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1
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∇-flat functions on manifolds

100%
Kozłowski W.
Annales Polonici Mathematici
|
2004
|
tom 84
|
nr 2
177-180
EN
We investigate ∇-flat and pointwise-∇-flat functions on affine and Riemannian manifolds. We show that the set of all ∇-flat functions on (M,∇) is a ring which has interesting properties similar to the ring of polynomial functions.
2
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L²-data Dirichlet problem for weighted form Laplacians

100%
Kozłowski W.
Colloquium Mathematicum
|
2008
|
tom 112
|
nr 1
73-88
EN
We solve the L²-data Dirichlet boundary problem for a weighted form Laplacian in the unit Euclidean ball. The solution is given explicitly as a sum of four series.
3
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Hodge type decomposition

100%
Kozłowski W.
Annales Polonici Mathematici
|
2007
|
tom 90
|
nr 2
99-104
EN
In the space $Λ^p$ of polynomial p-forms in ℝⁿ we introduce some special inner product. Let $H^p$ be the space of polynomial p-forms which are both closed and co-closed. We prove in a purely algebraic way that $Λ^p$ splits as the direct sum $d*(Λ^{p+1}) ⊕ δ*(Λ^{p-1}) ⊕ H^p$, where d* (resp. δ*) denotes the adjoint operator to d (resp. δ) with respect to that inner product.
4
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A remark on semi-∇-flat functions

100%
Kozłowski W.
Annales Polonici Mathematici
|
2006
|
tom 89
|
nr 2
147-149
EN
We give a pointwise characterization of semi-∇-flat functions on an affine manifold (M,∇).
5
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Recognition of atherosclerotic plaques and their extended dimensioning with computerized tomography angiography imaging

45%
Markiewicz T., Dziekiewicz M., Maruszyński M., Bogusławska-Walecka R., Kozłowski W.
International Journal of Applied Mathematics and Computer Science
|
2014
|
tom 24
|
nr 1
33-47
EN
In this paper the authors raise the issue of automatic discrimination of atherosclerotic plaques within an artery lumen based on numerical and statistical thresholding of Computerized Tomography Angiographic (CTA) images and their advanced dimensioning as a support for preoperative vessel assessment. For the study, a set of tomograms of the aorta, as well as the ilio-femoral and femoral arteries were examined. In each case a sequence of about 130-480 images of the artery cutoff planes were analyzed prior to their segmentation based on morphological image transformation. A crucial step in the staging of atherosclerotic alteration is recognition of the plaque in the CTA image. To solve this problem, statistical and linear fitting methods, including the least-squares approximation by polynomial and spline polynomial functions, as well as the error fitting function were used. Also, new descriptors of atherosclerotic changes, such as the lumen decrease factor, the circumference occupancy factor, and the convex plaque area factor, are proposed as a means of facilitating preoperative vessel examination. Finally, ways to reduce the computational time are discussed. The proposed methods can be very useful for automatic quantification of atherosclerotic changes visualized by CTA imaging.
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