Need for Analysis, . . . Symmetry: a . . . Algorithmic Aspects of . . . Algorithmic Aspects of Example: How to . . . Algorithmic Aspects of . . . Analysis, Prediction, and Example: Selecting . . . Control in Science and Acknowledgements Home Page Engineering: Symmetry- Title Page Based Approach ◭◭ ◮◮ ◭ ◮ Jaime Nava Page 1 of 45 Department of Computer Science Go Back University of Texas at El Paso El Paso, TX 79968 Full Screen jenava@miners.utep.edu Close Quit
Need for Analysis, . . . 1. Need for Analysis, Prediction, and Control in Symmetry: a . . . Science and Engineering Algorithmic Aspects of . . . • Prediction is one of the main objectives of science and Example: How to . . . engineering. Algorithmic Aspects of . . . Example: Selecting . . . • Example: in Newton’s mechanics, we want to predict Acknowledgements the positions and velocities of different objects. Home Page • Once we predict events, a next step is to influence these events, i.e., to control the corresponding systems. Title Page ◭◭ ◮◮ • In this step, we should select a control that leads to the best possible result. ◭ ◮ • To be able to predict and control a system, we need to Page 2 of 45 have a good description of this system, so that we can: Go Back – use this description to analyze the system’s behav- Full Screen ior and Close – extract the desired prediction and control algorithms from this analysis. Quit
Need for Analysis, . . . 2. Symmetry: a Fundamental Property of the Phys- Symmetry: a . . . ical World Algorithmic Aspects of . . . • One of the main objectives of science: prediction. Example: How to . . . Algorithmic Aspects of . . . • Basis for prediction: we observed similar situations in Example: Selecting . . . the past, and we expect similar outcomes. Acknowledgements • In mathematical terms: similarity corresponds to sym- Home Page metry , and similarity of outcomes – to invariance. Title Page • Example: we dropped the ball, it fall down. ◭◭ ◮◮ • Symmetries: shift, rotation, etc. ◭ ◮ • In this example, we used geometric symmetries, i.e., Page 3 of 45 symmetries that have a direct geometric meaning. Go Back Full Screen Close Quit
Need for Analysis, . . . 3. Example: Discrete Geometric Symmetries Symmetry: a . . . • In the above example, the corresponding symmetries Algorithmic Aspects of . . . form a continuous family. Example: How to . . . Algorithmic Aspects of . . . • In some other situations, we only have a discrete set of Example: Selecting . . . geometric symmetries. Acknowledgements • Molecules such as benzene or cubane are invariant with Home Page respect to , e.g., rotation by 60 ◦ . molecule. Title Page 1 1 ◭◭ ◮◮ t � ✒ �❅ ❅ �❅ �❅ � � ❘ ❅ � � ❅ ❅ ❅ ◭ ◮ � ❅ � ❅ � ❅ 2 6 2 t ✻ ⇒ ⇒ . . . ❄ Page 4 of 45 5 3 ❅ � ❅ � ❅ � � � � ■ ❅ ❅ � ❅ ❅ Go Back ❅ ❅� ✠ � ❅� ❅� 4 b 1 b 2 Full Screen Figure 1: Benzene – rotation by 60 ◦ Close Quit
Need for Analysis, . . . 4. More General Symmetries Symmetry: a . . . • Symmetries can go beyond simple geometric transfor- Algorithmic Aspects of . . . mations. Example: How to . . . Algorithmic Aspects of . . . • Example: the current simplified model of an atom. Example: Selecting . . . • Originally motivated by an analogy with a Solar sys- Acknowledgements tem. Home Page • The operation has a geometric aspect: it scales down Title Page all the distances. ◭◭ ◮◮ • However, it goes beyond a simple geometric transfor- ◭ ◮ mation. Page 5 of 45 • In addition to changing distances, it also changes masses, Go Back velocities, replaces masses with electric charges, etc. Full Screen Close Quit
Need for Analysis, . . . 5. Basic Symmetries: Scaling and Shift Symmetry: a . . . • To understand real-life phenomena, we must perform Algorithmic Aspects of . . . appropriate measurements. Example: How to . . . Algorithmic Aspects of . . . • We get a numerical value of a physical quantity, which Example: Selecting . . . depends on the measuring unit. Acknowledgements • Scaling: if we use a new unit which is λ times smaller, Home Page numerical values are multiplied by λ : x → λ · x . Title Page • Example: x meters = 100 · x cm. ◭◭ ◮◮ • Another possibility: change the starting point. ◭ ◮ • Shift: if we use a new starting point which is s units Page 6 of 45 before, then x → x + s (example: time). Go Back • Together, scaling and shifts form linear transforma- tions x → a · x + b . Full Screen • Invariance: physical formulas should not depend on Close the choice of a measuring unit or of a starting point. Quit
Need for Analysis, . . . 6. Example of Using Symmetries: Pendulum Symmetry: a . . . • Problem: find how period T depends on length L and Algorithmic Aspects of . . . on free fall acceleration g on the corresponding planet. Example: How to . . . Algorithmic Aspects of . . . • Originally found using Newton’s equations. Example: Selecting . . . • The same dependence (modulo a constant) can be ob- Acknowledgements tained only using symmetries. Home Page • There is no fixed length, so we assume that the physics Title Page don’t change if we change the unit of length. ◭◭ ◮◮ • If we change a unit of length to a one λ times smaller, ◭ ◮ we get new numerical value L ′ = λ · L . Page 7 of 45 • If we change a unit of time to one µ times smaller, we get a new numerical value for the period T ′ = µ · T . Go Back Full Screen • Under these transformations, the numerical value of the acceleration changes as g → g ′ = λ · µ − 2 · g . Close Quit
Need for Analysis, . . . 7. Pendulum Example (cont-d) Symmetry: a . . . Algorithmic Aspects of . . . • The physics does not change by simply changing the units. Example: How to . . . Algorithmic Aspects of . . . • Thus, it makes sense to require that if T = f ( L, g ), then T ′ = f ( L ′ , g ′ ). Example: Selecting . . . Acknowledgements • Substituting T ′ = µ · T , L ′ = λ · L , and g ′ = λ · µ − 2 · g Home Page into T ′ = f ( L ′ , g ′ ), we get f ( λ · L, λ · µ − 2 · g ) = µ · f ( L, g ). Title Page • From this formula, we can find the explicit expression ◭◭ ◮◮ for the desired function f ( L, g ). ◭ ◮ • Indeed, let us select λ and µ for which λ · L = 1 and λ · µ − 2 · g = 1. Page 8 of 45 � • Thus, we take λ = L − 1 and µ = √ λ · g = g/L . Go Back • For these values λ and µ , the above formula takes the Full Screen � form f (1 , 1) = µ · f ( L, g ) = g/L · f ( L, g ). Close � • Thus, f ( L, g ) = const · L/g (for the constant f (1 , 1)). Quit
Need for Analysis, . . . 8. What is the Advantage of Using Symmetries? Symmetry: a . . . • What is new is that we derived it without using any Algorithmic Aspects of . . . specific differential equations. Example: How to . . . Algorithmic Aspects of . . . • We only used the fact that these equations do not have Example: Selecting . . . any fixed unit of length or fixed unit of time. Acknowledgements • Thus, the same formula is true not only for Newton’s Home Page equations, but also for any alternative theory. Title Page • Physical theories need to be experimentally confirmed. ◭◭ ◮◮ • We do not need the whole Newton’s mechanics theory ◭ ◮ to derive the pend. formula – only need symmetries. Page 9 of 45 • This shows that: Go Back – if we have an experimental confirmation of the pen- dulum formula, Full Screen – this does not necessarily mean that we have con- Close firmed Newton’s equations – just the symmetries. Quit
Need for Analysis, . . . 9. Basic Nonlinear Symmetries Symmetry: a . . . • Sometimes, a system also has nonlinear symmetries. Algorithmic Aspects of . . . Example: How to . . . • If a system is invariant under f and g , then: Algorithmic Aspects of . . . – it is invariant under their composition f ◦ g , and Example: Selecting . . . – it is invariant under the inverse transformation f − 1 . Acknowledgements • In mathematical terms, this means that symmetries Home Page form a group . Title Page • In practice, at any given moment of time, we can only ◭◭ ◮◮ store and describe finitely many parameters. ◭ ◮ • Thus, it is reasonable to restrict ourselves to finite- Page 10 of 45 dimensional groups. Go Back • Question (N. Wiener): describe all finite-dimensional Full Screen groups that contain all linear transformations. Close • Answer (for real numbers): all elements of this group are fractionally-linear x → ( a · x + b ) / ( c · x + d ) . Quit
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