More Like this

1. The numerical unified transform method for the nonlinear Schrödinger equation on the half-line

   https://doi.org/10.1098/rspa.2021.0481  

   Yang, Xin; Deconinck, Bernard; Trogdon, Thomas (December 2021, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   We implement the numerical unified transform method to solve the nonlinear Schrödinger equation on the half-line. For the so-called linearizable boundary conditions, the method solves the half-line problems with comparable complexity as the numerical inverse scattering transform solves whole-line problems. In particular, the method computes the solution at any x and t without spatial discretization or time stepping. Contour deformations based on the method of nonlinear steepest descent are used so that the method’s computational cost does not increase for large x , t and the method is more accurate as x , t increase. Our ideas also apply to some cases where the boundary conditions are not linearizable. 

   more » « less
2. Eclogite thermobarometry: The consistency between conventional thermobarometry and forward phase‐equilibrium modelling

   https://doi.org/10.1111/jmg.12747  

   Hernández‐Uribe, David; Holder, Robert M.; Hernández‐Montenegro, Juan D. (October 2023, Journal of Metamorphic Geology)

   Abstract Eclogite thermobarometry is crucial for constraining the depths and temperatures to which oceanic and continental crust subduct. However, obtaining the pressure and temperature (P–T) conditions of eclogites is complex as they commonly display high‐variance mineral assemblages, and the mineral compositions only vary slightly withP–T. In this contribution, we present a comparison between two independent and commonly used thermobarometric approaches for eclogites: conventional thermobarometry and forward phase‐equilibrium modelling. We assess how consistent the thermobarometric calculations are using the garnet–clinopyroxene–phengite barometer and garnet–clinopyroxene thermometer with predictions from forward modelling (i.e. comparing the relative differences between approaches). Our results show that the overall mismatch in methods is typically ±0.2–0.3 GPa and ±29–42°C although differences as large as 80°C and 0.7 GPa are possible for a few narrow ranges ofP–Tconditions in the forward models. Such mismatch is interpreted as the relative differences among methods, and not as absolute uncertainties or accuracies for either method. For most of the investigatedP–Tconditions, the relatively minor differences between methods means that the choice in thermobarometric method itself is less important for geological interpretation than careful sample characterization and petrographic interpretation for derivingP–Tfrom eclogites. Although thermobarometry is known to be sensitive to the assumedX_Fe³⁺of a rock (or mineral), therelativedifferences between methods are not particularly sensitive to the choice of bulk‐rockX_Fe³⁺, except at high temperatures (>650°C, amphibole absent) and for very large differences in assumedX_Fe³⁺(0–0.5). We find that the most important difference between approaches is the activity–composition (a–x) relations, as opposed to the end‐member thermodynamic data or other aspects of experimental calibration. When equivalenta–xrelations are used in the conventional barometer,Pcalculations are nearly identical to phase‐equilibrium models (ΔP < 0.1). To further assess the implications of these results for real rocks, we also evaluate common mathematical optimizations of reaction constants used for obtaining the maximumP–Twith conventional thermobarometric approaches (e.g. using the highestaGrs² × aPrp in garnet and Si content in phengite, and the lowestaDi in clinopyroxene). These approaches should be used with caution, because they may not represent the compositions of equilibrium mineral assemblages at eclogite facies conditions and therefore systematically biasP–Tcalculations. Assuming method accuracy, geological meaningfulP_maxat a typical eclogite facies temperature of ~660°C will be obtained by using the greatestaDi,aCel, andaPrp and lowestaGrs andaMs; garnet and clinopyroxene with the lowest Fe²⁺/Mg ratios may yield geological meaningfulT_maxat a typical eclogite facies pressure of 2.5 GPa. 

   more » « less
3. Accurate and efficient open-source implementation of domain-based local pair natural orbital (DLPNO) coupled-cluster theory using a t1-transformed Hamiltonian

   https://doi.org/10.1063/5.0219963  

   Jiang, Andy; Glick, Zachary L; Poole, David; Turney, Justin M; Sherrill, C David; Schaefer, Henry F (August 2024, The Journal of Chemical Physics)

   We present an efficient, open-source formulation for coupled-cluster theory through perturbative triples with domain-based local pair natural orbitals [DLPNO-CCSD(T)]. Similar to the implementation of the DLPNO-CCSD(T) method found in the ORCA package, the most expensive integral generation and contraction steps associated with the CCSD(T) method are linear-scaling. In this work, we show that the t1-transformed Hamiltonian allows for a less complex algorithm when evaluating the local CCSD(T) energy without compromising efficiency or accuracy. Our algorithm yields sub-kJ mol−1 deviations for relative energies when compared with canonical CCSD(T), with typical errors being on the order of 0.1 kcal mol−1, using our TightPNO parameters. We extensively tested and optimized our algorithm and parameters for non-covalent interactions, which have been the most difficult interaction to model for orbital (PNO)-based methods historically. To highlight the capabilities of our code, we tested it on large water clusters, as well as insulin (787 atoms). 

   more » « less
4. More on codes for combinatorial composite DNA

   https://doi.org/10.1007/s10623-025-01634-8  

   Ye, Zuo; Sabary, Omer; Gabrys, Ryan; Yaakobi, Eitan; Elishco, Ohad (May 2025, Designs, Codes and Cryptography)

   Abstract In this paper, we focus on constructing unique-decodable and list-decodable codes for the recently studied (t, e)-composite-asymmetric error-correcting codes ((t, e)-CAECCs). Let$$\mathcal {X}$$ $X$be an$$m \times n$$ $m\times n$binary matrix in which each row has Hamming weightw. If at mosttrows of$$\mathcal {X}$$ $X$contain errors, and in each erroneous row, there are at mosteoccurrences of$$1 \rightarrow 0$$ $1\to 0$errors, we say that a (t, e)-composite-asymmetric error occurs in$$\mathcal {X}$$ $X$. For general values ofm, n, w, t, ande, we propose new constructions of (t, e)-CAECCs with redundancy at most$$(t-1)\log (m) + O(1)$$ $(t-1)log\left(m\right)+O\left(1\right)$, whereO(1) is independent of the code lengthm. In particular, this yields a class of (2, e)-CAECCs that are optimal in terms of redundancy. Whenmis a prime power, the redundancy can be further reduced to$$(t-1)\log (m) - O(\log (m))$$ $(t-1)log\left(m\right)-O(log(m\left)\right)$. To further increase the code size, we introduce a combinatorial object called a weak$$B_e$$ $B_e$-set. When$$e = w$$ $e=w$, we present an efficient encoding and decoding method for our codes. Finally, we explore potential improvements by relaxing the requirement of unique decoding to list-decoding. We show that when the list size ist! or an exponential function oft, there exist list-decodable (t, e)-CAECCs with constant redundancy. When the list size is two, we construct list-decodable (3, 2)-CAECCs with redundancy$$\log (m) + O(1)$$ $log\left(m\right)+O\left(1\right)$. 

   more » « less
5. Non-perturbative Solution of the 1d Schrödinger Equation Describing Photoemission from a Sommerfeld Model Metal by an Oscillating Field

   https://doi.org/10.1007/s00220-023-04771-0  

   Costin, Ovidiu; Costin, Rodica; Jauslin, Ian; Lebowitz, Joel L (September 2023, Communications in Mathematical Physics)

   Springer (Ed.)

   We analyze non-perturbatively the one-dimensional Schrödinger equation describing the emission of electrons from a model metal surface by a classical oscillating electric field. Placing the metal in the half-space x 0, the Schrödinger equation of the system is i∂t ψ = − 2 1 ∂x 2 ψ + (x)(U − E x cos ωt)ψ, t > 0, x ∈ R, where (x) is the Heaviside function and U > 0 is the effective confining potential (we choose units so that m = e = = 1). The amplitude E of the external electric field and the frequency ω are arbitrary. We prove existence and uniqueness of classical solutions of the Schrödinger equation for general initial conditions ψ(x, 0) = f (x), x ∈ R. When 2the initial condition is in L the evolution is unitary and the wave function goes to zero at any fixed x as t → ∞. To show this we prove a RAGE type theorem and show that the discrete spectrum of the quasienergy operator is empty. To obtain positive electron current we consider non-L 2 initial conditions containing an incoming beam from the left. The beam is partially reflected and partially transmitted for all t > 0. For these initial conditions we show that the solution approaches in the large t limit a periodic state that satisfies an infinite set of equations formally derived, under the assumption that the solution is periodic by Faisal et al. (Phys Rev A 72:023412, 2005). Due to a number of pathological features of the Hamiltonian (among which unboundedness in the physical as well as the spatial Fourier domain) the existing methods to prove such results do not apply, and we introduce new, more general ones. The actual solution exhibits a very complex behavior, as seen both analytically and numerically. It shows a steep increase in the current as the frequency passes a threshold value ω = ωc , with ωc depending on the strength of the electric field. For small E, ωc represents the threshold in the classical photoelectric effect, as described by Einstein’s theory. 

   more » « less