Are definitions of functions appearing in literature oftentimes ambiguous?


Is it just me not understanding some implicit rules or most of definitions of functions appearing in literature are ambiguous (in part. in physics)? I’m especially interested in the ambiguity of the sign of equality (see the explanations below) i.e. is it really ambiguous or I don’t understand something?

The following definition of a function seem ambiguous, at least to me: y = 5; it is because the definition neither explicitly says nor implies that it is dependent on some variable. I admit that normally such a definition is given in a context e.g. x and y axes, etc. However, even then one still can argue that y = 5 is not dependent on x and merely represents one point at mark 5 on the y axis. The given example is maybe to trivial to explain the ambiguity. Let’s concern a function in R^2 which is given as follows f = < 2t, sin(t) >. The assumption is that it is a vector function in 2D which depends only on t (i.e. f(t) = = < 2t, sin(t) >) and therefore represents a curve (a set of points in 2D). However, as dependencies are not indicated, it also can be that it depends on further variables e.g. on x, y and t, which means that the function f(x,y,t), for instance, represents a time dependent, 2D vector field (an infinite set of vectors for each given t). Equally, t may be not a parameter but one of two space coordinates i.e. f(t,u) which means that it represents a constant, 2D vector field (an infinite set of vectors). Furthermore, the very sign of equality when “defining” functions may be ambiguous. Consider a generalised position vector in R^3 e.g. r = < x,y,z >. When I say g = r do I define a vector field g or do I simply define another position vector g? To my taste g(r) = r would rather indicate definition of the vector field, whereas g = r would rather define another position vector. Or maybe, for some obscure reason, it is assumed that only one “generalised” position vector (“the” infinite set of vectors pointing from the origin to all possible locations in R^3) may exist so if you define r as a generalised position vector, each time you say e.g. u = r you automatically define a vector filed?

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Is there literature on a de Rham analogue of the Mumford-Tate group or ell-adic monodromy group?

Let $ X$ be a smooth projective variety over $ \mathbb{Q}$ . The theory of motives predicts that for each cohomology theory, there should be a distinguished Zariski closed subgroup of $ GL(H^k_{\bullet}(X))$ , the motivic Galois group. This group has conjectural descriptions for Betti and $ \ell$ -adic cohomology, and I am wondering if there is any literature on this group for de Rham cohomology.

The Betti cohomology $ H^k_B(X)$ is a rational Hodge structure, hence there is a representation $ \mathrm{Res}_{\mathbb{C}/\mathbb{R}}\mathbb{G}_m\to GL(H^k_B(X)\otimes\mathbb{R})$ . The Mumford-Tate group $ MT^k(X)\leq GL(H^k_B(X))$ is defined to be the smallest Zariski-closed subgroup containing the image of the representation. The Hodge conjecture would imply that $ MT^k(X)$ has finite index in the Betti motivic Galois group.

The $ \ell$ -adic cohomology $ H^k(X;\mathbb{Q}_\ell)$ is a representation of the absolute Galois group, and the $ \ell$ -adic monodromy group $ G_\ell^k(X)\leq GL(H^k(X;\mathbb{Q}_\ell))$ is defined to be the smallest Zariski-closed subgroup containing the image of the representation. The Tate conjecture would imply that $ G_\ell^k(X)$ is the full $ \ell$ -adic motivic Galois group.

Is there any literature on an analogue of the Mumford-Tate group or the $ \ell$ -adic monodromy group inside $ GL(H^k_{dR}(X))$ ?

I believe that a conjecture of Ogus would imply that the de Rham motivic Galois group is smallest Zariski closed subgroup whose $ \mathbb{Q}_p$ points contain $ F_p$ for all sufficiently large $ p$ (where $ F_p\in GL(H^k_{dR}(X))(\mathbb{Q}_p)$ is the crystalline Frobenius). However I have some doubt about this, because André’s book on motives states the relationship between the Hodge conjecture and the Mumford-Tate group (Proposition 7.2.2.1), and the relationship between the Tate conjecture and the $ \ell$ -adic monodromy group (Proposition 7.3.2.1), but does not give an analogous statement for the Ogus conjecture.

I also believe that the period conjecture of Grothendieck would imply that the de Rham motivic Galois group is the smallest Zariski closed subgroup containing all elements $ \varphi\in GL(H^k_{dR}(X))(\mathbb{Q})$ whose image in $ GL(H^k_{B}(X))(\mathbb{C})$ under the Betti-de Rham comparison isomorphism is contained in $ GL(H^k_{B}(X))(\mathbb{Q})$ . However, I also cannot find a statement like this in André’s book.

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