In many areas of imaging science, it is difficult to measure the phase of linear measurements. As such, one often wishes to reconstruct a signal from intensity measurements, that is, perform phase retrieval. In this paper, we provide a novel measurement design which is inspired by interferometry and exploits certain properties of expander graphs. We also give an efficient phase retrieval procedure, and use recent results in spectral graph theory to produce a stable performance guarantee which rivals the guarantee for PhaseLift in Voroninski, PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming, preprint, arXiv:1109.4499, 2011]. We use numerical simulations to illustrate the performance of our phase retrieval procedure, and we compare reconstruction error and runtime with a common alternating-projections-type procedure.

Finite frame theory has a number of real-world applications. In applications like sparse signal processing, data transmission with robustness to erasures, and reconstruction without phase, there is a pressing need for deterministic constructions of frames with the following property: every size-M subcollection of the M -dimensional frame elements is a spanning set. Such frames are called full spark frames, and this paper provides new constructions using the discrete Fourier transform. Later, we prove that full spark Parseval frames are dense in the entire set of Parseval frames, meaning full spark frames are abundant, even if one imposes an additional tightness constraint. Finally, we prove that testing whether a given matrix is full spark is hard for NP under randomized polynomial-time reductions, indicating that deterministic full spark constructions are particularly significant because they guarantee a property which is otherwise difficult to check.2000 Mathematics Subject Classification. 42C15, 68Q17.

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Abstract. The results of Strassen [Str73] and Raz [Raz10] show that good enough tensor rank lower bounds have implications for algebraic circuit/formula lower bounds.We explore tensor rank lower and upper bounds, focusing on explicit tensors. For odd d, we construct field-independent explicit 0/1 tensors T : [n] d → F with rank at least 2n ⌊d/2⌋ + n − Θ(d lg n). This matches (over F2) or improves (all other fields) known lower bounds for d = 3 and improves (over any field) for odd d > 3.We also explore a generalization of permutation matrices, which we denote permutation tensors. We show, by counting, that there exists an order-3 permutation tensor with super-linear rank. We also explore a natural class of permutation tensors, which we call group tensors. For any group G, we define the group tensor. We give two upper bounds for the rank of these tensors. The first uses representation theory and works over large fields F, showing (among other things) that rank. We also show that if this upper bound is tight, then super-linear tensor rank lower bounds would follow. The second upper bound uses interpolation and only works for abelian G, showing that over any field F that rankIn either case, this shows that many permutation tensors have far from maximal rank, which is very different from the matrix case and thus eliminates many natural candidates for high tensor rank.We also explore monotone tensor rank. We give explicit 0/1 tensors T : [n] d → F that have tensor rank at most dn but have monotone tensor rank exactly n d−1 . This is a nearly optimal separation.

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Quantum solitons are discovered with the help of generalized quantum hydrodynamics (GQH). The solitons have the character of the stable quantum objects in the self consistent electric field. These effects can be considered as explanation of the existence of lightning balls. The delivered theory demonstrates the great possibilities of the generalized quantum hydrodynamics in investigation of the quantum solitons. The paper can be considered also as comments and prolongation of the materials published in the known author`s monograph (Boris V. Alexeev, Generalized Boltzmann Physical Kinetics. Elsevier. 2004). The theory leads to solitons as typical formations in the generalized quantum hydrodynamics.Key words: Foundations of the theory of transport processes; The theory of solitons; Generalized hydrodynamic equations; Foundations of quantum mechanics; The theory of lightning balls.

This paper addresses the fundamental principles of generalized Boltzmann physical kinetics, as a part of non-local physics. It is shown that the theory of transport processes (including quantum mechanics) can be considered in the frame of unified theory based on the non-local physical description. Generalized Boltzmann physical kinetics leads to the strict approximation of non-local effects in space and time and after transmission to the local approximation leads to parameter of non-localityτ , which on the quantum level corresponds to the uncertainty principle "time- energy". Schrödinger equation is consequence of the Liouville equation as result of the local approximation of non-local equations. Generalized quantum hydrodynamics leads to Schrödinger equation as a deep particular case of the generalized Boltzmann physical kinetics and therefore of non-local hydrodynamics. Generalized quantum hydrodynamics can be used for solution of fundamental problems in nanoelectronics. Key words: Foundations of the theory of transport processes; The theory of solitons; Generalized hydrodynamic equations; Foundations of quantum mechanics PACS: 67.55.Fa, 67.55.Hc 1. Elementary introduction in the basic principles of the Generalized Boltzmann Physical Kinetics. 1 PhSV Moreover the particles starting after the last collision near the boundary between two mentioned volumes can change the distribution functions in the neighboring volume. The f KE nl B J J Dt Df + = , (1.4) where is the non-local integral term incorporating the time delay effect. The generalized Boltzmann physical kinetics, in essence, involves a local approximation nl J ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = Dt Df Dt D J nl 1 τ (1.5)for the second collision integral, here τ being the mean time between the particle collisions. We can draw here an analogy with the Bhatnagar -Gross -Krook (BGK) approximation for ,which popularity as a means to represent the Boltzmann collision integral is due to the huge simplifications it offers. In other words -the local Boltzmann collision integral admits approximation via the BGK algebraic expression, but more complicated non-local integral can be expressed as differential form (1.5).The ratio of the second to the first term on the right-hand side of Eqn (1.4) is given to an order of magnitude as )and at large Knudsen numbers (defining as ratio of mean free path of particles to the character hydrodynamic length) these terms become of the same order of magnitude. It would seem that at small Knudsen numbers answering to hydrodynamic description the contribution from the second term on the right-hand side of Eqn (1.4) is negligible. This is not the case, however. When one goes over to the hydrodynamic approximation (by multiplying the kinetic equation by collision invariants and then integrating over velocities), the Boltzmann integral part vanishes, and the second term on the right-hand side of Eqn (1.4)

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