complex analysis – Confusion in a proof: “Crude upper bound in Zeta Zeros.” And reference request.

I have been trying to understand the following proof for a while and I think I understand most of it. I however have just one query pertaining to the proof.

Consider the following theorem:enter image description here

Now the only query I have is:

What exactly is meant by “the discrete nature of zeros of a meromorphic function”?

I do not know about this property of meromorphic functions. Can someone kindly explain this?

I understand the rest of the argument. I just do not know what this property of meromorphic functions is.

Also, can someone suggest any relatively simple notes on complex analysis for the purposes of Analytic Number Theory?

Thanks.

Upper bound for the Gradient in terms of rank of a matrix

Assume $f$ is a convex function, for instance: $f(A) = lVert AX – B rVert_F^2, A in mathbb{R}^{n times n}, rank(A) = r < n$. Is there any proved relationship between an upper bound on $lVert nabla f(A) rVert$ in terms of the rank of $A$?

Upper bound for chromatic number ($chi(G) leq Delta(G)+1$)

i am new in graph theory. I saw in some text book this "obvious" upper bound $$chi(G) leq Delta(G)+1$$ i can’t see how to demonstrate it

control theory – Upper bound on trace of solution to discrete lyapunov or discrete matrix ricatti equation

Take a discrete Lyapunov equation
$$
P = R + beta A’ P A
$$

for $beta in (0,1)$, $R$ and $A$ symmetric.

Quetsion: Can I put an upper bound on $tr(P)$ given properties of the primitives? In particular, I have the $tr(R)$ and bounds on the maximum eigenvalue of $A$.

I can probably deal with something that doesn’t generate the tightest bound exploiting the $A$ as long as the $tr(R)$ is involved.

Alternatively: I am happy to use similar results for the solution to the Discrete matrix ricatti equation that comes out of optimal LQ control if that is available.

inequalities – Polynomial Markov versus Chernoff Bound for random variables

Suppose that $Xgeq0$, and that the moment generating function of $X$ exists in an interval around 0. Given some $delta>0$ and integer $k=1,2,…$, show that
$$inf_{k=0,1,…}frac{E(|X|^k)}{delta^k} leq inf_{lambda>0} frac{E(e^{lambda X})}{e^{lambda delta}}. $$

Consequently, an optimized bound based on polynomial moments is always at least as good
as the Chernoff upper bound. Could anyone enlighten me how to prove this?

complexity theory – Statement of the Goldwasser-Sipser Set Low Bound Protocol

I’m trying to understand the statement of the Goldwasser-Sipser Set Low Bound Protocol as presented in “Computational Complexity: A Modern Approach” by Arora and Barak.

In particular, I’m trying to understand why the following claims yields what we would want:

$S subseteq {0,1}^m$ is a set such that membership can be certified. Both parties know a number $K$. Let $k$ be an integer such that $2^{k-2} < K le 2^{k-1}$.

Claim: Let $S subseteq {0,1}^m$ satisfy $|S| le 2^{k}/2.$ Then for $p=|S|/2^k$ we have $p ge mathbf{P}_{h in_{_R} mathcal{H}_{m,k}, y in_{_R} {0,1}^k}left(exists x in S: h(x)=y right) ge frac{3p}{4}-frac{p}{2^k}.$

My understanding is that we want:

  • If $|S| ge K$ then the verifier should accept (corresponding to the prover finding such an $x$ in the claim) with “high” probability;
  • If $|S| le K/2$ then the verifier will accept with probability at most 1/2.

I’m getting thrown off by a few things here:

  • the claim assumes that $|S| le 2^k/2 = 2^{k-1} = 2times2^{k-2} < 2 K$, so how does this place us in one of the relevant cases?
  • How do we actually use those left ($p$) and right ($3p/4 – p/2^k$) bounds to conlude what we want?

Overall, I’m understand bits and pieces of this setup, but I’m having trouble seeing how it all fits together. Can someone walk through the logic/algebra here?

adjacency matrix – Min-eigenvalue bound for a random d-regular graph

I need help proving the following fact: Let $G$ be a random $d$-regular graph with adjacency matrix $A$. The smallest eigenvalue $lambda_n$ of $A$ should satisfy $|lambda_n| = o_d(d)$. (In particular, I think $|lambda_n| = O(sqrt{d})$.)

Exercise 7 in (1) is to prove this fact, so it seems like there should be a simple proof. However, I cannot think of a reasonably simple proof or find one anywhere, despite a lot of searching. In (2), they prove that $max{|lambda_2|, |lambda_n|} = o(d)$, but their proof is very complicated and only focused on the $lambda_2$ part — further indicating that there is some simple proof I’m missing for the $lambda_n$ part.

My attempts always seem to come to proving something harder than we started, like, $mathbb{E}(x_ix_j) ge -o(1)/n$ for adjacent $i, j$. I have no ideas that seem to actually move me forward.

(1) https://www.sumofsquares.org/public/lec02-1_maxcut.pdf

(2) https://www.researchgate.net/publication/4355188_On_the_second_eigenvalue_of_random_regular_graphs

operating systems – Algorithm favours CPU bound or I/O bound processes

Some scheduling algorithms favour CPU bound processes while others favour I/O bound processes.

Which algorithm favours which type of process?
My answers:

  1. SJF favours I/O bound processes.
    Explanation: Short CPU burst processes(I/O bound processes) are given higher priority.
  2. SRTF favours I/O bound processes.
    Explanation: Same as SJF
  3. Round Robin? Not able to decide categorically
  4. LRTF favours CPU bound processes.
    Explanation: Large CPU burst processes(CPU bound) are given higher priority.
  5. HRRN? Not able to decide categorically
  6. Multilevel feedback Queue favours I/O bound processes.
    Explanation: Short CPU burst processes(I/O bound processes) are given higher priority. While processes that have large CPU bursts(CPU bound) will slowly have decrement in priority. A process in lower priority queue can only execute when higher priority queues are empty. A process running in a lower priority queue is interrupted by a process arriving in a higher priority queue.
  7. Multilevel Queue favours I/O bound processes.
    Explanation: Similar to multilevel feedback.
  8. Priority Scheduling? Not able to decide categorically
  9. FCFS? Not able to decide categorically

Please verify above answers and explanations.
Correct me if I’m wrong somewhere and provide answers with explanation for the omitted ones.

bipartite matching – Flow graph with non zero lower bound or 0 capacity

I am afraid the question title might not be sufficiently accurate but I could not come up with something more accurate

Here is the problem

Given ‘n’ machines

  • Each machine has a set of capabilities
  • Each machine has max availability (A(m))

Given ‘m’ tasks

  • Each task requires a set of capabilities
  • Each task takes a certain time (D(t))
  • A task has to be performed on one machine only

The problem is to determine whether all tasks can be completed.

I get stuck with the ‘one machine only’ requirement. The only flow graphs I can come up with do not guarantee a task is not linked to more than one machine.

It’s sort of a bipartite matching problem but with capacities > 1

I also ran into XOR-like behavior in flow networks which is similar but has the ‘xor’ requirement on the ‘source’ end where I would need it on the target end.

Would anyone have any tips? Is it at all possible to model this as a flow graph?

Tx!

Peter

analytic number theory – Do we have any result proving a strong upper bound on the cardinality of set $P_{alpha}(x)$ for some large parameter $x$?

Define $x_0=0$ and $x_{i+1} = P(x_i)$ for all integers $i ge 0$.
Let $l(p)$ to be the least positive integer such that $p|x_{l(p)}$ for some prime $p$.

Then if we let
$$P_{(0<alpha<1)} = { pin mathbb{P}mid l(p)<p^{alpha}}$$ where $mathbb{P}$ is the set of primes, do we have any result proving a strong upper bound on the cardinality of set $P_{alpha}(x)$ for some large parameter $x$?