The Shortest-Known Paper Published in a Serious Math Journal: Two Succinct Sentences

My contribution to this topic:

Short post, short papers, enjoy! LOG#170. The shortest papers ever: the list.

http://www.thespectrumofriemannium.com/2015/04/13/log170-the-shortest-papers-ever-the-list/

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MATHEMATICS AND EDUCATION

ABSTRACT
]. In these approaches one may derive covariant versions of the Fokker-Planck equation of Brownian motion in curved spaces. The mathematical approach to path integrals uses similar techniques [5]. The inherent ambiguities can be removed by demanding a certain form for Schrödinger equation of the system, which in curved space is have the Laplace-Beltrami operator as an operator for the kinetic energy [2], without an additional curvature scalar.

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GAUSS AND EULER
ABSTRACT
Let’s approach Leonhard Euler and his work the same way. It will make things a whole lot easier.

If one is not a mathematician (and except for a few of you out there, who is?), it’s going to be impossible to actually understand why Euler was such a great man. Other people will have to tell us, and we should probably believe them.

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PROBLEM OF OPEN CULTURE AND APPLICATIONS
ABSTRACT
In general, partial differential equations are difficult to solve, but techniques have been developed for simpler classes of equations called linear, and for classes known loosely as “almost” linear, in which all derivatives of an order higher than one occur to the first power and their coefficients involve only the independent variables.
Many physically important partial differential equations are second-order and linear. For example:
uxx + uyy = 0 (two-dimensional Laplace equation)
uxx = ut (one-dimensional heat equation)
uxx − uyy = 0 (one-dimensional wave equation)

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PARTIAL DIFFERENTIAL EQUATIONS AND ANALYTICAL CALCULUS
ABSTRACT
is very pedagogical thes questions and Partial differential equation, in mathematics, equation relating a function of several variables to its partial derivatives. A partial derivative of a function of several variables expresses how fast the function changes when one of its variables is changed, the others being held constant (compare ordinary differential equation). The partial derivative of a function is again a function, and, if f(x, y) denotes the original function of the variables x and y, the partial derivative with respect to x—i.e., when only x is allowed to vary—is typically written as fx(x, y) or ∂f/∂x. The operation of finding a partial derivative can be applied to a function that is itself a partial derivative of another function to get what is called a second-order partial derivative.In general, partial differential equations are difficult to solve, but techniques have been developed for simpler classes of equations called linear, and for classes known loosely as “almost” linear, in which all derivatives of an order higher than one occur to the first power and their coefficients involve only the independent variables.
Many physically important partial differential equations are second-order and linear. For example:

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Effect of limited T
(3) Dose DFT give for every f ?
No! only discrete frequencies.
DFT as an estimate for X(f): even worse than due to the limited frequency resolution.

1. Effect of sampling frequency (or number of points) on accuracy when T is given: Example
use for
2. Effect of T (window size)
Compare and for

there are many questions in History or science that are observed by prof dr mircea orasanu and prof drd horia orasanu and as followed for establised of important situations and chapter of scleronoomous developments and other so a simple pendulum is a system composed of a weight and a string. The string is attached at the top end to a pivot and at the bottom end to a weight. Being inextensible, the string’s length is a constant. Therefore, this system is scleronomous; it obeys scleronomic constraint
x 2 + y 2 − L = 0 , {\displaystyle {\sqrt {x^{2}+y^{2}}}-L=0\,\!,} {\sqrt {x^{2}+y^{2}}}-L=0\,\!,
where ( x , y ) {\displaystyle (x,y)\,\!} (x,y)\,\! is the position of the weight and L {\displaystyle L\,\!} L\,\! is length of the string.
A simple pendulum with oscillating pivot point
Take a more complicated example. Refer to the next figure at right, Assume the top end of the string is attached to a pivot point undergoing a simple harmonic motion
x t = x 0 cos ⁡ ω t , {\displaystyle x_{t}=x_{0}\cos \omega t\,\!,} x_{t}=x_{0}\cos \omega t\,\!,
where x 0 {\displaystyle x_{0}\,\!} x_{0}\,\! is amplitude, ω {\displaystyle \omega \,\!} \omega \,\! is angular frequency, and t {\displaystyle t\,\!} t\,\! is time.

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for most important aspects are presented other situations so that observed prof dr mircea and prof dr d horia orasanu that remarked the form of applications and followed that then so lead to CONSTRAINTS problem and thus . Locomotion of a snake-like structure in accordance with the serpenoid curve, i.e. lateral undulation, is achieved if the joints of the robot move according to the reference joint trajectories in the form of a sinusoidal function with specified amplitude, frequency, and phase shift. In particular, using the foregoing defined new states, we define a constraint function for the ith joint of the snake robot by
Φi = αsin(η + (i − 1)δ) + ϕo
(45)
where i∈{1,…,N−1}, α denotes the amplitude of the sinusoidal joint motion, and δ is a phase shift that is used to keep the joints out of phase. Moreover, ϕo is an offset value that is identical for all of the joints. It was illustrated in [16] how the offset value ϕo affects the orientation of the snake robot in the plane. Building further on this insight, we consider the second-order time derivative of ϕo in the form of a dynamic compensator, which will be used to control the orientation of the robot. In particular, through this control term, we modify the orientation of the robot in accordance with a reference orientation. This will be done by adding an offset angle to the reference trajectory of each joint. We will show that this will steer the position of the CM of the robot towards the desired path. The constraint function (45) is dynamic, since it depends on the solution of a dynamic compensator.
Virtual holonomic constraint for the head link angle
In this subsection, we define a constraint function for the head angle of the robot. In particular, we use a line-of-sight (LOS) guidance law as the reference angle for the head link. LOS guidance is a much-used method in marine control systems (see, e.g. [27]). In general, guidance-based control strategies are based on defining a reference heading angle for the vehicle through a guidance law and designing a controller to track this angle [27]. Motivated by marine control literature, in [17] based on a simplified model of the snake robot, using cascade systems theory, it was proved that if the heading angle of the snake robot was controlled to the LOS angle, then also the position of the CM of the robot would converge to the desired path. We will show that a similar guidance-based control strategy can successfully steer the robot towards the desired path. However, we perform the model-based control design based on a more accurate model of the snake robot which does not contain the simplifying assumptions of [17] which are valid for small joint angles.
To define the guidance law, without loss of generality, we assign the global coordinate system such that the global x-axis is aligned with the desired path. Consequently, the position of the CM of the robot along the y-axis, denoted by py, defines the shortest distance between the robot and the desired path, often referred to as the cross-track error. In order to solve the path following problem, we use the LOS guidance law as a virtual holonomic constraint, which defines the desired head angle as a function of the cross-track error as
ΦN=−tan−1(pyΔ)

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