Papers by Vasiliy Knyshev
There is a well-known theorem that represents an interval as a finite or countable sum of pairwis... more There is a well-known theorem that represents an interval as a finite or countable sum of pairwise disjoint intervals. The sum of the lengths of these
On the integration of the derivative of a function. DEFINITION. There is a well-known theorem abo... more On the integration of the derivative of a function. DEFINITION. There is a well-known theorem about representing an interval as a finite or countable sum of pairwise disjoint intervals. The sum of the lengths of these adjacent intervals is equal to the length of the interval. Points that do not belong to intervals are junction points of adjacent intervals or condensation points. Let there be a sum such that on each interval the function is strictly monotone and has an inverse. Then we call such a function non-degenerate. If for any such sum of intervals on any of its intervals the function does not have an inverse or is not strictly monotone, then we call it degenerate. We call the remaining functions semidegenerate. COMMENT. If the function is continuous on intervals, then it is sufficient to require either strict monotonicity or invertibility. EXAMPLE. A constant is a degenerate function. THEOREM. A non-degenerate function has a finite non-zero derivative almost everywhere. PROOF. According to Lebesgue's theorem, a non-degenerate function, as a strictly monotone function on intervals, has a finite derivative almost everywhere. Let it be equal to zero almost everywhere. By definition, it has the opposite. Then its derivative will also be monotone and not finite almost everywhere. This contradicts Lebesgue's theorem. INVESTIGATION. As the negation of a non-degenerate function, a degenerate function will almost everywhere be either non-differentiable, or its derivative will not be finite, or equal to zero. COMMENT. The primitive of a non-degenerate derivative of a function is not degenerate. If a primitive function is degenerate, then its derivative cannot coincide with its non-degenerate derivative function. EXAMPLE. The example of a linear function shows that the derivative of a non-degenerate function can be degenerate. THEOREM. Let the non-degenerate derivative function be defined and finite at each point of each interval of the sum of intervals. Then it is Riemann integrable. PROOF. A non-degenerate derivative on intervals is strictly monotone, like a derivative, it takes all values, is continuous, and is integrable by the Riemann indefinite integral. Since the derivative is finite, its primitive is continuous. At points not belonging to intervals, it must be further defined by continuity. These are the junction points of intervals and the points of their condensation. GENERALIZATION. A non-degenerate function on intervals is strictly monotone and can have simple discontinuities of the first kind. Since a derivative function takes on all intermediate values, such a function may not be a derivative. At the same time, being monotone, it is Riemann integrable on intervals. There is a statement that the derivative of such an indefinite Riemann integral almost everywhere coincides with the integrable function. At points not belonging to intervals, if possible, it is necessary to further define the Riemann indefinite integral by continuity. Об интегрировании производной функции. ОПРЕДЕЛЕНИЕ. Известна теорема о представлении интервала конечной или счетной суммой попарно не пересекающихся интервалов. Сумма длин этих
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Annotation: We introduce the notion of a non-degenerate function. Thanks to this received Newton'... more Annotation: We introduce the notion of a non-degenerate function. Thanks to this received Newton's second law of the third order. As well as the equation of motion of a particle in a central field.
Annotation: We introduce the notion of a non-degenerate function. Thanks to this received Newton&... more Annotation: We introduce the notion of a non-degenerate function. Thanks to this received Newton's second law of the third order. As well as the equation of motion of a particle in a central field.
Galilean motion for Newton's second law of the third order.
What is a non-degenerate function?
Definition of a non-degenerate function. This allows us to differentiate the direct and inverse f... more Definition of a non-degenerate function. This allows us to differentiate the direct and inverse functions at intervals.
On second Newton law, 2015
We introduce the notion of a non-degenerate function. Thanks to this received Newton's second law... more We introduce the notion of a non-degenerate function. Thanks to this received Newton's second law of the third order. As well as the equation of motion of a particle in a central field.
О втором законе Ньютона, 2015
Второй закон Ньютона третьего порядка.
Newton's second law of the third order, 2018
Annotation: We introduce the notion of a non-degenerate function. Thanks to this received Newton'... more Annotation: We introduce the notion of a non-degenerate function. Thanks to this received Newton's second law of the third order. As well as the equation of motion of a particle in a central field.
Annotation: The introduction of notion of
a non-degenerate function. Thanks to this received
... more Annotation: The introduction of notion of
a non-degenerate function. Thanks to this received
Newton's second law of the third order. As well as
the equation of motion of a particle in a
central field.
Drafts by Vasiliy Knyshev
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Papers by Vasiliy Knyshev
a non-degenerate function. Thanks to this received
Newton's second law of the third order. As well as
the equation of motion of a particle in a
central field.
Drafts by Vasiliy Knyshev
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https://sites.google.com/site/knyshus1/home/ver2
a non-degenerate function. Thanks to this received
Newton's second law of the third order. As well as
the equation of motion of a particle in a
central field.
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https://sites.google.com/site/knyshus1/home/ver2