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In mathematical physics, spacetime algebra (STA) is the application of Clifford algebra Cl 1,3 (R), or equivalently the geometric algebra G(M 4) to physics. Spacetime algebra provides a "unified, coordinate-free formulation for all of relativistic physics, including the Dirac equation, Maxwell equation and General Relativity" and "reduces the mathematical divide between classical, quantum and ...
The satisfiability problem becomes more difficult if both "for all" and "there exists" quantifiers are allowed to bind the Boolean variables. An example of such an expression would be ∀x ∀y ∃z (x ∨ y ∨ z) ∧ (¬x ∨ ¬y ∨ ¬z); it is valid, since for all values of x and y, an appropriate value of z can be found, viz. z=TRUE if ...
Each algebraic expression or equation entered at the main prompt is stored in an equation space. Mathomatic is written to do most symbolic manipulations with memory moves, like an assembly language program. This causes Mathomatic to crash when used with the new LLVM backend, which doesn't seem to like the standard C library function memmove(3 ...
The expression on the left represents the curvature of spacetime as determined by the metric; the expression on the right represents the stress–energy–momentum content of spacetime. The EFE can then be interpreted as a set of equations dictating how stress–energy–momentum determines the curvature of spacetime.
An expression is often used to define a function, by taking the variables to be arguments, or inputs, of the function, and assigning the output to be the evaluation of the resulting expression. [5] For example, x ↦ x 2 + 1 {\displaystyle x\mapsto x^{2}+1} and f ( x ) = x 2 + 1 {\displaystyle f(x)=x^{2}+1} define the function that associates ...
In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualizing and understanding relativistic effects, such as how different observers perceive where and when events ...
In control engineering and system identification, a state-space representation is a mathematical model of a physical system that uses state variables to track how inputs shape system behavior over time through first-order differential equations or difference equations. These state variables change based on their current values and inputs, while ...
For example, the Schwarzschild solution has a Killing algebra of dimension four (three spatial rotational vector fields and a time translation), whereas the Friedmann–Lemaître–Robertson–Walker metric (excluding the Einstein static subcase) has a Killing algebra of dimension six (three translations and three rotations). The Einstein ...