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Register a new userThe Exterior Derivative - A direct approach
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Gopala Krishna Srinivasan Dr
Publication date
2018
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and research's language is
English
Authors
Gopala Krishna Srinivasan
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In this note we provide a direct approach to the most basic operator in this theory namely the exterior derivative. The crucial ingredient is a calculus lemma based on determinants. We maintain the view that in a first course at least this direct approach is preferable to the more abstract one based on characterization of the exterior derivative in terms of its properties.
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Let $A$ be Banach algebra over commutative ring $D$. The map $f:Arightarrow A $ is called differentiable in the Gateaux sense, if $$f(x+a)-f(x)=partial f(x)circ a+o(a)$$ where the Gateaux derivative $partial f(x)$ of map $f$ is linear map of increment $a$ and $o$ is such continuous map that $$ lim_{arightarrow 0}frac{|o(a)|}{|a|}=0 $$ Assuming that we defined the Gateaux derivative $partial^{n-1} f(x)$ of order $n-1$, we define $$ partial^n f(x)circ(a_1otimes...otimes a_n) =partial(partial^{n-1} f(x)circ(a_1otimes...otimes a_{n-1}))circ a_n $$ the Gateaux derivative of order $n$ of map $f$. Since the map $f(x)$ has all derivatives, then the map $f(x)$ has Taylor series expansion $$ f(x)=sum_{n=0}^{infty}(n!)^{-1}partial^n f(x_0)circ(x-x_0)^n $$
In this article we propose a new fractional derivative without singular kernel. We consider the potential application for modeling the steady heat-conduction problem. The analytical solution of the fractional-order heat flow is also obtained by means of the Laplace transform.
In this paper we suggest how the mathematical concept of hyperstructures may be a useful tool in the study of the higher, hierachical structure of languages.
Quark orbital angular momentum (OAM) in the proton can be calculated directly given a Wigner function encoding the simultaneous distribution of quark transverse positions and momenta. This distribution can be accessed via proton matrix elements of a quark bilocal operator (the separation in which is Fourier conjugate to the quark momentum) featuring a momentum transfer (which is Fourier conjugate to the quark position). To generate the weighting by quark transverse position needed to calculate OAM, a derivative with respect to momentum transfer is consequently required. This derivative is evaluated using a direct derivative method, i.e., a method in which the momentum derivative of a correlator is directly sampled in the lattice calculation, as opposed to extracting it a posteriori from the numerical correlator data. The method removes the bias stemming from estimating the derivative a posteriori that was seen to afflict a previous exploratory calculation. Data for Ji OAM generated on a clover ensemble at pion mass $m_{pi } = 317, mbox{MeV} $ are seen to agree with the result obtained via the traditional Ji sum rule method. By varying the gauge connection in the quark bilocal operator, also Jaffe-Manohar OAM is extracted, and seen to be enhanced significantly compared to Ji OAM.
The decay rates of quasistable states in quantum field theories are usually calculated using instanton methods. Standard derivations of these methods rely in a crucial way upon deformations and analytic continuations of the physical potential, and on the saddle point approximation. While the resulting procedure can be checked against other semi-classical approaches in some one-dimensional cases, it is challenging to trace the role of the relevant physical scales, and any intuitive handle on the precision of the approximations involved are at best obscure. In this paper, we use a physical definition of the tunneling probability to derive a formula for the decay rate in both quantum mechanics and quantum field theory directly from the Minkowski path integral, without reference to unphysical deformations of the potential. There are numerous benefits to this approach, from non-perturbative applications to precision calculations and aesthetic simplicity.
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