We introduce a pictorial approach to quantum information, called holographic software. Our software captures both algebraic and topological aspects of quantum networks. It yields a bi-directional dictionary to translate between a topological approach and an algebraic approach. Using our software, we give a topological simulation for quantum networks. The string Fourier transform (SFT) is our basic tool to transform product states into states with maximal entanglement entropy. We obtain a pictorial interpretation of Fourier transformation, of measurements, and of local transformations, including the n-qudit Pauli matrices and their representation by Jordan-Wigner transformations. We use our software to discover interesting new protocols for multipartite communication. In summary, we build a bridge linking the theory of planar para algebras with quantum information.

Let <i>X, Y</i> be compact Hausdorff spaces and <i>E, F</i> be Banach spaces. A linear map <i>T: C(X, E) C(Y, F)</i> is separating if <i>Tf, Tg</i> have disjoint cozeroes whenever <i>f, g</i> have disjoint cozeroes. We prove that a biseparating linear bijection <i>T</i> (that is, <i>T</i> and <i>T</i>^-1 are separating) is a weighted composition operator <i>Tf = h f</i> o . Here, <i>h</i> is a function from <i>Y</i> into the set of invertible linear operators from <i>E</i> onto <i>F</i>, and , is a homeomorphism from <i>Y</i> onto <i>X</i>. We also show that <i>T</i> is bounded if and only if <i>h(y)</i> is a bounded operator from <i>E</i> onto <i>F</i> for all <i>y</i> in <i>Y</i>. In this case, <i>h</i> is continuous with respect to the strong operator topology.

Our aim is to study fractional order differential inclusions with infinite delays in Banach spaces. We impose the regularity condition on multivalued nonlinearity in terms of measures of noncompactness to get the existence result. Some properties of the solution map are proved.

We demonstrate a one-dimensional magnetic system can exhibit a Cantor-type spectrum using an example of a chain graph with coupling at the vertices exposed to a magnetic field perpendicular to the graph plane and varying along the chain. If the field grows linearly with an irrational slope, measured in terms of the flux through the loops of the chain, we demonstrate the character of the spectrum relating it to the almost Mathieu operator.

We analyze two-dimensional Schrdinger operators with the potential |{xy}{|}^{p}-\lambda {({x}^{2}+{y}^{2})}^{p/(p+ 2)} where |{xy}{|}^{p}-\lambda {({x}^{2}+{y}^{2})}^{p/(p+ 2)} and |{xy}{|}^{p}-\lambda {({x}^{2}+{y}^{2})}^{p/(p+ 2)} which exhibit an abrupt change of spectral properties at a critical value of the coupling constant . We show that in the supercritical case the spectrum covers the whole real axis. In contrast, for below the critical value the spectrum is purely discrete and we establish a LiebThirring-type bound on its moments. In the critical case where the essential spectrum covers the positive halfline while the negative spectrum can only be discrete, we demonstrate numerically the existence of a ground-state eigenvalue.