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Physics: 5D Geometry . . . The Physical Model is . . . Geometry Needed Natural Idea and Its . . . What We Suggest From Fuzzification and Resulting Formalism: Idea K -Vectors Towards K -Covectors Intervalization to K -Covectors K -Tensors:

  1. Physics: 5D Geometry . . . The Physical Model is . . . Geometry Needed Natural Idea and Its . . . What We Suggest From Fuzzification and Resulting Formalism: Idea K -Vectors Towards K -Covectors Intervalization to K -Covectors K -Tensors: Definitions Anglification: A New 5D K -Tensors: Main Result Explaining the . . . Geometric Formalism for Differential Formalism . . . Coordinate . . . Potential Applications . . . Physics and Data Processing Acknowledgments Title Page Scott A. Starks and Vladik Kreinovich ◭◭ ◮◮ NASA Pan-American Center for Earth and Environmental Studies (PACES) ◭ ◮ University of Texas at El Paso, El Paso, TX 79968, USA Page 1 of 18 sstarks@utep.edu, vladik@utep.edu Go Back Full Screen Close Quit

  2. Physics: 5D Geometry . . . The Physical Model is . . . 1. Data Processing: Geometric Interpretation is Needed Geometry Needed Natural Idea and Its . . . • Data to be processed : several real numbers x 1 , . . . , x n . What We Suggest • Geometric interpretation: the sequence ( x 1 , . . . , x n ) is an n -D vector – an Resulting Formalism: Idea element of an n -D space. K -Vectors Towards K -Covectors • Resulting visualization: K -Covectors – level sets of Gaussian distribution are ellipsoids; K -Tensors: Definitions K -Tensors: Main Result – linear relation is a plane, etc. Explaining the . . . • Problem: we can only use geometric intuition for ≤ 3 (or 4). Differential Formalism . . . Coordinate . . . • Objective: to have similar geometric techniques for larger n . Potential Applications . . . • Idea: look at physics where multi-dimensional geometries are currently used. Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 2 of 18 Go Back Full Screen Close Quit

  3. Physics: 5D Geometry . . . The Physical Model is . . . 2. Physics: 5D Geometry is Useful Geometry Needed Natural Idea and Its . . . • General relativity (GRT) explained gravitation by combining space and time What We Suggest into a 4D space. Resulting Formalism: Idea • Question: can other dimensions explain other physics? K -Vectors Towards K -Covectors • Success (Th. Kaluza, O. Klein, 1921): 5D GRT K -Covectors – gravitation for 4 × 4 components g ij of the metric, K -Tensors: Definitions – g 5 i satisfy Maxwell’s equations (if g 55 = const). K -Tensors: Main Result Explaining the . . . • Problem: no physical explanation of 5-th dimension. Differential Formalism . . . Coordinate . . . • Solution (A. Einstein, P. Bergmann, 1938): 5th dimension forms a tiny circle, Potential Applications . . . so we don’t notice it. Acknowledgments • This is still relevant: this idea is standard in particle physics, where Title Page – space is 10- or 11-dimensional, ◭◭ ◮◮ – all dimensions except the first four are tiny. ◭ ◮ Page 3 of 18 Go Back Full Screen Close Quit

  4. Physics: 5D Geometry . . . The Physical Model is . . . 3. The Physical Model is Unusual, But This Un-Usualness Geometry Needed is Appropriate for Data Processing Natural Idea and Its . . . What We Suggest • Problem: the standard multi-D physical model is unusual geometrically: Resulting Formalism: Idea K -Vectors – the space is a cylinder, Towards K -Covectors – not a plane anymore. K -Covectors K -Tensors: Definitions • Observation: this feature is, however, interestingly related to data processing: K -Tensors: Main Result – some measured data are angles, and Explaining the . . . – angles do form a circle. Differential Formalism . . . Coordinate . . . • Conclusion: these geometric ideas can be directly applied to data processing. Potential Applications . . . Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 4 of 18 Go Back Full Screen Close Quit

  5. Physics: 5D Geometry . . . The Physical Model is . . . 4. Geometry Needed Geometry Needed Natural Idea and Its . . . • Problem: Kaluza-Klein theory requires several additional physical formulas What We Suggest w/o geometric meaning. Resulting Formalism: Idea • Objective: we show that these formulas can be geometrically explained. K -Vectors Towards K -Covectors • First, the assumption g 55 = const is artificial. K -Covectors • Second, since only 4 coordinates have a physical sense, the terms g 5 i · ∆ x 5 · ∆ x i K -Tensors: Definitions in the distance K -Tensors: Main Result 5 5 Explaining the . . . ∆ s 2 = � � g ij · ∆ x i · ∆ x j Differential Formalism . . . i =1 j =1 Coordinate . . . are not physical. Potential Applications . . . • Third, the observed values of physical fields do not depend on x 5 ( cylindric- Acknowledgments ity ). Title Page � • Rumer interpreted x 5 as action S = ◭◭ ◮◮ L dx dt . ◭ ◮ • Fourth, action transformations S → S + f ( x i ) should be geometrically mean- ingful. Page 5 of 18 Go Back Full Screen Close Quit

  6. Physics: 5D Geometry . . . The Physical Model is . . . 5. Natural Idea and Its Problems Geometry Needed Natural Idea and Its . . . • Main difference: What We Suggest – in Einstein-Bergmann’s 5D model we have a cylinder K = R 4 × S 1 ( K Resulting Formalism: Idea for Kaluza) K -Vectors Towards K -Covectors – in a standard 4D space, we have a linear space. K -Covectors • Idea: modify standard geometry by substituting K instead of R 4 into all K -Tensors: Definitions definitions. K -Tensors: Main Result Explaining the . . . • Problem: we need linear space structure, i.e., addition and multiplication by Differential Formalism . . . a scalar. Coordinate . . . • We still have addition in K . Potential Applications . . . Acknowledgments • However, multiplication is not uniquely defined for angle-valued variables: Title Page – we can always interpret an angle as a real number modulo the circum- ference, ◭◭ ◮◮ – but then, e.g., 0 ∼ 2 π while 0 . 6 · 0 �∼ 0 . 6 · 2 π . ◭ ◮ Page 6 of 18 Go Back Full Screen Close Quit

  7. Physics: 5D Geometry . . . The Physical Model is . . . 6. What We Suggest Geometry Needed Natural Idea and Its . . . • We need: a real-number representation of an angle variable. What We Suggest • Natural idea: an angle is not as a single real number. Resulting Formalism: Idea K -Vectors • It is a set { α + n · 2 π } of all possible real numbers that correspond to the Towards K -Covectors given angle. K -Covectors • Similar ideas: interval and fuzzy arithmetic. K -Tensors: Definitions K -Tensors: Main Result • Natural definition: element-wise operations, e.g., Explaining the . . . Differential Formalism . . . A + B = { a + b | a ∈ A, b ∈ B } . Coordinate . . . Potential Applications . . . • Other ideas: Acknowledgments – tensors are linear mappings that preserve the structure of such sets; Title Page – a tensor field is differentiable if its derivatives are also consistent with this structure. ◭◭ ◮◮ ◭ ◮ Page 7 of 18 Go Back Full Screen Close Quit

  8. Physics: 5D Geometry . . . The Physical Model is . . . 7. Resulting Formalism: Idea Geometry Needed Natural Idea and Its . . . • In mathematical terms, the resulting formalism is equivalent to the following: What We Suggest • We start with the space K which is not a vector space (only an Abelian Resulting Formalism: Idea group). K -Vectors Towards K -Covectors • We reformulate standard definitions of vector and tensor algebra and tensor K -Covectors analysis and apply them to K : K -Tensors: Definitions – K -vectors are defined as elements of K ; K -Tensors: Main Result Explaining the . . . – K -covectors as elements of the dual group, Differential Formalism . . . – etc. Coordinate . . . • All physically motivated conditions turn out to be natural consequences of Potential Applications . . . this formalism. Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 8 of 18 Go Back Full Screen Close Quit

  9. Physics: 5D Geometry . . . The Physical Model is . . . 8. K -Vectors Geometry Needed Natural Idea and Its . . . • In the traditional 4-D space-time R 4 , we can define a vector as simply an What We Suggest element of R 4 . Resulting Formalism: Idea def • In our case, instead of 4-D space-time R 4 , we have a 5-D space-time K = K -Vectors R 4 × S 1 . Towards K -Covectors K -Covectors • S 1 is a circle of a small circumference h > 0 – i.e., equivalently, a real line K -Tensors: Definitions in which two numbers differing by a multiple of h describe the same point: ( x 1 , . . . , x 4 , x 5 ) ∼ ( x 1 , . . . , x 4 , x 5 + k · h ). K -Tensors: Main Result Explaining the . . . • Thus, it is natural to define K -vectors as simply elements of K . Differential Formalism . . . • On R 4 , there are two operations: a + b and λ : a → λ · a . Thus, R 4 is a linear Coordinate . . . Potential Applications . . . space . Acknowledgments • On K we only have addition, so K is only an Abelian group . Title Page ◭◭ ◮◮ ◭ ◮ Page 9 of 18 Go Back Full Screen Close Quit

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