Dissipative Operators Research Articles

Real-Valued Spectral Shift Functions for Contractions and Dissipative Operators

Real-Valued Spectral Shift Functions for Contractions and Dissipative Operators

Phase-locked high-brightness interband cascade laser array with multimode interferometer couplers.

We report on the design, fabrication, and characterization of an interband cascade laser (ICL) array incorporating multimode interference (MMI) couplers for phase-locking emitters. The ICL array, emitting at 3-4 µm, was developed to address the challenges of heat dissipation and in-phase operation in mid-infrared lasers. The array features a 7.5-µm-wide ridge design for fundamental transverse mode propagation and employs MMI structures to realize an in-phase operation of multiple emitters. The far-field patterns, characterized by periodic and symmetrical interference fringes, confirm the coherent operation of the array and the efficacy of MMI couplers in achieving phase-locking. The single-ridge side of the array exhibits a single-lobe far-field profile, with higher-order transverse modes effectively suppressed, showcasing a nearly diffraction-limited beam quality (M2 ≈ 1.31) at high output powers (390 mW from the 1 × 4 array). The robust performance and scalable design of the ICL array, validated by experimental results and theoretical simulations, indicate its potential for high-power mid-infrared applications and as an optical phased array.

FPGA implementation of sobel edge detection algorithm

Sobel Edge detection algorithm is used to extract the edges (region of maximum variation) from an image. It is based on the concept that the edges of an image contains maximum information whose computation depends on multipliers and square root. As multipliers consume more logic, a modified sobel edge detection algorithm which does not employ multipliers and square root function is proposed. A mathematical model of the proposed sobel edge algorithm was first developed and MATLAB was used to verify the model. On comparing with the original model, the proposed model has a SSIM of 96.43%. To analyse the hardware complexity, Verilog model of the modified sobel edge detection algorithm was developed using Quartus II. The chosen evaluation board is Cylone III FPGA EP3C120F780. The performance metrics such has Logic Elements utilization, Power dissipation and Maximum Operating Frequency were obtained. Open-Source toolchain (Yosys, OpenVPR, and Google Skywater 130nm PDK) was used to obtain the RTL Netlist and Synthesis reports. Verilog Modules for the Camera (CMOS OV7670) interface and FIFO Buffer were synthesized. The modified algorithm was integrated with them. An HSMC (HSMB) breakout board was connected to the FPGA Development board to increase the number of I/O ports. Thus in real time, the proposed modified Sobel Edge detection system can be used as a pre-processor to reduce the amount of computations and power consumption.

The degradation mechanism of high power S-band AlGaN/GaN HEMTs under high temperature

Abstract The radio frequency (RF) degradation mechanism under high temperature is proposed in this paper. The saturated output power (P sat), power added efficiency (PAE) and gain of the HEMTs deteriorate significantly under high power dissipation operation conditions, which contribute to an elevation of the junction temperature (T j). The high T j causes a negative shift of the threshold voltage and a reduction of electron mobility, as well as enhancing the trapping effects of the devices. As a result, the P sat and PAE of the high-power HEMTs (P sat = 152 W) decreased by 0.76 dB and 8.3%, respectively, which is more severe compared to low-power HEMTs (P sat = 20 W), where P sat and PAE reduced by 0.16 dB and 3.1% when the operation temperature was increased from 25 °C to 150 °C. By increasing the gate-to-drain distance from 1.625 to 2.175 μm, drain current stability and high temperature reverse bias reliability are improved. The results demonstrate that the trapping effect can be suppressed by decreasing the electric field under the gate; therefore, the RF output characteristics are boosted by 1.2 W, 1 dB and 2.6% for a device with gate width of 11.52 mm.

On a class of third-order dissipative differential operator with distributional potentials

This paper aims to investigate a class of boundary value problems generated by a third-order differential equation with distributional potentials and some dissipative boundary conditions. We first construct an operator related to the problem, then prove that the operator is dissipative and show some eigenvalue properties of this operator. Further using Krein’s theorem we prove the completeness theorems of the operator.

Allosteric stem-loop multicomponent DNAzyme: A versatile stimuli-responsive switch for nanotweezer operations and multifunctional biosensing

DNAzyme, possessing the capabilities of both encoding information and catalyzing chemical reactions, holds significant value in advancing the widespread application of dynamic nanotechnology. Particularly, multicomponent DNAzyme (MNAzyme) has attracted considerable attention due to its remarkable versatility and practicality. However, the inevitable signal leakage of MNAzyme constrains the applications in complex nucleic acid reaction networks. Herein, we present an allosteric stem-loop multicomponent DNAzyme (ASLM) that incorporates a blocking domain to mitigate leakage resulting from secondary structures of MNAzyme while simultaneously stabilizing binding affinity during complex formation. As the sequence of blocking domain serves as an extension independent of the functional region in MNAzyme, this modulation structure can be flexibly assembled with conformation-switchable nanotweezer to form a time-responsive molecular device, enabling dynamic and dissipative operations through nucleic acid and RNase H. More importantly, the ASLM structure can be directly incorporated into DNA catalytic circuits for detecting multiple biomolecules (nucleic acid, protease, and small molecule). With its scalability and versatility, this effector-activated allosteric regulator will serve as a universal stimuli-responsive switch in complex and hierarchical DNA networks, expanding the scope of DNA nanotechnology in molecular device construction and diagnostic biosensing.

Noise-Aware Variational Eigensolvers: A Dissipative Route for Lattice Gauge Theories

We propose a novel variational ansatz for the ground-state preparation of the Z2 lattice gauge theory (LGT) in quantum simulators. It combines dissipative and unitary operations in a completely deterministic scheme with a circuit depth that does not scale with the size of the considered lattice. We find that, with very few variational parameters, the ansatz can achieve >99% precision in energy in both the confined and deconfined phase of the Z2 LGT. We benchmark our proposal against the unitary Hamiltonian variational ansatz showing a reduction in the required number of variational layers to achieve a target precision. After performing a finite-size scaling analysis, we show that our dissipative variational ansatz can predict accurate critical exponents without requiring a number of layers that scales with the system size, which is the standard situation for unitary ansätze. Furthermore, we investigate the performance of this variational eigensolver subject to circuit-level noise, determining variational error thresholds that fix the error rate below which it would be beneficial to increase the number of layers. In light of these quantities and for typical gate errors p in current quantum processors, we provide a detailed assessment of the prospects of our scheme to explore the Z2 LGT on near-term devices. Published by the American Physical Society 2024

Stabilization of approximate GHZ state with quasi-local couplings

We propose a reservoir design, composed of fixed dissipation operators acting each on few local subsystems, to stabilize an approximate GHZ state on n qubits. The main idea is to work out how a previously proposed sequence of two stabilization steps can be applied instead in appropriate (probabilistic) superposition. We examine various alternatives to synchronize the superposition using local couplings only, thanks to a chain of ‘clock’ ancillas or to additional levels on the data subsystems. The practical value of these alternatives depends on experimental constraints. They all feature a design tradeoff between approximate stabilization fidelity and protection against perturbations. These proposals illustrate how simple autonomous automata can be implemented in quantum reservoir engineering to replace sequential state preparation procedures. Encoding automaton actions via additional data levels only, appears particularly efficient in this context. Our analysis method, reducing the Lindblad master equation to a Markov chain on virtual output signals, may be of independent interest.

Quaternionic spherical sectorial and dissipative operators via S-resolvent kernels

In this paper, we treat the quaternionic functional calculus for right linear quaternionic operators whose components do not necessarily commute and develop a theory of quaternionic non-negative operator, spherical sectorial operator and dissipative operator via S-resolvent kernels in quaternionic locally convex spaces (short for q.l.c.s.). The notions of quaternionic non-negative operators and quaternionic (m-)dissipative operators are introduced via S-resolvent operators and H-valued inner product on Hilbert H-bimodule. By choosing the suitable spherical sector, the spherical sectorial operator is introduced to establish the relationship with the quaternionic non-negative operator. It is crucial to note that the quaternionic operators we consider do not necessarily commute.

ЗАВИСИМОСТЬ УДАРНО-АБРАЗИВНОЙ ИЗНОСОСТОЙКОСТИ СЛОИСТЫХ КОМПОЗИЦИОННЫХ МАТЕРИАЛОВ ОТ ДЕФОРМИРОВАННОГО СОСТОЯНИЯ РЕЗИНОВОЙ ПРОСЛОЙКИ

It is established that during impact-abrasive testing of layered composite materials made of 40X, P40X and rubber steels, the amount of absorbed energy depends not only on the physical and mechanical properties of the materials, but also on the geometric parameters of the elastic-dissipative layer. The impact and abrasive wear resistance of layered composite materials is 1.5-2 times greater than when using wear-resistant steels. A method for determining the deformed state of a rubber elastic-dissipative substrate of layered composite materials is proposed and the effect of stress and strain intensity on the dissipation operation when testing structural steels for impact and abrasive wear resistance is shown. The influence of the heat treatment regime of the 40X steel working layer on the impact and abrasive wear of the SCM is investigated. The mechanism of destruction of the surface of sintered structural steels during impact and abrasive wear has been revealed.

Quantum simulation of dissipation for Maxwell equations in dispersive media

The dissipative character of an electromagnetic medium breaks the unitary evolution structure that is present in lossless, dispersive optical media. In dispersive media, dissipation appears in the Schrödinger representation of classical Maxwell equations as a sparse diagonal operator occupying an r-dimensional subspace. A first order Suzuki-Trotter approximation for the evolution operator enables us to isolate the non-unitary operators (associated with dissipation) from the unitary operators (associated with lossless media). The unitary operators can be implemented through qubit lattice algorithm (QLA) on n qubits, based on the discretization and the dimensionality of the pertinent fields. However, the non-unitary-dissipative part poses a challenge both physically and computationally on how it should be implemented on a quantum computer. In this paper, two probabilistic dilation algorithms are considered for handling the dissipative operators. The first algorithm is based on treating the classical dissipation as a linear amplitude damping-type completely positive trace preserving (CPTP) quantum channel where an unspecified environment interacts with the system of interest and produces the non-unitary evolution. Therefore, the combined system-environment is now closed, and must undergo unitary evolution in the dilated space. The unspecified environment can be modeled by just one ancillary qubit, resulting in an implementation scaling of O(2n−1n2) elementary gates for the total system-environment unitary evolution operator. The second algorithm approximates the non-unitary operators by the Linear Combination of Unitaries (LCU). On exploiting the diagonal structure of the dissipation, we obtain an optimized representation of the non-unitary part, which requires O(2n) elementary gates. Applying the LCU method for a simple dielectric medium with homogeneous dissipation rate, the implementation scaling can be further reduced into O[poly(n)] basic gates. For the particular case of weak dissipation we show that our proposed post-selective dilation algorithms can efficiently delve into the transient evolution dynamics of dissipative systems by calculating the respective implementation circuit depth. A connection of our results with the non-linear-in-normalization-only (NINO) quantum channels is also presented.

Low power nanoscale S-FED based single ended sense amplifier applied in integrate and fire neuron circuit

Current advancements in neuromorphic computing systems are focused on decreasing power consumption and enriching computational functions. Correspondingly, state-of-the-art system-on-chip developers are encouraged to design nanoscale devices with minimum power dissipation and high-speed operation. This paper deals with designing a sense amplifier based on side-contacted field-effect diodes to reduce the power-delay product (PDP) and the noise susceptibility, as critical factors in neuron circuits. Our findings reveal that both static and dynamic power consumption of the S-FED-based sense amplifier, equal to 1.86 μW and 1.92 fW/GHz, are × 243.03 and × 332.83 lower than those of the conventional CMOS counterpart, respectively. While the sense-amplifier circuit based on CMOS technology undergoes an output voltage deviation of 170.97 mV, the proposed S-FED-based one enjoys a minor output deviation of 27.31 mV. Meanwhile, the superior HIGH-level and LOW-level noise margins of the S-FED-based sense amplifier to the CMOS counterparts (∆NMH = 70 mV and ∆NML = 120 mV), respectively, can ensure the system-level operation stability of the former one. Subsequent to the attainment of an area-efficient, low-power, and high-speed S-FED-based sense amplifier (PDP = 187.75 × 10–18 W s) as a fundamental building block, devising an innovative integrate-and-fire neuron circuit based on S-FED paves the way to realize a new generation of neuromorphic architectures. To shed light on this context, an S-FED-based integrate-and-fire neuron circuit is designed and analyzed utilizing a sense amplifier and feedback loop to enhance spiking voltage and subsequent noise immunity in addition to an about fourfold increase in firing frequency compared to CMOS-based ones.

The spectral form of the functional model for maximally dissipative operators: A Lagrange identity approach

This paper is a contribution to the theory of functional models. In particular, it develops the so-called spectral form of the functional model where the selfadjoint dilation of the operator is represented as the operator of multiplication by an independent variable in some auxiliary vector-valued function space. With the help of a Lagrange identity, in the present version the relationship between this auxiliary space and the original Hilbert space will be explicit. A simple example is provided.

Advancing Eddy Parameterizations: Dynamic Energy Backscatter and the Role of Subgrid Energy Advection and Stochastic Forcing

AbstractViscosity in the momentum equation is needed for numerical stability, as well as to arrest the direct cascade of enstrophy at grid scales. However, a viscous momentum closure tends to over‐dissipate eddy kinetic energy. To return excessively dissipated energy to the system, the viscous closure is equipped with what is called dynamic kinetic energy backscatter. The amplitude of backscatter is based on the amount of unresolved kinetic energy (UKE). This energy is tracked through space and time via a prognostic equation. Our study proposes to add advection of UKE by the resolved flow to that equation to explicitly consider the effects of nonlocality on the subgrid energy budget. UKE can consequently be advected by the resolved flow before it is reinjected via backscatter. Furthermore, we suggest incorporating a stochastic element into the UKE equation to account for missing small‐scale variability, which is not present in the purely deterministic approach. The implementations are tested on two intermediate complexity setups of the global ocean model FESOM2: an idealized channel setup and a double‐gyre setup. The impacts of these additional terms are analyzed, highlighting increased eddy activity and improved flow characteristics when advection and carefully tuned, stochastic sources are incorporated into the UKE budget. Additionally, we provide diagnostics to gain further insights into the effects of scale separation between the viscous dissipation operator and the backscatter operator responsible for the energy injection.

Second‐order trace formulas

AbstractKoplienko [Sib. Mat. Zh. 25 (1984), 62–71; English transl. in Siberian Math. J. 25 (1984), 735–743] found a trace formula for perturbations of self‐adjoint operators by operators of Hilbert–Schmidt class . Later, Neidhardt introduced a similar formula in the case of pairs of unitaries via multiplicative path in [Math. Nachr. 138 (1988), 7–25]. In 2012, Potapov and Sukochev [Comm. Math. Phys. 309 (2012), no. 3, 693–702] obtained a trace formula like the Koplienko trace formula for pairs of contractions by answering an open question posed by Gesztesy, Pushnitski, and Simon [Zh. Mat. Fiz. Anal. Geom. 4 (2008), no. 1, 63–107, 202; Open Question 11.2]. In this paper, we supply a new proof of the Koplienko trace formula in the case of pairs of contractions , where the initial operator is normal, via linear path by reducing the problem to a finite‐dimensional one as in the proof of Krein's trace formula by Voiculescu [Oper. Theory Adv. Appl. 24 (1987) 329–332] and Sinha and Mohapatra [Proc. Indian Acad. Sci. Math. Sci. 104 (1994), no. 4, 819–853] and [Integral Equations Operator Theory 24 (1996), no. 3, 285–297]. Consequently, we obtain the Koplienko trace formula for a class of pairs of contractions using the Schäffer matrix unitary dilation. Moreover, we also obtain the Koplienko trace formula for a pair of self‐adjoint operators and maximal dissipative operators using the Cayley transform. At the end, we extend the Koplienko–Neidhardt trace formula for a class of pairs of contractions via multiplicative path using the finite‐dimensional approximation method.

Conservative correction procedures utilizing artificial dissipation operators

The conservative correction procedure of Abgrall [1] is studied from the perspective of filter-based artificial dissipation methods, which motivates the ability to tailor the behavior of the method in both physical and spectral space. Compared to the original formulation, employing diffusion operators biases the correction towards smaller scales and better controls discretization errors when seeking to enforce auxiliary conservation relations. Effective entropy-stable regularization of sharp gradients is furthermore shown to be attainable. Calculations of the Sod shock tube problem as governed by the one-dimensional Euler equations are used to highlight the utility of considering alternate filters within the original correction framework, where the notion of entropy conservation/stability is leveraged for improving non-linear scheme robustness.

Inverse problems for general parabolic systems and application to Ornstein-Uhlenbeck equation

We investigate the link between inverse problems and final state observability for a general class of parabolic systems. We generalize a stability result for initial data known for the case of self-adjoint dissipative operators. More precisely, we consider a system governed by an analytic semigroup. Under the assumption of final state observability, we prove a logarithmic stability estimate depending on the analyticity angle of the semigroup. This is done by using a general logarithmic convexity result. The abstract result is illustrated by considering the Ornstein–Uhlenbeck equation.

ON A CLASS OF NONSTATIONARY CURVES IN HILBERT SPACE

Stationary random processes have been studied quite well over recent years starting with the works of A. N. Kolmogorov. The possibility of building nonstationary random process correlation theory was considered in the monographs by M. S. Livshits, A. A. Yantsevich, V. A. Zolotarev and others. Some classes of nonstationary curves were investigated by V. E. Katsnelson. In this paper nonstationary random processes are represented as curves in Hilbert space which "slightly deviate" from random processes with the correlation function of special kind. The infinitesimal correlation function has been introduced; in essence, this function characterizes the deviation from the correlation process with the given correlation function. The paper discusses the cases of nonstationary random processes, the operator of which has one‑dimensional imaginary component. Cases of a dissipative operator with descrete spectrum are also considered in this work. It is shown that the nonstationarity of the random process is closely related to the deviation of the operator from its conjugated operator. Using the triangle and universal models of non‑self‑ajoint operators it is possible to obtain the representation for the correlation function in the case of nonstationary process which replaces the Bochner – Khinchin representation for stationary random processes. The expresson for the infinitesimal correlation function was obtained for different cases of operator spectrum: for the descrete spectrum placed in the upper half‑plane and for the contrast‑free spectrum at zero. In the case of dissipative operator with descrete spectrum the infinitesimal function can be found in terms of special lambda function. For Lebesque spaces of compex‑valued squared integratable functions the expresson of infinitesimal function was found in terms of special zero order modified Bessel function. It was shown that a similar approach can be applied for the evolutionarily represented sequences in Hilbert spaces.

Fabrication of amorphous nanoporous ZrO2/SiO2 aerogel enabling nonlinear optical properties

Amorphous oxides have unique physicochemical properties with extensive opto-electronic applications such as the thin-film transistor, light-emitting diode backplanes, and supercontinuum generation. In this contribution, we synthesize the amorphous ZrO2/SiO2 nanoporous aerogel with high structural integrity. With the femtosecond excitation laser at 800–1,064 nm, the broadband second harmonic generation is observed. The nonlinear optical properties of the as-prepared ZrO2/SiO2 aerogel are investigated at 1.0 μm and 1.5 μm for the first time. Subsequently, the amorphous ZrO2/SiO2 saturable absorber is originally applied in the Yb-doped and Er-doped fiber lasers to realize the mode-locking operations. In the Yb-doped fiber laser, the dissipative soliton resonance mode-locking operation is demonstrated with the largest pulse duration of 22 ns at a repetition rate of 7.8 MHz and a high signal-to-noise ratio of 64 dB. In the Er-doped fiber laser, a conventional soliton mod-locking regime is observed with an ultrashort pulse width of 960 fs, a repetition frequency of 6.55 MHz, and a time-bandwidth production of 0.347. Our work shows the good ability of the ZrO2/SiO2 aerogel in generating ultrafast pulses and extends the saturable absorber into the amorphous material realm.

On the dissipative conformable fractional singular Sturm-Liouville operator

In this study, we consider the dissipative conformable fractional singular Sturm-Liouville operator. Using Krein’s theorem, we proved a completeness theorem of the system of eigenvectors and associated vectors (or root vectors) of this operator.