testing – What is the current definition of continuous integration?

It seems that there are at least two possible definitions of continuous integration:

  1. Frequent merging of a codebase to a common codebase (e.g. daily merge to the main branch of a VCS server).
  2. Frequent building and testing of a codebase (e.g. build and test at each push to a VCS server).

Both 1 and 2 can be automated. 1 does not imply 2, and 2 does not imply 1.

What is the current definition of continuous integration? (Please back up your claim with a reference.)

Does there exist an almost surely continuous martingale?

Does there exist a continuous time martingale $X_t$ not a.s. constant in $t$ that is almost surely everywhere differentiable?

notation – What is the meaning of $$ in Lipschitz continuous?

In Lipschitz continuous, there is a Theorem that

If $f$ is Lipschitz continuous gradient on $mathbb{R}^n$. Then for any $x,y in mathbb{R}^n$, we have

$$|f(y) – f(x) – <f'(x),y-x>| leq frac{L}{2} lVert y-x rVert ^2$$

I would ask what is the meaning of $<f'(x),y-x>$.

If T is a linear operator from banach space X to banach Y, if for all $f in Y^*$, foT is continuous then T is continuous

If T is a linear operator from banach space X to banach Y, if for all $f in Y^*$, foT is continuous then T is continuous.

This is an assignment that I am struggling for one hour at least. Only a little hint please!

plotting – Smooth continuous curve from a ListPlot

I have to plot a set of numerical data with a smooth continuous curve (and I’m not interested in its functional expression).
If I use ListPlot function without the Joined option, the result is the following:

ListPlot({data}, Joined -> False, Frame -> True, FrameLabel -> {"x"})

enter image description here

The simplest option I’ve thought is to switch on the Joined option, but the plot I get is not smooth at all:

ListPlot({data}, Joined -> True, Frame -> True, FrameLabel -> {"x"})

enter image description here

How can I obtain a smoother plot?

continuity – Let $f :R→R$ be a continuous function satisfying $f(x)=f(x+1) forall x∈R$. then which of the following is true.

Let $f :R→R$ be a continuous function satisfying $f(x)=f(x+1) forall x∈R$.Then

(A) f is not necessarily bounded above.

(B) there exists a unique $x_{0} ∈ R$ such that $f(x_{0} + π) = f(x_{0}).$

(C)there is no $x_{0} ∈R$ such that $f(x_{0} +π)=f(x_{0}).$

(D) there exist infinitely many $x_{0} ∈ R$ such that $f(x_{0} + π) = f(x_{0}).$

I know the every continuous periodic function is bounded above and uniformly continuous
so option (A) is wrong, but can’t figure out the other options.

Any hint/ Solution will be very helpful

Thank You

Select continuous time intervals in SQL

I have table with datetimes and i need to select continuous time intervals

My table:

Id Time
1 2021-01-01 10:00:00
1 2021-01-01 10:01:00
1 2021-01-01 10:02:00
1 2021-01-01 10:04:00
2 2021-01-01 10:03:00
2 2021-01-01 10:04:00
2 2021-01-01 10:06:00
2 2021-01-01 10:07:00

Result i need:

id date_from date_to
1 2021-01-01 10:00:00 2021-01-01 10:02:00
1 2021-01-01 10:04:00 2021-01-01 10:04:00
2 2021-01-01 10:03:00 2021-01-01 10:04:00
2 2021-01-01 10:06:00 2021-01-01 10:07:00

I tried like this, but can’t do that right

select id, 
  min(date_from) over 
    (partition by id, date_to 
       order by id) 
    as date_from, 
    max(date_to) over 
      (partition by _id, date_from 
      order by id) 
    as date_to
    from (
      select user_id, channel_id, 
      MIN(time) over 
        (PARTITION by id, 
        diff2 between 0 and 60
        ORDER BY id, time) 
      as date_from, 
      max(MINUTE) over 
        (PARTITION by id, 
        diff between 0 and 60
        ORDER BY id, time) 
      as date_to 
        from (
           select *, 
           unix_timestamp(date_lead) - unix_timestamp(time) 
              as diff, 
           unix_timestamp(time) - unix_timestamp(date_lag) 
              as diff2 
                    from (
                        select id, time,
                            NVL(LEAD(time) over 
                                (PARTITION by id 
                                ORDER BY id, time), time) 
                            as date_lead,
                            NVL(LAG(time) over 
                                (PARTITION by id 
                                ORDER BY id, time), time) 
                            as date_lag
                        from my_table) 
                    )
                )

continuity – Let $A = {(x,y) : x^2+y^2 = 1}$ and let $f : A to Bbb R$ be a continuous function. Then prove or disprove the following: 1. $f$ is one to one. .

Let $A = {(x,y) : x^2+y^2 = 1}$ and let $f : A to Bbb R$ be a continuous function. Then prove or disprove the following:

  1. $f$ is one to one.

  2. $f$ is onto.

If I take $f(x,y) = e^{x+y}$ which is continuous function but $nexists x,y$ s.t. $f(x,y) =0 in Bbb R$ so $f$ is not onto.

We know that if $f$ is one to one then $f(x_1,y_1) = f(x_2,y_2) implies x = y$, where $x = (x_1,x_2), y = (y_1,y_2)$. Also if $f$ is onto then for each $y in Bbb R, exists x in A$ such that $f(x) = y$
But I have no idea how to use conditions of one to one and onto function. Help me.

Is a continuous probability density function strictly positive on the interior of the support?

In Wikipedia, the support of a density function is defined as the closed set of all points $x$, where, all open balls $B_{epsilon}(x)$ that are in the sigma algebra with $epsilon> 0$ have a probability measure greater than 0:
begin{align}
supp(A):={x | P(B_{epsilon}(x))> 0}
end{align}

That is, taking the example of the pdf $f$ with support $supp(A)=(-frac 3 2, frac 3 2)$ with
begin{equation}
f(x)|_{A}:=x^2
end{equation}

$Int(A)$, the interior of $supp(A)$ is $(-frac 32 ,0)$ and $(0,frac 32)$.

Now my question: Is this statement generel true: If a pdf $f$ is continuous on $supp(A)$, then is $f>0$ on $Int(A)$?

algorithm – Finding a path to a circular zone in continuous space

I need to find the shortest path for an agent that wants to use a ranged attack against the specified target. This means that path’s ending point should be a) inside range of the attack and b) have line-of-sight on target, which can be blocked by various static obstacles. This part of the game is turn-based, so I only have to worry about finding a path for one agent at a time, and can allow myself to take a small performance hit (anything under 100ms is probably acceptable).

I’m operating in a continuous space, but even if I constrain it with an overlaid grid, the problem doesn’t get much easier, because the only way I can think of to solve it is to find a path (using A*) to every grid cell inside the range that has LOS on target and compare their length to find the shortest one – which is prohibitively expensive, given longer ranges (e.g. calculation of path to target 3000 cells takes up to 600ms). What’s worse, LOS check is also somewhat expensive, so even determining a complete set of cells which have LOS is out of question.

But before we even consider LOS, I can find no good way to find the shortest path to the target circle. Of course, I can sample a number of points on the circumference and find a path to each one, but this sounds too unreliable and also costly enough for long ranges. And of course, this approach breaks down completely if we take LOS into account, because it is possible that no point on circumference has LOS, but some point inside circle does.

I was unable to find any algorithms or even research papers on the topic (of finding a path to a circular zone in continuous space), so maybe I’m using wrong keywords or missing something. Are there any known algorithms for this problem? Or is finding path to every (sampled) point inside circle is my best be, and I should work on optimizing this approach somehow?