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Proofing – How to prove the function of a recursive big theta without using repeated substitution, mastering the sentence, or having the closed form?

I have defined a function: $ V (j, k) $ Where $ j, k in mathbb {N} $ and $ t> 0 in mathbb {N} $ and $ 1 leq q leq j – 1 $, Note $ mathbb {N} $ includes $ 0 $,

$ V (j, k) = begin {cases} tj & k leq 2 tk & j leq 2 tjk + V (q, k / 2) + T (j – q, k / 2) & j , k> 2 end {cases} $

I am not allowed to use repeated substitution and I want to prove this by induction. I can not use the main clause because the recursive part is not in that form. Any ideas how I can solve it with given limitations?

When I start induction: I fix $ j, q $ and introduce $ k $, Then the base case $ k = 0 $, Then $ V (j, 0) = tj $, The question indicated that the function may be $ Theta (jk) $ or maybe $ Theta (j ^ 2k ^ 2) $ (but it does not necessarily have to be either).

I choose $ Theta (j, k) $, In the base case, this would mean that I had to prove that $ tj = theta (j, k) $ when $ j = 0 $, But if I start with the big-oh, I have to show it $ km leq mn = m cdot0 = 0 $ which I currently do not think possible.

I am not sure if I did the basic case wrong or if there is another approach.

Matrices – Derivation of $ big (x ^ {T} A – beta y ^ {T} big) big (x ^ {T} A – beta y ^ {T} big) ^ {T} $

Suppose that $ beta $ is scalar, $ x, y $ are column vectors $ (m times 1) $ and $ A $ is a $ (m times m) $ Matrix. We want to differentiate the expression
$$ big (x ^ {T} A – beta y ^ {T} big) big (x ^ {T} A – beta y ^ {T} big) ^ {T} $$
in memory of $ x $, My idea:

To let $ f (x) = big (x ^ {T} A – beta y big) big (x ^ {T} A – beta y big) ^ {T} $ defined on a suitable finite dimensional space over $ mathbb {R}. $ We have:

begin {align}
f (x + h) & = big ((x + h) ^ {T} A – beta y ^ {T} big) big ((x + h) ^ {T} A – beta y ^ {T} big) ^ {T} \
& = big ( big (x ^ {T} + h ^ {T} big) A – beta y ^ {T} big) big ( big (x ^ {T} + h ^ {T } big) A – beta y ^ {T} big) ^ {T} \
& = big ( big (x ^ {T} A – beta y ^ {T} big) + h ^ {T} A big) big ( big (x ^ {T} A – beta y ^ {T} big) + h ^ {T} A big) ^ {T} \
& = big (x ^ {T} A – beta y ^ {T} big) big (x ^ {T} A – beta y ^ {T} big) ^ {T} + big ( x ^ {T} A – beta y ^ {T} big) hA ^ {T} + big (x ^ {T} A – beta y ^ {T} big) h ^ {T} A + hh ^ {T} A \
& = f (x) + big (x ^ {T} A – beta y ^ {T} big) hA ^ {T} + big (x ^ {T} A – beta y ^ {T} big) h ^ {T} A + hh ^ {T} AA ^ {T}.
end

Therefore,

begin {align}
f (x + h) – f (x) & = big (x ^ {T} A – beta y ^ {T} big) hA ^ {T} + big (x ^ {T} A – beta y ^ {T} big) h ^ {T} A + hh ^ {T} AA ^ {T} \
& = big (x ^ {T} A – beta y ^ {T} big) hA ^ {T} + big (x ^ {T} A – beta y ^ {T} big) h ^ {T} A + | h | ^ {2} AA ^ {T} \
& = 2 big (x ^ {T} A – beta y ^ {T} big) hA ^ {T} + | h | ^ {2} AA ^ {T},
end

which finally results (by dividing by $ h $ and for rent $ h rightarrow 0 $) that the derivative is $ 2 big (x ^ {T} A – beta y ^ {T} big) A ^ {T} $, What do you think ?

Exchanges – Big buy, but the price has not changed

I was wondering what happened yesterday at Kraken or other stock exchanges.

At a certain point in time (which I marked yellow in the graph), there was a big buy from BTC, which raised the "price" by 700 EUR within a few seconds, which equates to + 8.5% (see chart below)). ,

However, the price in the chart above did not follow the same direction.

I just watched this event live and the price was the same after this "short break". How can that be?

What happened here? Someone made a big purchase and handled many orders in the order book. But then the price would have to be increased.

Enter image description here

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SQL Server – Big archive table with "unimportant" data, how can you improve / redesign it?

We have an SQL database that collects analog and digital sensor data from hundreds of devices. The archive table is quite simple, just a (date / time) timestamp, an (int) station ID, an (int) data point ID, and a (double) value (plus 3 bool flags).

The DB collects about 12 million entries per month. This table should grow larger, no rollbacks, updates, or deletions are required.

The main purpose of this table is to display archive values ​​in our web visualization, sorted by timestamp and filtered by this or that ID. That's why I've created the relevant indexes for the fields I need. It works, but the performance is pretty bad (about 5 seconds when filtering by start / end time, station ID and data point ID).

How could such data be better stored and queried? Is SQL Server Standard 2016 not the right DBMS for such an archive? The original decision for SQL Server was several years ago, when the amount of data was much smaller than it is today. Is there something magical in SQL Server that I can apply to trade reliability for performance?

Java – Android Studio make the big slider image

Hello, I have a slider with img loaded from the base, and I have to click the selected img when I click.

Here I load the image that is in the base, but I do not know how to determine the position from which it was selected.

SliderItem sliderItems() = new SliderItem(imgs.size());
                    for(int i = 0; i <=imgs.size() -1; i++)
                    {
                        sliderItems(i) = (new SliderItem("",imgs.get(i)));
                    }

                    easySlider.setPages(Arrays.asList(sliderItems));
                    easySlider.setTimer(0);