An approximation of Daubechies wavelet matrices by perfect reconstruction filter banks with rational coefficients

It is described how the coefficients of Daubechies wavelet matrices can be approximated by rational numbers in such a way that the perfect reconstruction property of the filter bank be preserved exactly.     

On Approximate Spectral Factorization of Matrix Functions

It is proved that if positive definite matrix functions (i.e. matrix spectral densities) S _n , n=1,2,… , are convergent in the L ₁-norm, ||S_n-S||_L1? 0\|S_{n}-S\|_{L_{1}}\to 0, and ò₀^2plogdetS_n(e^iq) dq?ò₀^2plogdetS(e^iq) dq\int_{0}^{2\pi}\log \mathop{\mathrm{det}}S_{n}(e^{i\theta})\,d\theta\to\int_{0}^{2\pi}\log \mathop{\mathrm{det}}S(e^{i\theta})\,d\theta, then the corresponding (canonical) spectral factors are convergent in L ₂, ||S⁺_n-S⁺||_L2? 0\|S^{+}_{n}-S^{+}\|_{L_{2}}\to 0. The formulated logarithmic condition is easily seen to be necessary for the latter convergence to take place.     

Black Hole Information Problem and Wave Bursts

By reexamination of the boundary conditions of wave equation on a black hole horizon it is found not harmonic, but real-valued exponentially time-dependent solutions. This means that quantum particles probably do not cross the Schwarzschild horizon, but are absorbed and some are reflected by it, what potentially can solve the famous black hole information paradox. To study this strong gravitational lensing we are introducing an effective negative cosmological constant between the Schwarzschild and photon spheres. It is shown that the reflected particles can obtain their additional energy in this effective AdS space and could explain properties of some unusually strong signals, like LIGO events, gamma ray and fast radio bursts.     

On the uniqueness of maximal functions

The uniqueness theorem for the one-sided maximal operator has been proved.     

Generalized framework for auxiliary extra dimensions

The theory of gravity with an auxiliary extra dimension is known to give the ghost-free cubic completion of the Fierz-Pauli mass term in the decoupling limit. Our work generalizes the boundary condition in the auxiliary dimension that avoids ghosts order-by-order, and to all orders, in the decoupling limit. Furthermore, we extend the formalism to the case of many auxiliary dimensions, and we show that the multi-dimensional extension with the rotationally invariant boundaries of the bulk, is equivalent to the model with a single auxiliary dimension. The above constructions require the appropriate adjustment of the boundary condition, which we discuss in detail. The other possible extension of the original model by the Gauss-Bonnet term is studied as well.     

On Compact Wavelet Matrices of Rank m and of Order and Degree N

A new parametrization (one-to-one onto map) of compact wavelet matrices of rank $m$ and of order and degree $N$ is proposed in terms of coordinates in the Euclidian space $\mathbb {C}^{(m-1)N}$ . The developed method depends on Wiener–Hopf factorization of corresponding unitary matrix functions and allows to construct compact wavelet matrices efficiently. Some applications of the proposed method are discussed.     

Measuring randomness with periodic media

We show that the sensitivity to disorder of certain physical properties of periodic media depends on whether the disorder is truly random or not. This allows to utilize periodic media for testing sequences of random numbers and to quantify their departure from true randomness via simple transmission and/or reflection data analysis. This physics-based model shows promises for device applications to test random data.     

Justification of the nonlinear Schr?dinger equation in spatially periodic media

The dynamics of the envelopes of spatially and temporarily oscillating wave packets advancing in spatially periodic media can approximately be described by solutions of a Nonlinear Schr?dinger equation. Here we prove estimates for the error made by this formal approximation using Bloch wave analysis, normal form transformations, and Gronwall’s inequality.     

Supersymmetric grand unification with light color-triplet

We construct a natural model of the supersymmetric SU(6)$\mathit{SU}(6)$ unification, in which the symmetry breaking, down to the standard model gauge group, results in the number of pseudo-Nambu–Goldstone superfields with interesting properties. Namely, besides the Higgs doublet–antidoublet pair which is responsible for the electroweak phase transition, the Nambu–Goldstone sector consists of multiplets in the anti- and fundamental representations of SU(5)$\mathit{SU}(5)$. While being strictly massless in the supersymmetric limit, they acquire the weak scale masses as a result of its breaking. The color-triplet components of this light sector could, in principle, mediate an unacceptably fast proton decay; however, because of the natural TeV/_MGUT$\text{TeV}/M_{\mathrm{GUT}}$ suppression of the Yukawa couplings to the light quarks and leptons, their existence is compatible with the experimental bound on proton lifetime. This suppression is made further interesting, since it results in the lifetime, of the lightest of the above-mentioned colored particles from 1 s to 1 day$1\text{day}$, long enough for it to appear stable in the detector. Furthermore, we argue that the accommodation of the color-triplet pseudo-Nambu–Goldstones, without fine-tuning or contradicting observations, implies SU(6)$\mathit{SU}(6)$ unification.