A sample efficient sparse FFT for arbitrary frequency candidate sets in high dimensions

In this paper, a sublinear time algorithm is presented for the reconstruction of functions that can be represented by just few out of a potentially large candidate set of Fourier basis functions in high spatial dimensions, a so-called high-dimensional sparse fast Fourier transform. In contrast to many other such algorithms, our method works for arbitrary candidate sets and does not make additional structural assumptions on the candidate set. Our transform significantly improves upon the other approaches available for such a general framework in terms of the scaling of the sample complexity. Our algorithm is based on sampling the function along multiple rank-1 lattices with random generators. Combined with a dimension-incremental approach, our method yields a sparse Fourier transform whose computational complexity only grows mildly in the dimension and can hence be efficiently computed even in high dimensions. Our theoretical analysis establishes that any Fourier s-sparse function can be accurately reconstructed with high probability. This guarantee is complemented by several numerical tests demonstrating the high efficiency and versatile applicability for the exactly sparse case and also for the compressible case.

Quantum-assisted distortion-free audio signal sensing

Quantum sensors are keeping the cutting-edge sensitivities in metrology. However, for high-sensitive measurements of arbitrary signals, limitations in linear dynamic range could introduce distortions when sensing the frequency, magnitude and phase of unknown signals. Here, we overcome these limitations with advanced sensing protocol that combines quantum phase-sensitive detection with the heterodyne readout. We present theoretical and experimental investigations using nitrogen-vacancy centers in diamond, showing the ability to sense audio signals with a 98 dB linear dynamic range, a 31 pT/Hz1/2 sensitivity, and arbitrary frequency resolution. Further, we perform the quantum-assisted distortion-free audio signal (melody, speech) sensing with high fidelity. The methods developed here could broaden the horizon for quantum sensors towards applications in telecommunication, where high-fidelity and low-distortion at multiple frequency bands within small sensing volumes are required.

Analytical Solution of Problems about the Radiative and Radiative–Conductive Stationary Heat Transfer in a Medium with an Arbitrary Dependence of the Scattering and Absorption on Frequency Boundary Conditions

We defined a method for the analytical solution of problems on stationary radiative and radiative–conductive heat transfer in a medium with an arbitrary frequency dependence of absorption and scattering near its boundary. We obtained formulas for the heat conductance of the remote surface and the thickness of the radiative–conductive relaxation of the medium. We determined characteristics of radiant heat transfer from the medium to free space such as the radiation spectrum, the radiation temperature and the medium outer boundary temperature. In addition, we solved the problem on the radiative–conductive heat transfer from one of two parallel surfaces to another with a medium between them.

Inverse Nonlinear Eigenvalue Problem Framework for the Synthesis of Coupled-Resonator Filters With Non-Resonant Nodes and Arbitrary Frequency-Variant Couplings

A novel, general circuit-level description of coupled-resonator microwave filters is introduced in this paper. Unlike well-established coupling-matrix models based on frequency-invariant couplings or linear frequency-variant couplings (LFVCs), a model with arbitrary frequency-variant coupling (AFVC) coefficients is proposed. The engineered formulation is more general than prior-art ones and can be treated as an extension of previous synthesis models, since constant or linear couplings are special cases of arbitrary frequency dependence. The suggested model is fully general, allows for AFVCs with highly nonlinear (even singular) characteristics, loaded or unloaded non-resonating nodes (NRNs), frequency-dependent source-load coupling, multiple frequency-variant cross-couplings, and{/}or multiple dispersive couplings for connecting the source and load to the filter network. The model is accompanied by a powerful synthesis technique that is based on the zeros and poles of the admittance or scattering parameters and the eigenvalues of properly defined eigenproblems. In the most general case, the zeros and poles of the admittance or scattering parameters are related to solutions of nonlinear eigenvalue problems. The synthesis is defined as an inverse nonlinear eigenvalue problem (INEVP) where the matrix is constructed from three sets of eigenvalues. This is accomplished by optimization using an iterative nonlinear least-squares solver with excellent convergence property. Finally, third- and fifth-order examples of bandpass filter topologies involving AFVCs are shown, and the experimental validation of the proposed theory is presented through the manufacturing and characterization of a microstrip filter prototype with transmission zeros (TZs)

Inverse Nonlinear Eigenvalue Problem Framework for the Synthesis of Coupled-Resonator Filters With Non-Resonant Nodes and Arbitrary Frequency-Variant Couplings

A novel, general circuit-level description of coupled-resonator microwave filters is introduced in this paper. Unlike well-established coupling-matrix models based on frequency-invariant couplings or linear frequency-variant couplings (LFVCs), a model with arbitrary frequency-variant coupling (AFVC) coefficients is proposed. The engineered formulation is more general than prior-art ones and can be treated as an extension of previous synthesis models, since constant or linear couplings are special cases of arbitrary frequency dependence. The suggested model is fully general, allows for AFVCs with highly nonlinear (even singular) characteristics, loaded or unloaded non-resonating nodes (NRNs), frequency-dependent source-load coupling, multiple frequency-variant cross-couplings, and{/}or multiple dispersive couplings for connecting the source and load to the filter network. The model is accompanied by a powerful synthesis technique that is based on the zeros and poles of the admittance or scattering parameters and the eigenvalues of properly defined eigenproblems. In the most general case, the zeros and poles of the admittance or scattering parameters are related to solutions of nonlinear eigenvalue problems. The synthesis is defined as an inverse nonlinear eigenvalue problem (INEVP) where the matrix is constructed from three sets of eigenvalues. This is accomplished by optimization using an iterative nonlinear least-squares solver with excellent convergence property. Finally, third- and fifth-order examples of bandpass filter topologies involving AFVCs are shown, and the experimental validation of the proposed theory is presented through the manufacturing and characterization of a microstrip filter prototype with transmission zeros (TZs)

The Modigliani–Miller Theory with Arbitrary Frequency of Payment of Tax on Profit

The main purpose of the current study is the generalization and further development of the Modigliani–Miller theory taking into account one of the conditions of the real functioning of companies for the case of paying income tax with an arbitrary frequency (monthly, quarterly, semi-annual or annual payments). While a return is not required more than once a year, businesses may be responsible for filing estimated taxes based on profits earned. This requirement is dependent on showing a profit. For example, sole proprietors must file estimated taxes on profits quarterly, on the 15th day of April, June, September and January. In Russia, tax on profit payments could be made annually, quarterly, or monthly. We suppose, that more frequent payment of income tax impacts on all main financial indicators of the company and leads to some important consequences. We use analytical and numerical methods: we derive all main formulas of the modified Modigliani–Miller theory theoretically and then use them to obtain all main financial indicators of company and their dependences on different parameters by MS Excel. We show that: (1) all Modigliani–Miller theorems, statements and formulas change; (2) all main financial indicators, such as the weighted average cost of capital (WACC), company value, V, and equity cost, ke, depend on the frequency of tax on profit payments; (3) in the case of income tax payments more than once per year (at p ≠ 1), as takes place in practice, the WACC, company value, V and equity cost, and ke start depend on debt cost, kd, while in ordinary (classical) Modigliani–Miller theory all these values do not depend on kd; (4) obtained results allow a company to choose the number of payments of tax on profit per year (of course, within actual tax legislation): more frequent payments of income tax are beneficial for both parties, for the company and for the tax regulator.