Defining a Function programmatically

I need to create a function programmatically. For example, suppose I’ve got:

  1. mon – a Symbol
  2. mons – a List of Symbols
  3. vars – another List of Symbols (same Length)

and want to make a function as:

Function[{«mon»},  Function[«mons»,   Internal`InheritedBlock[«vars»,    «vars[[1]]» =.;    «vars[[2]]» =.;    ...    «vars[[-1]]» =.;    «vars[[1]]» = «mons[[1]]»;    «vars[[2]]»[t] = «mons[[2]]»;    ...    «vars[[-1]]»[t] = «mons[[-1]]»;    «mon» ]]] 

where «» denotes injecting from the given mon, mons, and vars.

So the input

mon = Unique[NDSolve`Monitor]; mons = Table[Unique[mon], {3}]; vars = {t, x, y}; 

would result in the desired output:

Function[{NDSolve`Monitor$  3080},   Function[{NDSolve`Monitor$  3080$  3081, NDSolve`Monitor$  3080$  3082, NDSolve`Monitor$  3080$  3083},   Internal`InheritedBlock[{t, x, y},    t =.;    x =.;    y =.;    t = NDSolve`Monitor$  3080$  3081;    x[t] = NDSolve`Monitor$  3080$  3082;     y[t] = NDSolve`Monitor$  3080$  3083;    NDSolve`Monitor$  3080 ]]] 

One possible solution involves building up a String, then using ToExpression:

str = "Function[{" <> ToString[mon] <> "},   Function[" <> ToString[mons] <> ",   Internal`InheritedBlock[" <> ToString[vars] <> ", "; Do[   str = str <> ToString[var] <> "=.;\n" , {var, vars}]; str = str <> "t=" <> ToString[mons[[1]]] <> ";\n"; Do[   str = str <> ToString[vars[[i]]] <> "[t]=" <> ToString[mons[[i]]] <> ";\n" , {i, 2, Length[vars]}]; str = str <> ToString[mon] <> "]]]\n"; 

but this is kind of inelegant and can be slow for large lists.

Are there any nicer and/or faster alternatives?

not able to configure a provider with token in angular without defining the string

I am trying to create a provider in the module as shown below but getting an error saying that MY_TOKEN is not defined. Is it necessary to put quotes around this literal as I think that following code snippet is legal. Does it have it have to be an import or a string constant if I have to use it. Will it not automatically treat MY_TOKEN as a string literal?

 providers: [{provide : MY_TOKEN, useValue:'dfd'}, 

On Defining the Fourier Transform and Performing Changes of Variable on Quotient Subgroups of $\mathbb{Q}/$

Much to my dismay, in my work the more number-theoretic side of harmonic analysis (ex: the fourier transform on the adeles, on the profinite integers, etc.), I have found myself struggling with technicalities that emerge from frustratingly simple issues—so simple (and yet, so technical) that I haven’t been able to find anything that might shine any light on the matter.

Let $ L_{\textrm{loc}}^{2}\left(\mathbb{Q}\right)$ be the space of functions $ f:\mathbb{Q}\rightarrow\mathbb{C}$ which satisfy $ \sum_{t\in T}\left|f\left(t\right)\right|^{2}<\infty$ for all bounded subsets $ T\subseteq\mathbb{Q}$ . Letting $ \mu$ be any positive integer, it is easy to show that any functions which is both $ \mu$ -periodic (an $ f:\mathbb{Q}\rightarrow\mathbb{C}$ such that $ f\left(t+\mu\right)=f\left(t\right)$ for all $ t\in\mathbb{Q})$ and in $ L_{\textrm{loc}}^{2}\left(\mathbb{Q}\right)$ is necessarily an element of $ L^{2}\left(\mathbb{Q}/\mu\mathbb{Z}\right)$ , the complex hilbert space of functions $ f:\mathbb{Q}/\mu\mathbb{Z}\rightarrow\mathbb{C}$ so that: $ $ \sum_{t\in\mathbb{Q}/\mu\mathbb{Z}}\left|f\left(t\right)\right|^{2}<\infty$ $ Equipping $ \mathbb{Q}/\mu\mathbb{Z}$ with the discrete topology, we can utilize Pontryagin duality to obtain a Fourier transform: $ \mathscr{F}_{\mathbb{Q}/\mu\mathbb{Z}}$ . The ideal case is when $ \mu=1$ . There, $ \mathscr{F}_{\mathbb{Q}/\mathbb{Z}}$ is an isometric hilbert space isomorphism from $ L^{2}\left(\mathbb{Q}/\mathbb{Z}\right)$ to $ L^{2}\left(\overline{\mathbb{Z}}\right)$ , where: $ $ \overline{\mathbb{Z}}\overset{\textrm{def}}{=}\prod_{p\in\mathbb{P}}\mathbb{Z}_{p}$ $ is the ring of profinite integers, where $ \mathbb{P}$ is the set of prime numbers, and where $ L^{2}\left(\overline{\mathbb{Z}}\right)$ is the space of functions $ \check{f}:\overline{\mathbb{Z}}\rightarrow\mathbb{C}$ which are square-integrable with respect to the haar probability measure $ d\mathfrak{z}=\prod_{p\in\mathbb{P}}d\mathfrak{z}_{p}$ on $ \overline{\mathbb{Z}}$ .

The first sign of trouble was when I learned that, for any integers $ \mu,\nu$ , the (additive) quotient groups $ \mathbb{Q}/\mu\mathbb{Z}$ and $ \mathbb{Q}/\nu\mathbb{Z}$ are group-isomorphic to one another, and thus, that both have $ \overline{\mathbb{Z}}$ as their Pontryagin dual. From my point of view, however, this isomorphism seems to only cause trouble. In my work, I am identifying $ L^{2}\left(\mathbb{Q}/\mu\mathbb{Z}\right)$ with the set of $ \mu$ -periodic functions $ f:\mathbb{Q}\rightarrow\mathbb{C}$ which are square integrable with respect to the counting measure on $ \mathbb{Q}\cap\left[0,\mu\right)$ . As such, $ \mathbb{Q}/\mu\mathbb{Z}$ and $ \mathbb{Q}/\nu\mathbb{Z} $ cannot be “the same” from my point of view, because $ f\left(t\right)\in L^{2}\left(\mathbb{Q}/\mu\mathbb{Z}\right)$ need not imply that $ f\left(t\right)\in L^{2}\left(\mathbb{Q}/\nu\mathbb{Z}\right)$ .

In my current work, I am dealing with a functional equation of the form:$ $ \sum_{n=0}^{N-1}g_{n}\left(t\right)f\left(\frac{a_{n}t+b_{n}}{d_{n}}\right)=0$ $ where $ N$ is an integer $ \geq2$ , where the $ g_{n}$ s are known periodic functions, where $ f$ is an unknown function in $ L^{2}\left(\mathbb{Q}/\mathbb{Z}\right)$ (i.e., $ f\left(t\right)\in L_{\textrm{loc}}^{2}\left(\mathbb{Q}\right)$ and $ f\left(t+1\right)=f\left(t\right)$ for all $ t\in\mathbb{Q})$ , and where $ a_{n},b_{n},d_{n}$ are integers with $ \gcd\left(a_{n},d_{n}\right)=1$ for all $ n$ . For brevity, I’ll write: $ $ \varphi_{n}\left(t\right)\overset{\textrm{def}}{=}\frac{a_{n}t+b_{n}}{d_{n}}$ $ Because $ \varphi_{n}\left(t+1\right)$ need not equal $ \varphi_{n}\left(t\right)+1$ , the individual functions $ f\circ\varphi_{n}$ , though periodic and in $ L_{\textrm{loc}}^{2}\left(\mathbb{Q}\right)$ , are not necessarily going to be of period $ 1$ . Letting $ p$ denote the least common multiple of the periods of $ g_{n}$ and the $ f\circ\varphi_{n}$ s, I can view the functional equation as existing in $ L^{2}\left(\mathbb{Q}/p\mathbb{Z}\right)$ , and as such, I hope to be able to simplify it by applying the fourier transform.

Letting $ e^{2\pi i\left\langle t,\mathfrak{z}\right\rangle }$ denote the duality pairing between elements $ t\in\mathbb{Q}$ (or $ \mathbb{Q}/\mu\mathbb{Z}$ ) and $ \mathfrak{z}\in\overline{\mathbb{Z}}$ , the idea is to multiply the functional equation by $ e^{2\pi i\left\langle t,\mathfrak{z}\right\rangle }$ , sum over an appropriate domain of $ t$ (ideally, $ \mathbb{Q}/p\mathbb{Z}$ ), make a change of variables in $ t$ to move one of the $ \varphi_{n}\left(t\right)$ s out of $ f$ and into $ e^{2\pi i\left\langle t,\mathfrak{z}\right\rangle }$ , pull out terms from this character, and then invert the fourier transform to return to $ L^{2}\left(\mathbb{Q}/p\mathbb{Z}\right)$ with a vastly simpler equation. My main difficulty can be broken into three parts:

(1) Is taking the least common multiple of the periods to reformulate the functional equation as one over $ L^{2}\left(\mathbb{Q}/p\mathbb{Z}\right)$ legal?

(2) I know that the fourier transform $ \mathscr{F}_{\mathbb{Q}/\mathbb{Z}}:L^{2}\left(\mathbb{Q}/\mathbb{Z}\right)\rightarrow L^{2}\left(\overline{\mathbb{Z}}\right)$ is given by:$ $ \mathscr{F}_{\mathbb{Q}/\mathbb{Z}}\left\{ f\right\} \left(\mathfrak{z}\right)\overset{\textrm{def}}{=}\sum_{t\in\mathbb{Q}/\mathbb{Z}}f\left(t\right)e^{2\pi i\left\langle t,\mathfrak{z}\right\rangle }$ $ and that the inverse transform is: $ $ \mathscr{F}_{\mathbb{Q}/\mathbb{Z}}^{-1}\left\{ \check{f}\right\} \left(t\right)\overset{\textrm{def}}{=}\int_{\overline{\mathbb{Z}}}\check{f}\left(\mathfrak{z}\right)e^{-2\pi i\left\langle t,\mathfrak{z}\right\rangle }d\mathfrak{z}$ $ However, I am at a loss as to what formula to use for $ \mathscr{F}_{\mathbb{Q}/p\mathbb{Z}}$ and its inverse, and for two reasons. On the one hand, because $ \mathbb{Q}/\mathbb{Z}$ and $ \mathbb{Q}/p\mathbb{Z}$ are group-isomorphic, what is to stop me from using the same formula for their fourier transforms? On the other hand, if I use a modified formula—say:$ $ \mathscr{F}_{\mathbb{Q}/p\mathbb{Z}}\left\{ f\right\} \left(\mathfrak{z}\right)=\sum_{t\in\mathbb{Q}/p\mathbb{Z}}f\left(t\right)e^{2\pi i\left\langle \frac{t}{p},\mathfrak{z}\right\rangle }$ $ does the fact that $ \mathscr{F}_{\mathbb{Q}/p\mathbb{Z}}\left\{ f\right\} \left(\mathfrak{z}\right)\in L^{2}\left(\overline{\mathbb{Z}}\right)$ then mean that I can recover $ f$ by applying $ \mathscr{F}_{\mathbb{Q}/\mathbb{Z}}^{-1}$ , or do I have to also modify it in order to make everything consistent? Knowing the correct formula for the fourier transform on $ L^{2}\left(\mathbb{Q}/p\mathbb{Z}\right)$ and its inverse is essential.

(3) I would like to think that performing a change-of-variables for a sum of the form:$ $ \sum_{\mathbb{Q}/r\mathbb{Z}}f\left(\alpha t+\beta\right)$ $ (where $ \alpha,\beta,r\in\mathbb{Q}$ , with $ \alpha\neq0$ and $ r=\frac{p}{q}>0$ ) would be a relatively simple matter, but, that doesn’t appear to be the case. For example, if $ t\in\mathbb{Q}\rightarrow f\left(\alpha t+\beta\right)\in\mathbb{C}$ is not a $ r$ -periodic function, then this sum is not well-defined over the quotient group $ \mathbb{Q}/r\mathbb{Z}$ . Worse yet—supposing $ f$ is $ r$ -periodic take a look at this: write elements of $ \mathbb{Q}/r\mathbb{Z}$ in co-set form: $ t+r\mathbb{Z}$ , where $ t\in\mathbb{Q}$ . Then, make the change-of-variable $ \tau=\alpha t+\beta$ . Consequently, the set of all $ \tau$ is:$ $ \alpha\left(\mathbb{Q}/r\mathbb{Z}\right)+\beta=\left\{ \alpha\left(t+r\mathbb{Z}\right)+\beta:t\in\mathbb{Q}\right\} =\left\{ \tau+\alpha r\mathbb{Z}:t\in\mathbb{Q}\right\}$ $ . Here is where things get loopy.

(1) Since $ \alpha,\beta\in\mathbb{Q}$ with $ \alpha\neq0$ , the map $ \varphi\left(t\right)\overset{\textrm{def}}{=}\alpha t+\beta$ is a bijection of $ \mathbb{Q}$ . As such, I would think that:$ $ \left\{ \tau+\alpha r\mathbb{Z}:t\in\mathbb{Q}\right\} =\left\{ \tau+\alpha r\mathbb{Z}:\varphi^{-1}\left(\tau\right)\in\mathbb{Q}\right\} =\left\{ \tau+\alpha r\mathbb{Z}:\tau\in\varphi\left(\mathbb{Q}\right)\right\}$ $ and hence:$ $ \alpha\left(\mathbb{Q}/r\mathbb{Z}\right)+\beta=\left\{ \tau+\alpha r\mathbb{Z}:\tau\in\mathbb{Q}\right\} =\mathbb{Q}/\alpha r\mathbb{Z}$ $ Using this approach, I obtain:$ $ \sum_{t\in\mathbb{Q}/r\mathbb{Z}}f\left(\alpha t+\beta\right)=\sum_{\tau\in\mathbb{Q}/\alpha r\mathbb{Z}}f\left(\tau\right)$ $

(2) Since $ r=\frac{p}{q}$ , decompose $ \mathbb{Z}$ into its equivalence classes mod $ q$ :$ $ \left\{ \tau+\alpha r\mathbb{Z}:\tau\in\mathbb{Q}\right\} =\bigcup_{k=0}^{q-1}\left\{ \tau+\alpha r\left(q\mathbb{Z}+k\right):\tau\in\mathbb{Q}\right\}$ $ and so:$ $ \sum_{t\in\mathbb{Q}/r\mathbb{Z}}f\left(\alpha t+\beta\right)=\sum_{k=0}^{q-1}\sum_{\tau\in\mathbb{Q}/\alpha rq\mathbb{Z}}f\left(\tau+\alpha rk\right)$ $ On the other hand, for each $ k$ :$ $ \left\{ \tau+\alpha r\left(q\mathbb{Z}+k\right):\tau\in\mathbb{Q}\right\} =\left\{ \tau+\alpha rk+\alpha rq\mathbb{Z}:\tau\in\mathbb{Q}\right\}$ $ and, since $ \tau\mapsto\tau+\alpha rk$ is a bijection of $ \mathbb{Q}$ , the logic of (1) would suggest that:$ $ \left\{ \tau+\alpha r\left(q\mathbb{Z}+k\right):\tau\in\mathbb{Q}\right\} =\left\{ \tau+\alpha rq\mathbb{Z}:\tau\in\mathbb{Q}\right\}$ $ for all $ k$ . But then, that gives:$ $ \sum_{k=0}^{q-1}\sum_{\tau\in\mathbb{Q}/\alpha rq\mathbb{Z}}f\left(\tau+\alpha rk\right)=\sum_{t\in\mathbb{Q}/r\mathbb{Z}}f\left(\alpha t+\beta\right)=q\sum_{\tau\in\mathbb{Q}/\alpha rq\mathbb{Z}}f\left(\tau\right)$ $ which hardly seems right.

missing ‘=’ etcd when defining service file

I’m struggling while following Kelsey Hightower’s “Kubernetes the Hard Way” tutorial. I’ve gone off script, because I’m trying to bootstrap k8s on a local server.

I’ve got the point where I’m bootstrapping etcd, however, when I’m creating the service I’m getting an error:

Failed to start etcd.service: Unit is not loaded properly: Bad message. See system logs and 'systemctl status etcd.service' for details. 

Checking the logs and I get:

Jun 21 20:16:49 controller-0 systemd[1]: [/etc/systemd/system/etcd.service:9] Missing '='. Jun 21 20:16:49 controller-0 systemd[1]: [/etc/systemd/system/etcd.service:9] Missing '='. Jun 21 20:17:25 controller-0 systemd[1]: [/etc/systemd/system/etcd.service:9] Missing '='. 

Here’s the etcd.service file:

Description=etcd service Documentation=https://github.com/coreos/etcd  [Service] User=etcd Type=notify ExecStart=/usr/local/bin/etcd \  --name $  {ETCD_NAME} \  --data-dir /var/lib/etcd \  --initial-advertise-peer-urls http://$  {ETCD_HOST_IP}:2380 \  --listen-peer-urls http://$  {ETCD_HOST_IP}:2380 \  --listen-client-urls http://$  {ETCD_HOST_IP}:2379,http://127.0.0.1:2379 \  --advertise-client-urls http://$  {ETCD_HOST_IP}:2379 \  --initial-cluster-token etcd-cluster-1 \  --initial-cluster etcd-1=http://192.168.0.7:2380 \  --initial-cluster-state new \  --heartbeat-interval 1000 \  --election-timeout 5000 Restart=on-failure RestartSec=5  [Install] WantedBy=multi-user.target 

Can the defining rep of $E_7$ split over a finite subgroup while the adjoint remains simple?

Does the (simply connected compact) Lie group $ E_7$ contain a finite subgroup $ G \subset E_7$ such that the $ 56$ -dimensional irrep of $ E_7$ splits over $ G$ as $ 28 \oplus \overline{28}$ , but the $ 133$ -dimensional adjoint representation remains simple when restricted to $ G$ ? By “$ 28 \oplus \overline{28}$ ” I mean of course a $ 28$ -dimensional complex irrep plus its dual.

Such a subgroup is definitely Lie primitive (meaning it doesn’t fit inside a proper Lie subgroup $ L \subset E_7$ ), since an imprimitive subgroup would split $ 133 = \mathfrak{l} \oplus (\dots)$ . According to Griess and Ryba, of the (quasi)simple subgroups of $ E_7$ , only three are primitive, and none of them work. ($ 133$ splits over $ SL_2(29)$ and $ SL_2(37)$ , and $ 56$ remains simple over $ SU_3(8)$ .) But I don’t know about nonsimple subgroups.

This feels like a good homework problem, but in fact came up in my research: such a subgroup would allow me to build a superconformal field theory with some nice properties.

complexity of system of equations defining affine variety

Say you have an affine variety $ X$ in $ n$ -dimensional affine space. (You can even assume we are over $ \mathbb{C}$ , but I believe the nature of my question is algebraic).

I want to bound from above the complexity of a system of equations defining $ X$ . My guess is that the following parameters should be enough:

  1. $ n$ – dimension of ambient space.

  2. $ \mu$ – dimension of $ X$

  3. $ d$ – degree of $ X$ in say the standart projective compactification of $ \mathbb{A}^n$ .

I don’t mind what concept of complexity to take, say the max power in which any variable is taken from any equation.

Thank you very much!

Defining computable functions on arbitrary sets

Turing machines take inputs that are strings of symbols from some alphabet, and they give outputs that are strings of symbols from the same alphabet. To show that a function is computable, we have to exhibit a Turing machine that computes it. In the case where the function is not from strings to strings, something about this bothers me.

I will introduce the problem with a simple example. Suppose that I define a function from binary trees to binary trees, and I want to show, formally, that it’s computable. To do that I would have to exhibit a Turing machine that computes it. Since binary trees are not strings, I first have to design a string representation of binary trees. For example, I could represent the tree

enter image description here

as ((*,*),(*)). Having done this, I can then pass binary trees to Turing machines in string form, and get trees back in string form, no problem.

But here’s my problem: the representation of binary trees as strings is itself a function (in fact a bijection) from binary trees to strings. How can I know formally that this function is a computable one?

On an intuitive level I have no problem seeing that it is, but on a formal level I can’t exhibit a Turing machine to verify it, because then I would need a string representation of binary trees, and we quickly get into an infinite regress. For simple things like binary trees the computability of the representation is obvious, but for more complicated sets it might not be.

Can this circle be squared — is there a formal definition of computability for functions of arbitrary countable sets that might not admit an obvious mapping to strings of symbols?

Additional thoughts

Here is another way of explaining my question: we might choose to define a model of computation that operates inherently on trees or some other set rather than strings. Indeed, many such models of computation have been proposed, including the lambda calculus and combinator calculi (which are arguably more naturally seen as operating on trees rather than strings), the recursive functions (which operate on integers), and various cellular automata (which operate on infinite strings or two dimensional lattice configurations, rather than finite strings). It’s only really historical accident that led the string-based Turing machines to become the gold standard against which other models of computation are judged.

The problem is that without being able to talk about computable mappings between these sets, it’s not strictly formally possible to say whether these other models of computation are equivalent to Turing machines or not. While it usually doesn’t raise any practical problems to just think of everything in terms of string representations, it seems dissatisfying that our theory of computation doesn’t treat all models of computation equally, but instead privileges those whose input and output are from a particular kind of set.

So really my question is whether a theory of computation exists has been formulated that is model-agnostic, in precisely this sense of not privileging one particular kind of set for input and output. It’s quite an abstract question, motivated more by a desire for formal niceness than by any practical concerns.

One suspects, for example, that the category theorists might have done some work in this direction. This is because category theory tends to talk about the properties of functions without talking much about the sets they map, so a “category of computable functions” would probably not know or care whether its underling objects were sets of strings or something else. This is an approach I would be particularly keen to read about if it exists. (I’ve asked another question, specifically about the category theory aspect, over at MathOverflow.)

Defining the standard model of PA so that a space alien could understand

First, some context. In one of the comments to an answer to the recent question Why not adopt the constructibility axiom V=L? I was directed to some papers of Nik Weaver at this link, on conceptualism. Many of the ideas in those papers appeal to me, especially the idea (put in my own words, but hopefully accurate) that the power set of the natural numbers is a work in progress and not a completed infinity like $ \mathbb{N}$ .

In some of those papers the idea of a supertask is used to argue for the existence of the completed natural numbers. One could think of performing a supertask as building a machine that does an infinite computation in a finite amount of time and space, say by doing the $ n$ th step, and then building a machine of half the size that will work twice as fast to do the $ (n+1)$ th step and also recurse. (We will suppose that the concept of a supertask machine is not unreasonable, although I think this point can definitely be argued.)

The way I’m picturing such a machine is that it would be a $ \Sigma_1$ oracle, able to answer certain questions about the natural numbers. I suppose we would also have machines that do “super-supertasks”, and so forth, yielding higher order oracles.

To help motivate my question, suppose that beings from outer space came to earth and taught us how to build such machines. I suppose that some of us would start checking the validity of our work as it appears in the literature. Others would turn to the big questions: P vs. NP, RH, Goldbach, twin primes. With sufficient iterations of “super” we could even use the machines to start writing our proofs for us. Some would stop bothering.

Others would want to do quality control to check that the machines were working as intended. Suppose that the machine came back with: “Con(PA) is false.” We would go to our space alien friends and say, “Something is wrong. The machines say that PA is not consistent.” The aliens respond, “They are only saying that Con(PA) is false.”

We start experimenting and discover that the machines also tell us that the shortest proof that “Con(PA) is false” is larger than BB(1000). It is larger than BB(BB(BB(1000))), and so forth. Thus, there would be no hope that we could ever verify by hand (or even realize in our own universe with atoms) a proof that $ 0=1$ .

One possibility would be that the machines were not working as intended. Another possibility, that we could simply never rule out (but could perhaps verify to our satisfaction if we had access to many more atoms), is that these machines were giving evidence that PA is inconsistent. But a third, important possibility would be that they were doing supertasks on a nonstandard model of PA. We would then have the option of defining natural numbers as those things “counted” by these supertask machines. And indeed, suppose our alien friends did just that–their natural numbers were those expressed by the supertask machines. From our point of view, with the standard model in mind, we might say that there were these “extra” natural numbers that the machines had to pass through in order to finish their computations–something vaguely similar to those extra compact dimensions that many versions of string theory posit. But from the aliens’ perspective, these extra numbers were not extra–they were just as actual to reality as the (very) small numbers we encounter in everyday life.

So, here (finally!) come my questions.

Question 1: How would we communicate to these aliens what we mean, precisely, by “the standard model”?

The one way I know to define the standard model is via second order quantification over subsets. But we know that the axiom of the power set leads to all sorts of different models for set theory. Does this fact affect the claim that the standard model is “unique”? More to the point:

Question 2: To assert the existence of a “standard model” we have to go well beyond assuming PA (and Con(PA)). Is that extra part really expressible?

Defining a function through an ODE containing unspecified operators

I want to do some algebra using a function only defined through a DE containing unspecified operators. The DE is $ $ \partial_zu(z) = \left[\hat{D}+\hat{N}(z,u)\right] u(z). $ $ Here $ u$ lives in some function space (it is bounded, integrable, continuously differentiable,…) and $ \hat{D},\hat{N}$ are bounded operators on said function space that don’t necessarilly commute and only $ \hat{D}$ is linear. How do I define $ u$ in Mathematica?

My reason for asking this question: I derived that for $ u_I(z):=e^{-(z-z’)\hat{D}}\cdot u(z)$ that \begin{align} \partial_zu_I(z) &= \hat{N}_I(z,u_I)\cdot u_I(z),\ \hat{N}_I(z,u_I) :&= e^{-(z-z’)\hat{D}}\cdot \hat{N}(z,u_I)\cdot e^{(z-z’)\hat{D}}. \end{align} I was wondering how to do such derivations with Mathematica.