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      Questions tagged [derivations]

      The tag has no usage guidance.

      2
      votes
      0answers
      62 views

      How can I prove that the following function is increasing according to x1?

      Suppose that $0 \le {X_1} < {X_2} < {X_3}$ . How is it possible to prove the following function is increasing based on ${X_1}$ in the range of $0 \le {X_1} < {X_2}$ ? $f({X_1},{X_2},{X_3})...
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      1answer
      65 views

      Continuity of the derivations from semisimple Banach algebras

      Let $A$ be a Banach algebra and $X$ a Banach $A$-bimodule. It is known that if $A$ is a $C^*$-algebra, then by Ringrose theorem every derivation $D:A\rightarrow X$ is continuous. Also, a famous ...
      1
      vote
      1answer
      61 views

      Locally nilpotent derivations on rings with zero divisors

      Almost all books that I have found deal with derivation on several types of rings (or algebras) (for instance, commutative, noncommutative, domains, non-domains etc). However, each paper about ...
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      votes
      2answers
      76 views

      Why we use Caputo fractional derivative in application?

      I'm working on some papers which use Caputo fractional evolution equation as application for thier main result: For example: $$\left\{\begin{matrix} ^CD^{\sigma}_tx(t)+Ax(t)=&f(t,x(t),\int_{...
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      votes
      2answers
      694 views

      “Insanely increasing” $C^\infty$ function with upper bound

      Let $C^\infty$ denote the collection of functions $f:\mathbb{R}\to\mathbb{R}$ such that for every positive integer $n$, the $n$-th derivative of $f$ exists. For $f\in C^\infty$ we set $f^{(0)} = f$, ...
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      votes
      2answers
      195 views

      Find formula for recurrence relation with two function and two variables

      $f(n,k) = 2g(n-2,k-1)+f(n-1,k)$ $g(n,k) = g(n-1,k-1)+f(n,k)$ when $n\le0$ or $k\le0: \quad f(n,k) = 0$ when $n < k:\quad f(n,k) = 0$ when $n-k<-1:\quad g(n,k) = 0$ when $k=0:\quad g(n,k) = 1$ $...
      16
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      0answers
      442 views

      Does every real function have this weak derivation property?

      After this question : Does every real function have this weak continuity property? Natrualy there are an other (more difficult) : Is it true that for every real function $f:\mathbb{R}\to\mathbb{R}...
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      votes
      3answers
      945 views

      Why should we study derivations of algebras?

      Some authors have written that derivation of an algebra is an important tools for studying its structure. Could you give me a specific example of how a derivation gives insight into an algebra's ...
      1
      vote
      1answer
      487 views

      Partial derivatives of spherical harmonics

      Is there any closed form formula (or some procedure) to find all $n$-th partial derivatives of a spherical harmonic?
      3
      votes
      1answer
      79 views

      Locally nilpotent derivation on $A[X,Y]$ whose kernel is $A$; where $A$ is an affine $k$ domain, $char k=0$

      Let $k$ be a field of characteristic zero and $A$ be a $k$-algebra. A derivation on $A$ is a $k$-linear map $D: A \to A$ such that $D(ab)=aD(b)+bD(a), \forall a,b \in A$. A derivation is called ...
      2
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      0answers
      46 views

      Lie algebra of derivations for a transcendental field extension and intersection fields

      Suppose that $L$ is a finite Galois extension of the field $K$. If $L_1$ and $L_2$ are subfields of $L$ containing $K$ then $L_1\cap L_2=L^H$ where $H$ is the group generated by ${\rm Aut}_{L_1}(L)$ ...
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      0answers
      102 views

      Bijective correspondence between $\mathbb G_a$ actions on affine varieties and exponential maps on affine $k$-domains

      Let $A$ be an integral domain which is a finitely generated algebra over an algebraically closed field $k$. Let $\phi :A \to A^{[1]}$ be a $k$-algebra homomorphism and let us write $\phi_t : A \to A[...
      1
      vote
      2answers
      119 views

      Prove a $C^{\infty}$ multivariable function is lipchitz via Jacobian matrix [closed]

      I would like to prove a real $C^{\infty}$(polynomial) multivariable function $F : (a_1,a_2,...a_n) \rightarrow (b_1,b_2,...b_n) $ is lipchitz of parameter $l$ is it sufficient to prove the norm of ...
      2
      votes
      0answers
      142 views

      Derivative with multiple summation operators

      I have a defined utility function as Eq.(1), and I am seeking the minimized utility subjects to some constraints. The notation used is as following: \linebreak $V$ is the set of nodes, $v_i\in V$; $O$...
      6
      votes
      1answer
      494 views

      Semantics of derivations as derivatives

      My understanding of how derivations on commutative rings are like derivatives is that a derivation on $R$ is differentiation with respect to a vector field on $\text{Spec}(R)$. But derivations are ...

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