If alghorithm solves NP problem, for what f(n) can we claim that R belongs to TIME(f(n))?

This is my problem:

Suppose that for the problem R belongs to NP the algorithm of solution check M(x,y) runs in time O(n^3) and uses additional information y, which is long ≤5 log n bits. For what f(n) can we claim that R belongs to TIME(f(n))?

I have no idea how can I know what must to be f(n) that R belongs to TIME(f(n)). Every suggestions are very welcome!

Excel : Summation of data which belongs to certain id

This is my first post in stack exchange. please bear with me and I accept any advice for questioning ethic.

It is easier to illustrate the problem based on the following example:

I have 2 sheets : mapping and data.

1. This Mapping sheet contains a mapping table between a country name and the corresponding ID

2. This table (from Data sheet) contains the country name with the corresponding GDP

The objective is to create a summary table which maps the group id with the SUM of GDP in a seperate sheet

I tried to use the SUM IFS formula as recommended here. They propose the following formula based on their own example.

=SUM(SUMIFS(sum_range,criteria_range,{“blue,”red”}))

However, the challenge arises due to 2 reasons:

  1. I cannot list down the ID in an array form as I have many ID variations. Also, the actual ID consists of both numbers and alphabets.
  2. I must follow a guideline not to merge the tables into a single sheet.

Thank you in advance!

Could we always find a curve on the manifold whose tangent vector always belongs to a linear subspace?

Suppose we have a smooth manifold $ M$ and the tangent space of every point $ x \in M$ has non-empty intersection with a given linear subspace. Could we find a curve on $ M$ such that the tangent vector of point on thus curve always belongs to this linear subspace?

$ \textbf{My attempt:}$ If the dimension for the linear subspace is one or two, I think I can find the curve. But I don’t know if the dimension is bigger than two?

I will appreciate for any useful answers and comments

If $f$ belongs to $M^{+} $ and $c \ge 0$ then $cf$ belongs to $M^{+}$ and $ \int cf = c\int f$

If $ f$ belongs to $ M^{+} $ and $ c \ge 0$ then $ cf$ belongs to $ M^{+}$ and $ \int cf = c\int f$ .

I need to proove that, using the following observation:

if $ f\in M^{+}$ and $ c>0 $ , then the mapping $ \varphi \rightarrow \psi = c\varphi$ is a one-toone mapping between simple function $ \varphi \in M^{+}$ with $ \varphi \le f $ and simple functions $ \varphi$ in $ M^{+} $ with $ \psi \le cf $ .

I know that this question is already answer here:One-to-one mapping of simple functions $ \phi \to \psi = c\,\phi$ implies $ \int cf\,d\mu = c \int f\,d\mu$ ?

But I can’t follow the verbal explanation.

My original idea was to proove $ $ c \int f \le \int cf \le c\int f $ $ But I can’t… some idea?

Is the language { | p and n are natural numbers and there’s no prime number in [p,p+n]} belongs to NP class?

I was wondering if the following language belongs to NP class and if its complimentary belongs to NP class:

\begin{align} C=\left\{\langle p,n\rangle\mid\right.&\ \left. p \text{ and $ n$ are natural numbers}\right.\ &\left.\text{ and there’s no prime number in the range}\left[p,p+n\right]\right\} \end{align}

I am not sure, but here’s what I think: for each word $ \langle p,n\rangle \in C$ we know that the word belongs to C because there exists a primal certificate – an nontrivial divisor to any of the numbers between $ [p,p+n]$ , though I am not really sure it is in NP.

regarding the complement: I think it is in NP because the compliment compositeness can be decided by guessing a factor nondeterministically. But again I am not so sure about it and I don’t know how to correctly prove and show it.

Would really appreciate your input on that as I am quite unsure and also checked textbooks and internet (and this site) about it.