Unique Minimum Research Articles

Towards an information-theoretic framework for multiple sequence alignment sub-sampling

In this article we develop an information-theoretic framework of multiple sequence alignments (MSAs), based on sub-sampling. The key component of this framework is an information-theoretical potential defined on pairs of sites (links) within the MSA. This potential quantifies the expected drop in variation of information between the two constituent sites. The expectation is taken with respect to all possible sub-alignments, obtained by removing a finite, fixed number of rows. We show that the potential is zero for linked sites representing columns, for which symbols are in bijective correspondence and that it is strictly positive, otherwise. It is furthermore shown that the potential assumes its unique minimum for links at which each symbol pair appears with the same multiplicity. We then show that the established drop of the variation of information exceeds finite-size effects inherent to the construction of the potential. Finally, we provide as a proof of concept an application of our results to a specific MSA composed of the inverse fold solutions of three distinguished secondary structures.

Propagation of chaos in the random field Curie–Weiss model

Abstract We prove quenched propagation of chaos in the Random field mean-field Ising model, also known ad the Random field Curie–Weiss model. We show that in the paramagnetic phase, i.e. in the regime where temperature and distribution of the external field admit a unique minimizer of the expected Helmholtz free energy, quenched propagation of chaos holds. By the latter we mean that the finite-dimensional marginals of the Gibbs measure converge towards a product measure with the correct expectation as the system size goes to infinity. This holds independently of whether the system is in a high-temperature phase or at a phase transition point and alsmost surely with respect to the random external field. If the Helmholtz free energy possesses several minima, there are several possible equilibrium measures. In this case, we show that the system picks one of them at random (depending on the realization of the random external field) and propagation of chaos with respect to a product measure with the same marginals as the one randomly picked holds true. We illustrate our findings in a simple example.

Pose-Constrained Control of Proximity Maneuvering for Tracking and Observing Noncooperative Targets with Unknown Acceleration

This paper proposes a pose control scheme of for proximity maneuvering for tracking and observing noncooperative targets with unknown acceleration, which is an important prerequisite for on-orbit operations in space. It mainly consists of a finite-time extended state observer and constraint processing procedures. Firstly, relative pose-coupled kinematics and dynamics models with unknown integrated disturbances are established based on dual quaternion representations. Then, a finite-time extended state observer is designed using the super-twisting algorithm to estimate the integrated disturbances. Both observation field of view and collision avoidance pose-constrained models are constructed to ensure that the service spacecraft continuously and safely observes the target during proximity maneuvering. And the constraint models are further incorporated into the design of artificial potential function with a unique minimum. After that, the proportional–derivative-like pose-constrained tracking control law is proposed based on the estimated disturbances and the gradient of the artificial potential function. Finally, the effectiveness of the control scheme is verified through numerical simulations.

High moments of the SHE in the clustering regimes

We analyze the high moments of the Stochastic Heat Equation (SHE) via a transformation to the attractive Brownian Particles (BPs), which are Brownian motions interacting via pairwise attractive drift. In those scaling regimes where the particles tend to cluster, we prove a Large Deviation Principle (LDP) for the empirical measure of the attractive BPs. Under the delta(-like) initial condition, we characterize the unique minimizer of the rate function and relate the minimizer to the spacetime limit shapes of the Kardar–Parisi–Zhang (KPZ) equation in the upper tails. The results of this paper are used in the companion paper [75] to prove an n-point, upper-tail LDP for the KPZ equation and to characterize the corresponding spacetime limit shape.

Probing the Effects of Size and Charge on the Monohydration and Dihydration of SiF5- and SiF62- via Comparisons with BF4- and PF6.

This study systematically examines the interactions of the trigonal bipyramidal silicon pentafluoride and octahedral silicon hexafluoride anions with either one or two water molecules, (SiF5-(H2O)n and SiF62-(H2O)n, respectively, where n = 1, 2). Full geometry optimizations and subsequent harmonic vibrational frequency computations are performed using the CCSD(T) ab initio method with a triple-ζ correlation consistent basis set augmented with diffuse functions on all non-hydrogen atoms (cc-pVTZ for H and aug-cc-pVTZ for Si, O, and F; denoted as haTZ). Two monohydrate and six dihydrate minima have been identified for the SiF5-(H2O)n systems, whereas one monohydrate and five dihydrate minima have been identified for the SiF62-(H2O)n systems. Both monohydrated anions have a minimum in which the water molecule adopts a symmetric double ionic hydrogen bond (DIHB) motif with C2v symmetry. However, a second unique monohydrate minimum has been identified for SiF5- in which the water molecule adopts an asymmetric DIHB motif along the edge of the trigonal bipyramidal anion between one axial and one equatorial F atom. This Cs structure is more than 2 kcal mol-1 lower in energy than the C2v local minimum at the CCSD(T)/haTZ level of theory. While the interactions between the solvent and ionic solute are quite strong for the monohydrated anions (electronic dissociation energies of ≈12 and ≈24 kcal mol-1 for the SiF5-(H2O)1 and SiF62-(H2O)1 global minima, respectively), these values are nearly perfectly doubled for the dihydrates, with the lowest-energy SiF5-(H2O)2 and SiF62-(H2O)2 minima exhibiting dissociation energies of ≈24 and ≈47 kcal mol-1, respectively. Structures that form hydrogen bonds between the solvating water molecules also exhibit the largest shifts in the harmonic OH stretching frequencies for the waters of hydration. These shifts can exceed -100 cm-1 for the SiF5-(H2O)2 minimum and -300 cm-1 for the SiF62-(H2O)2 minimum relative to an isolated H2O molecule at the CCSD(T)/haTZ level of theory. This work also corrects the OH stretching frequency shifts for two dihydrate minima of PF6- that were previously erroneously reported ( J. Phys. Chem. A 2020, 124, 8744-8752, DOI: 10.1021/acs.jpca.0c06466).

Investigating two-dimensional adjoint QCD on the lattice

We present our investigations of SU(N) adjoint QCD in two dimensions with one Majorana fermion on the lattice. We determine the relevant parameter range for the simulations with Wilson fermions and present results for Polyakov loop, chiral condensate, and string tension. In the theory with massive fermions, all observables we checked show qualitative agreement between numerical lattice data and theory, while the massless limit is more subtle since chiral and non-invertible symmetry of the continuum theory are explicitly broken by lattice regularization. In thermal compactification, we observe N perturbative vacua for the holonomy potential at high-T with instanton events connecting them, and a unique vacuum at low-T. At finite-N, this is a cross-over and it turns to a phase transition at large-N thermodynamic limit. In circle compactification with periodic boundary conditions, we observe a unique center-symmetric minimum at any radius. In continuum, the instantons in the thermal case carry zero modes (for even N) and indeed, in the lattice simulations, we observe that chiral condensate is dominated by instanton centers, where zero modes are localized. We present lattice results on the issue of confinement vs. screening in the theory and comment on the roles of chiral symmetry and non-invertible symmetry.

Temperature upper bound of an ideal gas

We study thermodynamics of a heat-conducting ideal gas system. The study is based on i) the first law of thermodynamics from action formulation which expects heat-dependence of energy density and ii) the existence condition of a (local) Lorentz boost between an Eckart observer and a Landau-Lifshitz observer–a condition that extends the stability criterion of thermal equilibrium. The implications of these conditions include: i) Heat contributes to the energy density through the combination q/nΘ2 where q, n, and Θ represent heat, the number density, and the temperature, respectively. ii) The energy density has a unique minimum at q=0. iii) The temperature upper bound suppresses the heat dependence of the energy density inverse quadratically. This result explains why the expected heat dependence is difficult to observe in ordinary situation thermodynamics.

Minimizers for the de Gennes–Cahn–Hilliard energy under strong anchoring conditions

AbstractIn this article, we use the Nehari manifold and the eigenvalue problem for the negative Laplacian with Dirichlet boundary condition to analytically study the minimizers for the de Gennes–Cahn–Hilliard energy with quartic double‐well potential and Dirichlet boundary condition on the bounded domain. Our analysis reveals a bifurcation phenomenon determined by the boundary value and a bifurcation parameter that describes the thickness of the transition layer that segregates the binary mixture's two phases. Specifically, when the boundary value aligns precisely with the average of the pure phases, and the bifurcation parameter surpasses or equals a critical threshold, the minimizer assumes a unique form, representing the homogeneous state. Conversely, when the bifurcation parameter falls below this critical value, two symmetric minimizers emerge. Should the boundary value be larger or smaller from the average of the pure phases, symmetry breaks, resulting in a unique minimizer. Furthermore, we derive bounds of these minimizers, incorporating boundary conditions and features of the de Gennes–Cahn–Hilliard energy.

Best-in-class modeling: A novel strategy to discover constitutive models for soft matter systems

The ability to automatically discover interpretable mathematical models from data could forever change how we model soft matter systems. For convex discovery problems with a unique global minimum, model discovery is well-established. It uses a classical top-down approach that first calculates a dense parameter vector, and then sparsifies the vector by gradually removing terms. For non-convex discovery problems with multiple local minima, this strategy is infeasible since the initial parameter vector is generally non-unique. Here we propose a novel bottom-up approach that starts with a sparse single-term vector, and then densifies the vector by systematically adding terms. Along the way, we discover models of gradually increasing complexity, a strategy that we call best-in-class modeling. To identify and select successful candidate terms, we reverse-engineer a library of sixteen functional building blocks that integrate a century of knowledge in material modeling with recent trends in machine learning and artificial intelligence. Yet, instead of solving the NP hard discrete combinatorial problem with 216=65,536 possible combinations of terms, best-in-class modeling starts with the best one-term model and iteratively repeats adding terms, until the objective function meets a user-defined convergence criterion. Strikingly, for most practical purposes, we achieve good convergence with only one or two terms. We illustrate the best-in-class one- and two-term models for a variety of soft matter systems including rubber, brain, artificial meat, skin, and arteries. Our discovered models display distinct and unexpected features for each family of materials, and suggest that best-in-class modeling is an efficient, robust, and easy-to-use strategy to discover the mechanical signatures of traditional and unconventional soft materials. We anticipate that our technology will generalize naturally to other classes of natural and man made soft matter with applications in artificial organs, stretchable electronics, soft robotics, and artificial meat.

A uniqueness criterion and a counterexample to regularity in an incompressible variational problem

In this paper we consider the problem of minimizing functionals of the form E(u)=∫Bf(x,∇u)dx\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E(u)=\\int _B f(x,\ abla u) \\,dx$$\\end{document} in a suitably prepared class of incompressible, planar maps u:B→R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u: B \\rightarrow \\mathbb {R}^2$$\\end{document}. Here, B is the unit disk and f(x,ξ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(x,\\xi )$$\\end{document} is quadratic and convex in ξ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi $$\\end{document}. It is shown that if u is a stationary point of E in a sense that is made clear in the paper, then u is a unique global minimizer of E(u) provided the gradient of the corresponding pressure satisfies a suitable smallness condition. We apply this result to construct a non-autonomous, uniformly convex functional f(x,ξ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(x,\\xi )$$\\end{document}, depending smoothly on ξ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi $$\\end{document} but discontinuously on x, whose unique global minimizer is the so-called N-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N-$$\\end{document}covering map, which is Lipschitz but not C1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^1$$\\end{document}.

Cauchy data for Levin’s method

Abstract In this paper, we describe the Cauchy data that gives rise to slowly oscillating solutions to the Levin equation. We present a general result on the existence of a unique minimizer of $\|Bx\|$ subject to the constraint $Ax=y$, where $A,B$ are linear, but not necessarily bounded operators on a complex Hilbert space. This result is used to obtain the solution to the Levin equation, both in the univariate and multivariate case, which minimizes the mean-square of the derivative over the domain. The Cauchy data that generates this solution is then obtained, and this can be used to supplement the Levin equation in the computation of highly oscillatory integrals in the presence of stationary points.

An Overview of Sunspot Observations in the Early Maunder Minimum: 1645–1659

Abstract Within four centuries of sunspot observations, the Maunder Minimum (MM) in 1645–1715 has been considered a unique grand minimum with weak solar cycles in group numbers of sunspots and hemispheric asymmetry in sunspot positions. However, the early part of the MM (1645–1659) is poorly understood in terms of its source records and has accommodated diverse reconstructions of the contemporaneous group number. This study identified their source records, classidied them in three different categories (datable observations, general descriptions, and misinterpreted records), and revised their data. On this basis, we estimated the yearly mean group number using the brightest star method, derived the active day fraction (ADF), reconstructed the sunspot number based on ADF, and compared them with proxy reconstructions from the tree-ring data sets. Our results revised the solar activity in the early MM downward in yearly mean group numbers using the brightest star method and upward in the active day fraction and sunspot number estimates. Our results are consistent with the proxy reconstruction for 1645–1654 and show more realistic values for 1657–1659 (against the unphysical negative sunspot number). These records have paid little attention to sunspot positions, except for Hevelius' report on a sunspot group in the northern solar hemisphere in 1652 April. Therefore, slight caveats are required to discuss if the sunspot positions are located purely in the southern solar hemisphere throughout the MM.

Uniting Nesterov and heavy ball methods for uniform global asymptotic stability of the set of minimizers

We propose a hybrid control algorithm that guarantees fast convergence and uniform global asymptotic stability of the unique minimizer of a C1, convex objective function. The algorithm, developed using hybrid system tools, employs a uniting control strategy, in which Nesterov’s accelerated gradient descent is used “globally” and the heavy ball method is used “locally”, relative to the minimizer. Without knowledge of its location, the proposed hybrid control strategy switches between these accelerated methods to ensure convergence to the minimizer without oscillations, with a (hybrid) convergence rate that preserves the convergence rates of the individual optimization algorithms. We analyze key properties of the resulting closed-loop system including existence of solutions, uniform global asymptotic stability, and convergence rate. Additionally, stability properties of Nesterov’s method are analyzed, and extensions on convergence rate results in the existing literature are presented. Numerical results validate the findings.

Learning a Flexible Neural Energy Function With a Unique Minimum for Globally Stable and Accurate Demonstration Learning

Learning a Flexible Neural Energy Function With a Unique Minimum for Globally Stable and Accurate Demonstration Learning

Distributed optimization of high‐order nonlinear multi‐agent systems with disturbance under switching topologies

AbstractThis article investigates distributed optimization of high‐order nonlinear multi‐agent systems (MASs) with disturbance under switching topologies. First, we propose a class of new distributed control algorithms for high‐order MASs under switching topologies, such that all agents asymptotically reach consensus at the unique minimizer of the sum of cost functions. Then, by constructing disturbance observer, the proposed algorithms are extended to address distributed optimization problems for high‐order nonlinear MASs with disturbance under switching topologies. In particular, both of new presented methods do not require optimal reference signal generator, which can save the computational resources of systems. In final, simulations are conducted to validate the theoretical results.

A note on some non-local variational problems

We study two non-local variational problems that are characterized by the presence of a Riesz-like repulsive term that competes with an attractive term. The first functional is defined on the subsets of the Euclidean space and has the fractional perimeter as an attractive term. The second functional instead is defined on non-negative integrable and uniformly bounded densities and contains an attractive term of positive-power type. For both of the functionals, we prove that balls are the unique minimizers in the appropriate volume constraint range, generalizing the results already present in the literature for more specific energies.

Generic uniqueness of optimal transportation networks

We prove that for the generic boundary, in the sense of Baire categories, there exists a unique minimizer of the associated optimal branched transportation problem.

Graphs with Unique Minimum Specified Domination Sets

Graphs with Unique Minimum Specified Domination Sets

Disproof of a conjecture on the minimum spectral radius and the domination number

Let Gn,γ be the set of all connected graphs on n vertices with domination number γ. A graph is called a minimizer graph if it attains the minimum spectral radius among Gn,γ. Very recently, Liu, Li and Xie (2023) [17] proved that the minimizer graph over all graphs in Gn,γ must be a tree. Moreover, they determined the minimizer graph among Gn,⌊n2⌋ for even n, and posed the conjecture on the minimizer graph among Gn,⌊n2⌋ for odd n. In this paper, we disprove the conjecture and completely determine the unique minimizer graph among Gn,⌊n2⌋ for odd n.

On the Global Minimum of the Energy–Momentum Relation for the Polaron

For the Fröhlich model of the large polaron, we prove that the ground state energy as a function of the total momentum has a unique global minimum at momentum zero. This implies the non-existence of a ground state of the translation invariant Fröhlich Hamiltonian and thus excludes the possibility of a localization transition at finite coupling.