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Apr
29
comment approximating smooth functions by non-smooth ones, in the distribution topology
@JochenWengenroth: thank you! indeed, a trivial fact
Apr
29
comment approximating smooth functions by non-smooth ones, in the distribution topology
@LiviuNicolaescu: I referred to the usual operator norm of $T_i$'s (which are continuous functionals, so it's well-defined). Thank you, I will check the Schwartz' book.
Apr
29
comment approximating smooth functions by non-smooth ones, in the distribution topology
Dear Jochen, could you enlighten me about the fact you have mentioned? Suppose we have a sequence of continuous functionals $T_i$ converging to $T$, and we have a test function $\varphi$ such that $T_i \varphi$ converges to $T \varphi$. Then we need to show that $T_i \varphi^{(k)}$ converges to $T \varphi^{(k)}$ for all $k$. My guess is that using the bound on the norm of $T_i$-s (since they are continuous and converging to a continuous functional) one bounds $T \varphi^{(k)} - T_i \varphi^{(k)}$ and thus establishes the convergence. Is that so?
Apr
29
comment approximating smooth functions by non-smooth ones, in the distribution topology
Dear Jochen, I guess your remark makes my question trivial for an application that I have in mind. Unfortunately I cannot add much to motivate the question: I looking to work with Lagerberg's differential forms on polyhedral complexes, and I wanted for some reason to approximate arbitrary forms with piecewise affine ones. Since I want to compute differentials of these forms, I wanted the approximations to be compatible in the sense as above, hence this question.
Apr
29
revised approximating smooth functions by non-smooth ones, in the distribution topology
added 9 characters in body
Apr
29
asked approximating smooth functions by non-smooth ones, in the distribution topology
Apr
9
revised why use Crofton's formula in order to prove log-analyticity of volume?
added 1 character in body
Apr
9
asked why use Crofton's formula in order to prove log-analyticity of volume?
Mar
26
accepted Are definable sets in an o-minimal expansion of the real field locally analytic?
Mar
26
comment Are definable sets in an o-minimal expansion of the real field locally analytic?
Dear @ACL, thank you for your answer, which comes as a surprise for me: I really expected that the setting of o-minimality forces definable sets to be "nice", but as it turns out, analyticity is not a part of the "niceness".
Mar
23
comment Are definable sets in an o-minimal expansion of the real field locally analytic?
Dear Emil, would it help if one assumes that the real field is expanded only by locally analytic functions? (because in practice typical o-minimal expansions of reals, like $R_{an}$ and $R_{exp}$, are such)
Mar
23
asked Are definable sets in an o-minimal expansion of the real field locally analytic?
Feb
25
comment ω-categorical, ω-stable structure with trivial geometry not definable in the pure set
@SzymonToruńczyk I can also recommend (shameless self-promotion!) to have a look at my paper where the notion of a generilized imaginary sort is defined using the language of groupoid torsors, this definition might look slightly more natural to you than Hrushovski's
Feb
25
comment ω-categorical, ω-stable structure with trivial geometry not definable in the pure set
@SzymonToruńczyk One can associate an extra sort to a groupoid, and provided the groupoid is connected, this sort is interpretable in the base theory iff the groupoid is split (one can draw parallells with the notion of elimination of imaginaries here, the extra sort is like an imaginary sort). I believe one can find an example of an omega categorical omega-stable theory (perhaps even $(M,=)^{eq}$ will do) and a definable non-split groupoid in it, such that this theory with the generalized imaginary sort added has trivial geometry
Feb
25
comment ω-categorical, ω-stable structure with trivial geometry not definable in the pure set
@SzymonToruńczyk to facilitate your task, I'll sketch my idea of the construction. Suppose you have a definable groupoid G, i.e. a pair of definable sets G_1 and G_0, and a collection of morphisms (source, tagret, composition etc), such that G_1(M) and G_0(M) together with these morphisms constitute a groupoid for any model M. Hrushovski defines the notion of equivalence of groupoids; a groupoid is called split if it is equivalent to a groupoid with trivial isomorphism groups of objects.
Feb
23
comment ω-categorical, ω-stable structure with trivial geometry not definable in the pure set
Dear Szymon, I cannot remember now exactly why, but I accepted John Baldwin's anwer by mistake. I think one can construct the counterexample from my question as a non-split cover of the theory of $(M,=)$. This can be done for example using groupoids, as in this article of Hrushovski
Dec
19
comment Bogomolov-Beauville-Fujiki form, algebraically
I wonder if a purely algebraic definition of $q$ is possible (just like existence of Hodge decomposition on algebraic de Rham cohomology can be proved by the use of Leschetz principle, but there is an algebraic proof of this fact due to Deligne and Illusie). Of course it is interesting to know if the fact is true in positive characteristic.
Dec
19
revised Bogomolov-Beauville-Fujiki form, algebraically
edited body
Dec
19
asked Bogomolov-Beauville-Fujiki form, algebraically
Dec
1
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