Finite Length Research Articles

Decoupling by Local Random Unitaries without Simultaneous Smoothing, and Applications to Multi-user Quantum Information Tasks

We show that a simple telescoping sum trick, together with the triangle inequality and a tensorisation property of expected-contractive coefficients of random channels, allow us to achieve general simultaneous decoupling for multiple users via local actions. Employing both old (Dupuis et al. in Commun Math Phys 328:251–284, 2014) and new methods (Dupuis in IEEE Trans Inf Theory 69:7784–7792, 2023), we obtain bounds on the expected deviation from ideal decoupling either in the one-shot setting in terms of smooth min-entropies, or the finite block length setting in terms of Rényi entropies. These bounds are essentially optimal without the need to address the simultaneous smoothing conjecture, which remains unresolved. This leads to one-shot, finite block length, and asymptotic achievability results for several tasks in quantum Shannon theory, including local randomness extraction of multiple parties, multi-party assisted entanglement concentration, multi-party quantum state merging, and quantum coding for the quantum multiple access channel. Because of the one-shot nature of our protocols, we obtain achievability results without the need for time-sharing, which at the same time leads to easy proofs of the asymptotic coding theorems. We show that our one-shot decoupling bounds furthermore yield achievable rates (so far only conjectured) for all four tasks in compound settings, which are additionally optimal for entanglement of assistance and state merging.

Efficient quantum circuits based on the quantum natural gradient

Efficient preparation of arbitrary entangled quantum states is crucial for quantum computation. This is particularly important for noisy intermediate-scale quantum simulators relying on variational hybrid quantum-classical algorithms. To that end, we propose symmetry-conserving modified quantum approximate optimization algorithm (SCom-QAOA) circuits. The depths of these circuits depend not only on the desired fidelity to the target state but also on the amount of entanglement the state contains. The parameters of the SCom-QAOA circuits are optimized using the quantum natural gradient method based on the Fubini-Study metric. The SCom-QAOA circuit transforms an unentangled state into a ground state of a gapped one-dimensional Hamiltonian with a circuit depth that depends not on the system size but rather on the finite correlation length. In contrast, the circuit depth grows proportionally to the system size for preparing low-lying states of critical one-dimensional systems. Even in the latter case, SCom-QAOA circuits with depth less than the system size were sufficient to generate states with fidelity in excess of 99%, which is relevant for near-term applications. The proposed scheme enlarges the set of the initial states accessible for variational quantum algorithms and widens the scope of investigation of nonequilibrium phenomena in quantum simulators. Published by the American Physical Society 2024

Design and Performance Analysis of A Linear Switched Reluctance Motor Considering End Effects

Introduction: In recent years, linear actuators have gained significant attention due to their ability to provide direct linear movement without the need for motion transformation devices. One significant challenge in the design and modeling of linear actuators is the occurrence of longitudinal end-effects. The finite length of the translator stack in linear actuators causes an unbalanced phase force, leading to discrepancies between the phases at the end and center of the actuator. Neglecting these end-effects can lead to inaccuracies in the actuator's performance and control. Methods: The objective of this study is to develop a comprehensive approach for the design, sizing, and modeling of the actuator to ensure optimal performance. An energy conversion procedure calculates the thrust force and determines the actuator's geometrical parameters. A numerical model based on the finite element method is developed to analyze the actuator's magnetic behavior and establish its characteristics. Furthermore, a thorough analysis of the end effect is conducted using two-dimensional finite element analysis. Results: To accurately capture the actuator's dynamic response and enable precise control, a mathematical model is formulated, incorporating the nonlinear behavior of inductance and incorporating the end effect factor. The proposed model demonstrates high accuracy and provides a solid foundation for the control of the linear switched reluctance actuator. Conclusion: Overall, this study contributes to the advancement of direct drive systems for industrial applications by providing a detailed design procedure and an accurate modeling approach for the linear actuator, enabling more efficient and precise control strategies.

Electro-osmotic flow and heat transfer in Jeffery fluid: A multi-membrane microchannel model

Microscale heat transfer is vital for the performance of smart thermal devices like heat sinks, thermosyphons, and microheat pipes. This study introduces a biothermal pumping flow model based on a multi-membrane pumping mechanism that leverages microscale heat transfer. The model describes rhythmic contraction and relaxation of membranes, combined with electro-osmosis in Jeffery fluid flow within a vertical microchannel of finite length. Two membranes on the microchannel walls, with varying amplitudes, diameters, and phase lags, generate pressure that moves fluid in both directions through contraction and expansion cycles. The model is based on the conservation of mass and momentum, using a low Reynolds number approximation to capture microscale transport phenomena at biomedical scales. Dimensionless conservation equations are analytically solved under no-slip boundary conditions, with results computed in MATLAB for clarity. Axial velocity results are simulated and verified using the optimal homotopy analysis method. The model explores the influence of key parameters (UHS, me, λ, Gr, β) on pressure gradient, velocity distribution, volumetric flow rates, skin friction, Nusselt number, and stream function. The findings demonstrate that pressure from membrane motion is significantly affected by thermal effects and buoyancy forces, and flow and pumping characteristics are largely determined by the fluid's rheological qualities and the geometrical features of the membrane. This study provides novel ideas for enhancing the functionality and design of smart thermal devices while also advancing microscale heat transfer technology.

Evolutionary mechanism of Y-branches in acoustic Lichtenberg figures just below the water surface

The acoustic Lichtenberg figure (ALF) in an ultrasonic cleaner with a frequency of 28 kHz at different power levels was observed using high-speed photography. The nonlinear response of the cavitation structure was analyzed by the entropy spectrum in the ALF images, which showed the modulation influence of the primary acoustic field, exhibiting the fluctuations of the bubble distribution with time. Typical Y-branches predict the paths by which surrounding bubbles are attracted and converge into the structure, the branches are curved due to bubble-bubble interactions, and the curvature increases as the bubbles are approaching the main chain. The average travelling speed of bubbles along the branches is about 1.1 m/s, almost independent of power level of the ultrasonic cleaner. A theoretical model consisting of free bubbles and a straight bubble chain of finite length was developed to explore the evolutionary mechanism of branching. It was found that the bubble trajectories showed a bending tendency similar to the experimentally observed Y-branches, and the stationary straight bubble chain parallel to the main chain could evolve into a curved chain and eventually become a branch of the main chain. The theoretical predictions agree well with the experimental results, verifying the evolutionary mechanism of Y-branches in ALF.

Nonlinear saturation of the ion flow driven ion sound instability in a finite length plasma

The saturation mechanism and nonlinear evolution of the ion sound instability driven by the subsonic ion flow in a finite length plasma are studied by a one-dimensional hybrid model considering kinetic ions and Boltzmann electrons. Three regimes of the instability nonlinear behavior are identified as a function of the frequency of the ion-neutral charge exchange (CX) collision fcoll. In the first (collisionless-alike) regime when the CX frequency is low, the instability is saturated by ions trapping in wave potentials leading to the formation of phase space vortexes (PSVs). One of the PSVs subsequently expands and becomes system long in the steady state. The transition to the second (medium) regime occurs when fcoll≳vp/d, where vp is the PSV expansion velocity and d is the system length. In the second regime, CX collisions convert fraction of beam ions into slow ions that can be trapped in potentials of small scale ion sound eigenmodes fluctuations. The trapping of slow ions results in the formation of a chain of small scale PSVs and the disruption of the establishment of the single system long PSV. In the third (collision-dominated) regime when fcoll≳γ (γ is the instability growth rate), CX collisions transform all beam ions into slow ions and the instability is thereby eliminated.

A Genetic Algorithm to Search the Space of Ehrhart h*-Vectors

We describe a genetic algorithm to find candidates for h * -vectors satisfying given properties in the space of integers vectors of finite length. We use an implementation of such algorithm to find a 52-dimensional lattice polytope having a non-unimodal h * -vector which is the Cartesian product of two lattice polytopes having unimodal h * -vectors. This counterexample answers negatively to a question by Ferroni and Higashitani.

Entanglement in ( 1+1 )-dimensional free scalar field theory: Tiptoeing between continuum and discrete formulations

We review some classic works on ground state entanglement entropy in (1+1)-dimensional free scalar field theory. We point out identifications between the methods for the calculation of entanglement entropy and we show how the formalism developed for the discretized theory can be utilized in order to obtain results in the continuous theory. We specify the entanglement spectrum and we calculate the entanglement entropy for the theory defined on an interval of finite length L. Finally, we derive the modular Hamiltonian directly, without using the modular flow, via the continuum limit of the expressions obtained in the discretized theory. In a specific coordinate system, the modular Hamiltonian assumes the form of a free field Hamiltonian on the Rindler wedge. Published by the American Physical Society 2024

Stacking correlation length in single-stranded DNA.

Base stacking is crucial in nucleic acid stabilization, from DNA duplex hybridization to single-stranded DNA (ssDNA) protein binding. While stacking energies are tiny in ssDNA, they are inextricably mixed with hydrogen bonding in DNA base pairing, making their measurement challenging. We conduct unzipping experiments with optical tweezers of short poly-purine (dA and alternating dG and dA) sequences of 20-40 bases. We introduce a helix-coil model of the stacking-unstacking transition that includes finite length effects and reproduces the force-extension curves. Fitting the model to the experimental data, we derive the stacking energy per base, finding the salt-independent value $\Delta G_0^{ST}=0.14(3)$ kcal/mol for poly-dA and $\Delta G_0^{ST}=0.07(3)$ kcal/mol for poly-dGdA. Stacking in these polymeric sequences is predominantly cooperative with a correlation length of ∼4 bases at zero force . The correlation length reaches a maximum of ∼10 and 5 bases at the stacking-unstacking transition force of ∼10 and 20 pN for poly-dA and poly-dGdA, respectively. The salt dependencies of the cooperativity parameter in ssDNA and the energy of DNA hybridization are in agreement, suggesting that double-helix stability is primarily due to stacking. Analysis of poly-rA and poly-rC RNA sequences shows a larger stacking stability but a lower stacking correlation length of ∼2 bases.

Derived functors and Hilbert Polynomials over regular local rings

Abstract Let $(A,\mathfrak{m})$ be a regular local ring of dimension $d \geq 1$ , I an $\mathfrak{m}$ -primary ideal. Let N be a nonzero finitely generated A-module. Consider the functions \begin{equation*} t^I(N, n) = \sum_{i = 0}^{d}\ell(\text{Tor}^A_i(N, A/I^n)) \ \text{and}\ e^I(N, n) = \sum_{i = 0}^{d}\ell(\text{Ext}_A^i(N, A/I^n)) \end{equation*} of polynomial type and let their degrees be $t^I(N) $ and $e^I(N)$ . We prove that $t^I(N) = e^I(N) = \max\{\dim N, d -1 \}$ . A crucial ingredient in the proof is that $D^b(A)_f$ , the bounded derived category of A with finite length cohomology, has no proper thick subcategories.

In search of a higher Bochner theorem

We initiate the study of a natural generalisation of the classical Bochner–Krall problem asking which linear ordinary differential operators possess sequences of eigenpolynomials satisfying linear recurrence relations of finite length; the classical case corresponds to the 3-term recurrence relations with real coefficients subject to some extra restrictions. We formulate a general conjecture and prove it in the first non-trivial case of operators of order 3.

Scattering of Acoustic Waves by an Inhomogeneous Elastic Cylindrical Shell of Finite Length in a Half-Space

An analytical solution to the problem of diffraction of a plane acoustic wave on a radially inhomogeneous thick-walled elastic cylindrical shell of finite length is obtained. The cylindrical shell is located in an acoustic half-space filled with an ideal liquid. The boundary of the half-space is an acoustically rigid or acoustically soft surface. The results of calculations of the acoustic field in the far zone are presented.

Quantum key distribution with unbounded pulse correlations

Abstract Typical security proofs of quantum key distribution (QKD) require that the emitted signals are independent and identically distributed. In practice, however, this assumption is not met because intrinsic device flaws inevitably introduce correlations between the emitted signals. Although analyses addressing this issue have been recently proposed, they only consider a restrictive scenario in which the correlations have a finite and known maximum length that is much smaller than the total number of emitted signals. While it is expected that the magnitude of the correlations decreases as the pulse separation increases, the assumption that this magnitude is exactly zero after a certain point does not seem to have any physical justification. Concerningly, this means that the available analyses cannot guarantee the security of current QKD implementations. Here, we solve this pressing problem by developing a rigorous framework that, when combined with existing results, can guarantee security against pulse correlations of unbounded length. Our framework is rather general and could be applied to other situations for which the existing analyses consider a scenario that differs slightly from the actual one.

Application of Surface Ultrasonic Waves for Detection of Contact Fatigue Cracks in the Rail Surface

The integrity and durability of rails plays a key role in train safety. However, the interaction of the rail head surface with the wheels of the rolling stock leads to high damage to these areas. The surface of the rail head is subject to the greatest damage, in which fatigue cracks develop on the surface with subsequent development. Timely detection of fatigue cracks in the rolling surface is possible using the ultrasonic technique using surface waves. To do this, the work investigated the sensitivity of a surface ultrasonic wave to obstacles in the surface of rails in the form of extended grooves, notches of finite length and known depth, as well as real cracks. The studies were carried out using the pulse-echo method at frequencies of 1.25, 1.8, 2.5, 4 MHz. Research has shown that surface waves are sensitive to surface fatigue cracks. The amplitude of the reflected pulses depends on the wavelength and the depth of crack propagation. The influence of the distance to fatigue cracks and the orientation angle of the transducer relative to the rail axis on the sensitivity of the surface wave to these defects was also studied. The results obtained allow us to conclude that the use of surface ultrasonic waves makes it possible to detect fatigue cracks in the crack surface. Thus, the use of the pulse-echo technique and surface wave converters makes it possible to detect contact fatigue cracks in the rail head, which can be used to eliminate them in a timely manner. The determination of their depth by amplitude is influenced by many factors, which are difficult to take into account in real conditions.

Scale dependence and non-isochoric effects on the thermohydrodynamics of rarefied Poiseuille flow

The plane Poiseuille flow of a rarefied gas in a finite length channel, driven by an axial pressure gradient, is analysed numerically to probe (i) the role of ‘dilatation’ ( $\varDelta ={\boldsymbol \nabla }\boldsymbol {\cdot }{\boldsymbol u}\neq 0$ ) on its thermohydrodynamics as well as to clarify (ii) the possible equivalence with its well-studied ‘dilatation-free’ or ‘isochoric’ ( ${\rm D}\rho /{\rm D}t=0$ ) counterpart driven by a constant acceleration. Focussing on the mass flow rate ${\mathcal {M}}({Kn})$ , which is an invariant quantity for both pressure-driven and acceleration-driven Poiseuille flows, it is shown that while ${\mathcal {M}}\sim \log {{Kn}}$ at ${Kn}\gg 1$ in the acceleration-driven case, the mass flow saturates to a constant value ${\mathcal {M}}\sim {{Kn}}^0$ at ${Kn}\gg 1$ in the pressure-driven case due to the finite length ( $L_x<\infty$ ) of the channel. The latter result agrees with prior theory and recent experiments, and holds irrespective of the magnitude of the axial pressure gradient ( $G_p$ ). The pressure-dilatation cooling ( $\varPhi _p=-p\varDelta <0$ ) is shown to be responsible for the absence of the bimodal shape of the temperature profile in the pressure-driven Poiseuille flow. The dilatation-driven reduction of the shear viscosity and the odd signs of two normal stress differences ( ${\mathcal {N}}_1$ and ${\mathcal {N}}_2$ ) in the pressure-driven flow in comparison with those in its acceleration-driven counterpart are explained from the Burnett-order constitutive relations for the stress tensor. While both ${\mathcal {N}}_1$ and ${\mathcal {N}}_2$ appear at the Burnett order $O({{Kn}}^2)$ in the acceleration-driven flow, the leading term in ${\mathcal N}_1$ scales as $(\mu/p)\varDelta$ due to the non-zero dilatation in the pressure-driven Poiseuille flow which confirms that the two flows are not equivalent even at the Navier–Stokes–Fourier order $O({{Kn}})$ . The heat-flow rate ( ${{\mathcal {Q}}_q}_x=\int q_x(x,y) \,{{\rm d} y}$ ) of the tangential heat flux is found to be negative (i.e. directed against the axial pressure gradient), in contrast to its positive asymptotic value (at ${Kn}\gg 1$ ) in the acceleration-driven flow. Similar to the scale-dependence of the mass flow rate, ${{\mathcal {Q}}_q}_x({{Kn}}, L_x)$ is found to saturate to a constant value at ${Kn}\gg 1$ in finite length channels. The double-well shape of the $q_x(y)$ -profile in the near-continuum limit agrees well with predictions from a generalized Fourier law. On the whole, the dilatation-driven signatures (such as the pressure-dilatation work and the ‘normal’ shear-rate differences) are shown to be the progenitor for the observed differences between the two flows with regard to (i) the hydrodynamic fields, (ii) the rheology and (iii) the flow-induced heat transfer.

IDENTIFICATION OF AN ARBITRARY MOVING AXISYMMETRIC LOAD ACTING ON A CYLINDRICAL SHELL

Various types of external non-stationary loads can act on various structural elements and cylindrical shells of finite length in particular distributed and concentrated, stationary and moving load. When using various external load identification methods, the type of external load is usually known. In prac- tice this is not always the case. The purpose of the study is to develop a method for identifying an arbitrary axisymmetric load acting on an elastic cy- lindrical shell of finite length, which can be used to identify a moving load. To model the non-stationary load of a cylindrical shell, a system of differ- ential equations of the refined Timoshenko theory for medium thickness shells was used. The solution to this system of differential equations is ob- tained by expanding the unknown functions into Fourier series and applying the integral Laplace transform. The solution of the inverse problem was obtained using the theory of integral equations and the Tikhonov regularization method. As a result of the study, a solution of the inverse problem of solid mechanics to identify an arbitrary axisymmetric non-stationary load was obtained. A numerical experiment was carried out to use the developed method to identify a moving non-stationary load acting on a simply supported cylindrical shell of medium thickness. The simulation results indicate a fairly accurate identification of both the change in time and the distribution along the shell of a non-stationary axisymmetric moving load. A method has been developed for identifying an external non-stationary load randomly distributed along a cylindrical shell, which makes it possible to identify a moving load without preliminary information about the type of this load. The developed method allows us to reproduce moving non-stationary loads, which are often encountered in practice, and expand it to other types of structural elements.

Reduction of turbulence by enhanced low-frequency zonal flow-like structures in HL-2A edge plasmas

Abstract A low-frequency zonal flow-like (LFZF-like) structure peaking at f ≈ 2.0 kHz has been observed in HL-2A ohmically heated deuterium plasmas using a combined Langmuir probe array. This time-varying potential structure, which has axisymmetric characteristics (n = 0) and a finite radial correlation length (less than 1 cm), was identified to be generated by the three-wave interaction in small-scale turbulence. The results illustrate that the amplitude of the LFZF-like structure dramatically increases with the influence of impurity ions, which is mainly due to the increased strength in the nonlinear energy transfer by the turbulence vortex symmetry-breaking process. Consequently, the enhanced LFZF-like structure has the ability to stabilize the local turbulence via the shearing decorrelation mechanism as demonstrated in this experiment. The observed results given here reveal the essential role played by the LFZF-like structure in the reduction of turbulence levels, which could advance our understanding of the multi-scale physics governing turbulence and the resulting transport in magnetically confined plasmas.

A spectral theorem for compact representations and non-unitary cusp forms

We show that a compact representation of either a semisimple Lie group or a totally disconnected group has a filtration with irreducible subquotients of finite multiplicity. In the Lie group case we show the stronger assertion, that it has an orthogonal decomposition into summands of finite lengths. This generalises and simplifies a number of more special spectral theorems in Deitmar and Monheim (Math Z 284(3–4):1199–1210, 2016, https://doi.org/10.1007/s00209-016-1695-9), Müller (Int Math Res Not 9(2):2068–2109, 2011), Venkov (in: Proceedings of the Steklov Institute of Mathematics, no. 4(153), ix+163 pp. (1983), 1982). We apply it to the case of cusp forms, thus settling the spectral theory for the space of non-unitary twisted cusp forms. We finally show that the space of cusp forms is complemented by the space of Eisenstein series.

Electroanalysis of bromate at riboflavin impregnated Ketjen black porous electrode via an autocatalytic cycle with finite length diffusion

The emergence of elevated concentrations of bromate, classified as a 2B type carcinogen, in drinking water following the treatment of water supplies with ozonation and other oxidation methods has underscored the importance of developing methods for reliable and rapid screening of drinking water. In this work, we report the use of electro activated riboflavin (RF) impregnated Ketjen black (KB) modified screen-printed electrode (elRF@KB-SPE) for the determination of bromate. The sensing surfaces were subjected to morphological and structural characterization using scanning electron microscopy and Raman spectroscopy. These analyses confirmed the successful modification of the electrode with an extended elRF@KB porous coating. Based on cyclic voltammetry (CV) and electrochemical impedance spectroscopy measurements, the response of elRF@KB-SPE towards bromate can be described through a dual mechanism involving i) an autocatalytic chemical/electrochemical process that predominantly occurs in the finite length pores of KB, and ii) an RF-mediated electrocatalytic process. Notably, the data demonstrated that the advanced electrocatalytic properties of the otherwise electrocatalytically inactive absorbed RF molecules are endowed after a simple electro activation process (three CV scans). Amperometric measurements at −0.05 V vs. Ag/AgCl revealed a linear relationship between the current response and the concentration of bromates over the concentration range 0.50−18.0 μM, achieving a limit of detection (3σ/slope) of 0.04 μM. The interference effect of various electroactive species was investigated demonstrating that elRF@KB-SPE is highly selective to bromate. The analytical utility of the method was evaluated in spiked tap water samples. Recoveries were 91.2 − 106.4 %.

Gradient flows of generalized relative entropy and functional inequalities on graphs

We derive the discrete forms of the porous medium equation and fast diffusion equation with drift terms on a finite graph as gradient flows of the m-relative entropy, with the global existence of solutions. We show the Łojasiewicz inequality to prove the convergence of solutions and some important functional inequalities. More specifically, the Łojasiewicz inequality holds near the stationary solution with exponent 12, which leads to the exactly exponential convergence rate with a finite trajectory length. It can also be applied to show the Talagrand-type inequality and the log-Sobolev-type inequality directly.