On the Kolmogorov complexity of continuous real functions

This paper supplements an earlier one by the authors which constructed the Dedekind completion of the ring of continuous real functions on an arbitrary frame L in terms of partial continuous real functions on L. In the present paper, we provide three alternative views of it, in terms of (i) normal semicontinuous real functions on L, (ii) the Booleanization of L (in the case of bounded real functions) and the Gleason cover of L (in the general case), and (iii) Hausdorff continuous partial real functions on L. The first is the normal completion and extends Dilworth’s classical construction to the pointfree setting. The second shows that in the bounded case, the Dedekind completion is isomorphic to the lattice of bounded continuous real functions on the Booleanization of L, and that in the non-bounded case, it is isomorphic to the lattice of continuous real functions on the Gleason cover of L. Finally, the third is the pointfree version of Anguelov’s approach in terms of interval-valued functions. Two new classes of frames, cb-frames and weak cb-frames, emerge naturally in the first two representations. We show that they are conservative generalizations of their classical counterparts.

In computable analysis, sequences of rational numbers which effectively converge to a real number x are used as the (rho-) names of x. A real number x is computable if it has a computable name, and a real function f is computable if there is a Turing machine M which computes f in the sense that, M accepts any rho-name of x as input and outputs a rho-name of f(x) for any x in the domain of f. By weakening the effectiveness requirement of the convergence and classifying the converging speeds of rational sequences, several interesting classes of real numbers of weak computability have been introduced in literature, e.g., in addition to the class of computable real numbers (EC), we have the classes of semi-computable (SC), weakly computable (WC), divergence bounded computable (DBC) and computably approximable real numbers (CA). In this paper, we are interested in the weak computability of continuous real functions and try to introduce an analogous classification of weakly computable real functions. We present definitions of these functions by Turing machines as well as by sequences of rational polygons and prove these two definitions are not equivalent. Furthermore, we explore the properties of these functions, and among others, show their closure properties under arithmetic operations and composition.

Riemann zeta is defined as a function of a complex variable that analytically continues the sum of the Dirichlet series, when the real part is greater than unity. In this paper, the Riemann zeta associated with the finite energy possessed by a 2mm radius, free falling water droplet, crashing into a plane is considered. A modified zeta function is proposed which is incorporated to the spherical coordinates and real analysis has been performed. Through real analytic continuation, the single point of contact of the drop at the instant of touching the plane is analyzed. The zeta function is extracted at the point of destruction of the drop, where it defines a unique real function. A special property is assumed for some continuous functions, where the function’s first derivative and first integral combine together to a nullity at all points. Approximate reverse synthesis of such a function resulted in a special waveform named the dying-surge. Extending the proposed concept to general continuous real functions resulted in the synthesis of the corresponding function’s Dying-surge model. The Riemann zeta function associated with the water droplet can also be modeled as a dying–surge. The Dying- surge model corresponds to an electrical squeezing or compression of a waveform, which was originally defined over infinite arguments, squeezed to a finite number of values for arguments placed very close together with defined final and penultimate values. Synthesized results using simulation software are also presented, along with the analysis. The presence of surges in electrical circuits will correspond to electrical compression of some unknown continuous, real current or voltage function and the method can be used to estimate the original unknown function.

Extended real functions in pointfree topology

Let $L$ be a completely regular frame and $mathcal{R}L$ be the ring of real-valued continuous functions<br /> on $L$.<br /> We consider the set $$mathcal{R}_{infty}L = {varphi in mathcal{R} L : uparrow varphi( dfrac{-1}{n}, dfrac{1}{n})<br /> mbox{ is a compact frame for any $n in mathbb{N}$}}.$$<br /> Suppose that $C_{infty} (X)$ is the family of all functions $f in C(X)$ for which the<br /> set ${x in X: |f(x)|geq dfrac{1}{n} }$<br /> is compact, for every $n in mathbb{N}$.<br /> Kohls has shown that $C_{infty} (X)$ is precisely the intersection<br /> of all the free maximal ideals of $C^{*}(X)$.<br /> The aim of this paper is to<br /> extend this result to<br /> the real continuous functions on a<br /> frame and hence we show that $mathcal{R}_{infty}L$ is precisely the intersection<br /> of all the free maximal ideals of $mathcal R^{*}L$.<br /> This result is used to characterize the maximal ideals in $mathcal{R}_{infty}L$.

Suppose that Ω is a compact Hausdorff space with a preorder [les ] whose graph is closed, and let Ω∘ be an open subset of Ω. This paper provides conditions sufficient to allow every increasing bounded real continuous function on Ω∘ to be extended to an increasing real continuous function on Ω. These conditions are: (i) that Ω is a Stonian space, and (ii) that the set C↑(Ω, [les ]) of increasing real continuous functions on Ω is a regular Dedekind complete sublattice of C(Ω). Under these conditions it is also shown that C↑(Ω, [les ]) is generated by idempotents, and an extension theorem for idempotents is proved.

Spiking neural P systems (SN P systems), inspired by biological neurons, are introduced as symbolical neural-like computing models that encode information with multisets of symbolized spikes in neurons and process information by using spike-based rewriting rules. Inspired by neuronal activities affected by enzymes, a numerical variant of SN P systems called enzymatic numerical spiking neural P systems (ENSNP systems) is proposed wherein each neuron has a set of variables with real values and a set of enzymatic activation-production spiking rules, and each synapse has an assigned weight. By using spiking rules, ENSNP systems can directly implement mathematical methods based on real numbers and continuous functions. Furthermore, ENSNP systems are used to model ENSNP membrane controllers (ENSNP-MCs) for robots implementing wall following. The trajectories, distances from the wall, and wheel speeds of robots with ENSNP-MCs for wall following are compared with those of a robot with a membrane controller for wall following. The average error values of the designed ENSNP-MCs are compared with three recently fuzzy logical controllers with optimization algorithms for wall following. The experimental results showed that the designed ENSNP-MCs can be candidates as efficient controllers to control robots implementing the task of wall following.

The description of frames “by generators and defining relations” from IV.2.4.1 can be used to introduce the reals in a point-free way. André Joyal ([156]) was the first to emphasize the benefits of such a point-free approach to ℝ in the constructive context of topos theory; it was first explicitly carried out in [104]. The idea is to take the standard totally ordered set ℚ of rationals and to take the set of open intervals with rational endpoints for the basic generators. One can then deal with continuous real functions and even more general real functions as we will see in this chapter.KeywordsReal FunctionSemicontinuous FunctionNormal FrameContinuous Real FunctionRegular FrameThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

There are many properties of mathematical objects that satisfy what is sometimes called a 0-1 law, in the following sense. Under some natural probability measure on the set of objects, the measure of the subset of objects having the given property is either 0 or 1. In the latter case we can say that almost all the objects have the property. Familiar examples of this phenomenon are the following: almost all real numbers are transcendental (or normal to every base), almost all integers are composite, almost all continuous real functions are nondifferentiable, etc. It is often the case that the objects under consideration can be partitioned into a countable number of finite classes Cn, with the probability assigned to an object in Cn being just 1/ICn I. In this case, we say that a property Pn satisfies a 0-1 law if the fraction of the number of objects in Cn that satisfy Pn either tends to 0 or tends to 1 as n -x 0o. For example, almost all graphs on n vertices have maximum cliques and maximum independent sets of size at most 2 log n, almost all Boolean functions with n variables have circuit complexity (1 + o( 1 ))2n and almost all binary codes of length n with at most 2nR codewords (with R less than the binary symmetric channel capacity C) have arbitrarily small error probability (a special case of Shannon's coding theorem; see [S48]). One of the first general results of this type was the theorem of Fagin [F76] and Glebskii et al. [GKLT69], which asserts that every property of graphs that can be expressed in first-order logic satisfies a 0-1 law (see [SS88] for recent striking developments in this topic). One obvious method for finding explicit objects having some property Pn shared by almost all objects in Cn is simply to select one at random. With overwhelming probability (tending to 1 as n -x oc), the selected object will have property Pn. Unfortunately, it may be (and often is) extremely difficult to prove that any particular object does indeed satisfy Pn . It is our purpose in this paper to describe a method that can to a certain extent circumvent this difficulty. We will show that, for a variety of families, it is possible to identify a natural hierarchy of equivalence classes of properties, all of which are shared by almost all objects in the family. Any object satisfying

"Real continuous submodular functions, as a generalization of the corresponding discrete notion to the continuous domain, gained considerable attention recently. The analog notion for entropy functions requires additional properties: a real function defined on the non-negative orthant of $\R^n$ is entropy-like (EL) if it is submodular, takes zero at zero, non-decreasing, and has the Diminishing Returns property. Motivated by problems concerning the Shannon complexity of multipartite secret sharing, a special case of the following general optimization problem is considered: find the minimal cost of those EL functions which satisfy certain constraints. In our special case the cost of an EL function is the maximal value of the $n$ partial derivatives at zero. Another possibility could be the supremum of the function range. The constraints are specified by a smooth bounded surface $S$ cutting off a downward closed subset. An EL function is feasible if at the internal points of $S$ the left and right partial derivatives of the function differ by at least one. A general lower bound for the minimal cost is given in terms of the normals of the surface $S$. The bound is tight when $S$ is linear. In the two-dimensional case the same bound is tight for convex or concave $S$. It is shown that the optimal EL function is not necessarily unique. The paper concludes with several open problems."

We investigate the subsets of the Fr\'echet space $s$ of all sequences of real numbers equipped with the Fr\'echet metric $\rho$ from the Baire category point of view. In particular, we concentrate on the convergence sets of the series $\sum f_n \left(x_n\right)$ that is, sets of sequences $x=(x_n)$ for which the series converges, or has a sum (perhaps infinite), or oscillates. Provided all $f_n$ are continuous real functions, sufficient conditions are given for the convergence sets to be of the first Baire category or residual in $s$.

Closed and basic closed C*D-filters are used in a process similar to Wallman method for compactifications of the topological spaces Y, of which, there is a subset of C*(Y) containing a non-constant function, where C*(Y) is the set of bounded real continuous functions on Y. An arbitrary Hausdorff compactification (Z,h) of a Tychonoff space X can be obtained by using basic closed C*D-filters from in a similar way, where C(Z) is the set of real continuous functions on Z.

Jerison introduced the Banach spaces C (S) of continuous real or complex-valued odd functions with respect to an involuntory homeomorphism a : S • • P S of the compact Hausdorff space S. It has been conjectured that any Banach space of the type C (S) is isomorphic to a Banach space of all continuous functions on some compact Hausdorff space. This conjecture is shown to be true if either (1) S is a Cartesian product of compact metric spaces or (2) S is a linearly ordered compact Hausdorff space and a has at most one fixed point.

We prove several splitting theorems related to the following one: Every continuous real trigonometric rational function (of t, with period $2\pi $) can be written as the real part of a rational function of $z = e^{it} $ that is analytic for $|z| \leqq 1$. As an application, we obtain immediately a method for computing the conjugate function of a real trigonometric rational function.

To investigate if maternal levels of human placental lactogen (hPL), placental growth hormone (PGH) and insulin-like growth factor-1 (IGF-1) are associated with growth rate of the biparietal diameter (BPD) in the first half of pregnancy. Data on 8215 singleton fetuses from the Copenhagen First Trimester Study with measurements of BPD from ultrasound scans performed at weeks 11-14 and 17-21 of pregnancy were analyzed. Growth rate was defined as millimeters of growth/day of BPD between the two scans. Fetuses with growth rate below the 2.5(th) centile (low growth rate, n = 203) and above the 97.5(th) centile (high growth rate, n = 203) were identified. As a reference group 212 fetuses with growth rate around the median were identified (intermediate growth rate). Out of the 618 selected cases in the three growth rate groups a total of 463 cases had a blood sample taken at the time of first-trimester ultrasound (5.6% of the original sample size of 8215 pregnancies). The maternal blood serum concentrations of hPL, PGH and IGF-1 were determined in the three different growth-rate groups. Linear regression analysis without adjustment and with adjustment for known and potential confounders was used to compare serum levels between the groups. Simple linear regression showed a difference in serum level of log(10) PGH between the high and intermediate growth-rate groups (P = 0.037). When adjusted for maternal weight and crown-rump length, multiple linear regression analysis confirmed this difference, as fetuses with high growth rates had a 12% (95% confidence interval, 2-20%; P = 0.009) higher maternal serum level of PGH than those with intermediate growth rates. No differences in hPL and IGF-1 levels between the three different growth-rate groups were found after simple and multiple linear regression analysis. Maternal PGH levels are higher in women carrying fetuses with high first-trimester growth rates than in controls, both in a simple unadjusted analysis and in analyses adjusted for known and potential confounders. Thus, PGH may be involved in fetal growth regulation as early as in the first trimester of pregnancy.