Partial Derivatives of Regular Expressions over Alphabet-Invariant and User-Defined Labels

A perturbation formalism is presented which shows how the stationary distribution and fundamental matrix of a Markov chain containing a single irreducible set of states change as the transition probabilities vary. Expressions are given for the partial derivatives of the stationary distribution and fundamental matrix with respect to the transition probabilities. Semi-group properties of the generators of transformations from one Markov chain to another are investigated. It is shown that a perturbation formalism exists in the multiple subchain case if and only if the change in the transition probabilities does not alter the number of, or intermix the various subchains. The formalism is presented when this condition is satisfied.

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ON A NONLOCAL PROBLEM FOR PARTIAL DIFFERENTIAL EQUATIONS OF PARABOLIC TYPE

Bukovinian Mathematical Journal ◽

10.31861/bmj2020.02.03 ◽

2020 ◽

Vol 8 (2) ◽

pp. 24-39

Author(s):

V. Gorodetskiy ◽

R. Kolisnyk ◽

O. Martynyuk

Keyword(s):

Cauchy Problem ◽

Fundamental Solution ◽

Parabolic Type ◽

Generalized Functions ◽

Initial Condition ◽

Partial Derivatives ◽

Time Problem ◽

The Cauchy Problem ◽

Space Of Generalized Functions ◽

Derivatives Of

Spaces of $S$ type, introduced by I.Gelfand and G.Shilov, as well as spaces of type $S'$, topologically conjugate with them, are natural sets of the initial data of the Cauchy problem for broad classes of equations with partial derivatives of finite and infinite orders, in which the solutions are integer functions over spatial variables. Functions from spaces of $S$ type on the real axis together with all their derivatives at $|x|\to \infty$ decrease faster than $\exp\{-a|x|^{1/\alpha}\}$, $\alpha > 0$, $a > 0$, $x\in \mathbb{R}$. The paper investigates a nonlocal multipoint by time problem for equations with partial derivatives of parabolic type in the case when the initial condition is given in a certain space of generalized functions of the ultradistribution type ($S'$ type). Moreover, results close to the Cauchy problem known in theory for such equations with an initial condition in the corresponding spaces of generalized functions of $S'$ type were obtained. The properties of the fundamental solution of a nonlocal multipoint by time problem are investigated, the correct solvability of the problem is proved, the image of the solution in the form of a convolution of the fundamental solution with the initial generalized function, which is an element of the space of generalized functions of $S'$ type.

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Comment on "Finding Generalized Forces by Use of Partial Derivatives of Energy Functions"

IEEE Transactions on Education ◽

10.1109/te.1968.4320341 ◽

1968 ◽

Vol 11 (1) ◽

pp. 78-78

Keyword(s):

Energy Functions ◽

Partial Derivatives ◽

Derivatives Of

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Identification of principal screws of two- and three-screw systems

Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science ◽

10.1243/09544062jmes752 ◽

2007 ◽

Vol 221 (12) ◽

pp. 1701-1715 ◽

Cited By ~ 7

Author(s):

J-S Zhao ◽

F Chu ◽

Z-J Feng

Keyword(s):

Screw Theory ◽

Coordinate Systems ◽

Third Order ◽

Analytical Methodology ◽

Partial Derivatives ◽

Spatial Mechanisms ◽

Analysis Theory ◽

Definition Of ◽

Derivatives Of ◽

Homogeneous Linear

The current paper proposes a unified analytical methodology to identify the principal screws of two- and three-screw systems. Based on the definition of the pitch of a screw, it first obtains an identical homogeneous quadric equation. According to functional analysis theory, it is known that the partial derivatives of an identical quadric equation with respect to its variables must be zero. Therefore, the paper deduces a set of linear homogeneous equations that are made up of the partial derivatives of the quadric equation. With the existing criteria of non-zero solutions for homogeneous linear algebra equations, it ultimately obtains the formulas of the principal pitches and the associated principal screws of the system. The most outstanding contribution of this methodology is that it proposes a unified analytical approach to identify the principal pitches and the principal coordinate systems of the second-order and the third-order screw systems. This should be a new contribution to the screw theory and will boost its applications to the kinematics analysis of robots and spatial mechanisms.

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