Porous Sets Research Articles

Interpolymer complexes as modifiers of the structure and properties of polymeric materials

Systematic analysis of the effect of various factors on the behavior of polymer solutions used for forming the porous structure by impregnation of nonwoven fibrous substrates in the process of fabrication of synthetic composite materials and coatings is performed. Scientific foundations of a targeted control of the phase separation of polymer solutions on their modification with interpolymer complexes of different compositions and structures in order to obtain products and materials with a porous structure and set of performance characteristics required for various applications.

Unions and ideals of locally strongly porous sets

For subsets of $\mathbb R^+ = [0,∞)$ we introduce a notion of coherently porous sets as the sets for which the upper limit in the definition of porosity at a point is attained along the same sequence. We prove that the union of two strongly porous at $0$ sets is strongly porous if and only if these sets are coherently porous. This result leads to a characteristic property of the intersection of all maximal ideals contained in the family of strongly porous at $0$ subsets of $\mathbb R^+$. It is also shown that the union of a set $A \subseteq \mathbb R^+$ with arbitrary strongly porous at $0$ set is porous at $0$ if and only if $A$ is lower porous at $0$.

Structure of porous sets in Carnot groups

We show that any Carnot group contains a closed nowhere dense set which has measure zero but is not $\sigma $-porous with respect to the Carnot–Carathéodory (CC) distance. In the first Heisenberg group, we observe that there exist sets which are porous with respect to the CC distance but not the Euclidean distance and vice-versa. In Carnot groups, we then construct a Lipschitz function which is Pansu differentiable at no point of a given $\sigma $-porous set and show preimages of open sets under the horizontal gradient are far from being porous.

Cardinal invariants of strongly porous sets

In this work we study cardinal invariants of the ideal of strongly porous sets on $\tothe{2}{\omega}$. We prove that $\add{SP}{} = \omega_1$, $\cof{SP}{} = \mathfrak{c}$ and that it is consistent that $\non{SP}{} < \add{\mathcal{N}}{}$, answering questions of Hru\v{s}\'ak and Zindulka. We also find a connection between strongly porous sets on $\tothe{2}{\omega}$ and the Martin number for $\sigma$-linked partial orders, and we use this connection to construct a model where all the Martin numbers for $\sigma$-$k$-linked forcings are mutually different.

Some results on porous set relating to ratio sets

Some results on porous set relating to ratio sets

Porosity, Differentiability and Pansu’s Theorem

We use porosity to study differentiability of Lipschitz maps on Carnot groups. Our first result states that directional derivatives of a Lipschitz function act linearly outside a $$\sigma $$ -porous set. The second result states that irregular points of a Lipschitz function form a $$\sigma $$ -porous set. We use these observations to give a new proof of Pansu’s theorem for Lipschitz maps from a general Carnot group to a Euclidean space.

Comparison of Some Subfamilies of Functions Having The Baire Property

Abstract We prove that the family Q of quasi-continuous functions is a strongly porous set in the space Ba of functions having the Baire property. Moreover, the family DQ of all Darboux quasi-continuous functions is a strongly porous set in the space DBa of Darboux functions having the Baire property. It implies that each family of all functions having the A-Darboux property is strongly porous in DBa if A has the (∗)-property.

Generalized dimension estimates for images of porous sets in metric spaces

Generalized dimension estimates for images of porous sets in metric spaces

On some modification of Darboux property

AbstractWe introduce some family of functionsf: ℝ → ℝ modifying Darboux property analogously as it was done in GRANDE, Z.: On a subclass of the family of Darboux functions, Colloq. Math. 117 (2009), 95-104, and changing approximate continuity with 𝓘-approximate continuity, i.e. continuity with respect to the 𝓘-density topology. We prove that our family is a strongly porous set in the space of Darboux functions having the Baire property and that each function from our family is quasi-continuous.

Removable Sets for Orlicz-Sobolev Spaces

We study removable sets for the Orlicz-Sobolev space W1,Ψ, for functions of the form \({\Psi }(t)=t^{p}\log ^{\lambda }(e+t)\). We show that (p,λ)-porous sets lying in a hyperplane are removable and that this result is essentially sharp.

Generalized dimension estimates for images of porous sets under monotone Sobolev mappings

We give an essentially sharp estimate in terms of generalized Hausdorff measures for images of porous sets under monotone Sobolev mappings, satisfying suitable Orlicz-Sobolev conditions.

Some properties of porouscontinuous functions

Abstract The notion of porouscontinuous function will be introduced on the base of porous set and relations between porouscontinuous, continuous and quasicontinuous functions will be investigated.

Differentiability of Lipschitz functions in Lebesgue null sets

We show that if $$n>1$$ then there exists a Lebesgue null set in $${\mathbb {R}}^{n}$$ containing a point of differentiability of each Lipschitz function $$f:{\mathbb {R}}^{n} \rightarrow {\mathbb {R}}^{n-1}$$ ; in combination with the work of others, this completes the investigation of when the classical Rademacher theorem admits a converse. Avoidance of $$\sigma $$ -porous sets, arising as irregular points of Lipschitz functions, plays a key role in the proof.

Book Review: Fréchet differentiability of Lipschitz functions and porous sets in Banach spaces

Book Review: Fréchet differentiability of Lipschitz functions and porous sets in Banach spaces

Directional Lower Porosity

We investigate differences between upper and lower poros- ity. In finite dimensional Banach spaces every upper porous set is direc- tionally upper porous. We show the situation is very different for lower porous sets; there exists a lower porous set in R 2 which is not even a countable union of directionally lower porous sets.

On some subclasses of the family of Darboux Baire 1 functions

We introduce a subclass of the family of Darboux Baire 1 functions \(f:\mathbb{R}\rightarrow\mathbb{R}\) modifying the Darboux property analogously as it was done by Z. Grande in [On a subclass of the family of Darboux functions, Colloq. Math. 17 (2009), 95-104], and replacing approximate continuity with \(\mathcal{I}\)-approximate continuity, i.e. continuity with respect to the \(\mathcal{I}\)-density topology. We prove that the family of all Darboux quasi-continuous functions from the first Baire class is a strongly porous set in the space \(\mathcal{DB}_1\) of Darboux Baire 1 functions, equipped with the supremum metric.

Avoiding σ-porous sets in Hilbert spaces

We give a constructive proof that any σ-porous subset of a Hilbert space has Lebesgue measure zero on typical C1 curves. Further, we discover that this result does not extend to all forms of porosity; we find that even power-p porous sets may meet many C1 curves in positive measure.

Surfaces meeting porous sets in positive measure

Let n>2 and X be a Banach space of dimension strictly greater than n. We show there exists a directionally porous set P in X for which the set of C^1 surfaces of dimension n meeting P in positive measure is not meager. If X is separable this leads to a decomposition of X into a countable union of directionally porous sets and a set which is null on residually many C^1 surfaces of dimension n. This is of interest in the study of certain classes of null sets used to investigate differentiability of Lipschitz functions on Banach spaces.

Porous sets and lineability of continuous functions on locally compact groups

Let G be a non-compact locally compact group. In this paper we study the size of the set {(f,g)∈A×B:f∗g is well-defined on G} where A and B are normed spaces of continuous functions on G. We also consider the problem of the spaceability of the set (C0(G)∩(C0(G)∗C0(G)))∖C00(G) and (among other results) we show that, for G=Rn, the above set is strongly c-algebrable (and, therefore, algebrable and lineable) with respect to the convolution product.

Products of non-σ-lower porous sets

In the present article we provide an example of two closed non-σ-lower porous sets A,B ⊆ ℝ such that the product A × B is lower porous. On the other hand, we prove the following: Let X and Y be topologically complete metric spaces, let A ⊆ X be a non-σ-lower porous Suslin set and let B ⊆ Y be a non-σ-porous Suslin set. Then the product A × B is non-σ-lower porous. We also provide a brief summary of some basic properties of lower porosity, including a simple characterization of Suslin non-σ-lower porous sets in topologically complete metric spaces.