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Meaning of sum of powers in a multiplication

This is possibly a very basic question, but I’m still new to complexity theory. Let $F$ be the functional that maps $\mathcal{A}$ to the real numbers, where $\mathcal{A} = \{0,1\}^* \cup \{a\}$. Define a new function $F’$, where $F'(a) = 2$ and $F'(0) = 0$. Also, define $F'(\mu)$ by

$$

F'(\mu) = \sum_{i=0}^{k} \mu(a^i \cdot a^{k-i})

$$

where $\mu$ is the uniform distribution over the nonempty words $a^i \cdot a^{k-i} \in \mathcal{A}^*$.

Is there an easy way to see that $F'(\mu) \geq 0.5(1-1/k)$? (I’m not even sure that that’s right; I was thinking of using the Cauchy-Schwarz inequality, but the fact that $\mu$ is not a probability distribution seems to make that problematic.)

A:

You can get this by induction on $k$. It’s best to do it by reducing everything to a problem about sets of integers. For the induction base, suppose $k=1$, so that $\mu$ is the measure $$\mu(a\cdot 0) + \mu(0\cdot a)=1+0=1.$$ Now take any word $w=a^i\cdot 0$ with $0\le i\le k$, and let $j=|a^i|$. This means that $w$ has length $j+1-i$ and is uniquely of the form $a^{i-1}\cdot 0=a^{j-1}\cdot a$. The term on the left of the equation is $\mu(w)$, which is $1/(j+1)$ because that’s the measure of the set $A$ of integers of the form $j,

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PDFtoMusic Pro v1.6.5 – SeuPirate

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Last week, Alwaleed Bin Talal, an investor whose family owns 20 percent of the world’s top crude oil producer, agreed to pump at least $200 billion of cash into the largely state-owned Saudi Aramco over the next 25 years. It’s the biggest investment ever in the world’s biggest oil company, which holds some $760 billion in cash.

The deal, announced in Abu Dhabi, also dwarfs the most lucrative U.S. shale oil and gas deals, bringing together the wealthy prince whose $31 billion fortune is larger than Saudi Arabia’s entire annual budget in a partnership with the state-owned China National Petroleum Corp. and the China Offshore Oil Corp.

The plans are part of a broader Saudi strategy to diversify its revenues and boost government revenues. The official Saudi budget for fiscal year 2015 was $98.5 billion, which is its lowest since the kingdom began releasing detailed financial data in 2010. But much of the money allocated for the Aramco project will go to the state budget.

“The deals illustrate the strength of the Saudi economy and its ability to generate cash, even if the fuel demand for the Saudis is coming from other parts of the world,” said Ellie Geranmayeh, research fellow for Middle East studies at the European Council on Foreign Relations. “It’s about moving away from government cash and the burden of government debt.”

The Aramco deal, in which China is not expected to carry much of the financial burden, was made without a single bidder meeting the 11 conditions set by Saudi Arabia. The deal made little mention of Aramco’s deteriorating finances, nevermind how to close a sale at $100 a barrel oil, which is at a five-year low. A fall in oil prices has crimped Aramco’s revenues while also slashing government revenues

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