369 research outputs found
A sharp exceptional set estimate for visibility
No full textA Borel set BRn is visible from xRn, if the radial projection of B with base point x has positive Hn-1 measure. I prove that if dimB>n-1, then B is visible from every point xRn\E, where E is an exceptional set with dimension dimE2(n-1)-dimB. This is the sharp bound for all n2. Many parts of the proof were already contained in a recent previous paper by P. Mattila and the author, where a weaker bound for dimE was derived as a corollary from a certain slicing theorem. Here, no improvement to the slicing result is obtained; in brief, the main observation of the present paper is that the proof method gives the optimal result, when applied directly to the visibility problem.Peer reviewe
On restricted families of projections in ℝ3
No full textWe study projections onto non-degenerate one-dimensional families of lines and planes in R 3 . Using the classical potential theoretic approach of R. Kaufman, one can show that the Hausdorff dimension of at most 12 -dimensional sets [Math Processing Error] is typically preserved under one-dimensional families of projections onto lines. We improve the result by an ε , proving that if [Math Processing Error], then the packing dimension of the projections is almost surely at least [Math Processing Error]. For projections onto planes, we obtain a similar bound, with the threshold 12 replaced by 1 . In the special case of self-similar sets [Math Processing Error] without rotations, we obtain a full Marstrand-type projection theorem for 1-parameter families of projections onto lines. The [Math Processing Error] case of the result follows from recent work of M. Hochman, but the [Math Processing Error] part is new: with this assumption, we prove that the projections have positive length almost surely
Absolute continuity in families of parametrised non-homogeneous self-similar measures
No full textPeer reviewe
On the dimension of visible parts
No full textI prove that the visible parts of a compact set in Rn, n > 2, have Hausdorff dimension at most n -1 50 n from almost every direction.Peer reviewe
Detergent market research & market entry plan for Haarla Oy
No full textThis thesis was commissioned by Haarla Oy. The company has been in the chemistry business for decades, providing customers with raw materials, mostly in Finland and Europe. They have not had segmented detergent side for their chemistry operations, and now they wanted to differentiate the segment. The objective was to research the detergent industry market in Finland and make a market entry plan for Haarla.
The author works for Haarla himself, being the person hired to expand the chemical side. He has familiarized himself with Haarla’s operations and
customers by working for the company. The author has idea of the market’s trends, values, and perceptions.
Based on the research, it was found that the Finnish detergent manufacturing industry is competitive but standing on strong foundations.
The findings suggest that Haarla has capacity to enter the desired detergent manufacturing supplier market. By utilizing their strong economic situation, Haarla can expand to desired detergent market. To maximize their potential,
Haarla must work toward achieving representations and focusing their efforts where needed as they advance
Characterising the big pieces of Lipschitz graphs property using projections
No full textWe characterise the big pieces of Lipschitz graphs property in terms of projections. Roughly speaking, we prove that if a large subset of an n-Ahlfors-David regular set E subset of R-d has plenty of projections in L-2, then a large part of E is contained in a single Lipschitz graph. This is closely related to a question of G. David and S. Semmes.Peer reviewe
Boundedness of the density normalised Jones' square function does not imply 1-rectifiability
No full textRecently, M. Badger and R. Schul proved [4] that for a 1-rectifiable Radon measure mu, the density weighted Jones' square function. (J) over tilde (2)(x) = Sigma(Q is an element of D) (l(Q) is finite for mu-a.e Answering a question of Badger Schul, we show that the converse is not true. Given epsilon > 0, we construct a Radon probability measure on [0,1](2) subset of R-2 with the properties that (J) over tilde (2)(x)Peer reviewe
Curve packing and modulus estimates
No full textA family of planar curves is called a Moser family if it contains an isometric copy of every rectifiable curve in of length one. The classical ``worm problem'' of L. Moser from 1966 asks for the least area covered by the curves in any Moser family. In 1979, J. M. Marstrand proved that the answer is not zero: the union of curves in a Moser family always has area at least for some small absolute constant . We strengthen Marstrand's result by showing that for , the -modulus of a Moser family of curves is at least $ c_{p} > 0
The Assouad dimensions of projections of planar sets
No full textThe first named author is supported by a Leverhulme Trust Research Fellowship and the second named author is supported by the Academy of Finland through the grant Restricted families of projections and connections to Kakeya type problems, grant number 274512.We consider the Assouad dimensions of orthogonal projections of planar sets onto lines. Our investigation covers both general and self-similar sets. For general sets, the main result is the following: if a set in the plane has Assouad dimension s ∈ [0, 2], then the projections have Assouad dimension at least min{1, s} almost surely. Compared to the famous analogue for Hausdorff dimension – namely Marstrand’s Projection Theorem – a striking difference is that the words ‘at least’cannot be dispensed with: in fact, for many planar self-similar sets of dimension s < 1, we prove that the Assouad dimension of projections can attain both values sand 1 for a set of directions of positive measure. For self-similar sets, our investigation splits naturally into two cases: when the group of rotations is discrete, and when it is dense. In the ‘discrete rotations’ case we prove the following dichotomy for any given projection: either the Hausdorff measure is positive in the Hausdorff dimension, in which case the Hausdorff and Assouad dimensions coincide; or the Hausdorff measure is zero in the Hausdorff dimension,in which case the Assouad dimension is equal to 1. In the ‘dense rotations’ case we prove that every projection has Assouad dimension equal to one, assuming that the planar set is not a singleton. As another application of our results, we show that there is no Falconer’s Theorem for Assouad dimension. More precisely, the Assouad dimension of a self-similar (or self-affine) set is not in general almost surely constant when one randomises the translation vectors.Peer reviewe
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