NP-hardness does not imply lower bound, strictly speaking?

A problem is NP-hard iff every NP problem can be polynomially-time reduced to it. 

Hardness is often intuitively explained as a lower bound. But it isn’t, strictly speaking. For the sake of the argument, assume P=NP. Now the definition becomes:

A problem is P-hard iff every P problem can be polynomially-time reduced to it. 

Because the above definition uses the polynomial-time reduction, the overall running time is polynomial (reduction + solving the resulting problem), no matter how easy is the resulting problem. Hence we could get an absurd result: a problem, which runs in constant time (hence lower bound is constant), is P-hard.

The definition makes total sense if we assume that NP>P. Do people assume NP>P?

The same question arises for the definition of PSPACE-hardness, where book authors use polynomial-time reduction, rather than something strictly easier than PSPACE.

I guess the answer to this question is simply “yes”, sorry for the rant.

How to understand out of bound in the following theoretical context?

Consider the following initialization step of loop invariant for merge procedure

Initialization: Prior to the first iteration of the loop, we have $ k=p$ , so that the subarray $ A[p .. k – 1]$ is empty. This empty subarray contains the k- p= 0 smallest elements of $ L$ and $ R$ , and since $ i = j = 1$ , both $ L[i]$ and $ R[j]$ are the smallest elements of their arrays that have not been copied back into $ A$ .

I have doubt in the above statement that if $ k=p$ , then array $ A[p..p-1]$ is impossible and hence the further argument cannot proceed, which didn’t happen. Where am I going wrong?

How can an elemental be bound into an object or vehicle (e.g. lightning rails, elemental airships) in Eberron?

I will soon be DMing an Eberron campaign for my Dungeons & Dragons group. As part of the plot, I want a player or NPC to bind an air elemental to an elemental airship, and it may come up in the future. I recently bought the Wayfinder’s Guide to Eberron, and I used an example from the book as an NPC (a gnome artificer).

Is there a specific ritual for binding an Elemental to an object or vehicle (e.g. lightning rail or something similar), assuming they already have the elemental?

Example of convex functions fulfilling a (strange) lower bound

I am reading a preliminary version of a paper which focuses on some minimization problems connected to a class of integral functionals. Reading the assumptions of one of the theorems I cannot convince myself that a “concrete” example exists.

Question. Does there exist a function $ f \colon \mathbb R^N \to [0,+\infty)$ such that

  1. $ f$ is convex;
  2. $ f(\lambda x) = \lambda f(x)$ for any $ \lambda >0$ and $ \forall x \in \mathbb R^N$ ;
  3. there are $ a>0, b \ge 0$ and $ \gamma \in \mathbb R^N$ such that $ $ a|x| \le f(x) + \langle \gamma, x \rangle + b $ $ for any $ x \in \mathbb R^N$ ?

I have some problems in finding a function satisfying the three points… It does not have to be smooth, still I do not see an example.

Given least upper bound $\alpha$ for $\{\ f(x) : x \in [a,b] \ \}$, $\forall \epsilon > 0 \ \exists x$ s.t. $\alpha – f(x) < \epsilon$

I can’t figure out how all of this follows. Taken from Ch.8 of Spivak’s Calculus.

If $ \alpha$ is the least upper bound of $ \{\ f(x) : x \in [a,b] \ \}$ then, $ $ \forall \epsilon > 0 \ \exists x\in [a,b] \ \ \ \ \ \ \ \alpha – f(x) < \epsilon$ $ This, in turn, means that $ $ \frac{1}{\epsilon} < \frac{1}{\alpha – f(x)}$ $

Quantitative bound on irrational rotation recurrence time

Given an irrational $ a$ , the sequence $ b_n := na$ is dense and equidistributed in $ \mathbb S^1$ where we view $ \mathbb S^1$ as $ [0, 1]$ with its endpoints identified.

Given a point $ p$ in $ \mathbb S^1$ , can we obtain a quantitative upper bound (that can depend on $ a, p, e$ ) on the smallest $ n$ such that $ na$ is in $ B_e (p)$ ?

Proof of a lower bound of the recurrence relation (the CLRS’s 4.6-2 exercise)

I am trying to find a solution to the ex. 4.6-2 of the Introduction to Algorithms by Cormen, Leiserson, Rivest, Stein (the third edition). It requires, for recurrence relations $ T(n)=aT(n/b)+f(n)$ where $ a\geq 1, b > 1$ , $ n$ is an exact power of $ b$ and $ f(n)$ is an asymptotically positive function, to prove that if $ f(n) = \Theta(n^{\log_ba}lg^{k}n)$ , where $ k\geq0$ , then $ T(n)=\Theta(n^{\log_ba}lg^{k+1}n)$ .

Since $ T(n) = n^{\log_ba} + g(n)$ where $ g(n) = \sum_{j=0}^{\log_b n – 1} a^{j}f(n/b^{j})$ , I decided to consider the $ g(n)$ function at the first. And I have shown $ g(n) = O(n^{\log_ba}lg^{k+1}n)$ already (I think so).

But the doing the proof $ g(n) = \Omega(n^{\log_ba}lg^{k+1}n)$ became a challenge for me.

Below is my research (with the simplified assumption $ k$ is an integer). By condtition, there is such constant $ c$ that

$ g(n) \geq$

$ c \sum_{j=0}^{\log_b n – 1} a^{j}(n/b^{j})^{log_ba}log^{k}(n/b^{j}) =$

$ cn^{\log_ba}\sum_{j=0}^{\log_b n – 1} log^{k}(n/b^{j}) =$

$ cn^{\log_ba}\sum_{j=0}^{\log_b n – 1}(logn – logb^{j})^{k} =$

$ cn^{\log_ba}\sum_{j=0}^{\log_b n – 1}\sum_{i=0}^{k} {k \choose i}log^{k-i}n(-logb^{j})^{i} =$

$ cn^{\log_ba}log^{k}n\sum_{j=0}^{\log_b n – 1}\sum_{i=0}^{k} {k \choose i}(-logb^{j}/logn)^{i} =$

$ cn^{\log_ba}log^{k}n \biggl(log_bn + \sum_{j=0}^{\log_b n – 1}\sum_{i=1}^{k} {k \choose i}(-logb^{j}/logn)^{i} \biggr) \geq$

$ c’n^{\log_ba}log^{k+1}n – cn^{\log_ba}log^{k}n\sum_{j=0}^{\log_b n – 1}\sum_{i=1}^{k} {k \choose i}(logb^{j}/logn)^{i} =$

$ A(n) – B(n) = \Theta(n^{\log_ba}lg^{k+1}n) – B(n)$

Actually I am stuck with it. I can not show that $ B(n)$ grows slower than $ A(n)$ . For instance, since $ (logb^{j}/logn)^{i} \lt 1$ we are able to enhance our $ \geq$ condition by the substitution $ B(n)$ to some fucntion $ B'(n)$ with the sums of binominal coefficients only. But then finally $ B'(n)$ has $ n^{\log_ba}log^{k+1}n$ .

So how to prove $ g(n) = \Omega(n^{\log_ba}lg^{k+1}n)$ ?