A Modified Wallman Method of Compactification

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.

On the Kolmogorov complexity of continuous real functions

Extended real functions in pointfree topology

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.

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.

A topological space X X is Hindman if for every sequence ( x n ) n (x_n)_n in X X there exists an infinite D ⊆ N D\subseteq \mathbb {N} so that the sequence ( x n ) n ∈ F S ( D ) (x_n)_{n\in FS(D)} , indexed by all finite sums over D D , is IP-converging in X X . Not all sequentially compact spaces are Hindman. The product of two Hindman spaces is Hindman. Furstenberg and Weiss proved that all compact metric spaces are Hindman. We show that every Hausdorff space X X that satisfies the following condition is Hindman: \[ ( ∗ )\quad The closure of every countable set in X is compact and first-countable.\quad \text {($*$)\quad The closure of every countable set in $X$ is compact and first-countable.\quad } \] Consequently, there exist nonmetrizable and noncompact Hindman spaces. The following is a particular consequence of the main result: every bounded sequence of monotone (not necessarily continuous) real functions on [ 0 , 1 ] [0,1] has an IP-converging subsequences.

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.

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.

In this paper the differential equation (1) y″=q(t)y is considered where q(t) is a real continuous function with period π. There is proved a necessary and sufficient condition for the stability of the trivial solution of Equation (1) when the zeros of the characteristic equation λ2 - Aλ+1=0, coincide. Moreover, there is shown the construction of all Equations (1) admitting only periodic or half-periodic solutions with period π.

The allometric scaling of metabolic rate of organisms, the three-quarters power rule, has led to a questioning of the basis for the relation. We attacked this problem experimentally for the first time by employing the modular organism, the ascidian that forms a single layered flat colony, as a model system. The metabolic rate and colony size followed the three-quarters power relation, which held even after the colony size was experimentally manipulated. Our results established that the three-quarters power relation is a real continuous function, not an imaginary statistical regression. The fact that all the hypotheses failed to explain why the two-dimensional organism adhered to the three-quarters power relation led us to propose a new hypothesis, in which the allometric relation derives from the self-organized criticality based on local interaction between modulus-comprising organisms.

A given real continuous function f on [a, b] is approximated by polynomials P n {P_n} of degree n that are subject to certain restrictions. Let 1 ≦ k 1 &gt; ⋯ &gt; k p ≦ n 1 \leqq {k_1} &gt; \cdots &gt; {k_p} \leqq n be given integers, ε i = ± 1 {\varepsilon _i} = \pm 1 , given signs. It is assumed that P n ( k i ) ( x ) P_n^{({k_i})}(x) has the sign of ε i , i = 1 , … , p , a ≦ x ≦ b {\varepsilon _i},i = 1, \ldots ,p,a \leqq x \leqq b . Theorems are obtained which describe the polynomials of best approximation, and (for p = 1 p = 1 ) establish their uniqueness. Relations to Birkhoff interpolation problems are of importance. Another tool are the sets A, where | f ( x ) − P n ( x ) | |f(x) - {P_n}(x)| attains its maximum, and the sets B i {B_i} with P n ( k i ) ( x ) = 0 P_n^{({k_i})}(x) = 0 . Conditions are discussed which these sets must satisfy for a polynomial P n {P_n} of best approximation for f. Numbers of the points of sets A, B i {B_i} are studied, the possibility of certain extreme situations established. For example, if p = 1 , k 1 = 1 , n = 2 q + 1 p = 1,{k_1} = 1,n = 2q + 1 , it is possible that | A | = 3 , | B | = n |A| = 3,|B| = n .

"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 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.

Some new theorems about differentiable, continuously differentiable, or highly differentiable restrictions of continuous and measurable real functions are presented.