Weak Separation Property Research Articles

Intersections of the pieces of self-similar dendrites in the plane

We prove that each self-similar dendrite in the plane has the weak separation property and that the ramification orders of its points are bounded. We show that there are self-similar dendrites in the plane that satisfy the Open Set Condition, such that several pieces may intersect in the same non-trivial subdendrite.

Lossy compression of general random variables

AbstractThis paper is concerned with the lossy compression of general random variables, specifically with rate-distortion theory and quantization of random variables taking values in general measurable spaces such as, e.g. manifolds and fractal sets. Manifold structures are prevalent in data science, e.g. in compressed sensing, machine learning, image processing and handwritten digit recognition. Fractal sets find application in image compression and in the modeling of Ethernet traffic. Our main contributions are bounds on the rate-distortion function and the quantization error. These bounds are very general and essentially only require the existence of reference measures satisfying certain regularity conditions in terms of small ball probabilities. To illustrate the wide applicability of our results, we particularize them to random variables taking values in (i) manifolds, namely, hyperspheres and Grassmannians and (ii) self-similar sets characterized by iterated function systems satisfying the weak separation property.

Hausdorff measure and Assouad dimension of generic self-conformal IFS on the line

This paper considers self-conformal iterated function systems (IFSs) on the real line whose first level cylinders overlap. In the space of self-conformal IFSs, we show that generically (in topological sense) if the attractor of such a system has Hausdorff dimension less than 1 then it has zero appropriate dimensional Hausdorff measure and its Assouad dimension is equal to 1. Our main contribution is in showing that if the cylinders intersect then the IFS generically does not satisfy the weak separation property and hence, we may apply a recent result of Angelevska, Käenmäki and Troscheit. This phenomenon holds for transversal families (in particular for the translation family) typically, in the self-similar case, in both topological and in measure theoretical sense, and in the more general self-conformal case in the topological sense.

Assouad dimension and local structure of self-similar sets with overlaps in [formula omitted

For a self-similar set in Rd that is the attractor of an iterated function system that does not verify the weak separation property, Fraser, Henderson, Olson and Robinson showed that its Assouad dimension is at least 1. In this paper, it is shown that the Assouad dimension of such a set is the sum of the dimension of the vector space spanned by the set of overlapping directions and the Assouad dimension of the orthogonal projection of the self-similar set onto the orthogonal complement of that vector space. This result is applied to give sufficient conditions on the orthogonal parts of the similarities so that the self-similar set has Assouad dimension bigger than 2, and also to answer a question posed by Farkas and Fraser. The result is also extended to the context of graph directed self-similar sets. The proof of the result relies on finding an appropriate weak tangent to the set. This tangent is used to describe partially the topological structure of self-similar sets which are both attractors of an iterated function system not satisfying the weak separation property and of an iterated functions system satisfying the open set condition.

The Other Closure and Complete Sublocales

Sublocales of a locale (frame, generalized space) can be equivalently represented by frame congruences. In this paper we discuss, a.o., the sublocales corresponding to complete congruences, that is, to frame congruences which are closed under arbitrary meets, and present a “geometric” condition for a sublocale to be complete. To this end we make use of a certain closure operator on the coframe of sublocales that allows not only to formulate the condition but also to analyze certain weak separation properties akin to subfitness or $$T_1$$ . Trivially, every open sublocale is complete. We specify a very wide class of frames, containing all the subfit ones, where there are no others. In consequence, e.g., in this class of frames, complete homomorphisms are automatically Heyting.

Inhomogeneous self-similar sets with overlaps

It is known that if the underlying iterated function system satisfies the open set condition, then the upper box dimension of an inhomogeneous self-similar set is the maximum of the upper box dimensions of the homogeneous counterpart and the condensation set. First, we prove that this ‘expected formula’ does not hold in general if there are overlaps in the construction. We demonstrate this via two different types of counterexample: the first is a family of overlapping inhomogeneous self-similar sets based upon Bernoulli convolutions; and the second applies in higher dimensions and makes use of a spectral gap property that holds for certain subgroups of $\text{SO}(d)$ for $d\geq 3$. We also obtain new upper bounds, derived using sumsets, for the upper box dimension of an inhomogeneous self-similar set which hold in general. Moreover, our counterexamples demonstrate that these bounds are optimal. In the final section we show that if the weak separation property is satisfied, that is, the overlaps are controllable, then the ‘expected formula’ does hold.

Weak separation properties for closed subgroups of locally compact groups

Three separation properties for a closed subgroup $H$ of a locally compact group $G$ are studied: (1) the existence of a bounded approximate indicator for $H$, (2) the existence of a completely bounded invariant projection of $VN\left(G\right)$ onto $VN_{H}\left(G\right)$, and (3) the approximability of the characteristic function $\chi_{H}$ by functions in $M_{cb}A\left(G\right)$ with respect to the weak$^{*}$ topology of $M_{cb}A\left(G_{d}\right)$. We show that the $H$-separation property of Kaniuth and Lau is characterized by the existence of certain bounded approximate indicators for $H$ and that a discretized analogue of the $H$-separation property is equivalent to (3). Moreover, we give a related characterization of amenability of $H$ in terms of any group $G$ containing $H$ as a closed subgroup. The weak amenability of $G$ or that $G_{d}$ satisfies the approximation property, in combination with the existence of a natural projection (in the sense of Lau and Ulger), are shown to suffice to conclude (3). Several consequences of (2) involving the cb-multiplier completion of $A\left(G\right)$ are given. Finally, a convolution technique for averaging over the closed subgroup $H$ is developed and used to weaken a condition for the existence of a bounded approximate indicator for $H$.

Long time existence for the semi-linear beam equation on irrational tori of dimension two

We prove a long time existence result for the semi-linear beam equation with small and smooth initial data. We use a regularizing effect of the structure of beam equations and a very weak separation property of the spectrum of an irrational torus under a Diophantine assumption on the radius. Our approach is inspired from a paper by Zhang about the Klein–Gordon equation with a quadratic potential.

On the Assouad dimension of self-similar sets with overlaps

It is known that, unlike the Hausdorff dimension, the Assouad dimension of a self-similar set can exceed the similarity dimension if there are overlaps in the construction. Our main result is the following precise dichotomy for self-similar sets in the line: either the weak separation property is satisfied, in which case the Hausdorff and Assouad dimensions coincide; or the weak separation property is not satisfied, in which case the Assouad dimension is maximal (equal to one). In the first case we prove that the self-similar set is Ahlfors regular, and in the second case we use the fact that if the weak separation property is not satisfied, one can approximate the identity arbitrarily well in the group generated by the similarity mappings, and this allows us to build a weak tangent that contains an interval. We also obtain results in higher dimensions and provide illustrative examples showing that the ‘equality/maximal’ dichotomy does not extend to this setting.

A topological separation condition for fractal attractors

We consider finite systems of contractive homeomorphisms of a complete metric space, which satisfy the minimality property. In general this separation condition is weaker than the strong open set condition and is not equivalent to the weak separation property. We prove that this separation condition is equivalent to the strong Markov property (see definition below). We also show that the set of N -tuples of contractive homeomorphisms, having the minimality property, is a G_\delta set in the topology of pointwise convergence of every component mapping with an additional requirement that the supremum of contraction coefficients of mappings in the sequence be strictly less than one. We find a class of N -tuples of d\times d invertible contraction matrices, which define systems of affine mappings in \mathbb R^d having the minimality property for almost every N -tuple of fixed points with respect to the Nd -dimensional Lebesgue measure.

The thinnest molecular separation sheet by graphene gates of single-walled carbon nanohorns.

Graphene is possibly the thinnest membrane that could be used as a molecular separation gate. Several techniques including absorption, cryogenic distillation, adsorption, and membrane separation have been adopted for constructing separation systems. Molecular separation using graphene as the membrane has been studied because large area synthesis of graphene is possible by chemical vapor deposition. Control of the gate sizes is necessary to achieve high separation performances in graphene membranes. The separation of molecules and ions using graphene and graphene oxide layers could be achieved by the intrinsic defects and defect donation of graphene. However, the controllability of the graphene gates is still under debate because gate size control at the picometer level is inevitable for the fabrication of the thinnest graphene membranes. In this paper, the controlled gate size in the graphene sheets in single-walled carbon nanohorns (NHs) is studied and the molecular separation ability of the graphene sheets is assessed by molecular probing with CO2, O2, N2, CH4, and SF6. Graphene sheets in NHs with different sized gates of 310, 370, and >500 pm were prepared and assessed by molecular probing. The 310 pm-gates in the graphene sheets could separate the molecules tested, whereas weak separation properties were observed for 370 pm-gates. The amount of CO2 that penetrated the 310 pm-gates was more than 35 times larger than that of CH4. These results were supported by molecular dynamics simulations of the penetration of molecules through 300, 400, and 700 pm-gates in graphene sheets. Therefore, a gas separation membrane using a 340-pm-thick graphene sheet has high potential. These findings provide unambiguous evidence of the importance of graphene gates on the picometer level. Control of the gates is the primary challenge for high-performance separation membranes made of graphene.

Preparation of Porous Polysulfone Membranes Using Ionic Liquids with Different Alkyl Chain as Additives

1-alky-3-methylimidazolium hexfluorophosphate ([n-Cnmim][PF6]) and 1-alky-3- methylimidazolium thiocyanate ([n-Cnmim]SCN) (n=4,8,16) were firstly introducing into the casting solution (PSF/NMP), porous polysulfone (PSF) membranes were successfully prepared by the wet-phase-inversion. The scanning electron microscope was utilized to visualize the cross-sections of the membranes, in order to study the influence of ionic liquids with different alkyl chain on the porous structures. The results indicate that the PSF membranes have different structures and separation properties due to ionic liquids with different alky chain. For ionic liquids with the short alkyl chain, the prepared membranes have the porous structure on cross-sections, but no or weak separation properties. However, for ionic liquids with the long alkyl chain, the prepared membranes have both porous structure and good separation properties. According to the structure of a hydrophobic straight-long alkyl and a hydrophilic limidazolium ring, the ionic liquid with long alkyl chain is considered as a surfactant, it can change not only the interface properties between the casting solution and coagulation bath, but also induce the space structures of the polymer chains.

SEPARATION PROPERTIES FOR ITERATED FUNCTION SYSTEMS OF BOUNDED DISTORTION

In this paper we study a general separation property for subsystems G, whose attractor KG is a sub-self-similar set. This is a generalization of the Lau-Ngai weak separation property for the bounded distortion case. For subsystems with positive Hausdorff measure in its similarity dimension, we characterize the subsets of KG with positive measure where the separation property may fail. We exhibit two examples of fractal sets, one not satisfying the weak separation property and whose existence was questioned by Zerner, the other having positive Hausdorff measure in its dimension and with the separation property failing on a subset of positive measure.

Finite type, open set conditions and weak separation conditions

We study graph-directed iterated function systems of finite type. We show that such an IFS of finite type induces another graph-directed IFS of finite type where every strongly connected component satisfies the open set condition. We introduce the notions of topological and geometric weak separation properties, and summarize the relationship between the different separation conditions. For the induced IFS, the similarity, growth, box and Hausdorff dimensions coincide. Finally we prove that the generalized finite-type condition for graphs implies the geometric weak separation property.

Separation properties for closures of toric orbits

A subset of a vector space is said to have the `separation property' if it separates linear forms in the following sense: for each pair of linearly independent forms on there exists a point such that and ; equivalently, each homogeneous hyperplane is linearly spanned by its intersection with . For orbit closures in representation spaces of an algebraic torus a criterion for the separation property is obtained. Strong and weak separation properties are also considered.

A generalized finite type condition for iterated function systems

We study iterated function systems (IFSs) of contractive similitudes on R d with overlaps. We introduce a generalized finite type condition which extends a more restrictive condition in [S.-M. Ngai, Y. Wang, Hausdorff dimension of self-similar sets with overlaps, J. London Math. Soc. (2) 63 (3) (2001) 655–672] and allows us to include some IFSs of contractive similitudes whose contraction ratios are not exponentially commensurable. We show that the generalized finite type condition implies the weak separation property. Under this condition, we can identify the attractor of the IFS with that of a graph-directed IFS, and by modifying a setup of Mauldin and Williams [R.D. Mauldin, S.C. Williams, Hausdorff dimension in graph directed constructions, Trans. Amer. Math. Soc. 309 (1988) 811–829], we can compute the Hausdorff dimension of the attractor in terms of the spectral radius of certain weighted incidence matrix.

Managing Credit Risk with Credit and Macro Derivatives

We use the industrial organization approach to the microeconomics of banking, augmented by uncertainty and risk aversion, to examine credit derivatives and macro derivatives as instruments to hedge credit risk for a large commercial bank. In a partial-analytic framework we distinguish between the probability of default and the loss given default, model different forms of derivatives, and derive hedge rules and strong and weak separation properties between deposit and loan decisions on the one hand and hedging decisions on the other. We also suggest how bank-specific macro derivatives could be designed from common macro indexes which serve as underlyings of recently introduced financial products.

Separation properties for graph-directed self-similar fractals

Examples of “separation properties” for iterated function systems of similitudes include: the open set condition, the weak separation property, finite type. Alternate descriptions for these properties and relations among these properties have been worked out. Here we consider the same situation for “graph-directed” iterated function systems, and provide the definitions and proofs for that setting. We deal with the case of strongly connected graphs. In many cases the definitions (and proofs) are much like the one-node case. But sometimes we have found changes were needed.

Managing Credit Risk with Credit and Macro Derivatives

We use the industrial organization approach to the microeconomics of banking, augmented by uncertainty and risk aversion, to examine credit derivatives and macro derivatives as instruments to hedge credit risk for a large commercial bank. In a partial-analytic framework we distinguish between the probability of default and the loss given default, model different forms of derivatives, and derive hedge rules and strong and weak separation properties between deposit and loan decisions on the one hand and hedging decisions on the other. We also suggest how bank-specific macro derivatives could be designed from common macro indexes which serve as underlyings of recently introduced financial products.

On the separation property of orbits in representation spaces

A subset X of a vector space V is said to have the “Separation Property” if it separates linear forms in the following sense: given a pair ( α , β ) of linearly independent linear forms on V there is a point x on X such that α ( x )=0 and β ( x )≠0. A more geometric way to express this is the following: every linear subspace H ⊂ V of codimension 1 is linearly spanned by its intersection with X . The separation property was first asked for conjugacy classes in simple Lie algebras, in connection with some classification problems. We give a general answer for orbits in representation spaces of algebraic groups and discuss in detail some special cases. We also introduce a strong and a weak separation property which come up very naturally in our setting. It turns out that these separation properties have a number of very nice features. For example, we discovered the surprising fact that in an irreducible representation of a connected semisimple group every linear hyperplane meets every orbit, and we show that a generic orbit always has the separation property.