and C∞I(X) of C(X) are introduced and studied, and I-pseudocompact spaces which are generalizations of pseudocompact spaces are presented as well as some of their properties.

Noncoercive and noncontinuous equilibrium problems: existence theorem in infinite-dimensional spaces

In this paper, we extend the definition of the qx-asymptotic functions, for an extended real-valued function defined on the infinite-dimensional topological normed spaces without lower semicontinuity or quasi-convexity condition. As the main result, by using some asymptotic conditions, we obtain sufficient optimality conditions for the existence of solutions to equilibrium problems, under weaker assumptions of continuity and convexity, when the feasible set is an unbounded subset of infinite-dimensional space. Also, as a corollary, we obtain necessary and sufficient optimality conditions for the existence of solutions to equilibrium problems with an unbounded feasible set. Finally, as an application, we establish a result for the existence of solutions to minimization problems.

Measurable functions on σ-frames

We study (semi)measurable functions on σ-frames, extending the theory of real-valued functions from frames to σ-frames. The new objects of study are the σ-frame homomorphisms ▪ from the usual frame of reals into the congruence lattice of a σ-frame L, and its subclass of measurable functions L(R)→L. The desired extension faces two obstacles: (1) in general, σ-frames have no uncountable joins and are not pseudocomplemented; (2) σ-sublocales, that is, the subobjects in the dual category of the category of σ-frames and σ-frame homomorphisms, cannot be described as concrete subsets of the σ-frame L, unlike their counterpart in the category of locales, forcing us to work in the congruence lattice of L.Nevertheless, it is shown that, despite (1), the familiar method for generating real functions on frames via scales can be extended to arbitrary σ-frames. This is achieved by the new notions of σ-scale and finite σ-scale. It is also shown that, despite (2), the familiar insertion, extension and separation results for real-valued functions in several classes of frames (normal, extremally disconnected, G-perfect, F-perfect, perfectly normal) can be proved without involving uncountable joins and pseudocomplements, thus allowing their extension to measurable and semimeasurable functions.

Some connections between the generalized Hukuhara derivative and the fuzzy derivative based on strong linear independence

This article investigates some connections between the notions of the generalized Hukuhara derivative and the Ψ-derivative of fuzzy number-valued functions. The concept of Ψ-differentiability is defined on a fuzzy number-valued function φ in the form of φ(x)=ϱ1(x)A1+…+ϱn(x)An, where ϱ1,…,ϱn are n real-valued functions defined on an interval [a,b] and {A1,…,An} is a strongly linearly independent subset of fuzzy numbers. The Ψ-derivative of φ at some x is φ′(x)=ϱ1′(x)A1+…+ϱn′(x)An whenever the derivatives ϱ1′(x),…,ϱn′(x) exist. This article provides conditions for these two notions of derivatives of such functions to coincide. Moreover, under some weak conditions, we show that an arbitrary continuously gH-differentiable function defined on an interval [a,b] and its gH-derivative can be uniformly approximated as closely as desired by a Ψ-differentiable function and its Ψ-derivative, respectively. Finally, we apply these results to obtain numerical and analytical solutions of a simple fuzzy decay model described by a fuzzy initial value problem under gH-derivative.

Measuring small displacements of an optical point source with digital holography.

The image of an optical point source is blurred due to light diffraction so that estimating small displacements of the point source with direct imaging demands elaborate processing on the observation data of a camera. Using quantum parameter estimation, we show that for the imaging systems with a real point spread function, any measurement basis constituted by a complete set of real-valued spatial-mode functions is optimal for estimating the displacement. For small displacements, we can concentrate the information about the value of displacement to the measurement of a few spatial modes, which can be selected in terms of the Fisher information distribution. We use digital holography with a phase-only spatial light modulator to implement two simple estimation strategies that are mainly based on the projection measurement of two spatial modes and the readout of a single pixel of a camera.

Just Tilt Your Head: A Graphical Technique for Classifying Fixed Points

Summary Iterations of the form are used throughout mathematics, and convergence (or not) of an iteration to a fixed point is always a central question. We present an informal graphical technique for resolving whether a fixed point in one dimension is attracting or repelling. Specifically, given a real-valued function f, we are determining whether or at a point for which . The former condition on classifies the fixed point as attracting, and the iteration as locally convergent; the latter corresponds to a repulsive fixed point, in which the iterations get further away from the fixed point no matter how close you start them off. The technique requires only that we be able to compute points on the graph of and not necessarily on an explicit functional form or its derivative.

Spectral shift for relative Schatten class perturbations

We affirmatively settle the question on existence of a real-valued higher order spectral shift function for a pair of self-adjoint operators H and V such that V is bounded and V(H-iI)^{-1} belongs to a Schatten–von Neumann ideal \mathcal{S}^n of compact operators in a separable Hilbert space. We also show that the function satisfies the same trace formula as in the known case of V\in\mathcal{S}^n and that it is unique up to a polynomial summand of order n-1 . Our result significantly advances earlier partial results where counterparts of the spectral shift function for noncompact perturbations lacked real-valuedness and aforementioned uniqueness as well as appeared in more complicated trace formulas for much more restrictive sets of functions. Our result applies to models arising in noncommutative geometry and mathematical physics.

Extensions of continuous increasing functions

A topological preordered space is a topological space with a preorder ⪯, that is, a binary relation which is reflexive and transitive. Nachbin generalized Tietze's theorem and Urysohn's theorem for continuous increasing functions on a normally preordered space, where a real-valued function f is called increasing if f(x)≤f(x′) whenever x⪯x′. In this paper, we provide an increasing version of Gillman-Jerison theorem characterizing C⁎-embedding. Namely, we show that every bounded continuous increasing function on a subspace A of a topological preordered space X can be extended over X if and only if every pair of completely order separated subsets of A can be completely order separated in X. Various theorems and several counterexamples of extensions of continuous increasing functions are also given.

Graphs of continuous functions and fractal dimensions

In this paper, we show that, for any β∈[1,2], a given strictly positive (or strictly negative) real-valued continuous function on [0,1] whose graph has the upper box dimension less than or equal to β can be decomposed as a product of two real-valued continuous functions on [0,1] whose graphs have upper box dimensions equal to β. We also obtain a formula for the upper box dimension of every element of a ring of polynomials in a finite number of continuous functions on [0,1] over the field R.

The Simultaneous Fractional Dimension of Graph Families

For a connected graph G with vertex set V, let RG{x, y} = {z ∈ V: dG(x, z) ≠ dG(y, z)} for any distinct x, y ∈ V, where dG(u, w) denotes the length of a shortest uw-path in G. For a real-valued function g defined on V, let g(V) = ∑s∈Vg(s). Let $${\cal C} = \{{G_1},{G_2}, \ldots ,{G_k}\} $$ be a family of connected graphs having a common vertex set V, where k ≥ 2 and ∣V∣≥ 3. A real-valued function h: V → [0, 1] is a simultaneous resolving function of $${\cal C}$$ if h(RG{x, y}) ≥ 1 for any distinct vertices x, y ∈ V and for every graph $$G \in {\cal C}$$ . The simultaneous fractional dimension, $${\rm{S}}{{\rm{d}}_f}({\cal C})$$ , of $${\cal C}$$ is min{h(V): h is a simultaneous resolving function of $${\cal C}$$ }. In this paper, we initiate the study of the simultaneous fractional dimension of a graph family. We obtain $${\max _{1 \le i \le k}}\{{\dim _f}({G_i})\} \le {\rm{S}}{{\rm{d}}_f}({\cal C}) \le \min \{\sum\nolimits_{i = 1}^k {{{\dim}_f}({G_i}),{{|V|} \over 2}} $$ , where both bounds are sharp. We characterize $${\cal C}$$ satisfying $${\rm{S}}{{\rm{d}}_f}({\cal C}) = 1$$ , examine $${\cal C}$$ satisfying $${\rm{S}}{{\rm{d}}_f}({\cal C}) = {{|V|} \over 2}$$ , and determine $${\rm{S}}{{\rm{d}}_f}({\cal C})$$ when $${\cal C}$$ is a family of vertex-transitive graphs. We also obtain some results on the simultaneous fractional dimension of a graph and its complement.

P-Adic statistical field theory and convolutional deep Boltzmann machines

Abstract Understanding how deep learning architectures work is a central scientific problem. Recently, a correspondence between neural networks (NNs) and Euclidean quantum field theories has been proposed. This work investigates this correspondence in the framework of p-adic statistical field theories (SFTs) and neural networks. In this case, the fields are real-valued functions defined on an infinite regular rooted tree with valence p, a fixed prime number. This infinite tree provides the topology for a continuous deep Boltzmann machine (DBM), which is identified with a statistical field theory on this infinite tree. In the p-adic framework, there is a natural method to discretize SFTs. Each discrete SFT corresponds to a Boltzmann machine with a tree-like topology. This method allows us to recover the standard DBMs and gives new convolutional DBMs. The new networks use O(N) parameters while the classical ones use O(N2) parameters.

Ordinal optimization through multi-objective reformulation

We analyze combinatorial optimization problems with ordinal, i.e., non-additive, objective functions that assign categories (like good, medium and bad) rather than cost coefficients to the elements of feasible solutions. We review different optimality concepts for ordinal optimization problems and discuss their similarities and differences. We then focus on two prevalent optimality concepts that are shown to be equivalent. Our main focus lies on the investigation of a bijective linear transformation that transforms ordinal optimization problems to associated standard multi-objective optimization problems with binary cost coefficients. Since this transformation preserves all properties of the underlying problem, problem-specific solution methods remain applicable. A prominent example is dynamic programming and Bellman’s principle of optimality, that can be applied, e.g., to ordinal shortest path and ordinal knapsack problems. We investigate the interrelation between scalarization techniques and methods based on the hypervolume indicator when applied to the ordinal and the transformed problem, respectively. Furthermore, we extend our results to multi-objective optimization problems that combine ordinal and real-valued objective functions.

Stochastic zeroth-order gradient and Hessian estimators: variance reduction and refined bias bounds

Abstract We study stochastic zeroth-order gradient and Hessian estimators for real-valued functions in $\mathbb{R}^n$. We show that, via taking finite difference along random orthogonal directions, the variance of the stochastic finite difference estimators can be significantly reduced. In particular, we design estimators for smooth functions such that, if one uses $ \varTheta \left ( k \right ) $ random directions sampled from the Stiefel manifold $ \text{St} (n,k) $ and finite-difference granularity $\delta $, the variance of the gradient estimator is bounded by $ \mathscr{O} \left ( \left ( \frac{n}{k} - 1 \right ) + \left ( \frac{n^2}{k} - n \right ) \delta ^2 + \frac{ n^2 \delta ^4} { k } \right ) $, and the variance of the Hessian estimator is bounded by $\mathscr{O} \left ( \left ( \frac{n^2}{k^2} - 1 \right ) + \left ( \frac{n^4}{k^2} - n^2 \right ) \delta ^2 + \frac{n^4 \delta ^4 }{k^2} \right ) $. When $k = n$, the variances become negligibly small. In addition, we provide improved bias bounds for the estimators. The bias of both gradient and Hessian estimators for smooth function $f$ is of order $\mathscr{O} \big( \delta ^2 \varGamma \big )$, where $\delta $ is the finite-difference granularity, and $ \varGamma $ depends on high-order derivatives of $f$. Our results are evidenced by empirical observations.

Isomorphisms of {mathcal {C}}(K, E) Spaces and Height of K

Let K_1, K_2 be compact Hausdorff spaces and E_1, E_2 be Banach spaces not containing a copy of c_0. We establish lower estimates of the Banach–Mazur distance between the spaces of continuous functions {mathcal {C}}(K_1, E_1) and {mathcal {C}}(K_2, E_2) based on the ordinals ht(K_1), ht(K_2), which are new even for the case of spaces of real-valued functions on ordinal intervals. As a corollary we deduce that {mathcal {C}}(K_1, E_1) and {mathcal {C}}(K_2, E_2) are not isomorphic if ht(K_1) is substantially different from ht(K_2).