Solving Quantum Equations with Gauge Fields: How Explicit Integrators Based on a Bipartite Lattice and Affine Transformations Can Help

2021 ◽  
Vol 1 ◽  
Keyword(s):  
Time Evolution ◽  
Affine Transformation ◽  
Evolution Equations ◽  
Gauge Fields ◽  
Affine Transformations ◽  
Spatial Discretization ◽  
Quantum Vortices ◽  
Numerical Integrator ◽  
Quantum Phenomena ◽  
Quantum Equations

We proposed an explicit numerical integrator consisting of affine transformation pairs resulting from the checkerboard lattice for spatial discretization. It can efficiently solve time evolution equations that describe dynamical quantum phenomena under gauge fields, e.g., generation, motion, interaction of quantum vortices in superconductors or superfluids.

The Standard Metric

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Affine Transformation ◽  
Convex Sets ◽  
Finite Dimension ◽  
Affine Subspace ◽  
Affine Transformations ◽  
Real Vector ◽  
Euclidean Metric ◽  
Affine Hyperplane ◽  
Tits Building ◽  
Convex Closure

This chapter presents some results about groups generated by reflections and the standard metric on a Bruhat-Tits building. It begins with definitions relating to an affine subspace, an affine hyperplane, an affine span, an affine map, and an affine transformation. It then considers a notation stating that the convex closure of a subset a of X is the intersection of all convex sets containing a and another notation that denotes by AGL(X) the group of all affine transformations of X and by Trans(X) the set of all translations of X. It also describes Euclidean spaces and assumes that the real vector space X is of finite dimension n and that d is a Euclidean metric on X. Finally, it discusses Euclidean representations and the standard metric.

scholarly journals A Remark on the Principle of Zero Utility

1982 ◽  
Vol 13 (2) ◽  
pp. 133-134 ◽  
Author(s):  
Hans U. Gerber
Keyword(s):  
Risk Aversion ◽  
Utility Function ◽  
Affine Transformation ◽  
Random Variable ◽  
Affine Transformations ◽  
Premium Calculation ◽  
Image Position

Let u(x) be a utility function, i.e., a function with u′(x)>0, u″(x)<0 for all x. If S is a risk to be insured (a random variable), the premium P = P(x) is obtained as the solution of the equationwhich is the condition that the premium is fair in terms of utility. It is clear that an affine transformation of u generates the same principle of premium calculation. To avoid this ambiguity, one can standardize the utility function in the sense thatfor an arbitrarily chosen point y. Alternatively, one can consider the risk aversionwhich is the same for all affine transformations of a utility function.Given the risk aversion r(x), the standardized utility function can be retrieved from the formulaIt is easily verified that this expression satisfies (2) and (3).The following lemma states that the greater the risk aversion the greater the premium, a result that does not surprise.

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