On planar sets with prescribed packing dimensions of line sections

2001 ◽  
Vol 130 (3) ◽  
pp. 523-539 ◽  
Author(s):  
MARIANNA CSÖRNYEI
Keyword(s):  
Measurable Function ◽  
Packing Dimension ◽  
Borel Measurable Function ◽  
Packing Dimensions ◽  
Borel Measurable

We prove that for an arbitrary Borel measurable function f on the space of all planar lines there exists a set which intersects almost every line [lscr ] in a set of packing dimension f([lscr ]).

The identifiability of mixtures of distributions

1969 ◽  
Vol 6 (02) ◽  
pp. 389-398 ◽  
Author(s):  
G. M. Tallis
Keyword(s):  
Distribution Function ◽  
Measurable Function ◽  
Borel Measurable Function ◽  
Mixtures Of Distributions ◽  
Image Position ◽  
Borel Measurable

This paper considers aspects of the following problem. Let F(x, θ) be a distribution function, d.f., in x for all θ and a Borel measurable function of θ. Define the mixture (Robbins (1948)), where Φ is a d.f., then it is of interest to determine conditions under which F(x) and F(x, θ) uniquely determine Φ. If there is only one Φ satisfying (1), F is said to be an identifiable mixture. Usually a consistency assumption is used whereby it is presumed that there exists at least one solution to (1).

scholarly journals Estimates for the green function and existence of positive solutions for higher-order elliptic equations

2006 ◽  
Vol 2006 ◽  
pp. 1-22
Author(s):  
Imed Bachar
Keyword(s):  
Green Function ◽  
Boundary Conditions ◽  
Measurable Function ◽  
Elliptic Equations ◽  
Nonlinear Problems ◽  
Borel Measurable Function ◽  
Continuous Solutions ◽  
Existence Of Positive Solutions ◽  
The Green Function ◽  
Borel Measurable

We establish a3G-theorem for the iterated Green function of(−∆)pm, (p≥1,m≥1), on the unit ballBofℝn(n≥1)with boundary conditions(∂/∂ν)j(−∆)kmu=0on∂B, for0≤k≤p−1and0≤j≤m−1. We exploit this result to study a class of potentials𝒥m,n(p). Next, we aim at proving the existence of positive continuous solutions for the following polyharmonic nonlinear problems(−∆)pmu=h(‧,u), inD(in the sense of distributions),lim|x|→1((−∆)kmu(x)/(1−|x|)m−1)=0, for0≤k≤p−1, whereD=BorB\{0}andhis a Borel measurable function onD×(0,∞)satisfying some appropriate conditions related to𝒥m,n(p).

The identifiability of mixtures of distributions

1969 ◽  
Vol 6 (2) ◽  
pp. 389-398 ◽  
Author(s):  
G. M. Tallis
Keyword(s):  
Distribution Function ◽  
Measurable Function ◽  
Borel Measurable Function ◽  
Mixtures Of Distributions ◽  
Image Position ◽  
Borel Measurable

This paper considers aspects of the following problem. Let F(x, θ) be a distribution function, d.f., in x for all θ and a Borel measurable function of θ. Define the mixture (Robbins (1948)), where Φ is a d.f., then it is of interest to determine conditions under which F(x) and F(x, θ) uniquely determine Φ. If there is only one Φ satisfying (1), F is said to be an identifiable mixture. Usually a consistency assumption is used whereby it is presumed that there exists at least one solution to (1).

Uniform approximation by universal series on arbitrary sets

2008 ◽  
Vol 144 (1) ◽  
pp. 207-216 ◽  
Author(s):  
VANGELIS STEFANOPOULOS
Keyword(s):  
Uniform Approximation ◽  
Measurable Function ◽  
Continuous Functions ◽  
Partial Sums ◽  
Borel Measurable Function ◽  
Universal Series ◽  
Space Of Continuous Functions ◽  
Almost Everywhere ◽  
Bounded Functions ◽  
Borel Measurable

AbstractBy considering a tree-like decomposition of an arbitrary set we prove the existence of an associated series with the property that its partial sums approximate uniformly all elements in a relevant space of bounded functions. In a topological setting we show that these sums are dense in the space of continuous functions, hence in turn any Borel measurable function is the almost everywhere limit of an appropriate sequence of partial sums of the same series. The coefficients of the series may be chosen in c0, or in a weighted ℓp with 1 < p < ∞, but not in the corresponding weighted ℓ1.

CANONICAL BEHAVIOR OF BOREL FUNCTIONS ON SUPERPERFECT RECTANGLES

2001 ◽  
Vol 01 (02) ◽  
pp. 173-220 ◽  
Author(s):  
OTMAR SPINAS
Keyword(s):  
Equivalence Relation ◽  
Measurable Function ◽  
Borel Measurable Function ◽  
Canonical Function ◽  
Borel Functions ◽  
Borel Measurable

We describe a list of canonical functions from (ωω)2 to ℝ such that every Borel measurable function from (ωω)2 to ℝ, on some superperfect rectangle, induces the same equivalence relation as some canonical function.

scholarly journals Packing dimension and measure of homogeneous Cantor sets

2006 ◽  
Vol 74 (3) ◽  
pp. 443-448 ◽  
Author(s):  
H.K. Baek
Keyword(s):  
Lower Bound ◽  
Explicit Formula ◽  
Cantor Sets ◽  
Packing Dimension ◽  
Packing Measure ◽  
Hausdorff Measures ◽  
Packing Dimensions

For a class of homogeneous Cantor sets, we find an explicit formula for their packing dimensions. We then turn our attention to the value of packing measures. The exact value of packing measure for homogeneous Cantor sets has not yet been calculated even though that of Hausdorff measures was evaluated by Qu, Rao and Su in (2001). We give a reasonable lower bound for the packing measures of homogeneous Cantor sets. Our results indicate that duality does not hold between Hausdorff and packing measures.

Projection theorems for box and packing dimensions

1996 ◽  
Vol 119 (2) ◽  
pp. 287-295 ◽  
Author(s):  
K. J. Falconer ◽  
J. D. Howroyd
Keyword(s):  
Orthogonal Projection ◽  
Packing Dimension ◽  
Analytic Subset ◽  
Box Counting ◽  
Packing Dimensions ◽  
Image Position ◽  
Almost All ◽  
Projection Theorems

AbstractWe show that if E is an analytic subset of ℝn thenfor almost all m–dimensional subspaces V of ℝn, where projvE is the orthogonal projection of E onto V and dimp denotes packing dimension. The same inequality holds for lower and upper box counting dimensions, and these inequalities are the best possible ones.

Packing dimensions of sections of sets

1999 ◽  
Vol 125 (1) ◽  
pp. 89-104 ◽  
Author(s):  
K. J. FALCONER ◽  
M. JÄRVENPÄÄ
Keyword(s):  
Packing Dimension ◽  
Packing Dimensions ◽  
Local Projections

We obtain a formula for the essential supremum of the packing dimensions of the sections of sets parallel to a given subspace. This depends on a variant of packing dimension defined in terms of local projections of sets.

scholarly journals Pointwise measurable functions

2004 ◽  
Vol 95 (2) ◽  
pp. 305
Author(s):  
Herman Render ◽  
Lothar Rogge
Keyword(s):  
Large Class ◽  
Metric Space ◽  
Measurable Function ◽  
Polish Space ◽  
Topological Spaces ◽  
Range Space ◽  
Compact Metric Space ◽  
Delta Set ◽  
Measurable Functions ◽  
Borel Measurable

We introduce the new concept of pointwise measurability. It is shown in this paper that a measurable function is measurable at each point and that for a large class of topological spaces the converse also holds. Moreover it can be seen that a function which is continuous at a point is Borel-measurable at this point too. Furthermore the set of measurability points is considered. If the range space is a $\sigma$-compact metric space, then this set is a $G_{\delta}$-set; if the range space is only a Polish space this is in general not true any longer.

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