What is behind the physical phenomenon of the velocity; of the force; there is the mathematical concept of the vector. This is a new concept, since force has direction, sense, and magnitude, and we accept the physical principle that the forces exerted on a body can be added to the rule of the parallelogram. This is the first axiom of Newton. Newton essentially requires that the power is a " vectorial " size , without writing clearly , and Galileo that applies the principle of the independence of forces .
Using Mathematical Foundations To Study The Equivalence Between Mass And Ener...QUESTJOURNAL
Abstract:This paper study the equivalence between mass and energy in special relativity, using mathematical methods to connect this work by de-Broglie equation, in this work found the relation between the momentum and energy, It has also been connect the mass and momentum and the speed of light in the energy equation, moreover it has been found that the relative served as an answer to a logical relationship de-Broglie through equivalence relationship between mass and energy.
The Gödel incompleteness can be modeled on the alleged incompleteness of quantum mechanics
Then the proved and even experimentally confirmed completeness of quantum mechanics can be reversely interpreted as a strategy of completeness as to the foundation of mathematics
Infinity is equivalent to a second and independent finiteness
Two independent Peano arithmetics as well as one single Hilbert space as an unification of geometry and arithmetic are sufficient to the self-foundation of mathematics
Quantum mechanics is inseparable from the foundation of mathematics and thus from set theory particularly
A talk presented at the University of New South Wales on the occasion of Ian Sloan's 80th birthday, remembering our work together and thinking about how math is used in science.
Using Mathematical Foundations To Study The Equivalence Between Mass And Ener...QUESTJOURNAL
Abstract:This paper study the equivalence between mass and energy in special relativity, using mathematical methods to connect this work by de-Broglie equation, in this work found the relation between the momentum and energy, It has also been connect the mass and momentum and the speed of light in the energy equation, moreover it has been found that the relative served as an answer to a logical relationship de-Broglie through equivalence relationship between mass and energy.
The Gödel incompleteness can be modeled on the alleged incompleteness of quantum mechanics
Then the proved and even experimentally confirmed completeness of quantum mechanics can be reversely interpreted as a strategy of completeness as to the foundation of mathematics
Infinity is equivalent to a second and independent finiteness
Two independent Peano arithmetics as well as one single Hilbert space as an unification of geometry and arithmetic are sufficient to the self-foundation of mathematics
Quantum mechanics is inseparable from the foundation of mathematics and thus from set theory particularly
A talk presented at the University of New South Wales on the occasion of Ian Sloan's 80th birthday, remembering our work together and thinking about how math is used in science.
2-1 POSITION, DISPLACEMENT, AND AVERAGE VELOCITY
After reading this module, you should be able to …
2.01 Identify that if all parts of an object move in the same direction and at the same rate, we can treat the object as if it
were a (point-like) particle. (This chapter is about the motion of such objects.)
2.02 Identify that the position of a particle is its location as
read on a scaled axis, such as an x-axis.
2.03 Apply the relationship between a particle’s
displacement and its initial and final positions.
2.04 Apply the relationship between a particle’s average
velocity, its displacement, and the time interval for that
displacement.
2.05 Apply the relationship between a particle’s average
speed, the total distance it moves, and the time interval for
the motion.
2.06 Given a graph of a particle’s position versus time,
determine the average velocity between any two particular
times.
2-1 POSITION, DISPLACEMENT, AND AVERAGE VELOCITY
After reading this module, you should be able to . . .
2.07 Given a particle’s position as a function of time,
calculate the instantaneous velocity for any particular time.
2.08 Given a graph of a particle’s position versus time, determine the instantaneous velocity for any particular time.
2.09 Identify speed as the magnitude of the instantaneous
velocity.
etc......
3-1 VECTORS AND THEIR COMPONENTS
After reading this module, you should be able to . . .
3.01 Add vectors by drawing them in head-to-tail arrangements, applying the commutative and associative laws.
3.02 Subtract a vector from a second one.
3.03 Calculate the components of a vector on a given coordinate system, showing them in a drawing.
3.04 Given the components of a vector, draw the vector
and determine its magnitude and orientation.
3.05 Convert angle measures between degrees and radians.
3-2 UNIT VECTORS, ADDING VECTORS BY COMPONENTS
After reading this module, you should be able to . . .
3.06 Convert a vector between magnitude-angle and unit vector notations.
3.07 Add and subtract vectors in magnitude-angle notation
and in unit-vector notation.
3.08 Identify that, for a given vector, rotating the coordinate
system about the origin can change the vector’s components but not the vector itself.
etc...
What is quantum information? Information symmetry and mechanical motionVasil Penchev
The concept of quantum information is introduced as both normed superposition of two orthogonal subspaces of the separable complex Hilbert space and invariance of Hamilton and Lagrange representation of any mechanical system. The base is the isomorphism of the standard introduction and the representation of a qubit to a 3D unit ball, in which two points are chosen.
The separable complex Hilbert space is considered as the free variable of quantum information and any point in it (a wave function describing a state of a quantum system) as its value as the bound variable.
A qubit is equivalent to the generalization of ‘bit’ from the set of two equally probable alternatives to an infinite set of alternatives. Then, that Hilbert space is considered as a generalization of Peano arithmetic where any unit is substituted by a qubit and thus the set of natural number is mappable within any qubit as the complex internal structure of the unit or a different state of it. Thus, any mathematical structure being reducible to set theory is representable as a set of wave functions and a subspace of the separable complex Hilbert space, and it can be identified as the category of all categories for any functor represents an operator transforming a set (or subspace) of the separable complex Hilbert space into another. Thus, category theory is isomorphic to the Hilbert-space representation of set theory & Peano arithmetic as above.
Given any value of quantum information, i.e. a point in the separable complex Hilbert space, it always admits two equally acceptable interpretations: the one is physical, the other is mathematical. The former is a wave function as the exhausted description of a certain state of a certain quantum system. The latter chooses a certain mathematical structure among a certain category. Thus there is no way to be distinguished a mathematical structure from a physical state for both are described exhaustedly as a value of quantum information. This statement in turn can be utilized to be defined quantum information by the identity of any mathematical structure to a physical state, and also vice versa. Further, that definition is equivalent to both standard definition as the normed superposition and invariance of Hamilton and Lagrange interpretation of mechanical motion introduced in the beginning of the paper.
Then, the concept of information symmetry can be involved as the symmetry between three elements or two pairs of elements: Lagrange representation and each counterpart of the pair of Hamilton representation. The sense and meaning of information symmetry may be visualized by a single (quantum) bit and its interpretation as both (privileged) reference frame and the symmetries of the Standard model.
On the Mathematical Structure of the Fundamental Forces of NatureRamin (A.) Zahedi
The main idea of this article is based on my previous articles (references [1], [2], [3]). In this work by introducing a new mathematical approach based on the algebraic structure of integers (the domain of integers), and assuming the “discreteness” of physical quantities such as the components of the relativistic n-momentum, we derive all the mathematical laws governing the fundamental forces of nature. These obtained laws that are unique, distinct and in the form of the complex tensor equations, represent the force of gravity, the electromagnetic (including electroweak) force, and the (strong) nuclear force (and only these three kinds of forces, for all dimensions D ≥2). Each derived tensor equation contains the term of the mass m_0 (as the invariant mass of the supposed force carrier particle), as well as the term of the external current (as the external source of the force field). In some special cases, these tensor equations are turned into the wave equations that are similar to the Pauli and Dirac equations. In fact, the mathematical laws obtained in this paper, are the corrected and generalized forms of the current field equations including Maxwell equations, Yang-Mils equations and Einstein equations, as well as (in some special conditions) Pauli equation, Dirac equation, and so on. A direct proof of the absence of magnetic monopoles in nature is one of the outcomes of this research, according to the unique formulations of the laws of the fundamental forces that we have derived.
Keywords: Foundations of Physics, Ontology, Discrete Physics, Discrete Mathematics, The Fundamental Forces of Nature.
Comments: 51 Pages. Expanded version of my previous articles:
Ramin (A.) Zahedi, "Linearization Method in the Ring Theory," Bulletin of the Lebedev Physics Institute, Springer-Verlag, No. 5-6, 1997;
Ramin (A.) Zahedi, "On the Connection Between Methods of the Ring Theory and the Group Approach", Bulletin of the Lebedev Physics Institute, Springer-Verlag, No. 7-8, 1997.
PACS Classifications: 04.20.Cv, 04.50.Kd, 04.90.+e, 04.62.+v, 02.10.Hh, 02.10.Yn, 02.20.Bb, 02.90.+p, 03.50.-z, 03.65.Fd, 03.65.Pm, 03.50.Kk, 12.40.-y, 12.60.-i, 12.10.Dm, 12.10.-g.
External URL: http://arXiv.org/abs/1501.01373. (arXiv:1501.01373 [physics.gen-ph])
Copyright: CC Attribution-NonCommercial-NoDerivs 4.0 International
License URL: https://creativecommons.org/licenses/by-nc-nd/4.0/
The Ancient-Greek Special Problems, as the Quantization Moulds of SpacesScientific Review SR
The Special Problems of E-geometry consist the , Mould Quantization , of Euclidean
Geometry in it , to become → Monad , through mould of Space –Anti-space in itself , which is the
material dipole in inner monad Structure as the Electromagnetic cycloidal field → Linearly , through
mould of Parallel Theorem [44-45], which are the equal distances between points of parallel and line →
In Plane , through mould of Squaring the circle [46] , where the two equal and perpendicular monads
consist a Plane acquiring the common Plane-meter → and in Space (volume) , through mould of the
Duplication of the Cube [46] , where any two Unequal perpendicular monads acquire the common
Space-meter to be twice each other , as analytically all methods are proved and explained . [39-41]. The
Unification of Space and Energy becomes through [STPL] Geometrical Mould Mechanism of Elements ,
the minimum Energy-Quanta , In monads → Particles , Anti-particles , Bosons , Gravity –Force , Gravity -Field , Photons , Dark Matter , and Dark-Energy ,consisting Material Dipoles in inner monad Structures
i.e. the Electromagnetic Cycloidal Field of monads. Euclid’s elements consist of assuming a small set of
intuitively appealing axioms , proving many other propositions . Because nobody until [9] succeeded to
prove the parallel postulate by means of pure geometric logic , many self-consistent non - Euclidean
geometries have been discovered , based on Definitions , Axioms or Postulates , in order that none of them
contradicts any of the other postulates . In [39] the only Space-Energy geometry is Euclidean , agreeing
with the Physical reality , on unit AB = Segment which is The Electromagnetic field of the Quantized on
AB Energy Space Vector , on the contrary to the General relativity of Space-time which is based on the
rays of the non-Euclidean geometries to the limited velocity of light and Planck`s cavity . Euclidean
geometry elucidated the definitions of geometry-content ,{ for Point , Segment , Straight Line , Plane ,
Volume, Space [S] , Anti-space [AS] , Sub-space [SS] , Cave, Space-Anti-Space Mechanism of the Six-Triple-Points-Line , that produces and transfers Points of Spaces , Anti -Spaces and Sub-Spaces in a
Common Inertial Sub-Space and a cylinder ,Gravity field [MFMF] , Particles } and describes the Space-Energy beyond Plank´s length level [ Gravity Length 3,969.10 ̄ 62 m ] , reaching the Point = L
v
=
e
i.
Nπ
2
b=10 N= − ∞
m = 0 m , which is nothing and zero space .[43-46] -The Geometrical solution of the
Special Problems is now presented
Dimensional analysis offers a method for reducing complex physical problems to the simplest (that is, most economical) form prior to obtaining a quantitative answer.
Fundamentals of Physics "MOTION IN TWO AND THREE DIMENSIONS"Muhammad Faizan Musa
4-1 POSITION AND DISPLACEMENT
After reading this module, you should be able to . . .
4.01 Draw two-dimensional and three-dimensional position
vectors for a particle, indicating the components along the
axes of a coordinate system.
4.02 On a coordinate system, determine the direction and
magnitude of a particle’s position vector from its components, and vice versa.
4.03 Apply the relationship between a particle’s displacement vector and its initial and final position vectors.
4-2 AVERAGE VELOCITY AND INSTANTANEOUS VELOCITY
After reading this module, you should be able to . . .
4.04 Identify that velocity is a vector quantity and thus has
both magnitude and direction and also has components.
4.05 Draw two-dimensional and three-dimensional velocity
vectors for a particle, indicating the components along the
axes of the coordinate system.
4.06 In magnitude-angle and unit-vector notations, relate a particle’s initial and final position vectors, the time interval between
those positions, and the particle’s average velocity vector.
4.07 Given a particle’s position vector as a function of time,
determine its (instantaneous) velocity vector. etc...
2-1 POSITION, DISPLACEMENT, AND AVERAGE VELOCITY
After reading this module, you should be able to …
2.01 Identify that if all parts of an object move in the same direction and at the same rate, we can treat the object as if it
were a (point-like) particle. (This chapter is about the motion of such objects.)
2.02 Identify that the position of a particle is its location as
read on a scaled axis, such as an x-axis.
2.03 Apply the relationship between a particle’s
displacement and its initial and final positions.
2.04 Apply the relationship between a particle’s average
velocity, its displacement, and the time interval for that
displacement.
2.05 Apply the relationship between a particle’s average
speed, the total distance it moves, and the time interval for
the motion.
2.06 Given a graph of a particle’s position versus time,
determine the average velocity between any two particular
times.
2-1 POSITION, DISPLACEMENT, AND AVERAGE VELOCITY
After reading this module, you should be able to . . .
2.07 Given a particle’s position as a function of time,
calculate the instantaneous velocity for any particular time.
2.08 Given a graph of a particle’s position versus time, determine the instantaneous velocity for any particular time.
2.09 Identify speed as the magnitude of the instantaneous
velocity.
etc......
3-1 VECTORS AND THEIR COMPONENTS
After reading this module, you should be able to . . .
3.01 Add vectors by drawing them in head-to-tail arrangements, applying the commutative and associative laws.
3.02 Subtract a vector from a second one.
3.03 Calculate the components of a vector on a given coordinate system, showing them in a drawing.
3.04 Given the components of a vector, draw the vector
and determine its magnitude and orientation.
3.05 Convert angle measures between degrees and radians.
3-2 UNIT VECTORS, ADDING VECTORS BY COMPONENTS
After reading this module, you should be able to . . .
3.06 Convert a vector between magnitude-angle and unit vector notations.
3.07 Add and subtract vectors in magnitude-angle notation
and in unit-vector notation.
3.08 Identify that, for a given vector, rotating the coordinate
system about the origin can change the vector’s components but not the vector itself.
etc...
What is quantum information? Information symmetry and mechanical motionVasil Penchev
The concept of quantum information is introduced as both normed superposition of two orthogonal subspaces of the separable complex Hilbert space and invariance of Hamilton and Lagrange representation of any mechanical system. The base is the isomorphism of the standard introduction and the representation of a qubit to a 3D unit ball, in which two points are chosen.
The separable complex Hilbert space is considered as the free variable of quantum information and any point in it (a wave function describing a state of a quantum system) as its value as the bound variable.
A qubit is equivalent to the generalization of ‘bit’ from the set of two equally probable alternatives to an infinite set of alternatives. Then, that Hilbert space is considered as a generalization of Peano arithmetic where any unit is substituted by a qubit and thus the set of natural number is mappable within any qubit as the complex internal structure of the unit or a different state of it. Thus, any mathematical structure being reducible to set theory is representable as a set of wave functions and a subspace of the separable complex Hilbert space, and it can be identified as the category of all categories for any functor represents an operator transforming a set (or subspace) of the separable complex Hilbert space into another. Thus, category theory is isomorphic to the Hilbert-space representation of set theory & Peano arithmetic as above.
Given any value of quantum information, i.e. a point in the separable complex Hilbert space, it always admits two equally acceptable interpretations: the one is physical, the other is mathematical. The former is a wave function as the exhausted description of a certain state of a certain quantum system. The latter chooses a certain mathematical structure among a certain category. Thus there is no way to be distinguished a mathematical structure from a physical state for both are described exhaustedly as a value of quantum information. This statement in turn can be utilized to be defined quantum information by the identity of any mathematical structure to a physical state, and also vice versa. Further, that definition is equivalent to both standard definition as the normed superposition and invariance of Hamilton and Lagrange interpretation of mechanical motion introduced in the beginning of the paper.
Then, the concept of information symmetry can be involved as the symmetry between three elements or two pairs of elements: Lagrange representation and each counterpart of the pair of Hamilton representation. The sense and meaning of information symmetry may be visualized by a single (quantum) bit and its interpretation as both (privileged) reference frame and the symmetries of the Standard model.
On the Mathematical Structure of the Fundamental Forces of NatureRamin (A.) Zahedi
The main idea of this article is based on my previous articles (references [1], [2], [3]). In this work by introducing a new mathematical approach based on the algebraic structure of integers (the domain of integers), and assuming the “discreteness” of physical quantities such as the components of the relativistic n-momentum, we derive all the mathematical laws governing the fundamental forces of nature. These obtained laws that are unique, distinct and in the form of the complex tensor equations, represent the force of gravity, the electromagnetic (including electroweak) force, and the (strong) nuclear force (and only these three kinds of forces, for all dimensions D ≥2). Each derived tensor equation contains the term of the mass m_0 (as the invariant mass of the supposed force carrier particle), as well as the term of the external current (as the external source of the force field). In some special cases, these tensor equations are turned into the wave equations that are similar to the Pauli and Dirac equations. In fact, the mathematical laws obtained in this paper, are the corrected and generalized forms of the current field equations including Maxwell equations, Yang-Mils equations and Einstein equations, as well as (in some special conditions) Pauli equation, Dirac equation, and so on. A direct proof of the absence of magnetic monopoles in nature is one of the outcomes of this research, according to the unique formulations of the laws of the fundamental forces that we have derived.
Keywords: Foundations of Physics, Ontology, Discrete Physics, Discrete Mathematics, The Fundamental Forces of Nature.
Comments: 51 Pages. Expanded version of my previous articles:
Ramin (A.) Zahedi, "Linearization Method in the Ring Theory," Bulletin of the Lebedev Physics Institute, Springer-Verlag, No. 5-6, 1997;
Ramin (A.) Zahedi, "On the Connection Between Methods of the Ring Theory and the Group Approach", Bulletin of the Lebedev Physics Institute, Springer-Verlag, No. 7-8, 1997.
PACS Classifications: 04.20.Cv, 04.50.Kd, 04.90.+e, 04.62.+v, 02.10.Hh, 02.10.Yn, 02.20.Bb, 02.90.+p, 03.50.-z, 03.65.Fd, 03.65.Pm, 03.50.Kk, 12.40.-y, 12.60.-i, 12.10.Dm, 12.10.-g.
External URL: http://arXiv.org/abs/1501.01373. (arXiv:1501.01373 [physics.gen-ph])
Copyright: CC Attribution-NonCommercial-NoDerivs 4.0 International
License URL: https://creativecommons.org/licenses/by-nc-nd/4.0/
The Ancient-Greek Special Problems, as the Quantization Moulds of SpacesScientific Review SR
The Special Problems of E-geometry consist the , Mould Quantization , of Euclidean
Geometry in it , to become → Monad , through mould of Space –Anti-space in itself , which is the
material dipole in inner monad Structure as the Electromagnetic cycloidal field → Linearly , through
mould of Parallel Theorem [44-45], which are the equal distances between points of parallel and line →
In Plane , through mould of Squaring the circle [46] , where the two equal and perpendicular monads
consist a Plane acquiring the common Plane-meter → and in Space (volume) , through mould of the
Duplication of the Cube [46] , where any two Unequal perpendicular monads acquire the common
Space-meter to be twice each other , as analytically all methods are proved and explained . [39-41]. The
Unification of Space and Energy becomes through [STPL] Geometrical Mould Mechanism of Elements ,
the minimum Energy-Quanta , In monads → Particles , Anti-particles , Bosons , Gravity –Force , Gravity -Field , Photons , Dark Matter , and Dark-Energy ,consisting Material Dipoles in inner monad Structures
i.e. the Electromagnetic Cycloidal Field of monads. Euclid’s elements consist of assuming a small set of
intuitively appealing axioms , proving many other propositions . Because nobody until [9] succeeded to
prove the parallel postulate by means of pure geometric logic , many self-consistent non - Euclidean
geometries have been discovered , based on Definitions , Axioms or Postulates , in order that none of them
contradicts any of the other postulates . In [39] the only Space-Energy geometry is Euclidean , agreeing
with the Physical reality , on unit AB = Segment which is The Electromagnetic field of the Quantized on
AB Energy Space Vector , on the contrary to the General relativity of Space-time which is based on the
rays of the non-Euclidean geometries to the limited velocity of light and Planck`s cavity . Euclidean
geometry elucidated the definitions of geometry-content ,{ for Point , Segment , Straight Line , Plane ,
Volume, Space [S] , Anti-space [AS] , Sub-space [SS] , Cave, Space-Anti-Space Mechanism of the Six-Triple-Points-Line , that produces and transfers Points of Spaces , Anti -Spaces and Sub-Spaces in a
Common Inertial Sub-Space and a cylinder ,Gravity field [MFMF] , Particles } and describes the Space-Energy beyond Plank´s length level [ Gravity Length 3,969.10 ̄ 62 m ] , reaching the Point = L
v
=
e
i.
Nπ
2
b=10 N= − ∞
m = 0 m , which is nothing and zero space .[43-46] -The Geometrical solution of the
Special Problems is now presented
Dimensional analysis offers a method for reducing complex physical problems to the simplest (that is, most economical) form prior to obtaining a quantitative answer.
Fundamentals of Physics "MOTION IN TWO AND THREE DIMENSIONS"Muhammad Faizan Musa
4-1 POSITION AND DISPLACEMENT
After reading this module, you should be able to . . .
4.01 Draw two-dimensional and three-dimensional position
vectors for a particle, indicating the components along the
axes of a coordinate system.
4.02 On a coordinate system, determine the direction and
magnitude of a particle’s position vector from its components, and vice versa.
4.03 Apply the relationship between a particle’s displacement vector and its initial and final position vectors.
4-2 AVERAGE VELOCITY AND INSTANTANEOUS VELOCITY
After reading this module, you should be able to . . .
4.04 Identify that velocity is a vector quantity and thus has
both magnitude and direction and also has components.
4.05 Draw two-dimensional and three-dimensional velocity
vectors for a particle, indicating the components along the
axes of the coordinate system.
4.06 In magnitude-angle and unit-vector notations, relate a particle’s initial and final position vectors, the time interval between
those positions, and the particle’s average velocity vector.
4.07 Given a particle’s position vector as a function of time,
determine its (instantaneous) velocity vector. etc...
Quantum Geometry: A reunion of math and physicsRafa Spoladore
Caltech's professor Anton Kapustin "describes the relationship between mathematics and physics, mathematicians and physicists, and so on. He focuses on the noncommutative character of algebras of observables in quantum mechanics." via http://motls.blogspot.com.br/2014/11/anton-kapustin-quantum-geometry-reunion.html
Physics and Measurement. VECTORS. IntroductionAikombi
Like all other sciences, physics is based on experimental observations and quantitative measurements. The main objectives of physics are to identify a limited number of fundamental laws that govern natural phenomena and use them to develop theories that can predict the
results of future experiments. The fundamental laws used in developing theories are expressed in the language of mathematics, the tool that provides a bridge between theory and experiment.
Euclidean Equivalent of Minkowski’s Space-Time Theory and the Corresponding M...Premier Publishers
This document communicates some of the main results obtained from a theoretical work which performs a type of Wick’s rotation, where Lorentz’s group is connected in the resulting Euclidean metric, and as a consequence models the particles with rest mass as photons in a compacted additional dimension (for a photon of the ordinary 3-dimensional space, they do not go through the 4-dimension due to null angle in this dimension). Among its reported results are new explanations, much more elegant than the current ones, of the material waves of De Broglie, the uncertainty principle, the dilation of the proper time, the Higgs field, the existence of the antiparticles and specifically of the electron-positron annihilation, among others. It also leaves open the possibility of unifying at least three of the fundamental forces and the different types of particles under a single model of photon and compact dimension. Additionally, two experimental results are proposed that can only currently be explained by this theory.
dSolution The concept of Derivative is at th.pdftheaksmart2011
Dry Ice., is a manufacturer of air conditioners that has seen its demand grow significantly. The
companyanticipates nationwide demand for the next year to be 180,000 units in the South,
120,000 units inin the Midwest, 110,000 units in the East, and 100,000 units in the West.
Managers at DryIce are designingthe manufacturing network and have selected four potential
sites-- New York, Atlanta, Chicago, and San DiegoPlants could have a capacity of either 200,000
or 400,000 units. The annual fixed costs are at the four locations areshown in the Table, along
with the cost of producing and shipping an air conditioner to each of the four markets.Where
should DryIce build its factories and how large should they be?Dry Ice., is a manufacturer of air
conditioners that has seen its demand grow significantly. The companyanticipates nationwide
demand for the next year to be 180,000 units in the South, 120,000 units inin the Midwest,
110,000 units in the East, and 100,000 units in the West. Managers at DryIce are designingthe
manufacturing network and have selected four potential sites-- New York, Atlanta, Chicago,
and San DiegoPlants could have a capacity of either 200,000 or 400,000 units. The annual fixed
costs are at the four locations areshown in the Table, along with the cost of producing and
shipping an air conditioner to each of the four markets.Where should DryIce build its factories
and how large should they be?
Solution
If the fixed cost is not taken into consideration then,
Dry Ice has the maximum demand in the south region as: 180,000 units
The company should thus build a plant size of 400,000 units (maximum possible) in order to
satisfy the demand of the regions and earn economies of scale.
The site of the company should be chosen from the factors like: proximity to the markets,
perishable or nonperishable goods, nearness to the warehouse, suppliers convenient etc..
International Refereed Journal of Engineering and Science (IRJES)irjes
International Refereed Journal of Engineering and Science (IRJES) is a leading international journal for publication of new ideas, the state of the art research results and fundamental advances in all aspects of Engineering and Science. IRJES is a open access, peer reviewed international journal with a primary objective to provide the academic community and industry for the submission of half of original research and applications
Slide 1: Title Slide
Extrachromosomal Inheritance
Slide 2: Introduction to Extrachromosomal Inheritance
Definition: Extrachromosomal inheritance refers to the transmission of genetic material that is not found within the nucleus.
Key Components: Involves genes located in mitochondria, chloroplasts, and plasmids.
Slide 3: Mitochondrial Inheritance
Mitochondria: Organelles responsible for energy production.
Mitochondrial DNA (mtDNA): Circular DNA molecule found in mitochondria.
Inheritance Pattern: Maternally inherited, meaning it is passed from mothers to all their offspring.
Diseases: Examples include Leber’s hereditary optic neuropathy (LHON) and mitochondrial myopathy.
Slide 4: Chloroplast Inheritance
Chloroplasts: Organelles responsible for photosynthesis in plants.
Chloroplast DNA (cpDNA): Circular DNA molecule found in chloroplasts.
Inheritance Pattern: Often maternally inherited in most plants, but can vary in some species.
Examples: Variegation in plants, where leaf color patterns are determined by chloroplast DNA.
Slide 5: Plasmid Inheritance
Plasmids: Small, circular DNA molecules found in bacteria and some eukaryotes.
Features: Can carry antibiotic resistance genes and can be transferred between cells through processes like conjugation.
Significance: Important in biotechnology for gene cloning and genetic engineering.
Slide 6: Mechanisms of Extrachromosomal Inheritance
Non-Mendelian Patterns: Do not follow Mendel’s laws of inheritance.
Cytoplasmic Segregation: During cell division, organelles like mitochondria and chloroplasts are randomly distributed to daughter cells.
Heteroplasmy: Presence of more than one type of organellar genome within a cell, leading to variation in expression.
Slide 7: Examples of Extrachromosomal Inheritance
Four O’clock Plant (Mirabilis jalapa): Shows variegated leaves due to different cpDNA in leaf cells.
Petite Mutants in Yeast: Result from mutations in mitochondrial DNA affecting respiration.
Slide 8: Importance of Extrachromosomal Inheritance
Evolution: Provides insight into the evolution of eukaryotic cells.
Medicine: Understanding mitochondrial inheritance helps in diagnosing and treating mitochondrial diseases.
Agriculture: Chloroplast inheritance can be used in plant breeding and genetic modification.
Slide 9: Recent Research and Advances
Gene Editing: Techniques like CRISPR-Cas9 are being used to edit mitochondrial and chloroplast DNA.
Therapies: Development of mitochondrial replacement therapy (MRT) for preventing mitochondrial diseases.
Slide 10: Conclusion
Summary: Extrachromosomal inheritance involves the transmission of genetic material outside the nucleus and plays a crucial role in genetics, medicine, and biotechnology.
Future Directions: Continued research and technological advancements hold promise for new treatments and applications.
Slide 11: Questions and Discussion
Invite Audience: Open the floor for any questions or further discussion on the topic.
A brief information about the SCOP protein database used in bioinformatics.
The Structural Classification of Proteins (SCOP) database is a comprehensive and authoritative resource for the structural and evolutionary relationships of proteins. It provides a detailed and curated classification of protein structures, grouping them into families, superfamilies, and folds based on their structural and sequence similarities.
(May 29th, 2024) Advancements in Intravital Microscopy- Insights for Preclini...Scintica Instrumentation
Intravital microscopy (IVM) is a powerful tool utilized to study cellular behavior over time and space in vivo. Much of our understanding of cell biology has been accomplished using various in vitro and ex vivo methods; however, these studies do not necessarily reflect the natural dynamics of biological processes. Unlike traditional cell culture or fixed tissue imaging, IVM allows for the ultra-fast high-resolution imaging of cellular processes over time and space and were studied in its natural environment. Real-time visualization of biological processes in the context of an intact organism helps maintain physiological relevance and provide insights into the progression of disease, response to treatments or developmental processes.
In this webinar we give an overview of advanced applications of the IVM system in preclinical research. IVIM technology is a provider of all-in-one intravital microscopy systems and solutions optimized for in vivo imaging of live animal models at sub-micron resolution. The system’s unique features and user-friendly software enables researchers to probe fast dynamic biological processes such as immune cell tracking, cell-cell interaction as well as vascularization and tumor metastasis with exceptional detail. This webinar will also give an overview of IVM being utilized in drug development, offering a view into the intricate interaction between drugs/nanoparticles and tissues in vivo and allows for the evaluation of therapeutic intervention in a variety of tissues and organs. This interdisciplinary collaboration continues to drive the advancements of novel therapeutic strategies.
Professional air quality monitoring systems provide immediate, on-site data for analysis, compliance, and decision-making.
Monitor common gases, weather parameters, particulates.
Comparing Evolved Extractive Text Summary Scores of Bidirectional Encoder Rep...University of Maribor
Slides from:
11th International Conference on Electrical, Electronics and Computer Engineering (IcETRAN), Niš, 3-6 June 2024
Track: Artificial Intelligence
https://www.etran.rs/2024/en/home-english/
Introduction:
RNA interference (RNAi) or Post-Transcriptional Gene Silencing (PTGS) is an important biological process for modulating eukaryotic gene expression.
It is highly conserved process of posttranscriptional gene silencing by which double stranded RNA (dsRNA) causes sequence-specific degradation of mRNA sequences.
dsRNA-induced gene silencing (RNAi) is reported in a wide range of eukaryotes ranging from worms, insects, mammals and plants.
This process mediates resistance to both endogenous parasitic and exogenous pathogenic nucleic acids, and regulates the expression of protein-coding genes.
What are small ncRNAs?
micro RNA (miRNA)
short interfering RNA (siRNA)
Properties of small non-coding RNA:
Involved in silencing mRNA transcripts.
Called “small” because they are usually only about 21-24 nucleotides long.
Synthesized by first cutting up longer precursor sequences (like the 61nt one that Lee discovered).
Silence an mRNA by base pairing with some sequence on the mRNA.
Discovery of siRNA?
The first small RNA:
In 1993 Rosalind Lee (Victor Ambros lab) was studying a non- coding gene in C. elegans, lin-4, that was involved in silencing of another gene, lin-14, at the appropriate time in the
development of the worm C. elegans.
Two small transcripts of lin-4 (22nt and 61nt) were found to be complementary to a sequence in the 3' UTR of lin-14.
Because lin-4 encoded no protein, she deduced that it must be these transcripts that are causing the silencing by RNA-RNA interactions.
Types of RNAi ( non coding RNA)
MiRNA
Length (23-25 nt)
Trans acting
Binds with target MRNA in mismatch
Translation inhibition
Si RNA
Length 21 nt.
Cis acting
Bind with target Mrna in perfect complementary sequence
Piwi-RNA
Length ; 25 to 36 nt.
Expressed in Germ Cells
Regulates trnasposomes activity
MECHANISM OF RNAI:
First the double-stranded RNA teams up with a protein complex named Dicer, which cuts the long RNA into short pieces.
Then another protein complex called RISC (RNA-induced silencing complex) discards one of the two RNA strands.
The RISC-docked, single-stranded RNA then pairs with the homologous mRNA and destroys it.
THE RISC COMPLEX:
RISC is large(>500kD) RNA multi- protein Binding complex which triggers MRNA degradation in response to MRNA
Unwinding of double stranded Si RNA by ATP independent Helicase
Active component of RISC is Ago proteins( ENDONUCLEASE) which cleave target MRNA.
DICER: endonuclease (RNase Family III)
Argonaute: Central Component of the RNA-Induced Silencing Complex (RISC)
One strand of the dsRNA produced by Dicer is retained in the RISC complex in association with Argonaute
ARGONAUTE PROTEIN :
1.PAZ(PIWI/Argonaute/ Zwille)- Recognition of target MRNA
2.PIWI (p-element induced wimpy Testis)- breaks Phosphodiester bond of mRNA.)RNAse H activity.
MiRNA:
The Double-stranded RNAs are naturally produced in eukaryotic cells during development, and they have a key role in regulating gene expression .
Deep Behavioral Phenotyping in Systems Neuroscience for Functional Atlasing a...Ana Luísa Pinho
Functional Magnetic Resonance Imaging (fMRI) provides means to characterize brain activations in response to behavior. However, cognitive neuroscience has been limited to group-level effects referring to the performance of specific tasks. To obtain the functional profile of elementary cognitive mechanisms, the combination of brain responses to many tasks is required. Yet, to date, both structural atlases and parcellation-based activations do not fully account for cognitive function and still present several limitations. Further, they do not adapt overall to individual characteristics. In this talk, I will give an account of deep-behavioral phenotyping strategies, namely data-driven methods in large task-fMRI datasets, to optimize functional brain-data collection and improve inference of effects-of-interest related to mental processes. Key to this approach is the employment of fast multi-functional paradigms rich on features that can be well parametrized and, consequently, facilitate the creation of psycho-physiological constructs to be modelled with imaging data. Particular emphasis will be given to music stimuli when studying high-order cognitive mechanisms, due to their ecological nature and quality to enable complex behavior compounded by discrete entities. I will also discuss how deep-behavioral phenotyping and individualized models applied to neuroimaging data can better account for the subject-specific organization of domain-general cognitive systems in the human brain. Finally, the accumulation of functional brain signatures brings the possibility to clarify relationships among tasks and create a univocal link between brain systems and mental functions through: (1) the development of ontologies proposing an organization of cognitive processes; and (2) brain-network taxonomies describing functional specialization. To this end, tools to improve commensurability in cognitive science are necessary, such as public repositories, ontology-based platforms and automated meta-analysis tools. I will thus discuss some brain-atlasing resources currently under development, and their applicability in cognitive as well as clinical neuroscience.
Observation of Io’s Resurfacing via Plume Deposition Using Ground-based Adapt...Sérgio Sacani
Since volcanic activity was first discovered on Io from Voyager images in 1979, changes
on Io’s surface have been monitored from both spacecraft and ground-based telescopes.
Here, we present the highest spatial resolution images of Io ever obtained from a groundbased telescope. These images, acquired by the SHARK-VIS instrument on the Large
Binocular Telescope, show evidence of a major resurfacing event on Io’s trailing hemisphere. When compared to the most recent spacecraft images, the SHARK-VIS images
show that a plume deposit from a powerful eruption at Pillan Patera has covered part
of the long-lived Pele plume deposit. Although this type of resurfacing event may be common on Io, few have been detected due to the rarity of spacecraft visits and the previously low spatial resolution available from Earth-based telescopes. The SHARK-VIS instrument ushers in a new era of high resolution imaging of Io’s surface using adaptive
optics at visible wavelengths.
Multi-source connectivity as the driver of solar wind variability in the heli...Sérgio Sacani
The ambient solar wind that flls the heliosphere originates from multiple
sources in the solar corona and is highly structured. It is often described
as high-speed, relatively homogeneous, plasma streams from coronal
holes and slow-speed, highly variable, streams whose source regions are
under debate. A key goal of ESA/NASA’s Solar Orbiter mission is to identify
solar wind sources and understand what drives the complexity seen in the
heliosphere. By combining magnetic feld modelling and spectroscopic
techniques with high-resolution observations and measurements, we show
that the solar wind variability detected in situ by Solar Orbiter in March
2022 is driven by spatio-temporal changes in the magnetic connectivity to
multiple sources in the solar atmosphere. The magnetic feld footpoints
connected to the spacecraft moved from the boundaries of a coronal hole
to one active region (12961) and then across to another region (12957). This
is refected in the in situ measurements, which show the transition from fast
to highly Alfvénic then to slow solar wind that is disrupted by the arrival of
a coronal mass ejection. Our results describe solar wind variability at 0.5 au
but are applicable to near-Earth observatories.
Mammalian Pineal Body Structure and Also Functions
The mathematical and philosophical concept of vector
1. The mathematical and philosophical concept of vector 1
The mathematical and philosophical concept of vector
by George Mpantes www.mpantes .gr
Historical summary
The vectors
The transformation equations of coordinates
the transformation equations of vectors
Euclidean Geometry and Newtonian physics .
Philosophical comments, Aristotle
Historical summary
“There is un unspoken hypothesis which underlies all the physical theories so far
created, namely that behind physical phenomena lies a unique mathematical structure
which is the purpose of theory to reveal. According to this hypothesis , the
mathematical formulae of physics are discovered not invented, the Lorentz
transformation , for example ,being as much a part of physical reality as a table or a
chair”. ( RELATIVITY: THE SPECIAL THEORY J.L.Synge p.163)
Indeed in our example the physical phenomenon is the force, and the
underlying mathematical structure is vector analysis. But looking the historical
process, mathematics create their truths independently, discover new entities,
and their tendency for generalization goes ahead exceeding the initial physical
presuppositions. The new discoveries of the mathematical process return to the
physical theory where they create new unifications and generalizations, now for
the phenomena of the actual world. Can we trust them? Can mathematics lead
2. The mathematical and philosophical concept of vector 2
the physical theory? The answer seems to be positive, if the measurements
agree with the mathematical conjectures (electromagnetic waves!). that is that
mathematical structure extends the physical theory. So, for example, with the
support of the use of vector methods , we had a development of theoretical
physics and by the beginning of the twentieth century, vector analysis had
become firmly entrenched as a tool for the development of geometry and
theoretical physics.
As we look back on the nineteenth century it is apparent that a
mathematical theory in terms of which physical laws could be described and
their universality checked was needed. Figuratively speaking two men stepped
forward in this direction, Hamilton and Grassman. Hamilton was trying to find
the appropriate mathematical tools with which he could apply Newtonian
mechanics to various aspects of astronomy and physics. Grassman tried to
develop an algebraic structure on which geometry of any number of dimension
could be based. The quaternions of Hamilton and Grassmann’s calculus of
extension proved to be too complicated for quick mastery and easy application ,
but from them emerged the much more easily learned and more easily applied
subject of vector analysis. This work was due principally to the American
physicist John Willard Gibbs (1839-1903) and is encountered by every student
of elementary physics.
The vectors .
What is behind the physical phenomenon of the velocity; of the force;
there is the mathematical concept of the vector. This is a new concept, since
force has direction, sense, and magnitude, and we accept the physical principle
that the forces exerted on a body can be added to the rule of the
parallelogram. This is the first axiom of Newton. Newton essentially requires
that the power is a " vectorial " size , without writing clearly , and Galileo that
applies the principle of the independence of forces .
3. The mathematical and philosophical concept of vector 3
These are the basic physical indications for the mathematical treatment,
for “vector1 geometry”, where
the term vector denotes a translation or a displacement a in the space.2 The
statement that the displacement a transfers the point P to the point Q (“transforms” P
into Q ) may also be expressed by saying that Q is the end-point of the vector a whose
starting point is at P. if P and Q are ant two points then there is one and only one
displacement a which transforms P to Q. We shall cal it the vector defined by P and Q
.
and indicate it by PQ
There are two fundamental operations, which are subject to a system of
laws, viz. addition of two vectors (the translation which arises through two
successive translations (law of parallelogram), and multiplication of a vector by a
number (is defined through the addition). These laws are
A.Addition: a+b=c
with the properties
a+b=b+a
(a+b)+c=a+(b+c)
If a and c are any two vectors , then there is one and only one value of x
for which the equation a+x=c holds
B. Multiplication b=λ.a
with the properties
(λ+μ)a=(λa)+(μa)
λ(μa)=(λμ)a
1.a=a
λ(a+b)=(λa)+(λb)
In elementary physics , a vector is graphically regarded as a directed line
segment , or arrow. This is the translation or the displacement described by
Weyl. So in elementary physics , vector was something apparent, something
concrete and intuitively simple. It was geometrical. In theoretical physics it
became an idea, something cerebral, connected with algebra. The first was a
1 The term vector was introduced from Hamilton
2 This definition is from Weyl, (Space, time, matter)
4. The mathematical and philosophical concept of vector 4
sketch of the second. This is the course of mathematics. The formula for
algebraic vector was the old bold Cartesian binding of geometry with algebra viz
this of a picture with the abstract and compact truth of numbers, a good
combination between intuition and rigor, through concepts, in the center of
which was the well-known coordinate system, one of the more significant
generalization of mathematics.
By means of a coordinate system, a set of ordered triples of real
numbers can be put into one-to-one correspondence with the points of a three
dimensional Euclidean space. However many aspects of modern-day science
cannot be adequately described in terms of a three-dimensional Euclidean model.
The ideas of vector analysis when expressed in a notational fashion are
immediately extendable to n-dimensional space and their physical usage is amply
demonstrated in the development of special and general relativity theory.
With the change of the figurativeness of the points, change also the
description of the vector.
The set {A1, A2, A3} of all triples (A1, A2, A3), (A1΄, A2΄, A3΄) etc.,
determined by orthogonal projections of a common arrow representation on the
axes of the associated rectangular Cartesian coordinate system is said to be a
Cartesian vector. Many triples means many systems, but all these represent
the same Cartesian vector, which has a family of arrows as its geometrical
representative. The binding of orthogonal projections with the law of
parallelogram is the base of all the formalism of vector analysis.3
A Cartesian vector (A1, A2, A3), (3-tuple), can be represented
graphically by an arrow, with it’s initial point at the origin and it’s terminal
point at the position with coordinates (A1, A2, A3), but it is not the only possible
arrow representation. An arrow with initial and terminal points (a,b,c) and
(A,B,C) such that A1=A-a, A2=B-b, A3=C-c can be considered a representative
of a 3-tuple.
3 See my article “the mathema tical forms of nature, the tensors”
5. The mathematical and philosophical concept of vector 5
A Cartesian vector with respect to a coordinate system, is
characterized by a magnitude , a direction and a sense , and its components in
any coordinate system satisfy the algebraic laws of the triples, viz the laws 1
and 2 for the vectors, expressed algebraically , if we define a=(a1,a2,…..an)
b=(b1,b2,….bn)
i.e (a1,a2,…..an)+(b1,b2,….bn)=(a1+b1, a2+b2+……an,bn).
λ.a=λ(a1,a2,…..an)= (λa1,λa2,…..λan) .
Now an analytical treatment of vector geometry is possible, in which
every vector is represented by it’s components and every point by its
coordinates.
How all these triples, (A1, A2, A3), (A1΄, A2΄, A3΄) etc., are related?
The transformation equations of coordinates
A fundamental problem of theoretical physics is that formulating
universally valid laws relating natural phenomena. Because the transformation
idea is of such importance, the development of vector geometry and later of
vector analysis is build around this.
A rectangular Cartesian coordinate system4 imposes a one-to-one
correspondence between the points of Euclidean three-space and the set of all
ordered triples of real numbers. A second rectangular Cartesian system brings
about another correspondence of the same point . What is the nature of those
transformations that relate such coordinate representations of the three-space?
The specific transformations of coordinates for our example in the
development of vector analysis, are called translations and rotations. They are
linear transformations and they connect orthogonal Cartesian systems. All linear
transformations have the characteristic that the fundamental relations (A) and
(B) are not disturbed by the transformation viz they hold for the transformed
points and vectors :
4 We examine this particular case in our example.
6. The mathematical and philosophical concept of vector 6
α΄+b΄=c΄ b΄=λ .a΄………
DEFINITION 1. The transformation equations that relate the
coordinates ( , , ) and (x , , ) 1 2 3 1 2 3 x x x x x in rectangular coordinate systems , the
axes of which are parallels are
j j j x x x
.......... .......... ....(1) 0
where ( , , ) 3
1
0 x x x represent the unbarred coordinates of the origin of the
0
2
0
barred system O’. These are called equations of translation. The Cartesian
vector concept is employed in obtaining them.
DEFINITION 2.
The transformation equations that relate the coordinates
( , , ) and (x , , ) 1 2 3 1 2 3 x x x x x in rectangular coordinate systems, having a common
origin and such that there is no change of unit distance along coordinate axes,
are related by the transformation equations
j x c x
.......... ..(2) j k
k
where the coefficients of
transformations j
k a are direction
cosines satisfying the conditions
p
3
c j
c j
j
k p
1 k
7. The mathematical and philosophical concept of vector 7
These are called equations of rotation.
The transformations of the coordinates (2) are a subset of the linear or
affine transformations, with the general form
.........( 3)
1
2
3
c c c
3
3
c c c
3
2
3
1
2
3
2
2
2
1
1
3
1
2
1
1
1
2
3
x
x
x
c c c
x
x
x
where apply the conditions of orthogonality, they are the orthogonal
transformations that connect orthogonal Cartesian systems with common origin
and are produced from the vectorial behavior of the vectorial units (bases) in
the axes of the two systems
. Physically they describe, as we have mention, the rotation of an
orthogonal Cartesian system. The orthogonal transformations fulfill the first
unification of geometry (the Euclidean metrical geometry in every orthogonal
system) and as the geometry is a fundamental branch of physics, this unification
will be the model of the unification of physical laws in all the
systems.(universality)
But what about the vectors? What is their deepest behavior in the scene
of coordinate systems?
the transformation equations of vectors
We have seen that a Cartesian vector (A1, A2, A3) can be represented
graphically by an arrow, but
The components of this arrow, transform under rotation, as the coordinates .
Proof: If the transformation (2) is applied to the coordinates of P0 and P1 , the
coordinate differences { j j x x1 0 } satisfy
j j x x c x c x c x x
j k k
k
j k
k
j k
k
( )......... .......... ..(4) 1 0 1 0 1 0
that is the transformation (2).
A corresponding verification of the statement holds for translations, where the
vector components remain unaltered.
8. The mathematical and philosophical concept of vector 8
So we have the definition of the Cartesian vector under the light of both
transformations:
A Cartesian vector (A1, A2, A3), is a collection of ordered triples , each
associated with a rectangular Cartesian coordinate system and such that any
two satisfy the transformation law
.......... .......... .......... ......( 5) k
j
x
j A
k
x
A
where the partial derivatives are the coefficients j
i c of the linear
transformation (3), of coordinates.
We must notice that every component of the vector in the new system is
a linear combination of the components in the initial variables. So if all
components of the vector are zero in the initial system they will be also zero in
the new variables. This is the more important property of vectors: a vectorial
equation holds in every rectangular Cartesian system (for our paradigm), if it
holds in one! This is the root of the universality of the physical or geometrical
laws, as we see in the end of the article. Newton’s law is universal because it is
written in vectorial form. It’s invariance in translation is the mathematical
acceptance of the Newtonian principle of relativity.
The scalars .
A second concept which has evolved in the development of vector analysis is
that of the scalar. The definition of scalar states that it is a quantity
possessing magnitude but no direction. Such entities as mass, time, density and
temperature are given as examples. But for mathematics, the prize example is
the real number, as it does not have to be associated with magnitude. From a
historical point of view scalar is a quantity invariant under all transformations
of coordinates (Felix Klein). Whether a given algebraic form is invariant depends
on the group of transformations under consideration. Again the scalars , as
vectors, are associated with coordinate systems and transformations.
9. The mathematical and philosophical concept of vector 9
Euclidean Geometry and Newtonian physics .
The mathematical investigation showed that our Known geometrical
vector (arrow) has hidden qualities which are raised by their correlation with
coordinate systems: The laws of it’s transformation. The vector concept
received much of its impetus from this fact, so it plays a fundamental role in
many aspects of geometry and physics. This mathematical result underlies the
principles of relativity of Newton and Einstein, that would be ungrounded
without the mathematical discovery of the transformation theory of the vectors
and (later) of tensors.
Magnitude and angle are fundamental to the metric structure of
Euclidean space. They are scalar invariants under the transformations of the
orthogonal Cartesian set.
The inner product transforms
k
P j Q j
c j P r
c j Q s
c j
c j P r Q s
P r Q s
P k
Q
r
s
r
s
r
j
j
j
k
3
1
s
3
1
3
1
3
1
( )( )
1 0 , X j X (through 4)
and the distance of the points j
1
1 (x x ) (x x ) (x x ) (x x ) (x x ) (x x )
3 2
0
3
1
2 2
0
2
1
1 2
0
1
1
3 2
0
3
1
2
0
2
1
1 2
0
These formulas carry out the first unification of metrical Euclidean geometry.
An observer who measures a distance and an angle in a orthogonal Cartesian
system uses the same formulas and finds the same results with somebody else
who measures the same magnitudes in another orthogonal Cartesian system,
which subsists a translation or a rotation of the first. The question of finding
those entities (as distance and inner product) that have an absolute meaning
transcending the coordinate system, is of prime significance. This gives us a
direction as to which of the concepts considered in the framework of
rectangular Cartesian systems should be generalized as well as how to bring
about the generalizations. This is the criminal point of the universality of the
physical laws. Moreover, the availability of the Cartesian systems of reference
10. The mathematical and philosophical concept of vector 10
will be valuable when considering the special theory of relativity, in a later
article.
In vector formalism, we will now show the covariance of Newton's
law in linear systems with given origin ( rotation) .
In the system K we have
d
F
( ) k k m
Multiply by
k
r
dt
x
x
and summing with respect to k (from 5) we have
x
k
m
d
F
m
( ) ( ) k r r
x
r
F
k
k
x
x
r
dt
d
dt
So the form of the equation remains the same in the new system (
covariant ) , and the mathematical formalism demonstrates that the laws of
Newton have a universal application in Euclidean space, where we can adjust
orthogonal Cartesian systems. The physical laws are invariant in form in all the
orthogonal Cartesians systems (my article “covariance and invariance in physics”)
Philosophical comments, Aristotle .
The quotation of Synge about the discovery of mathematical structure
of vector analysis , where the vector is as much a part of physical reality as a
table, is very poetic, having construct a reality of visible and invisible objects,
as the vector and the table. Vector exists as the table but it is invisible!
Aristotle in his ideas of the theory of knowledge (Analytica posterioria)
says that “the knowledge of a fact differs from the knowledge of a reasoned fact” .
The theoretical foundations of the systems of this deductive reasoning,
account of first principles where are the bases of every science. …. The
scientific knowledge through reasoning is impossible if we do not know the first
principles. ….it is clear that in science of nature as elsewhere we should try first to
determine questions about the first principles …..αληθείς και πρωταρχικές και άμεσες και
πρωθύστερες και αιτίες του συμπεράσματος,the first basis from which a thing is
11. The mathematical and philosophical concept of vector 11
known……as regards their existence must be assumed for the principle ,(what a straight
line is , what a triangle is ..)but proved for the rest of the system, by logical reasoning.
How are these first principles to be established? ….they are arrived by
the repeated visual sensations, which leave their marks in the memory. Then we reflect
on these memories and arrive by a process of intuition (νους) at the first principles ….if
there in not something intelligible behind the phenomena, there is not science for
anything, science is not created from senses….
So vector is an intelligible creation, a first principle, that is a conviction,
a support of the deductive reasoning. This reasoning constructs a mental logical
reality in our brains, which is the human’s way of comprehension. Mathematics
are neither discoveries nor inventions. Mathematics are creations , as poems,
but logical creations, based on the rules of deductive reasoning.
But in fact, their first principles are founded in nature.
Sources
Herman Weyl (space,time, matter,Dover)
H.Eves (foundations and fundamental concepts of mathematics,Dover)
J.L Synge (Relativity: the special theory, Noth Holland publising Company
Amsterdam New York Oxford)
Robert C.Wrede (introduction to vector and tensor analysis, Dover)
Aristotle (Analytica posterioria, internet)
George Mpantes mathematics teacher www.mpantes.gr