Envelope Function Research Articles

Efficient ground motion conditional simulation method for nonstationary multivariate processes based on Kriging method and orthogonal discrete wavelet transform

Conditional ground motion simulation methods preserve more physical features of actual ground motions than unconditional methods by including actual earthquake records. This study developed a conditional multivariate nonstationary process simulation method based on the combination of the Kriging method and the orthogonal discrete wavelet transform (DWT). The Kriging method is used to ensure the spectral coherences between simulations as prescribed and is extended to nonstationary random processes by applying it to the average power spectral density (APSD) of ground motions. The ground motions' DWT wavelet coefficients are assumed to be uniformly time-modulated stationary processes whose time-modulating functions are the envelopes of the seed record's wavelet coefficients. The APSD and the envelopes together outline the energy distribution of the random field in both the time and frequency domain features. The power spectral densities of the stationary random processes' DWT coefficients are derived and used to estimate the APSD of the ground motions' DWT coefficients. Approaches to realizing uniformly time-modulated stationary processes with prescribed APSDs and envelope functions are developed. Monte Carlo simulations were conducted to verify the new method's effectiveness. Results show that both the target APSD and coherences can be well approximated, but precise approximation cannot always be realized in the high-frequency range. The new approach possesses a simple structure, involves simple algorithms, and applies to time-independent spectral coherence models, making it an efficient method compared with existing methods.

Combined hydrodynamic and control analysis on optimal kinematic parameters for bio-inspired autonomous underwater vehicle manoeuvring

To investigate the manoeuvring performance of a body-caudal fin robot fish, a numerical framework combining computational fluid dynamics and multi-body dynamics with a closed-loop control algorithm was established in this study. Within this framework, we modelled a body-caudal fin swimmer as a multi-body system with the shape of a NACA0012 hydrofoil. The manoeuvring performance was investigated by using different curvature magnitudes and distributions along the centre line (the curvature is defined by means of a curvature envelop function as part of the general body undulation equation). To characterize the turning performance, a new parameter named cost of manoeuvring (CoM) is proposed. This parameter provides a combined assessment of the turning radius, linear and angular velocity components, and power. It is found that when the body curvature is introduced, the swimmer switches from straight-line swimming to quasi-steady turning at a constant speed. Further investigations were conducted to study contributions of head and tail deformations on the turning performance by comparing predominantly head and tail curved envelopes. Results reveal that a tail-dominated envelope improves performance, whereas a head-dominated envelope has a negative effect.

Spin-phonon coupling in a double-stranded model of DNA.

We address the electron-spin-phonon coupling in an effective model Hamiltonian for DNA to assess its role in spin transfer involved in the Chiral-Induced Spin Selectivity (CISS) effect. The envelope function approach is used to describe semiclassical electron transfer in a tight-binding model of DNA at half filling in the presence of intrinsic spin-orbit coupling. Spin-phonon coupling arises from the orbital-configuration dependence of the spin-orbit interaction. We find spin-phonon coupling only for the acoustic modes, while the optical modes exhibit electron-phonon interaction without coupling to spin. We derive an effective Hamiltonian whose eigenstates carry spin currents that are protected by spin-inactive stretching optical modes. As optical phonons interact more strongly than acoustic phonons, side buckling and tilting optical base modes will be more strongly associated with decoherence, which allows for the two terminal spin filtering effects found in CISS.

A previously uncharacterized divisome-associated lipoprotein, DalA, is needed for normal cell division in Rhodobacterales.

The bacterial cell envelope is a key subcellular compartment with important roles in antibiotic resistance, nutrient acquisition, and cell morphology. We seek to gain a better understanding of proteins that contribute to the function of the cell envelope in Alphaproteobacteria. Using Rhodobacter sphaeroides, we show that a previously uncharacterized protein, RSP_1200, is an outer membrane (OM) lipoprotein that non-covalently binds peptidoglycan (PG). Using a fluorescently tagged version of this protein, we find that RSP_1200 undergoes a dynamic repositioning during the cell cycle and is enriched at the septum during cell division. We show that the position of RSP_1200 mirrors the location of FtsZ rings, leading us to propose that RSP_1200 is a newly identified component of the R. sphaeroides' divisome. Additional support for this hypothesis includes the co-precipitation of RSP_1200 with FtsZ, the Pal protein, and several predicted PG L,D-transpeptidases. We also find that a ∆RSP_1200 mutation leads to defects in cell division, sensitivity to PG-active antibiotics, and results in the formation of OM protrusions at the septum during cell division. Based on these results, we propose to name RSP_1200 DalA (for division-associated lipoprotein A) and postulate that DalA serves as a scaffold to position or modulate the activity of PG transpeptidases that are needed to form envelope invaginations during cell division. We find that DalA homologs are present in members of the Rhodobacterales order within Alphaproteobacteria. Therefore, we propose that further analysis of this and related proteins will increase our understanding of the macromolecular machinery and proteins that participate in cell division in Gram-negative bacteria. IMPORTANCE Multi-protein complexes of the bacterial cell envelope orchestrate key processes like growth, division, biofilm formation, antimicrobial resistance, and production of valuable compounds. The subunits of these protein complexes are well studied in some bacteria, and differences in their composition and function are linked to variations in cell envelope composition, shape, and proliferation. However, some envelope protein complex subunits have no known homologs across the bacterial phylogeny. We find that Rhodobacter sphaeroides RSP_1200 is a newly identified lipoprotein (DalA) and that loss of this protein causes defects in cell division and changes the sensitivity to compounds, affecting cell envelope synthesis and function. We find that DalA forms a complex with proteins needed for cell division, binds the cell envelope polymer peptidoglycan, and colocalizes with enzymes involved in the assembly of this macromolecule. The analysis of DalA provides new information on the cell division machinery in this and possibly other Alphaproteobacteria.

A method for fast calculating the electronic states in 2D quantum structures based on AIIIBV nitrides

The paper presents a method for fast calculating the electronic states in two-dimensional quantum structures based on AIIIBV nitrides. The method is based on the representation of electronic states in the form of a linear combination of bulk wave functions of materials, from which quantum structures are made. The parameters and criteria for the selection of bulk wave functions that provides fast convergence of the numerical procedures for calculating the eigenvalues of the quantum Hamiltonian have been considered. The results of the calculations have been given both for one polar InGaN/GaN quantum well and for a system of several quantum wells. Being based on the full band structure of AIIIBV nitrides with a wurtzite-type crystal lattice, the proposed approach takes into account the states far from the center of the Brillouin zone, while preserving the computational efficiency of traditional methods of envelope function in approximating the effective mass.

Advancing carrier transport models for InAs/GaSb type-II superlattice mid-wavelength infrared photodetectors

To provide the best possible performance, modern infrared photodetector designs necessitate extremely precise modeling of the superlattice absorber region. We advance the Rode's method for the Boltzmann transport equation in conjunction with the $\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{p}$ band structure and the envelope function approximation for a detailed computation of the carrier mobility and conductivity of layered type-II superlattice structures, using which we unravel two crucial insights. First, the significance of both elastic- and inelastic-scattering mechanisms, particularly the influence of the interface roughness and polar optical phonon scattering mechanisms in technologically relevant superlattice structures. Second, that the structure-specific Hall mobility and Hall scattering factor reveal that temperature and carrier concentrations significantly affect the Hall scattering factor, which deviates significantly from unity even for small magnetic fields. This reinforces the caution that should be exercised when employing the Hall scattering factor in experimental estimations of drift mobilities and carrier concentrations. Our research hence offers a comprehensive microscopic understanding of carrier dynamics in such technologically relevant superlattices. Our models also provide highly accurate and precise transport parameters beyond the relaxation-time approximation and thereby paving the way to develop physics-based device modules for mid-wavelength infrared photodetectors.

Improved seismic envelope full-waveform inversion

The envelope full-waveform-inversion method has the potential to greatly improve the estimation of long-wavelength components of subsurface seismic velocity, and simultaneously avoid or reduce inversion cycle skipping, by fitting the envelopes of observed and modeled seismic waveform data, rather than fitting the waveforms directly. However, some issues such as instability of the adjoint source and gradient term make it challenging to use in practice for many seismology applications. We examine the envelope inversion (EI) methods with different envelope function exponent p values from an adjoint source analysis perspective and reveal the adverse effects of the adjoint source weighting term on the data residuals, which degrade or mute important inversion gradient contributions, especially from reflected or scattered phases in the seismic data. We develop the theory and implementation for an improved EI (IEI) method by adding an upshift constant (zero-frequency direct current [DC] bias term) to the seismic waveform data before calculating the envelope functions, which can help solve the instability issue of the adjoint source of the EI with power value p = 1. Our approach is different than the common but simple addition of a “water-level” damping constant in the denominator of the data residual-weighting term, which can ignore important information contained in the data residuals. Our IEI method becomes a mixed envelope-waveform objective function, which introduces a new adjoint source weighting term that can preserve all of the information contained in the data residuals, by boosting the low-amplitude phases of the seismic waveform, while simultaneously preserving the correct envelope (energy shape) of the high-amplitude phases. We test the performance of all three methods in detail using the highly realistic SEAM4D model and data and find the advantages of our new IEI method compared with conventional EI and full-waveform inversion methods.

The role of CenKR in the coordination of Rhodobacter sphaeroides cell elongation and division.

Cell elongation and division are essential aspects of the bacterial life cycle that must be coordinated for viability and replication. The impact of misregulation of these processes is not well understood as these systems are often not amenable to traditional genetic manipulation. Recently, we reported on the CenKR two-component system (TCS) in the Gram-negative bacterium Rhodobacter sphaeroides that is genetically tractable, widely conserved in α-proteobacteria, and directly regulates the expression of components crucial for cell elongation and division, including genes encoding subunit of the Tol-Pal complex. In this work, we show that overexpression of cenK results in cell filamentation and chaining. Using cryo-electron microscopy (cryo-EM) and cryo-electron tomography (cryo-ET), we generated high-resolution two-dimensional (2D) images and three-dimensional (3D) volumes of the cell envelope and division septum of wild-type cells and a cenK overexpression strain finding that these morphological changes stem from defects in outer membrane (OM) and peptidoglycan (PG) constriction. By monitoring the localization of Pal, PG biosynthesis, and the bacterial cytoskeletal proteins MreB and FtsZ, we developed a model for how increased CenKR activity leads to changes in cell elongation and division. This model predicts that increased CenKR activity decreases the mobility of Pal, delaying OM constriction, and ultimately disrupting the midcell positioning of MreB and FtsZ and interfering with the spatial regulation of PG synthesis and remodeling.IMPORTANCEBy coordinating cell elongation and division, bacteria maintain their shape, support critical envelope functions, and orchestrate division. Regulatory and assembly systems have been implicated in these processes in some well-studied Gram-negative bacteria. However, we lack information on these processes and their conservation across the bacterial phylogeny. In R. sphaeroides and other α-proteobacteria, CenKR is an essential two-component system (TCS) that regulates the expression of genes known or predicted to function in cell envelope biosynthesis, elongation, and/or division. Here, we leverage unique features of CenKR to understand how increasing its activity impacts cell elongation/division and use antibiotics to identify how modulating the activity of this TCS leads to changes in cell morphology. Our results provide new insight into how CenKR activity controls the structure and function of the bacterial envelope, the localization of cell elongation and division machinery, and cellular processes in organisms with importance in health, host-microbe interactions, and biotechnology.

Atomic scale localization of Kohn–Sham wavefunction at SiO2/4H–SiC interface under electric field, deviating from envelope function by effective mass approximation

To clarify the cause of the low channel conductivity at the SiO2/4H–SiC interface, the wavefunction at the SiC conduction band minimum was calculated using density functional theory under an applied electric field. We found that the wavefunction for a 4H–SiC (0001) slab tends to be localized at the cubic site closest to the interface. Importantly, because the conduction electrons are distributed closer to the interface (<5 Å) than expected from the effective mass approximation (EMA), they are more frequently scattered by interface defects. This is expected to be the reason why the channel conductivity for the (0001) face is particularly low compared with that for other faces, such as (112¯0). The breakdown of the EMA for the (0001) interface is related to the long structural periodicity along the [0001] direction in 4H–SiC crystals.

P88 CARDIAC INVOLVEMENT IN FOUR NOVEL NESPRIN–1 GENETIC VARIANTS

Abstract Background Nesprins are proteins encoded by synaptic nuclear envelope (SYNE)–1 and –2 genes, that concur, together with lamin A/C and emerin, to nuclear envelope function. Almost 140 disease–related variants were identified, mostly resulting in central nervous system and skeletal muscle disorders; only a few were associated with cardiomyopathy, mostly with relatively late onset and no specific arrhythmic profile. Purpose To describe cardiac involvement in 4 novel SYNE1 variants. Methods 189 patients underwent genetic analysis in our center (2019–2022) for suspected genetic cardiomyopathy (TruSight Cardio panel, Illumina); 72% had a positive genetic report, including pathogenic/likely pathogenic or variants of undetermined significance (VUS). Among them, 4 (2%) males presented with a sole SYNE1 heterozygous VUS (figure). Results None had muscle or neurological involvement. Patients 1 and 2 had negative family history (FH) and underwent pacemaker implantation under 50 years old for high–grade atrioventricular block (AVB). Transthoracic echocardiography (TTE) and cardiac magnetic resonance (CMR) showed a mildly impaired left ventricular ejection fraction (LVEF) without oedema or fibrosis. During 3 and 6 years of follow–up (FU), several non–sustained episodes of fast ventricular tachycardia (VT) and of atrial tachycardia occurred. Patient 3 is a 55–year–old man with negative FH and a left bundle branch block (LBBB) known since the age of 45.In 2021, while playing football, he suffered a cardiac arrest due to ventricular fibrillation treated with two DC shocks; ECG showed first–degree AVB and LBBB; TTE and CMR revealed a mildly impaired LVEF without oedema or fibrosis. A subcutaneous ICD was implanted, which recorded 3 atrial fibrillation episodes in a 1–year FU. Patient 4 is a 49–year–old man, with FH of sudden death (mother’s brothers at 40 and 50 years) and of polycystic kidney, who underwent renal transplant. He had left anterior fascicular block and suffered 2 episodes of sustained monomorphic VT from the right ventricle treated with ablation. TTE and CMR showed mild LV concentric hypertrophy with normal LVEF, segmental abnormalities in right and LV strain, no fibrosis. Conclusions SYNE1 genetic variants were found to be associated with a complex cardiac phenotype, including early onset conduction disease, atrial and ventricular arrhythmias, and cardiomyopathy. Further studies are needed to determine the functional significance of the variants.

Inequalities for integral operators in Hölder–Morrey spaces on differential forms

The Hölder–Morrey spaces Lambda _{kappa}^{p,tau}(Omega ,wedge ^{l}) are proposed in this paper. The imbedding inequalities for homotopy operator are derived in Hölder–Morrey spaces on differential forms. The Hölder continuity for Riesz potential with envelope function is deduced. As application, some composite theorems, which are associated with conjugate A-harmonic equations on differential forms, are given.

Temporal coherence envelope function of field emission in electron microscopy

Imaging in electron microscopy is adversely affected by partial electron spatial and temporal coherence. Temporal coherence has been treated theoretically in the past using the method pioneered fifty years ago by Hanßen and Trepte, who assumed a Gaussian energy distribution. However, state-of-the-art instruments employ field emission (FE) sources that emit electrons with a non-Gaussian energy distribution. We have updated the treatment of temporal coherence to describe the effects of an arbitrary energy distribution on image formation. The updated approach is implemented in Fourier optics simulations to explore the effect of FE on image formation in conventional, non-aberration-corrected (NAC) and aberration-corrected (AC) low energy electron microscopy. It is found that the resolution that can be achieved for the FE distribution is only slightly degraded compared to a Gaussian distribution with the same energy spread. FE also produces a focus offset. These two effects are weaker for AC than for NAC microscopy. These and other insights may be relevant to the selection of the aperture size that optimizes resolution and to analyses that make use of focal image series. The approach developed here is also applicable to transmission electron microscopy.

Enhancement of Carrier Mobility in Semiconductor Nanostructures by Carrier Distribution Engineering

Two-dimensional (2D) van der Waals semiconductors are appealing for low-power transistors. Here, we show the feasibility in enhancing carrier mobility in 2D semiconductors through engineering the vertical distribution of carriers confined inside ultrathin channels via symmetrizing gate configuration or increasing channel thickness. Through self-consistently solving the Schrödinger–Poisson equations, the shapes of electron envelope functions are extensively investigated by clarifying their relationship with gate configuration, channel thickness, dielectric permittivity, and electron density. The impacts of electron distribution variation on various carrier scattering matrix elements and overall carrier mobility are insightfully clarified. It is found that the carrier mobility can be generally enhanced in the dual-gated configuration due to the centralization of carrier redistribution in the nanometer-thick semiconductor channels and the rate of increase reaches up to 23% in HfO2 dual-gated 10-layer MoS2 channels. This finding represents a viable strategy for performance optimization in transistors consisting of 2D semiconductors.

Level structure of hole two-center complex and related luminescence in semiconductors

Real semiconductors usually contain both donor and acceptor impurities. A finite compensation degree leads to the possibility of the single carrier to be bounded by two impurity centers. For such an electron complex (near a simple conduction band), the spectral problem resembles the hydrogen molecule ion problem (up to renormalization of the effective mass and dielectric screening). In $p$-type semiconductors, the spectral problem for a single hole in the field of two attracting centers (${A}_{2}^{\ensuremath{-}}$ complex) is more complicated due to the complex structure of the valence band. Here such a problem is presented for the case of the hole bounded at two shallow acceptors close to the ${\mathrm{\ensuremath{\Gamma}}}_{8}$ valence band edge (${\mathrm{A}}^{\mathrm{III}}{\mathrm{B}}^{\mathrm{V}}$ or group $\mathrm{IV}$ semiconductors). The multicomponent envelope functions are used to develop quantum chemistrylike approach (molecular orbital method). The variational approach is applied to calculate the level structure of the complex. The states of the complex are classified by the total angular momentum projection onto the intercenter axis and by the parity with respect to the intercenter permutation. The ground state has a total angular momentum projection of $\ifmmode\pm\else\textpm\fi{}1/2$ and a wave function that is symmetric with respect to the intercenter permutation. The energy levels are found as a function of the intercenter distance. A possible manifestation of ${A}_{2}^{\ensuremath{-}}$ complexes in acceptor-related luminescence is discussed.

On the intrinsic pinning and shape of charge-density waves in 1D Peierls systems

Within the standard perturbative approach of Peierls, a charge-density wave is usually assumed to have a cosine shape of weak amplitude. In nonlinear physics, we know that waves can be deformed. What are the effects of the nonlinearities of the electron-lattice models in the physical properties of Peierls systems? We study in details a nonlinear discrete model, introduced by Brazovskii, Dzyaloshinskii and Krichever. First, we recall its exact analytical solution at integrable points. It is a cnoidal wave, with a continuous envelope, which may slide over the lattice potential at no energy cost, following Fröhlich’s argument. Second, we show numerically that integrability-breaking terms modify some important physical properties. The envelope function may become discontinuous: electrons form stronger chemical bonds which are local dimers or oligomers. We show that an Aubry transition from the sliding phase to an insulating pinned phase occurs when the model is no longer integrable.

Weibull multiscale interlaminar fracture analysis of low-weight percentage CNT composites

Delamination due to interlaminar fracture has limited the application of composite materials in highly integrity-sensitive structural systems. The novelty of this experimental and statistical investigation is twofold: (i) the effects of the size of multi-wall carbon nanotube (MWCNT) bundles and their interphase with the surrounding matrix on the multimodal fracture toughness and crack propagation regime; (ii) the effectiveness of single lamina vs. multiple laminas on fracture toughness in carbon fiber nanocomposites. The Atomic Force Microscopy Peak Force Quantitative Nanomechanics Mapping (AFM PFQNM) technique was utilized to characterize the CNT bundle size and interphase. Weibull analysis of a myriad of 480 CNT bundle sizes and interphases confirmed a high scatter in the data with more occurrences at the larger size. Two different nanocomposite laminates were considered: (i) laminates with 1wt% CNT in all laminas (FCNT); (ii) laminates with 1wt% CNT in the middle layer in front of the pre-crack (CNT). The average size of the CNT bundles and the interphase thickness were 549.7 nm and 22.9 nm in the CNT laminates and 480.1 nm and 22.2 nm in the FCNT laminates. The ratio of interphase thickness to CNT bundle size was higher in FCNT than in CNT, confirming a better stiffening effect due to the nano reinforcement of FCNT laminates. Weibull moduli of mode I initiation and propagation and mode II initiation for laminates with CNT in all laminas, FCNT, are higher than reference and CNT laminates, confirming a higher crack energy consistency. The Mode I initiation and propagation and mode II initiation fracture energies of CNT and FCNT were (15% and 10%), (38% and 44%), and (1.7% and 9.6%) higher than those without CNT. Mixing 1wt% CNT with all laminas expanded the mixed-mode I-II fracture envelope function by 21%. Unlike FCNT laminates, CNT laminates do not considerably improve the mixed-mode I-II fracture toughness. Crack deflection and a more arduous crack path due to the rigid CNT bundles increased the crack energy and delayed the crack propagation.

Vector-valued envelope functions for constructing tool swept surfaces in continuous domains

Abstract This paper proposes the vector-valued envelope functions to describe form cutters’ swept profiles at the choice of particular motion instances. The general idea behind such a formulation is to embed the tangency constraint into the tool surface expression and form a single continuous function to describe the segmental curves whose elements lie on the envelopes. This results in skipping unnecessary calculations when no envelope-surface point is associated with the specified input data, reducing the sizeable computational burden due to repetitive constraint implementations, and most importantly, depending on the tool kinematics’ complexity, avoiding the tangency-constraint violations between successive envelope-surface points where the exact fit cannot be obtained. In this study, firstly, the NC tool surface models present in extant literature have been restructured using the four-parameter set of spheres to attain the required one-to-one correspondence between the domain and range sets of these functions, followed by excluding the tangency-constraint evaluations by introducing the systematic parameter-reduction procedures that led to the development of constraint-embedded envelope functions. Next, we introduced the branch-existence test, which allowed us to check whether these functions are continuous over closed domain intervals. Finally, we covered algorithms for implementing the functions.

Analysis of axial waves in viscoelastic complex structural-acoustic systems: Theory and experiment

An experimental and theoretical study of the spectral response of coupled viscoelastic bars subject to axial oscillations is performed. Novel closed formulas for the envelope function and its width are derived. These formulas explicitly show the role played by energy dissipation. They show that the internal friction does not affect the width of the envelope of the individual resonances. The formulation is based on the equations of classical mechanics combined with Voigt’s viscoelastic model. The systems studied consist of a sequence of one, two, or three coupled bars, with their central axes collinear. One of the bars is assumed to be much longer than the others. We discuss the connection between our results and the concept of the strength function phenomenon discovered for the first time in nuclear physics. Our formulation is an alternative and exact approach to the approximated studies based on the fuzzy structure theory that has been used by other authors to describe systems consisting of coupled bars. The analytical expressions describe the measurements in the laboratory very well.

Modern metal framed glazed façades refurbishment: conservation in the energy and ecological transition

Metal framed glazed façades and curtain walls can be considered one of the most important innovations of modern architecture but are certainly a critical point in the conservation of modern heritage. Next to the most important modern architecture masterpieces, which are under special conservation and safeguard regulations, are a lot of minor and less known architectures that are part of the diffuse modern heritage and risk to be compromised in refurbishment interventions. In most cases, modern metal framed glazed façades systems do not meet the current energy requirements, and it is difficult to preserve them in their original features, finding a compromise between conservation and energy and ecological transition. This contribution presents a critical overview of the problems relating to the conservation of metal framed glazed façades built in the second half of the twentieth century. Requirements and standards established by energy and environmental international protocols and Italian regulations are critically analysed. Through the description of case studies, different attitudes of intervention are presented and discussed: from conservation to total replacement, with a radical change of modern building envelope original features. The mere verification of the minimum performances required by the current regulations inevitably leads to the substitution of original curtain walls. Therefore - looking beyond but respecting the regulations - architects in charge of refurbishment interventions can find innovative envelope technologies and function distribution solutions to preserve curtain walls, limiting the intervention to the strictly needed.

Generalised uncertainty relations from finite-accuracy measurements

In this short note we show how the Generalised Uncertainty Principle (GUP) and the Extended Uncertainty Principle (EUP), two of the most common generalised uncertainty relations proposed in the quantum gravity literature, can be derived within the context of canonical quantum theory, without the need for modified commutation relations. A generalised uncertainty principle-type relation naturally emerges when the standard position operator is replaced by an appropriate Positive Operator Valued Measure (POVM), representing a finite-accuracy measurement that localises the quantum wave packet to within a spatial regionσg> 0. This length scale is the standard deviation of the envelope function,g, that defines the positive operator valued measure elements. Similarly, an extended uncertainty principle-type relation emerges when the standard momentum operator is replaced by a positive operator valued measure that localises the wave packet to within a regionσ̃g>0in momentum space. The usual generalised uncertainty principle and extended uncertainty principle are recovered by settingσg≃ℏG/c3, the Planck length, andσ̃g≃ℏΛ/3, where Λ is the cosmological constant. Crucially, the canonical Hamiltonian and commutation relations, and, hence, the canonical Schrödinger and Heisenberg equations, remain unchanged. This demonstrates that generalised uncertainty principle and extended uncertainty principle phenomenology can be obtained without modified commutators, which are known to lead to various pathologies, including violation of the equivalence principle, violation of Lorentz invariance in the relativistic limit, the reference frame-dependence of the “minimum” length, and the so-called soccer ball problem for multi-particle states.