American Institute of Mathematics
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Recent publications
The bacterial division apparatus catalyses the synthesis and remodelling of septal peptidoglycan (sPG) to build the cell wall layer that fortifies the daughter cell poles. Understanding of this essential process has been limited by the lack of native three-dimensional views of developing septa. Here, we apply state-of-the-art cryogenic electron tomography (cryo-ET) and fluorescence microscopy to visualize the division site architecture and sPG biogenesis dynamics of the Gram-negative bacterium Escherichia coli. We identify a wedge-like sPG structure that fortifies the ingrowing septum. Experiments with strains defective in sPG biogenesis revealed that the septal architecture and mode of division can be modified to more closely resemble that of other Gram-negative (Caulobacter crescentus) or Gram-positive (Staphylococcus aureus) bacteria, suggesting that a conserved mechanism underlies the formation of different septal morphologies. Finally, analysis of mutants impaired in amidase activation (ΔenvC ΔnlpD) showed that cell wall remodelling affects the placement and stability of the cytokinetic ring. Taken together, our results support a model in which competition between the cell elongation and division machineries determines the shape of cell constrictions and the poles they form. They also highlight how the activity of the division system can be modulated to help generate the diverse array of shapes observed in the bacterial domain.
For a class of functionals having the ( p , q )-growth, we establish an improved range of exponents p , q for which the Lavrentiev phenomenon does not occur. The proof is based on a standard mollification argument and Young convolution inequality. Our contribution is two-fold. First, we observe that it is sufficient to regularise only bounded functions. Second, we exploit the $$L^{\infty }$$ L ∞ bound on the function rather than the $$L^p$$ L p estimate on the gradient. Our proof does not rely on the properties of minimizers to variational problems but it is rather a consequence of the underlying Musielak–Orlicz function spaces. Moreover, our method works for unbounded boundary data, the variable exponent functionals and vectorial problems. In addition, the result seems to be optimal for $$p\leqq d$$ p ≦ d .
The tight upper bound pt+(G)≤⌈|V(G)|−Z+(G)2⌉ is established for the positive semidefinite propagation time of a graph in terms of its positive semidefinite zero forcing number. To prove this bound, two methods of transforming one positive semidefinite zero forcing set into another and algorithms implementing these methods are presented. Consequences of the bound, including a tight Nordhaus-Gaddum sum upper bound on positive semidefinite propagation time, are established.
We prove the existence of a weak solution to the compressible Navier–Stokes system with singular pressure that explodes when density achieves its congestion level. This is a quantity whose initial value evolves according to the transport equation. We then prove that the “stiff pressure” limit gives rise to the two-phase compressible/incompressible system with congestion constraint describing the free interface. We prescribe the velocity at the boundary and the value of density at the inflow part of the boundary of a general bounded C2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^2$$\end{document} domain. For the positive velocity flux, there are no restrictions on the size of the boundary conditions, while for the zero flux, a certain smallness is required for the last limit passage.
In this manuscript, we prove the existence of slow and fast traveling wave solutions in the original Gatenby–Gawlinski model. We prove the existence of a slow traveling wave solution with an interstitial gap. This interstitial gap has previously been observed experimentally, and here we derive its origin from a mathematical perspective. We give a geometric interpretation of the formal asymptotic analysis of the interstitial gap and show that it is determined by the distance between a layer transition of the tumor and a dynamical transcritical bifurcation of two components of the critical manifold. This distance depends, in a nonlinear fashion, on the destructive influence of the acid and the rate at which the acid is being pumped.
Sputnik Planitia basin, the dominant surface feature of the dwarf planet Pluto, is located very close to the far point of Pluto‐Charon tidal axis. This position is currently believed to be a result of whole body reorientation driven by the combination of (a) the uplift of a subsurface ocean in response to a basin‐forming impact and (b) the nitrogen layer accumulated inside the basin. Since an ice shell made of pure water ice cannot maintain the uplift on timescales of billions of years, the presence of an insulating and highly viscous layer of methane clathrates at the base of the shell has recently been proposed. In this study, we solve the thermo‐mechanical evolution of the ice shell in a 2D spherical axisymmetric geometry and evaluate the gravity anomaly associated with the evolving ice shell shape. Taking into account the effect of impact heating and stress‐dependent rheology of both ice and clathrates, we show that a thick shell (≥200 km) loses the impact heat slowly which leads to fast uplift relaxation of the order of hundreds of million years. On the contrary, a thin shell (∼100 km) cools down quickly (∼10 Myr), becoming rigid and more likely to preserve the ocean/shell interface uplift till the present. These results suggest that a thick ocean may be present beneath Pluto's ice shell.
We prove the analogue of the strong Szegő limit theorem for a large class of bordered Toeplitz determinants. In particular, by applying our results to the formula of Au-Yang and Perk (Physica A 144:44–104, 1987) for the next-to-diagonal correlations ⟨σ0,0σN-1,N⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\langle \sigma _{0,0}\sigma _{N-1,N} \rangle $$\end{document} in the square lattice Ising model, we rigorously justify that the next-to-diagonal long-range order is the same as the diagonal and horizontal ones in the low temperature regime. We also confirm the leading and subleading terms in an asymptotic formula of Cheng and Wu (Phys Rev 164:719–735, 1967) for ⟨σ0,0σM,N⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\langle \sigma _{0,0}\sigma _{M,N} \rangle $$\end{document} when M=N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M=N$$\end{document} and M=N-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M=N-1$$\end{document}, thereby establishing the anisotropy-dependence of the subleading term in the asymptotics of the next-to-diagonal correlations. We use Riemann-Hilbert and operator theory techniques, independently and in parallel, to prove these results.
By classical results of G.C. Evans and G. Choquet on “good” kernels G in potential theory, for every polar Kσ-set P, there exists a finite measure μ on P such that its potential Gμ is infinite on P, and a set P admits a finite measure μ on P such that Gμ is infinite exactly on P if and only if P is a polar Gδ-set. A known application of Evans’ theorem yields the solutions of the generalized Dirichlet problem for open sets by the Perron-Wiener-Brelot method using only harmonic upper and lower functions. It is shown that, by an elementary “metric sweeping” of measures and without using any potential theory, such results can be obtained for general kernels G satisfying a local triangle property, a property which amounts to G being locally equivalent to some negative power of some metric. The particular case, G(x,y) = |x − y|α−d on Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathbbm {R}^{d}}$\end{document}, 2 < α < d, solves a long-standing open problem.
Density gradient theory describes the evolution of diffuse interfaces in both mixtures and pure substances by minimization of the total free energy, which consists of a non-convex bulk part and an interfacial part. Minimization of the bulk free energy causes phase separation while building up the interfacial free energy (proportional to the square of gradients of the species’ densities) and it results in the equilibrium shape of the interface. However, direct minimization of the free energy is numerically unstable and the coefficients in the interfacial part of the free energy are often estimated from experimental data (not determined from the underlying physics). In this paper we develop a robust physics-based numerical approach that leads to the interface density profiles for both pure substances and mixtures. The model is free of fitting parameters and validated by available experimental data.
For a multivariate normal distribution, the sparsity of the covariance and precision matrices encodes complete information about independence and conditional independence properties. For general distributions, the covariance and precision matrices reveal correlations and so-called partial correlations between variables, but these do not, in general, have any correspondence with respect to independence properties. In this paper, we prove that, for a certain class of non-Gaussian distributions, these correspondences still hold, exactly for the covariance and approximately for the precision. The distributions—sometimes referred to as “nonparanormal”—are given by diagonal transformations of multivariate normal random variables. We provide several analytic and numerical examples illustrating these results.
For possibly discontinuous functions including, for instance, Sobolev functions, we present new Blaschke-Privaloff-type criteria for superharmonicity and harmonicity. This opens the way for introduction of a substantial generalization of the Laplace operator. These potential-theoretic considerations lead to a new kind of non-absolutely convergent integral where the integrand may be a highly oscillating pointwise function or even a distribution-valued function. In turn, this integral gives a precise meaning to some generalized Poisson equations with a wild right hand side.
We improve the current bounds for an inequality of Erdős and Turán from 1950 related to the discrepancy of angular equidistribution of the zeros of a given polynomial. Building upon a recent work of Soundararajan, we establish a novel connection between this inequality and an extremal problem in Fourier analysis involving the maxima of Hilbert transforms, for which we provide a complete solution. Prior to Soundararajan (2019), refinements of the discrepancy inequality of Erdős and Turán had been obtained by Ganelius (1954) and Mignotte (1992).
A bstract We propose a transformation between the off-shell field variables of Witten’s open bosonic string field theory and the traditional lightcone string field theory of Kaku and Kikkawa, based on Mandelstam’s interacting string picture. This is accomplished by deforming the Witten vertex into lightcone cubic and quartic vertices, followed by integrating out the ghost and lightcone oscillator excitations from the string field. Surprisingly, the last step does not alter the cubic and quartic interactions and does not generate effective vertices, and leads precisely to Kaku and Kikkawa’s lightcone string field theory.
Plain Language Summary The tiger stripes are four sub‐parallel, linear depressions in the south polar region (SPR) of Saturn's moon Enceladus, which are known for their vapor plumes containing organic molecules. The nature of the tiger stripes is not fully understood and remains a subject of intense debate. Here, we propose a new model of the tiger stripes in which Enceladus' ice shell is modeled as an elastic system with frictional interfaces subjected to periodic tidal loading. We find that the diurnal tides produce a complex pattern of stress anomalies, characterized by a length scale of tens of km and the peak values exceeding 100 kPa. Friction delays the system's response to tidal loading and leads to an asymmetry between the compression and extension phases. This asymmetry results in an additional stress, which is constant in time and comparable in magnitude to the cyclic stress. The static stress field is characterized by compression in the direction perpendicular to the faults, and its magnitude is large enough to influence the evolution of the SPR on geological time scales. The total heat flow generated by friction is 0.1–1 GW, accounting for only a small fraction of the heat power emitted from the tiger stripes.
We consider the Navier–Stokes–Fourier system in a bounded domain Ω⊂Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset R^d$$\end{document}, d=2,3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d=2,3$$\end{document}, with physically realistic in/out flow boundary conditions. We develop a new concept of weak solutions satisfying a general form of relative energy inequality. The weak solutions exist globally in time for any finite energy initial data and comply with the weak–strong uniqueness principle.
We investigate the creation and properties of eventual vacuum regions in the weak solutions of the continuity equation, in general, and in the weak solutions of compressible Navier–Stokes equations, in particular. The main results are based on the analysis of renormalized solutions to the continuity and pure transport equations and their inter-relations which are of independent interest.
Throttling addresses the question of minimizing the sum or the product of the resources used in a graph searching process and the time needed to complete the process. The study of throttling began with the study of sum throttling, and parameters that have been studied include various types of zero forcing, power domination, and cops and robbers. Recently two different definitions of product throttling have been introduced for cops and robbers and power domination. This chapter summarizes prior results for these two cases and introduces universal versions of the two definitions. Each of the definitions is then applied to each of the following parameters: standard zero forcing, positive semidefinite zero forcing, power domination, and cops and robbers.
We study the evolutionary compressible Navier–Stokes–Fourier system in a bounded two-dimensional domain with the pressure law p(ϱ,θ)∼ϱθ+ϱlogα(1+ϱ)+θ4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p(\varrho ,\theta ) \sim \varrho \theta + \varrho \log ^{\alpha }(1+ \varrho )+ \theta ^{4}$\end{document}. We consider the weak solutions with entropy inequality and total energy balance. We show the existence of this type of weak solutions without any restriction on the size of the initial conditions or the right-hand sides provided α>17+41716≅2.34\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha > \frac{17+\sqrt{417}}{16}\cong 2.34$\end{document}.
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