ON WEAK POINCARE PROBLEM

Quantum Weak Parity Problem

In this work we shall investigate quantum query complexity of weak PARITY problem. Weak Parity is the following natural query problem: What is the minimum number of queries necessary to compute PARITY(x1, x2, x3, . . . , xn) on at least 1 2 + fraction of the inputs. For randomized classical or exact quantum machines this relaxed PARITY problem remains as hard as original PARITY problem. Althoug...

Pattern Encoding on the Poincare Sphere

This paper presents a convenient graphical tool for encoding visual patterns (such as image patches and image atoms) as point constellations in a space spanned by perceptual features and with a clear geometrical interpretation. General theory and a practical pattern encoding scheme are presented, inspired by encoding polarization states of a light wave on the Poincaré sphere. This new pattern e...

Poincare Inequalities

Poincare inequalities are a simple way to obtain lower bounds on the distortion of mappings X into Y. These are shown below to be sharp when we consider the Lp spaces. A Poincare inequality is one of the following type: suppose Ψ : [0, ∞) → [0, ∞) is a nondecreasing function and that au,v, bu,v are finite arrays of real numbers (for u, v ∈ X, and not all of the numbers 0). We say that functions...

On Curvature and Euler-poincare Characteristic.

QUASI-∗ STRUCTURE ON q-POINCARE ALGEBRAS

We use braided groups to introduce a theory of ∗-structures on general inhomogeneous quantum groups, which we formulate as quasi-∗ Hopf algebras. This allows the construction of the tensor product of unitary representations up to a quantum cocycle isomorphism, which is a novel feature of the inhomogeneous case. Examples include q-Poincaré quantum group enveloping algebras in R-matrix form appro...