# Questions tagged [derivations]

The derivations tag has no usage guidance.

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### 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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**1**answer

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### 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**

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**1**answer

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 ...

**3**

votes

**2**answers

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_{...

**16**

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**2**answers

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
$f^{(0)} = f$, ...

**2**

votes

**2**answers

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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**0**answers

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}...

**9**

votes

**3**answers

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

**1**answer

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?

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**1**answer

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**

votes

**0**answers

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 ${\rm Aut}_{L_1}(L)$ ...

**1**

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**0**answers

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[...

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vote

**2**answers

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

**0**answers

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

**1**answer

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 ...