In a continuum, all converging series of locations ends in a limit that belongs to the continuum. The definition of change uses the limit of a converging series of locations. Change can be divergence, gradient, variation with time, or rotation (curl). A spherical shock front results in expansion of the continuum.
Deformation is part of the change of a continuum.
A continuum can have a real part and it can have a spatial part. In the real part, the real number calculation rules determine the arithmetic. In the spatial part, the spatial number calculation rules determine the arithmetic.
Differential calculus rules the change of a continuum.
Dear Leunen,
As you know a body (or solid) consists of several particles. Each particle can be subdivided into molecules, atoms and electrons etc. It is not feasible to solve an engineering problem by treating a body as a conglomeration of such discrete particles. Hence, the body is assumed to consist of a continuous distribution of matter. In other words, the body is treated as a continuum. Continuum always deals with deformable bodies, as opposed to rigid bodies.
A change in the configuration of a continuum body results in a displacement. The displacement of a body has two components: a rigid-body displacement and a deformation. A rigid-body displacement consists of a simultaneous translation and rotation of the body without changing its shape or size. Deformation implies the change in shape and/or size of the body from an initial or undeformed configuration to a current or deformed configuration.
Since we are talking about a solid body, deformation can occur because of external loads, body forces (such as gravity or electromagnetic forces), changes in temperature, etc. Therefore, the continuum is considered as deformable and not a rigid body displacement.
This is my interpretation. I hope you agree!
Regards,
Padalu
P. Pravin Kumar Venkat Rao
A set becomes a continuum when the number of elements of the set becomes uncountable. This can occur in a set of real numbers or in a set of spatial numbers. The spatial numbers can identify locations. I cannot imagine an uncountable set of deformable objects. The spatial continuum becomes deformed when it is affected by an isotropic pulse. Spatial continuums react in a special way to an isotropic pulse. They react with a spherical shock front.
Hans van Leunen
I have defined 'continuum' with respect to solid mechanics, where the fundamental assumption inscribed in the name is that materials are assumed to be homogeneous, isotropic, continuous, and independent of any coordinate system. This assumption is based on the intuitive perception that, although nanoscale studies established the discontinuous (atomic, granular, molecular) nature of matter, the macroscopic appearance of solid objects is continuous, i.e. their properties seem to vary gradually, without discontinuity.
Regards,
Padalu
P. Pravin Kumar Venkat Rao
The set of rational numbers is countable. This means that in sets that restrict to rational numbers always sufficient empty space exists in which locations that are identified by a rational number can be added. In contrast, the set of irrational numbers is uncountable. Real numbers combine rational numbers and irrational numbers. This means that in uncountable sets that restrict to real numbers never sufficient empty space exists in which locations that are identified by a rational number or an irrational number can be added. The empty space is consumed by convergent series of real numbers that end in a limit that belongs to the real numbers. In the uncountable set of real numbers, all members touch each other.
This also happens for spatial numbers. There the effect of touching is stronger because it acts in all directions. The continuum acts as a sticky medium.
This behavior is typical for sets of locations in space and does not apply to condensed matter. The phenomenon is counterintuitive. You can accept it or you can deny it. Set theorists took great trouble to prove the existence of the phenomenon. They failed, but they also could not disprove it. The existence of continuums is very handy because it enables differential calculus, which is the calculus of change of a continuum.
From mathematical point of view, a deformation of a continuum denoted by S( S is an uncountable subset of a space X) is a homeomorphism(a bicontinuous function) from S to U, where U is an uncountable subset of another space Y. Y may be X itself.
A bicontinuous function f from S to U is a continuous invertible function, with continuous inverse. We say that S was deformed in U.
The "deformation" f may have some other extra properties, as differentiability of a certain order, for example.
Mathematics offer a multitude of strange(counterintuitive) examples of continuums which can be deformed from each other.
PS Let R be the set of rationals from the interval [0,1], I the set of irrationals from [0,1] and m the Lebesgue measure on R(the real numbers set). We have m(R)=0, but m(I)=m([0,1])=1.
Dinu Teodorescu
The system of Hilbert spaces that share the same underlying vector space contains one background Hilbert space which owns a companion non-separable Hilbert space that embeds its separable partner. All other members of the system are separable. Only the non-separable Hilbert space supports continuums as eigenspaces of operators. One of these represents our universe. All other members of the system embed their geometric center into the universe. These embeddings represent elementary fermions. See" Advanced Hilbert space technology" in https://vixra.org/author/j_a_j_van_leunen
Hans van Leunen
Dear Professor Van Leunen,
Your Hilbert Book Model Project is very, very interesting. I have some question about!
I understand partially the affirmation that our universe can be represented as an eigenspace of a normal operator between two Hilbert spaces. Why the operator must be normal? What is the significance of the fact that the operator commutes with his adjoint(i.e. the operator is normal)? Can he be self-adjoint( self-adjoint operators are normal !)? What is the importance of the structure of spectrum, particullary of the structure of eigenvalues set of operator? For self-adjoint operators we have good properties of the set of eigenvalues.
At different moments in time, s>t, can we consider the universe at the moment t as the eigenspace(as Hilbert space) Ht(T) and the universe at the moment s as the eigenspace Hs(T) (T is the considered normal operator), Hs(T) being a deformation of Ht(T) in a mathematical sense?
This "book of Hilbert spaces" can be considered as a manifold( a complex manifold) - every "page" of book is homeomorphic with a special subset(closed) of some set(here, a Hilbert space)?
Your construction seems to be related to Riemann surfaces!
Sorry if my questions are inappropiate! That would mean I didn't understand anything regarding your project!
Dinu Teodorescu
A Hilbert space maps a bra-vector space onto a ket_Hilbert space. Operators manage eigenspaces that the Dirac's bra-ket combinations define. A self-adjoint operator is a normal operator that provides real eigenvalues. A normal operator can provide complex number valued eigenvalues and quaternionic eigenvalues. The eigenvectors of a normal operator form a basis of the complete Hilbert space. Only non-separable Hilbert spaces can support continuums in their eigenspaces. The paper"Advanced Hilbert space technology"; treats all aspects of Hilbert spaces that you need to understand the basics of the universe. See https://vixra.org/author/j_a_j_van_leunen
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