Dear Azamat, (01)
Can you explain the statements:
- "So, the second order/rank/degree tensors or higher are nothing but
properties of properties";
- "Any tensor is a variable quantity". (02)
I asserted that a tensor is a linear function (with appropriate
symmetries) over a space of tuples of one or more directions. Do you
disagree with this? (03)
At 16:20 11/07/2011, AzamatAbdoullaev wrote:
>David Leal wrote:
>"A theory of properties is interesting to me, and it may help to
>look in detail at a property which mentioned by Azamat."
>A small region of solid matter can have a property that is its
>stress tensor. A stress tensor is a single thing, and should not be thought
>>of as a "compound property" or a "property of properties".
>All physical tensors, stress-energy tensor, electromagnetic tensor,
>or stress/strain tensor, come under a second order property of
>different natural kinds. So, the second order/rank/degree tensors or
>higher are nothing by properties of properties, like as second order
>predicates, fourth order predicates, etc. If you wish, call them
>"physical predicates" or "geometric predicates". In the Reality
>book, its generally defined as "ontological predicates".
>By nature, tensors represent relationships/correspondances between
>properties (sets of vectors). Any tensor is a variable quantity, and
>it is often conflated with a matrix (as an array of elements/entries
>of rows and colums).
>In computer science, tensors are represented as multidimensional
>array/table data structure of values or variables, identified by two
>and more element indices.
>Hope it was helpful.
>----- Original Message ----- From: "David Leal"
>To: "[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx>;
>"[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx>
>Sent: Sunday, July 10, 2011 11:47 PM
>Subject: Re: [ontolog-forum] Theory of properties - stress tensor
>>Dear Azamat and others,
>>A theory of properties is interesting to me, and it may help to look
>>in detail at a property which mentioned by Azamat.
>>A small region of solid matter can have a property that is its stress
>>tensor. A stress tensor is a single thing, and should not be thought
>>of as a "compound property" or a "property of properties". The
>>following points should be noted about a stress tensor:
>>1) Inevitably stress varies from position to position within a solid
>>object. Sometimes when we choose a small enough neighbourhood of a
>>point P within a solid object, the stress within that neighbourhood
>>varies by only a small amount, so that we can say "Body B in the
>>neighbourhood of point P has stress S". But if the neighbourhood is
>>too small, then there is no concept of stress and instead only
>>molecular bonds. Hence sometimes, we cannot say that "Body B in the
>>neighbourhood of point P has stress S", because by the time we have
>>shrunk the neighbourhood sufficiently to get a stress value that does
>>not vary too much within the neighbourhood, the concept of stress has
>>disappeared. This situation can occur at a crack tip - a location of
>>particular interest to engineers.
>>Density is also a property that can only apply to "Body B in the
>>neighbourhood of a point P", if the neighbourhood is of a sufficient
>>size. If some neighbourhoods have one molecule in them and others
>>have two, then the variation of density from point to point is
>>somewhat extreme. :)
>>2) A stress tensor can be thought of as bilinear function from pairs
>>of direction vectors to force per unit area. The function can be
>>encoded as a matrix of force per unit area values. However, this
>>encoding is not the thing itself. The encoding depends upon the
>>coordinate system used to express the directions. If the coordinate
>>system is not Cartesian, then there are alternative co-variant and
>>contra-variant matrix encodings.
>>3) A stress tensor has "invariants". These are quantities that are
>>derived from the tensor, where the quantity is not dependent upon the
>>coordinate system chosen to encode the stress tensor. An example is
>>von Mises's equivalent stress.
>>It is not clear to me, how a stress tensor would be included in an
>>ontology for properties. Is a stress tensor regarded as a "quantity",
>>like a mass? Is a von Mises's equivalent stress regarded as a
>>"property" of a stress tensor?
>>At 19:44 07/07/2011, AzamatAbdoullaev wrote:
>>>AA: Again, we need to start from classification - what kinds of
>>>properties exist: formal properties (attributes and predicates) or
>>>substantial properties (real properties); intrinsic, mutual,
>>>permanent, transient, emergent, simple, complex properties, or
>>>compound properties as property of properties, like "stress tensor".
>>>By default, in any sound classification, the property to be used is
>>>a simple or basic property. The more basic property the more common
>>>it is. A theory of properties is better to develop as a theory of
>>>state space, an aggregate of properties.
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