Packing dimension and Ahlfors regularity of porous sets in metric spaces

Let X be a metric measure space with an s-regular measure μ. We prove that if A ì X{A\subset X} is r{\varrho} -porous, then dim_p(A) ￡ s-cr^s{{\rm {dim}_p}(A)\le s-c\varrho^s} where dim_p is the packing dimension and c is a positive constant which depends on s and the structure constants of μ. This is an analogue of a well known asymptotically sharp result in Euclidean spaces. We illustrate by an example that the corresponding result is not valid if μ is a doubling measure. However, in the doubling case we find a fixed N ì X{N\subset X} with μ(N) = 0 such that dim_p(A) ￡ dim_p(X)-c(log\tfrac1r)^-1r^t{{\rm {dim}_p}(A)\le{\rm {dim}_p}(X)-c(\log \tfrac1\varrho)^{-1}\varrho^t} for all r{\varrho} -porous sets A ì X\ N{A \subset X{\setminus} N} . Here c and t are constants which depend on the structure constant of μ. Finally, we characterize uniformly porous sets in complete s-regular metric spaces in terms of regular sets by verifying that A is uniformly porous if and only if there is t < s and a t-regular set F such that A ì F{A\subset F} .     

Nonlinear vibrations in a covered roll system with viscoelastic contact

A theoretical analysis on the contact vibration problem of a nip rolling system composed with two paper machine rolls is presented. It is modelled at the contact region by a two-degree-of-freedom (DOF) mass-spring-damper system. The rolling contact force in a case of viscoelastic polymer covered roll is formulated by integration of the contact stress. Regenerative chatter effects to the stability of the vibrations are considered. The perturbation technique is applied through the multi-scale method to calculate the nonlinear normal form of the governing equations to determine the stability behavior of the system. The numerical results in time domain are consistent with the bifurcation diagrams.     

Besicovitch-Federer projection theorem and geodesic flows on Riemann surfaces

We extend the Besicovitch-Federer projection theorem to transversal families of mappings. As an application we show that on a certain class of Riemann surfaces with constant negative curvature and with boundary, there exist natural 2D measures invariant under the geodesic flow having 2D supports such that their projections to the base manifold are 2D but the supports of the projections are Lebesgue negligible. In particular, the union of complete geodesics has Hausdorff dimension 2 and is Lebesgue negligible.     

Average homogeneity and dimensions of measures

We introduce the concept of average homogeneity of a measure by comparing the measure to the uniform distribution in a relatively simple way. This leads to a very general notion which may be regarded as an inverse of porosity. In this paper the emphasis is given to relations between homogeneity and dimensions of measures. First we consider the effect of homogeneity on dimensions by proving an upper bound to the Hausdorff dimension as a function of homogeneity and its order. The opposite question of how dimensions effect homogeneity is solved by giving an upper bound to homogeneity in terms of upper packing dimension. We also illustrate by examples that all our results are the best possible ones.Mathematics Subject Classification (2000): 28A75, 28A80MJ acknowledges the support of the Academy of Finland, project #48557.     

Singularity of Projections of 2-Dimensional Measures Invariant Under the Geodesic Flow

We show that on any compact Riemann surface with variable negative curvature there exists a measure which is invariant and ergodic under the geodesic flow and whose projection to the base manifold is 2-dimensional and singular with respect to the 2-dimensional Lebesgue measure.     

The use of supercritical fluid extraction for the determination of 4-deoxynivalenol in grains: The effect of the sample clean-up and analytical methods on quantitative results

Summary Deoxynivalenol (DON) is one of the trichothecene mycotoxins produced byFusarium molds in grains. Polar cosolvents in supercritical carbon dioxide (SC-CO₂) are needed to extract and isolate the polar DON moiety. This unfortunately results in the extraction of many interfering compounds from the grains into the extracts obtained by supercritical fluid extraction (SFE). Analysis of DON by high performance liquid chromatography (HPLC) using ultraviolet detection (UV) does not provide a specific detection method, although specific detection of DON can be enhanced by using purification steps after SFE. Alternatively, combining SFE with an immunoaffinity method can improve detection specificity and sample cleanup. In this study, SFE was employed to determine DON in grains and cereal products. The effectiveness of the SFE method was compared with two different solvent extraction methods. The extracted DON was quantitatively determined by HPLC-UV using external standardization or competitive enzymelinked immunosorbent assay (ELISA). In some cases, extracts were purified prior to quantitative analysis of the DON by using solvent partitioning, and/or solid phase extraction, or immunoaffinity columns. Therefore, this paper describes the analysis of DON in cereals using different extraction, cleanup and analysis methods. Names are necessary to report factually on available data: however, the USDA neither guarantees nor warrants the standard of the product, and the use of the name by USDA implies no approval of the products to the exclusion of others that may also be suitable.     

Hausdorff Dimension and Non-degenerate Families of Projections

We study parameterized families of orthogonal projections for which the dimension of the parameter space is strictly less than that of the Grassmann manifold. We answer the natural question of how much the Hausdorff dimension may decrease by verifying the best possible lower bound for the dimension of almost all projections of a finite measure. We also show that a similar result is valid for smooth families of maps from the n-dimensional Euclidean space to the m-dimensional one.     

(Non)regularity of Projections of Measures Invariant Under Geodesic Flow

We show that, unlike in the 2-dimensional case [LL], the Hausdorff dimension of a measure invariant under the geodesic flow is not necessarily preserved under the projection from the unit tangent bundle onto the base manifold if the base manifold is at least 3-dimensional. In the 2-dimensional case we reprove the preservation theorem due to Ledrappier and Lindenstrauss [LL] using the general projection formalism of Peres and Schlag [PS]. The novelty of our proof is that it illustrates the reason behind the failure of the preservation in the higher dimensional case. Finally, we show that the projected measure has fractional derivatives of order for all <(–2)/2 provided that the invariant measure has finite -energy for some >2 and the base manifold has dimension 2.MJ and ML acknowledge the support of the Academy of Finland, project #48557.Acknowledgement. We thank the referee for valuable comments clarifying the exposition.     

On Paper Machine Roll Contact with Beating Vibrations

Beating vibration phenomenon observed in paper machine roll units is studied. An analytical model of a roll system is described and an expression for the beating frequency is derived. Numerical results are compared to measurements and conclusions are made based on the observations. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)