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Yu D Iosifov - One of the best experts on this subject based on the ideXlab platform.

Yu Chaehyun - One of the best experts on this subject based on the ideXlab platform.

  • Proof of Cramer's rule with Dirac Delta Function
    2020
    Co-Authors: Ee June-haak, Lee Jungil, Yu Chaehyun
    Abstract:

    We present a new proof of Cramer's rule by interpreting a system of linear equations as transformation of $n$-dimensional Cartesian-Coordinate Vectors. To find the solution, we carry out the inverse transformation by convolving the original Coordinate Vector with Dirac delta functions and changing integration variables from the original Coordinates to new Coordinates. Our formulation of finding a transformation rule for multi-variable functions shall be particularly useful in changing a partial set of generalized Coordinates of a mechanical system.Comment: 3 pages, no figure

  • Proof of Cramer's rule with Dirac Delta Function
    'IOP Publishing', 2020
    Co-Authors: Ee June-haak, Lee Jungil, Yu Chaehyun
    Abstract:

    We present a new proof of Cramer's rule by interpreting a system of linear equations as a transformation of $n$-dimensional Cartesian-Coordinate Vectors. To find the solution, we carry out the inverse transformation by convolving the original Coordinate Vector with Dirac delta functions and changing integration variables from the original Coordinates to new Coordinates. As a byproduct, we derive a generalized version of Cramer's rule that applies to a partial set of variables, which is new to our best knowledge. Our formulation of finding a transformation rule for multi-variable functions shall be particularly useful in changing a partial set of generalized Coordinates of a mechanical system.Comment: 7 pages, 1 figure, version published in Eur. J. Phy

A L Shestakov - One of the best experts on this subject based on the ideXlab platform.

Andrea Pedrini - One of the best experts on this subject based on the ideXlab platform.

  • The Euler characteristic of a polyhedron as a valuation on its Coordinate Vector lattice
    arXiv: Metric Geometry, 2012
    Co-Authors: Andrea Pedrini
    Abstract:

    A celebrated theorem of Hadwiger states that the Euler-Poincar\'e characteristic is the the unique invariant and continuous valuation on the distributive lattice of compact polyhedra in R^n that assigns value one to each convex non-empty such polyhedron. This paper provides an analogue of Hadwiger's result for finitely presented unital Vector lattices (i.e. real Vector spaces with a compatible lattice order, also known as Riesz spaces). The Vector lattice of continuous and piecewise (affine) linear real-valued functions on a compact polyhedron, with operations defined pointwise from the Vector lattice R, is a finitely presented unital Vector lattice; and it is a non-trivial fact that all such Vector lattices arise in this manner, to within an isomorphism. Each function in such a Vector lattice can be written as a linear combination of a subset of distinguished elements that we call vl-Schauder hats. We prove here that the functional that assigns to each non-negative piecewise linear function on the polyhedron the Euler-Poincar\'e characteristic of its support is the unique vl-valuation (a special class of valuations on Vector lattices) that assigns one to each vl-Schauder hat of the Vector lattice.

Ee June-haak - One of the best experts on this subject based on the ideXlab platform.

  • Proof of Cramer's rule with Dirac Delta Function
    2020
    Co-Authors: Ee June-haak, Lee Jungil, Yu Chaehyun
    Abstract:

    We present a new proof of Cramer's rule by interpreting a system of linear equations as transformation of $n$-dimensional Cartesian-Coordinate Vectors. To find the solution, we carry out the inverse transformation by convolving the original Coordinate Vector with Dirac delta functions and changing integration variables from the original Coordinates to new Coordinates. Our formulation of finding a transformation rule for multi-variable functions shall be particularly useful in changing a partial set of generalized Coordinates of a mechanical system.Comment: 3 pages, no figure

  • Proof of Cramer's rule with Dirac Delta Function
    'IOP Publishing', 2020
    Co-Authors: Ee June-haak, Lee Jungil, Yu Chaehyun
    Abstract:

    We present a new proof of Cramer's rule by interpreting a system of linear equations as a transformation of $n$-dimensional Cartesian-Coordinate Vectors. To find the solution, we carry out the inverse transformation by convolving the original Coordinate Vector with Dirac delta functions and changing integration variables from the original Coordinates to new Coordinates. As a byproduct, we derive a generalized version of Cramer's rule that applies to a partial set of variables, which is new to our best knowledge. Our formulation of finding a transformation rule for multi-variable functions shall be particularly useful in changing a partial set of generalized Coordinates of a mechanical system.Comment: 7 pages, 1 figure, version published in Eur. J. Phy