## Valiant-Vazirani Theorem

Consider the following promise problem: Given a boolean formula that has at most one satisfying assignment, decide whether it is unsatisfiable or it has exactly one satisfying assignment. The Valiant-Vazirani theorem says that, if this problem is solvable in polynomial

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## Lower Bounds Against Circuits with Modular Gates

The modular gate MODm, for any integer $$m$$, outputs 0 if the sum of its inputs is 0 modulo $$m$$, and outputs 1 otherwise. An ACC[m] circuit is a circuit with NOT gates and unbounded fan-in AND, OR, and MODm

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## Combinatorial Design and Pseudorandom Generator

For a set of size $$l$$, choose $$m$$ subsets each of size $$n$$ such that each pair of subsets has intersection size at most $$d$$. This gives a combinatorial design with parameters $$(m,l,n,d)$$. That is, we have \(I_1,\ldots, I_m \subseteq

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## Embed Google Maps Image and Street View

Google Maps APIs provide easy ways to embed maps images. The following are static map image, satellite image and street view image of “Lincoln Memorial”. Static Maps API supports map images with custom styles, markers, satellite maps. The following is

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