Prove that the following inequalities are equivalent.

Prove that

If ‎$ ‎‎f:(0,+‎\infty‎)\to‎\mathbb{R}‎$ ‎be a‎ ‎‎continuous ‎function, then the following are equivalent (for every $ x\in(0,+‎\infty‎)$ ).

(1)‎$ ‎‎‎‎‎\frac{‎‎f(x_4)-f(x_3)}{{x_4}-x_3}‎\leq‎‎‎‎\frac{‎‎f(x_2)-f(x_1)}{{x_2}-x_1}\;\;;x_4‎>x_3‎>x_2‎>x_1;‎‎‎$

(2) ‎$ ‎\frac{1}{2}(f(x_2)+f(x_1))‎\leq‎‎‎\frac{1}{x_2-x_1}\int_{x_1}^{x_2}f(u)du‎‎\leq‎ f(‎\frac{x_2+x_1}{2}‎)‎‎.$

I know that if $ f$ has the first condition, it means that $ f$ is a concave function and it implies that $ f$ is a midpoint concave. So we have $ ‎\frac{1}{2}(f(x_2)+f(x_1))‎\leq‎ f(‎\frac{x_2+x_1}{2}‎)$ . Moreover I know that the condition (2) along with continuity imply that $ f$ is a convex function and it make the condition (1) be held. Also I know that if $ f$ is a continuous function then according to fundamental theorem of calculus, the integral of $ f$ is well-defined. In addition to these mentioned, I tried to prove the above assertion with Mean value theorem fo integrals. But I could not achieve the aim. Can anyone help me.

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Is there a functor which is equivalent to discriminant of number field?

Let $ K$ be a number field, i.e. a finite extension of $ \mathbb{Q}$ . The ring of integer $ O_K$ is a free $ \mathbb{Z}$ -module. Let $ \{ a_1, \cdots , a_n\}$ be a integral basis of $ O_K$ . Then, $ $ \Delta_{K/ \mathbb{Q}} = \det (\mathrm{Tr}(a_ia_j)_{i,j}) $ $ is independent of choice of integral basis. We call $ \Delta_{K/ \mathbb{Q}}$ a discriminant of number field $ K$ over $ \mathbb{Q}$ .

My question is: Is there some categories $ C,D$ and a functor $ F \colon C \to D$ such that you have a simple way to get the discriminant $ \Delta_{K/ \mathbb{Q}}$ from an object $ F(K)$ ? I want $ F$ to be a canonical one.

Thanks.

Is there a melee equivalent for Clustered Shots?

The Clustered Shots feat says:

You take a moment to carefully aim your shots, causing them all to strike nearly the same spot.

Prerequisites: Point-Blank Shot, Precise Shot, base attack bonus +6.

Benefit: When you use a full-attack action to make multiple ranged weapon attacks against the same opponent, total the damage from all hits before applying that opponent’s damage reduction.

Special: If the massive damage optional rule is being used, that rule applies if the total damage you deal with this feat is equal to or exceeds half the opponent’s full normal hit points (minimum 50 points of damage).

Is there a Paizo feat that does the same thing except for melee attacks instead of ranged?

Cheerio / jQuery equivalent of Descendants.LastOrDefault (HTML Agility pack)

I have web scraping code that is implemented in C# .NET Core 2 with the HTML Agility pack.

We ported most of our server over to node.js. The only remaining part is this web scraping bit. I’m currently trying to replicate this with Cheerio/jQuery, but i’m not too familiar with jQuery syntax.

What would be the equivalent of

DocumentNode.Descendants("table")             .LastOrDefault(t => t.InnerHtml.Contains("<td align=\"center\">Test</td>")) 

I tried doing:

$  ('table > tbody > tr > td > table > tbody') 

But this gives me a lot of tbodys still. I essentially want to grab the table’s rows. The table I want should contain that td field.

Thanks!

What is the equivalent of /etc/pam.d/system-auth on Ubuntu 18

I want to add a pam module (auth required pam_group.so use_first_pass) that is called when any user logs in via any method (at the desktop manager, ssh, su, tty, etc).

Ubuntu has common-auth, but it appears only be used as an @include and, as such, often does not get called after a sufficient pam permissions has been encountered.

The issue is related to This question

What are the web3py equivalent for communicating with bitcoin in Python?

I am fairly new to working with Python, I have a requirement where I need to perform some basic bitcoin operations on my local system rather than relying on some external API’s, those function mainly include :

1. Wallet Creation

2. Address Creation

3. Transaction signing

For other operations, I am willing to use external API’s like blockcypher. I need the above operations to be performed locally so to ensure some security. FOr ethereum I use web3.py which provides me these functions. Can anyone help me with the same for bitcoin?

Are the two LTL properties $GF(\psi_1 \land F\psi_2 )$ and $GF(\psi_2 \land F\psi_1 )$ equivalent?

Is $ GF(\psi_1 \land F\psi_2 )$ equivalent to the property $ GF(\psi_2 \land F\psi_1 )$ ?

Attempt:

In the first property each state must eventually see $ \psi_1$ and $ \psi_2$ , in the second property as well each state must eventually see $ \psi_1$ and $ \psi_2$ , as such the two properties must be equivalent. Is this correct?

Characterization of NFA whose equivalent (minimal) DFA has exponential number of states

(I don’t know if there are standard names for this, so) Let’s say that a Nondeterministic Finite Automaton (NFA) is $ n$ -expansive if it has $ n$ states and any Deterministic Finite Automaton (DFA) recognizing the same language has $ \Omega(2^n)$ states.

Let $ E_n$ be the set of all $ n$ -expansive NFAs.

Is there a characterization to $ E_n$ ?

Given any NFA $ A$ , is there a way to test if $ A \in E_n$ in polynomial time in $ n$ (therefore, without constructing the minimal DFA of $ A$ )?