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Tabulation of high accuracy gamma function values for some rational arguments, noting applications to Gaussian quadrature formulas, fractional Bessel functions and Airy functions
A theory of scaling for rational (transfer) functions in terms of transformation groups is developed. Two different four-parameter scaling groups which play natural roles in studying linear systems are identified and the effect of scaling on Fisher information and related statistical measures in system identification are studied. The scalings considered include change of time scale, feedback, exponential scaling, magnitude scaling, etc. The scaling action of the groups studied is tied to the geometry of transfer functions in a rather strong way as becomes apparent in the examination of the invariants of scaling. As a result, the scaling process also provides new insight into the parameterization question for rational functions.
Functional porous metal–organic frameworks (MOFs) have been explored for a number of potential applications in catalysis, chemical sensing, water capture, gas storage, and separation. MOFs are among the most promising candidates to address challenges facing our society related to energy and environment, but the successful implementation of functional porous MOF materials are contingent on their stability; therefore, the rational design of stable MOFs plays an important role towards the development of functional porous MOFs. In this Focus article, we summarize progress in the rational design and synthesis of stable MOFs with controllable pores and functionalities. The implementation of reticular chemistry allows for the rational top-down design of stable porous MOFs with targeted topological networks and pore structures from the pre-selected building blocks. We highlight the reticular synthesis and applications of stable MOFs: (1) MOFs based on high valent metal ions (e.g., Al 3+ , Cr 3+ , Fe 3+ , Ti 4+ and Zr 4+ ) and carboxylate ligands; (2) MOFs based on low valent metal ions (e.g., Ni 2+ , Cu 2+ , and Zn 2+ ) and azolate linkers. We envision that the synthetic strategies, including modulated synthesis and post-synthetic modification, can potentially be extended to other more complex systems like metal-phosphonate framework materials.
Polypeptoids (poly-N-substituent glycines) are a class of highly tailorable peptidomimetic polymers. Polypeptoids have identical backbones as polypeptides (poly-C-substituent glycines), but sidechains of polypeptoids are appended to backbone nitrogen rather than α-carbon of polypeptides. As a result, peptoid backbone lacks of chirality and hydrogen bond donors. This unique structure gives polypeptoids a combined merit of both high stability as synthetic polymers and biocompatibility as biopolymers. In addition, peptoid sequences can be engineered precisely to assemble specific crystalline patterns such as spheres, fibers, ribbons, tubes, and sheets, which shows promising potentials of polypeptoids for different applications such as antimicrobials, catalysts, drug delivery, and templating inorganic materials. In this review, we summarize recent investigations into hierarchical self-assembly pathways and molecular structures of peptoid crystals that are of interest as templates for fabricating functional materials for potential biomedical, biochemical, and bioengineering applications. Furthermore, this review provides a summary of recent experimental and computational studies of polypeptoid assembly in solution and solid-liquid interfaces, current achievements in the field, and discusses future challenges and opportunities for the rational design of self-assembled polypeptoid nanomaterials.
Improvements to the automated co-registration and change detection software package, AFIDS (Automatic Fusion of Image Data System) has recently completed development for and validation by NGA/GIAT. The improvements involve the integration of the AFIDS ultra-fine gridding technique for horizontal displacement compensation with the recently evolved use of Rational Polynomial Functions/ Coefficients (RPFs/RPCs) for image raster pixel position to Latitude/Longitude indexing. Mapping and orthorectification (correction for elevation effects) of satellite imagery defies exact projective solutions because the data are not obtained from a single point (like a camera), but as a continuous process from the orbital path. Standard image processing techniques can apply approximate solutions, but advances in the state-of-the-art had to be made for precision change-detection and time-series applications where relief offsets become a controlling factor. The earlier AFIDS procedure required the availability of a camera model and knowledge of the satellite platform ephemeredes. The recent design advances connect the spacecraft sensor Rational Polynomial Function, a deductively developed model, with the AFIDS ultrafine grid, an inductively developed representation of the relationship raster pixel position to latitude /longitude. As a result, RPCs can be updated by AFIDS, a situation often necessary due to the accuracy limits of spacecraft navigation systems. An example of precision change detection will be presented from Quickbird.
Closed form rational approximations to Tricomi psi function
We introduce a formalism for describing four-dimensional scattering amplitudes for particles of any mass and spin. This naturally extends the familiar spinor-helicity formalism for massless particles to one where these variables carry an extra SU(2) little group index for massive particles, with the amplitudes for spin Sparticles transforming as symmetric rank 2S tensors. We systematically characterise all possible three particle amplitudes compatible with Poincare symmetry. Unitarity, in the form of consistent factorization, imposes algebraic conditions that can be used to construct all possible four-particle tree amplitudes. This also gives us a convenient basis in which to expand all possible four-particle amplitudes in terms of what can be called “spinning polynomials”. Many general results of quantum field theory follow the analysis of four-particle scattering, ranging from the set of all possible consistent theories for massless particles, to spin-statistics, and the Weinberg-Witten theorem. We also find a transparent understanding for why massive particles of sufficiently high spin cannot be “elementary”. The Higgs and Super-Higgs mechanisms are naturally discovered as an infrared unification of many disparate helicity amplitudes into a smaller number of massive amplitudes, with a simple understanding for why this can’t be extended to Higgsing for gravitons. We illustrate a number of applications of the formalism at one-loop, giving few-line computations of the electron (g - 2) as well as the beta function and rational terms in QCD. “Off-shell” observables like correlation functions and form-factors can be thought of as scattering amplitudes with external “probe” particles of general mass and spin, so all these objects — amplitudes, form factors and correlators, can be studied from a common on-shell perspective.
This paper presents the development and experimental results of an automated on-orbit system identification method for large flexible spacecraft that yields estimated quantities to support on-line design and tuning of robust high performance control systems. The procedure consists of applying an input to the plant, obtaining an output, and then conducting nonparametric identification to yield the spectral estimate of the system transfer function. A parametric model is determined by curve fitting the spectral estimate to a rational transfer function. The identification method has been demonstrated experimentally on the Large Spacecraft Control Laboratory in JPL.
Turbulence simulation is computationally much simpler using rational spectra, but turbulence falls off as f exp -5/3 in frequency ranges of interest to aircraft response and as predicted by von Karman's model. Rational approximations to von Karman spectra should satisfy three requirements: (1) the rational spectra should provide a good approximation to the von Karman spectra in the frequency range of interest; (2) for stability, the resulting rational transfer function should have all its poles in the left half-plane; and (3) at high frequencies, the rational spectra must fall off as an integer power of frequency, and since the -2 power is closest to the -5/3 power, the rational approximation should roll off as the -2 power at high frequencies. Rational approximations to von Karman spectra that satisfy these three criteria are presented, along with spectra from simulated turbulence. Agreement between the spectra of the simulated turbulence and von Karman spectra is excellent.
Five novel Zintl phase solid solutions in the Ba 1–x Sr x Zn 2–y Cd y Sb 2 (0 ≤ x ≤ 0.13(1); 0 ≤ y ≤ 0.32(2)) system were successfully synthesized by the molten Pb metal-flux method, and the powder X-ray diffraction and single-crystal X-ray diffraction analyses proved that all five title compounds adopted the BaCu 2 S 2 -type phase having the orthorhombic Pnma space group (Z = 4, Pearson code oP20) with five crystallographically independent atomic sites. The previously studied BaCu 2 S 2 -type antimonides demonstrated a limited tolerance for doping in contrast to the CaAl 2 Si 2 -type antimonides. To understand the relatively narrower phase width and limited dopability of the title BaCu 2 S 2 -type phase than the CaAl 2 Si 2 -type phase in the overall Ba 1–x Sr x Zn 2–y Cd y Sb 2 system, the radius ratio of cations and anionic elements r + /r – for two structure types were thoroughly investigated. For the first time, the r + /r – ratio was identified as a critical factor for the phase selectivity: (1) r + /r – > 1 favored the BaCu 2 S 2 -type phase, and (2) r + /r – < 1 favored the CaAl 2 Si 2 -type phase. Further, we also revealed the structural transformation mechanism from the more widely observed CaAl 2 Si 2 -type phase to the title BaCu 2 S 2 -type phase as the relatively larger cationic elements were introduced to the system. A series of DFT calculations using the three hypothetical models indicated that a resonance peak near EF in the density of states curves was descended from the relatively flat band structure at several special symmetry points rationalizing the enhanced Seebeck coefficients of Ba 0.94(1) Sr 0.06 Zn 1.86(3) Cd 0.14 Sb 2 and Ba 0.96(1) Sr 0.04 Zn 1.68(2) Cd 0.32 Sb 2 . Electron localization function analysis rationalized the correlation between the polarity change of anionic Zn/Cd–Sb bonds and the charge carrier mobility on the anionic frameworks. Temperature-dependent thermoelectric properties were studied for the four title compounds, and the results proved that the Sr and Cd doping in the title Ba 1–x Sr x Zn 2–y Cd y Sb 2 system successfully enhanced the ZT values through the increased Seebeck coefficients and the reduced total thermal conductivities.
Building a training system for medical personnel to properly interpret fetal heart rate tracing requires developing accurate models that can relate various signal patterns to certain pathologies. In addition to modeling the fetal heart rate signal itself, the change of uterine pressure that bears strong relation to fetal heart rate and provides indications of maternal and fetal status should also be considered. In this work, we have developed a group of parametric models to simulate uterine contractions during labor and delivery. Through analysis of real patient records, we propose to model uterine contraction signals by three major components: regular contractions, impulsive noise caused by fetal movements, and low amplitude noise invoked by maternal breathing and measuring apparatus. The regular contractions are modeled by an asymmetric generalized Gaussian function and least squares estimation is used to compute the parameter values of the asymmetric generalized Gaussian function based on uterine contractions of real patients. Regular contractions are detected based on thresholding and derivative analysis of uterine contractions. Impulsive noise caused by fetal movements and low amplitude noise by maternal breathing and measuring apparatus are modeled by rational polynomial functions and Perlin noise, respectively. Experiment results show the synthesized uterine contractions can mimic the real uterine contractions realistically, demonstrating the effectiveness of the proposed algorithm.
Understanding the complex temporal and spatial correlations of ions in disordered perovskite oxides is critical to rationalize their functional properties. Here, in this study, we provide new insights into the longstanding controversy regarding the off-centering of transition metal ions in the archetypal ferroelectric alloy KTa 1-x Nb x O 3 (KTN). By mapping the full energy (E) and wavevector (Q) dependence of the dynamical structure factor S(Q, E) using neutron scattering, and rationalizing our observations with atomistic simulations leveraging machine learning, we fully resolve the static vs dynamic nature of diffuse scattering sheets, as well as their composition (x) and temperature dependence. Our first-principles simulations, extended with machine-learning molecular dynamics, reproduce both inelastic neutron spectra and diffuse features, and establish how dynamically-correlated transition metal off-centerings couple to phonons, unifying local and collective viewpoints. This study sheds new light into an exemplary ferroelectric systems and shows the importance of mapping the full S(Q, E) to reveal critical spatio-temporal correlations of atomic disorder from which functional properties emerge.
An optimal Routh approximant of a single-input single-output dynamic system is a reduced-order transfer function of which the denominator is obtained by the Routh approximation method while the numerator is determined by minimizing a time-response integral-squared-error (ISE) criterion. In this paper, a new elegant approach is presented for obtaining the optimal Routh approximants for linear time-invariant continuous-time systems. The approach is based on the Routh canonical expansion, which is a finite-term orthogonal series of rational basis functions, and minimization of the ISE criterion. A procedure for combining the above approach with the bilinear transformation is also presented in order to obtain the optimal bilinear Routh approximants of linear time-invariant discrete-time systems. The proposed technique is simple in formulation and is amenable to practical implementation.
A formal theory of the scattering of time-harmonic acoustic scalar waves from impenetrable, immobile obstacles is established. The time-independent formal scattering theory of nonrelativistic quantum mechanics, in particular the theory of the complete Green's function and the transition (T) operator, provides the model. The quantum-mechanical approach is modified to allow the treatment of acoustic-wave scattering with imposed boundary conditions of impedance type on the surface (delta-Omega) of an impenetrable obstacle. With k0 as the free-space wavenumber of the signal, a simplified expression is obtained for the k0-dependent T operator for a general case of homogeneous impedance boundary conditions for the acoustic wave on delta-Omega. All the nonelementary operators entering the expression for the T operator are formally simple rational algebraic functions of a certain invertible linear radiation impedance operator which maps any sufficiently well-behaved complex-valued function on delta-Omega into another such function on delta-Omega. In the subsequent study, the short-wavelength and the long-wavelength behavior of the radiation impedance operator and its inverse (the 'radiation admittance' operator) as two-point kernels on a smooth delta-Omega are studied for pairs of points that are close together.
The field of hybrid engineered living materials seeks to pair living organisms with synthetic materials to generate biocomposite materials with augmented function since living systems can provide highly-programmable and complex behavior. Engineered living materials have typically been fabricated using techniques in benign aqueous environments, limiting their application. In this work, biocomposite fabrication is demonstrated in which spores from polymer-degrading bacteria are incorporated into a thermoplastic polyurethane using high-temperature melt extrusion. Bacteria are engineered using adaptive laboratory evolution to improve their heat tolerance to ensure nearly complete cell survivability during manufacturing at 135 °C. Furthermore, the overall tensile properties of spore-filled thermoplastic polyurethanes are substantially improved, resulting in a significant improvement in toughness. The biocomposites facilitate disintegration in compost in the absence of a microbe-rich environment. Finally, embedded spores demonstrate a rationally programmed function, expressing green fluorescent protein. This research provides a scalable method to fabricate advanced biocomposite materials in industrially-compatible processes.
The analysis, design, and on-orbit tuning of robust controllers require more information about the plant than simply a nominal estimate of the plant transfer function. Information is also required concerning the uncertainty in the nominal estimate, or more generally, the identification of a model set within which the true plant is known to lie. The identification methodology that was developed and experimentally demonstrated makes use of a simple but useful characterization of the model uncertainty based on the output error. This is a characterization of the additive uncertainty in the plant model, which has found considerable use in many robust control analysis and synthesis techniques. The identification process is initiated by a stochastic input u which is applied to the plant p giving rise to the output. Spectral estimation (h = P sub uy/P sub uu) is used as an estimate of p and the model order is estimated using the produce moment matrix (PMM) method. A parametric model unit direction vector p is then determined by curve fitting the spectral estimate to a rational transfer function. The additive uncertainty delta sub m = p - unit direction vector p is then estimated by the cross spectral estimate delta = P sub ue/P sub uu where e = y - unit direction vectory y is the output error, and unit direction vector y = unit direction vector pu is the computed output of the parametric model subjected to the actual input u. The experimental results demonstrate the curve fitting algorithm produces the reduced-order plant model which minimizes the additive uncertainty. The nominal transfer function estimate unit direction vector p and the estimate delta of the additive uncertainty delta sub m are subsequently available to be used for optimization of robust controller performance and stability.
Material synthesis is one of the most important aspects in humankinds’ endeavor to discover and create new materials for energy applications. One strategy to tailor materials with desired functions in a rational way is by knowing how functions relate to structure, synthetic variables, arrangement of atoms and molecules, and how functions evolve during synthesis. In order to accelerate materials synthesis, discovery, and optimization by 10 times it is the right time now to integrate computational tools, synthesis, and characterization. One particular barrier to realizing this concept is the understanding of when and how phases form in real time during synthesis, which is challenging to asses by existing theoretical frameworks. In addition, transient or metastable phases with positive free energy above the lowest-free energy ground state can be revealed by such real time (in situ) measurements. Metastable materials are ubiquitous in condensed matter and can show superior properties compared to their equilibrium form. This work discusses the value of emerging multimodal in situ characterizations exemplified on hybrid halide perovskites. Finally, the ways in which the implementation of in situ measurements can advance the materials science synthesis field as well as their role to enable close-loop feedback control and autonomous synthesis are discussed.
Xylan O-acetyltransferase 1 (XOAT1) is involved in O-acetylating the backbone of hemicellulose xylan. Recent structural analysis of XOAT1 showed two unequal lobes forming a cleft that is predicted to accommodate and position xylan acceptors into proximity with the catalytic triad. Here, we used docking and molecular dynamics simulations to investigate the optimal orientation of xylan in the binding cleft of XOAT1 and identify putative key residues (Gln445 and Arg444 on Minor lobe & Asn312, Met311 and Asp403 on Major lobe) involved in substrate interactions. Site-directed mutagenesis coupled with biochemical analyses revealed the major lobe of XOAT1 is important for xylan binding. Mutation of single key residues yielded XOAT1 variants with various enzymatic efficiencies that are applicable to one-pot synthesis of xylan polymers with different degrees of O-acetylation. Taken together, our results demonstrate the effectiveness of computational modeling in guiding enzyme engineering aimed at modulating xylan and redesigning plant cell walls.