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Today's devices for quantum computing are still far from implementing useful and powerful quantum algorithms. Decoherence and the wish to resist the effects of errors in a system of quantum bits incurs a lot of overhead in the number of…

In this paper, we prove that a very general cubic threefold does not admit a universal codimension-two cycle and hence is stably irrational.

The strong, intermediate, and weak Turing impossibility properties are introduced. Some facts concerning Turing impossibility for stack machine programming are trivially adapted from previous work. Several intriguing questions are raised…

We study the computational complexity theory of smooth, finite-dimensional dynamical systems. Building off of previous work, we give definitions for what it means for a smooth dynamical system to simulate a Turing machine. We then show that…

SAT is not in P, is true and provable in a simply consistent extension B' of a first order theory B of computing, with a single finite axiom characterizing a universal Turing machine. Therefore, P is not equal to NP, is true and provable in…

A problem of universality in simulation of evolution of quantum system and in theory of quantum computations is related with the possibility of expression or approximation of arbitrary unitary transformation by composition of specific…

We consolidate two widely believed conjectures about tautologies -- no optimal proof system exists, and most require superpolynomial size proofs in any system -- into a $p$-isomorphism-invariant condition satisfied by all paddable…

We prove a Central Limit Theorem for the sequence of random compositions of a two-color randomly reinforced urn. As a consequence, we are able to show that the distribution of the urn limit composition has no point masses.

We investigate the truth-table degrees of (co-)c.e.\ sets, in particular, sets of random strings. It is known that the set of random strings with respect to any universal prefix-free machine is Turing complete, but that truth-table…

This short survey of recent work in tile self-assembly discusses the use of simulation to classify and separate the computational and expressive power of self-assembly models. The journey begins with the result that there is a single…

We derive the optimal curve satisfied by the reduction factors, in the case of universal disentangling machine which uses only local operations. Impossibility of constructing a better disentangling machine, by using non-local operations, is…

The no-masking theorem for quantum information proves that it is impossible to encode an arbitrary input state into a larger bipartite entangled state such that the full information is stored in the correlation but the individual subsystems…

Operating quantum sensors and quantum computers would make data in the form of quantum states available for purely quantum processing, opening new avenues for studying physical processes and certifying quantum technologies. In this…

For a few quadratic fields, the non-existence is proved of continuous irreducible mod 2 Galois representations of degree 2 unramified outside 2.

This paper shows that the programming model of Babbage's Analytical Engine, although unconventional, can be harnessed in order to simulate indirect addressing, a capability that was not included in the original instruction set. That is, in…

We supply a rigorous proof that an open dense set of all possible 2-qubit gates G has the property that if the quantum circuit model is restricted to only permit swap of qubits lines and the application of G to pairs of lines, then the…

To better understand quantum computation we can search for its limits or no-gos, especially if analogous limits do not appear in classical computation. Classical computation easily implements and extensively employs the addition of two bit…

It is shown that 'non-quantum systems', with anomalous statistical properties, would carry a distinctive experimental signature. Such systems can exist in deterministic hidden-variables theories (such as the pilot-wave theory of de Broglie…

This paper synthesizes a series of formal proofs to construct a unified theory on the logical limits of the Symbol Grounding Problem. We distinguish between internal meaning (sense), which formal systems can possess via axioms, and external…

Entanglement, one of the central mysteries of quantum mechanics, plays an essential role in numerous applications of quantum information theory. A natural question of both theoretical and experimental importance is whether universal…