Search results for "Tangent"

Viscosity of Polymer Solutions over the Full Range of Composition: A Thermodynamically Inspired Two-Parameter Approach

2015

The approach yields the following relation for the relative viscosity η rel as a function of polymer concentration c (mass/volume): ln ηrel = c/(1 + pc + qc2). Reduced concentrations c (defined as c = c[η], where [η] is the intrinsic viscosity) are used instead of c to incorporate thermodynamic information. The parameters p and q account for changes in the free volume of the solvent caused by the polymer. The analysis of literature data for seven very dissimilar systems discloses the following common feature: p > 0 and q < 0. This means that the curves in the plots of ln ηrel as a function of c are normally located below the tangent at low c and above it at high c. The values of p and q cor…

Differential structure associated to axiomatic Sobolev spaces

2020

The aim of this note is to explain in which sense an axiomatic Sobolev space over a general metric measure space (à la Gol’dshtein–Troyanov) induces – under suitable locality assumptions – a first-order differential structure. peerReviewed

Geometrical-mechanical artefacts for managing tangent concept

2012

Assessment of cross-flow filtration as microalgae harvesting technique prior to anaerobic digestion: Evaluation of biomass integrity and energy demand

2018

[EN] In the present study, the effect of cross-flow filtration (CFF) on the overall valorization of Chlorella spp. microalgae as biogas was assessed. The effect of CFF on microalgae cell integrity was quantified in terms of viability which was correlated with the anaerobic biodegradability. The viability dropped as the biomass concentration increased, whereas anaerobic biodegradability increased linearly with the viability reduction. It was hypothesized that a stress-induced release and further accumulation of organic polymers during CFF increased the flux resistance which promoted harsher shear-stress conditions. Furthermore, the volume reduction as the concentration increased entailed an …

Visuaalinen tangentti lukion pitkässä matematiikassa

2017

Opinnäytetyössä selvitellään lukion pitkän matematiikan opiskelijoiden käsityksiä visuaalisesta tangenttisuorasta (lyh. tangentista). Työ sisältää tietokoosteen tutkielman aihepiirin visuaalisesta tangentista. Lukio-opiskelijoiden käsityksiä tangentista esitellään Tallin 1980-luvun tietokoneavusteisesta opetuskokeilusta ja Bizan 2000-luvun tutkimuksesta. Empiirisessä tutkimuksessa tutkittiin lukion pitkän matematiikan oppikirjasarjan visuaalista tangenttia. Ihmiset ymmärtävät aistein havainnoitavat kohteet pääosin intuition avulla, kun taasen matemaattiset käsitteet voidaan ymmärtää aksioomien, määritelmien ja lauseiden pohjalta. Kuitenkin Bizan laaja tutkimus osoittaa, että lukio-opiskelij…

Geometric Aspects of Thermodynamics

2016

This chapter deals with mathematical aspects of thermodynamics most of which will be seen to be primarily of geometrical nature. Starting with a short excursion to differentiable manifolds we summarize the properties of functions, of vector fields and of one-forms on thermodynamic manifolds. This summary centers on exterior forms over Euclidean spaces and the corresponding differential calculus. In particular, one-forms provide useful tools for the analysis of thermodynamics. A theorem by Caratheodory is developed which is closely related to the second law of thermodynamics. The chapter closes with a discussion of systems which depend on two variables and for which there is an interesting a…

On one-dimensionality of metric measure spaces

2019

In this paper, we prove that a metric measure space which has at least one open set isometric to an interval, and for which the (possibly non-unique) optimal transport map exists from any absolutely continuous measure to an arbitrary measure, is a one-dimensional manifold (possibly with boundary). As an immediate corollary we obtain that if a metric measure space is a very strict $CD(K,N)$ -space or an essentially non-branching $MCP(K,N)$-space with some open set isometric to an interval, then it is a one-dimensional manifold. We also obtain the same conclusion for a metric measure space which has a point in which the Gromov-Hausdorff tangent is unique and isometric to the real line, and fo…

Differential of metric valued Sobolev maps

2020

We introduce a notion of differential of a Sobolev map between metric spaces. The differential is given in the framework of tangent and cotangent modules of metric measure spaces, developed by the first author. We prove that our notion is consistent with Kirchheim's metric differential when the source is a Euclidean space, and with the abstract differential provided by the first author when the target is $\mathbb{R}$.

Differentiaalimuodot ja niiden integrointi euklidisten avaruuksien alimonistoilla

2016

Differentiaalimuodot ovat oleellinen osa modernin matematiikan koneistoa. Niitä käytetään paitsi geometrian tutkimuksessa, myös teoreettisen fysiikan kentällä muun muassa elektrostatiikassa, mekaniikassa ja termodynamiikassa. Differentiaalimuodot elävät luonnollisesti sileillä monistoilla, jotka puolestaan esiintyvät kaikkialla, missä on tarve puhua siisteistä joukoista koordinaattien avulla. Tässä tutkielmassa tutustutaan differentiaalimuotojen perusteoriaan alkaen euklidisten avaruuksien alimonistoista. Tämän jälkeen määritellään monistojen tangentti- ja kotangenttiavaruudet, k-muotojen ulkoinen tulo, differentiaalimuotojen ulkoinen derivaatta sekä lopulta differentiaalimuodon integraali …

Algebraic singularities have maximal reductive automorphism groups

1989

LetX = On/ibe an analytic singularity where ṫ is an ideal of theC-algebraOnof germs of analytic functions on (Cn, 0). Letdenote the maximal ideal ofXandA= AutXits group of automorphisms. An abstract subgroupequipped with the structure of an algebraic group is calledalgebraic subgroupofAif the natural representations ofGon all “higher cotangent spaces”are rational. Letπbe the representation ofAon the first cotangent spaceandA1=π(A).