Continuous Functions with a Dense Set of Proper Local Maxima

1985 ◽  
Vol 92 (3) ◽  
pp. 209 ◽  
Author(s):  
Vladimir Drobot ◽  
Michael Morayne
Keyword(s):  
Continuous Functions ◽  
Local Maxima ◽  
Dense Set

Continuous Functions with a Dense Set of Proper Local Maxima

1985 ◽  
Vol 92 (3) ◽  
pp. 209-211 ◽  
Author(s):  
Vladimir Drobot ◽  
Michal Morayne
Keyword(s):  
Continuous Functions ◽  
Local Maxima ◽  
Dense Set

scholarly journals Large Function Algebras with Certain Topological Properties

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Artur Bartoszewicz ◽  
Szymon Głąb
Keyword(s):  
Pointwise Convergence ◽  
Continuous Functions ◽  
Local Maxima ◽  
Zero Function ◽  
Open Set ◽  
Function Algebras ◽  
Pointwise Topology ◽  
Bernstein Set ◽  
Dense Set ◽  
Convergence Topology

LetFbe a family of continuous functions defined on a compact interval. We give a sufficient condition so thatF∪{0}contains a densec-generated free algebra; in other words,Fis denselyc-strongly algebrable. As an application we obtain densec-strong algebrability of families of nowhere Hölder functions, Bruckner-Garg functions, functions with a dense set of local maxima and local minima, and nowhere monotonous functions differentiable at all but finitely many points. We also study the problem of the existence of large closed algebras withinF∪{0}whereF⊂RXorF⊂CX. We prove that the set of perfectly everywhere surjective functions together with the zero function contains a2c-generated algebra closed in the topology of uniform convergence while it does not contain a nontrivial algebra closed in the pointwise convergence topology. We prove that an infinitely generated algebra which is closed in the pointwise convergence topology needs to contain two valued functions and infinitely valued functions. We give an example of such an algebra; namely, it was shown that there is a subalgebra ofRRwith2cgenerators which is closed in the pointwise topology and, for any functionfin this algebra, there is an open setUsuch thatf-1(U)is a Bernstein set.

Measures Induced by Units

2013 ◽  
Vol 78 (3) ◽  
pp. 886-910
Author(s):  
Giovanni Panti ◽  
Davide Ravotti
Keyword(s):  
Propositional Logic ◽  
Continuous Functions ◽  
Unit Interval ◽  
Algebraic Semantics ◽  
Absolutely Continuous ◽  
Strong Unit ◽  
Borel Measures ◽  
Maximal Spectrum ◽  
Dense Set ◽  
Finitely Presented

AbstractThe half-open real unit interval (0,1] is closed under the ordinary multiplication and its residuum. The corresponding infinite-valued propositional logic has as its equivalent algebraic semantics the equational class of cancellative hoops. Fixing a strong unit in a cancellative hoop—equivalently, in the enveloping lattice-ordered abelian group—amounts to fixing a gauge scale for falsity. In this paper we show that any strong unit in a finitely presented cancellative hoopHinduces naturally (i.e., in a representation-independent way) an automorphism-invariant positive normalized linear functional onH. SinceHis representable as a uniformly dense set of continuous functions on its maximal spectrum, such functionals—in this context usually called states—amount to automorphism-invariant finite Borel measures on the spectrum. Different choices for the unit may be algebraically unrelated (e.g., they may lie in different orbits under the automorphism group ofH), but our second main result shows that the corresponding measures are always absolutely continuous w.r.t. each other, and provides an explicit expression for the reciprocal density.

The Complexity Of Nowhere Differentiable Continuous Functions

1989 ◽  
Vol 41 (1) ◽  
pp. 83-105 ◽  
Author(s):  
T. I. Ramsamujh
Keyword(s):  
Continuous Function ◽  
Continuous Functions ◽  
Set Of Points ◽  
Dense Set

It was not always clear that there could exist a continuous function which was differentiable at no point. (Such functions are now known as nowhere differentiable continuous functions. By “differentiable” we mean having a finite derivative.) In fact in 1806 M. Ampere [2] even tried to show that no such function could exist but his reasonings were later discovered to be fallacious. Of the early attempts at constructing a nowhere differentiable continuous function mention must be made of B. Bolzano. In a manuscript dated around 1830, (see [21]) he constructed a continuous function on an interval and showed that it was not differentiable on a dense set of points. (It was later shown by K. Rychlik [21] that this function was in fact nowhere differentiable.)

scholarly journals Dense and σ-Porous Subsets in Some Families of Darboux Functions

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 759
Author(s):  
Gertruda Ivanova ◽  
Irena Domnik
Keyword(s):  
Metric Space ◽  
Continuous Functions ◽  
Darboux Functions ◽  
The Family ◽  
Nowhere Dense Set ◽  
Supremum Metric ◽  
Dense Set

G. Ivanova and E. Wagner-Bojakowska shown that the set of Darboux quasi-continuous functions with nowhere dense set of discontinuity points is dense in the metric space of Darboux quasi-continuous functions with the supremum metric. We prove that this set also is σ-strongly porous in such space. We obtain the symmetrical result for the family of strong Świątkowski functions, i.e., that the family of strong Świątkowski functions with nowhere dense set of discontinuity points is dense (thus, “large”) and σ-strongly porous (thus, asymmetrically, “small”) in the family of strong Świątkowski functions.

scholarly journals Ideal and some applications of simply open sets

2017 ◽  
Vol 13 (3) ◽  
pp. 7264-7271
Author(s):  
Arafa A Nasefa ◽  
R Mareay
Keyword(s):  
Closed Subset ◽  
Continuous Functions ◽  
Logical Space ◽  
Fundamental Properties ◽  
New Type ◽  
Nowhere Dense Set ◽  
Dense Set ◽  
The Ideal

Recently there has been some interest in the notion of a locally closed subset of a topo- logical space. In this paper, we introduce a useful characterizations of simply open sets in terms of the ideal of nowhere dense set. Also, we study a new notion of functions in topo- logical spaces known as dual simply-continuous functions and some of their fundamental properties are investigated. Finally, a new type of simply open sets is introduced.

scholarly journals Mixing for suspension flows over skew-translations and time-changes of quasi-abelian filiform nilflows

2018 ◽  
Vol 39 (12) ◽  
pp. 3407-3436 ◽  
Author(s):  
DAVIDE RAVOTTI
Keyword(s):  
Fourier Coefficients ◽  
Continuous Functions ◽  
Dense Subspace ◽  
Dimensional Torus ◽  
Smooth Functions ◽  
Suspension Flows ◽  
Time Changes ◽  
Uniquely Ergodic ◽  
Dense Set ◽  
Roof Function

We consider suspension flows over uniquely ergodic skew-translations on a $d$-dimensional torus $\mathbb{T}^{d}$ for $d\geq 2$. We prove that there exists a set $\mathscr{R}$ of smooth functions, which is dense in the space $\mathscr{C}(\mathbb{T}^{d})$ of continuous functions, such that every roof function in $\mathscr{R}$ which is not cohomologous to a constant induces a mixing suspension flow. We also construct a dense set of mixing examples which is explicitly described in terms of their Fourier coefficients. In the language of nilflows on nilmanifolds, our result implies that, for every uniquely ergodic nilflow on a quasi-abelian filiform nilmanifold, there exists a dense subspace of smooth time-changes in which mixing occurs if and only if the time-change is not cohomologous to a constant. This generalizes a theorem by Avila, Forni and Ulcigrai [Mixing for time-changes of Heisenberg nilflows. J. Differential Geom.89(3) (2011), 369–410] for the classical Heisenberg group.

Long-term Cyclic Variations of Prominences, Green and Red Coronae over Solar Cycles

2000 ◽  
Vol 179 ◽  
pp. 201-204
Author(s):  
Vojtech Rušin ◽  
Milan Minarovjech ◽  
Milan Rybanský
Keyword(s):  
Emission Lines ◽  
High Latitudes ◽  
Green Line ◽  
Solar Cycles ◽  
The South ◽  
Coronal Emission ◽  
Local Maxima ◽  
The North ◽  
Cyclic Variations

AbstractLong-term cyclic variations in the distribution of prominences and intensities of green (530.3 nm) and red (637.4 nm) coronal emission lines over solar cycles 18–23 are presented. Polar prominence branches will reach the poles at different epochs in cycle 23: the north branch at the beginning in 2002 and the south branch a year later (2003), respectively. The local maxima of intensities in the green line show both poleward- and equatorward-migrating branches. The poleward branches will reach the poles around cycle maxima like prominences, while the equatorward branches show a duration of 18 years and will end in cycle minima (2007). The red corona shows mostly equatorward branches. The possibility that these branches begin to develop at high latitudes in the preceding cycles cannot be excluded.

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