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In this paper we study the asymptotic stability problem for KdV solitons on the half-line, with zero boundary condition and absence of the drift term, represented as ux. Unlike standard KdV, these are not exact solutions to the equation. In a previous result, we showed that these solitons are orbitally stable, provided they are placed sufficiently far from the origin. In this paper, we prove their asymptotic stability in the energy space, and provide decay properties for all remaining regions, except the "small soliton region". For the proof we follow the ideas by Martel and Merle for the big soliton part, and for the linearly dominated region we follow recent results on generalized KdV decay [C. Muñoz and G. Ponce, Comm. Math. Phys., 367 (2019), pp. 581-598, A. [ABSTRACT FROM AUTHOR]

In this paper we consider finding a second-order stationary point (SOSP) of nonconvex equality constrained optimization when a nearly feasible point is known. In particular, we first propose a new Newton-conjugate gradient (Newton-CG) method for finding an approximate SOSP of unconstrained optimization and show that it enjoys a substantially better complexity than the Newton-CG method in [C. W. Royer, M. O'Neill, and S. J. Wright, Math. Program., 180 (2020), pp. 451-488]. We then propose a Newton-CG based augmented Lagrangian (AL) method for finding an approximate SOSP of nonconvex equality constrained optimization, in which the proposed Newton-CG method is used as a subproblem solver. We show that under a generalized linear independence constraint qualification (GLICQ), our AL method enjoys a total inner iteration complexity of O(ε7/peration complexity of O(ε7/2 min\{ n, ε3/4\}) for finding an (ε, ε)-SOSP of nonconvex equality constrained optimization with high probability, which are significantly better than the ones achieved by the proximal AL method in [Y. Xie and S. J. Wright, J. Sci. Comput., 86 (2021), pp. 1-30]. In addition, we show that it has a total inner iteration complexity of O(ε11/2) and an operation complexity of O(ε11/2 min\{ n, ε5/4\}) when the GLICQ does not hold. To the best of our knowledge, all the complexity results obtained in this paper are new for finding an approximate SOSP of nonconvex equality constrained optimization with high probability. Preliminary numerical results also demonstrate the superiority of our proposed methods over the other competing algorithms. [ABSTRACT FROM AUTHOR]

In this paper we derive a two dimensional spray model with gyroscopic effects as the mean-field limit of a system modeling the interaction between an incompressible fluid and a finite number of solid particles. This spray model has been studied by Moussa and Sueur (Asymptotic Anal., 2013), in particular the mean-field limit was established in the case of W1,∞ interactions. First we prove the local in time existence and uniqueness of strong solutions of a monokinetic version of the model with a fixed point method. Then we adapt the proof of Duerinckx and Serfaty (Duke Math. J., 2020) to establish the mean-field limit to the spray model in the monokinetic regime in the case of Coulomb interactions. [ABSTRACT FROM AUTHOR]

In this paper, we construct time periodic doubly connected solutions for the threedimensional quasi-geostrophic model in the patch setting. More specifically, we prove the existence of nontrivial m-fold doubly connected rotating patches bifurcating from a generic doubly connected revolution shape domain with higher symmetry m > mo and mo large enough. The linearized matrix operator at the equilibrium state has variable and singular coefficients and its spectral analysis is performed via the approach devised in [C. Garcia, T. Hmidi, and J. Mateu, Comm. Math. Phys., 390 (2022), pp. 617--756], where a suitable symmetrization was introduced. New difficulties emerge due to the interaction between the surfaces making the spectral problem richer and involved. [ABSTRACT FROM AUTHOR]

Geodesics are of fundamental interest in mathematics, physics, computer science, and many other subjects. The so-called leapfrog algorithm was proposed in [L. Noakes, J. Aust. Math. Soc., 65 (1998), pp. 37--50] (but not named there as such) to find geodesics joining two given points x0 and x1 on a path-connected complete Riemannian manifold. The basic idea is to choose some junctions between x0 and x1 that can be joined by geodesics locally and then adjust these junctions. It was proved that the sequence of piecewise geodesics\{γk} k≥1 generated by this algorithm converges to a geodesic joining x0 and x1. The present paper investigates leapfrog's convergence rate τi,n of ith junction depending on the manifold M. A relationship is found with the maximal root λn of a polynomial of degree n-3, where n (n >3) is the number of geodesic segments. That is, the minimal τi,n is upper bounded by λn(1+c+), where c+ is a sufficiently small positive constant depending on the curvature of the manifold M. Moreover, we show that λn increases as n increases. These results are illustrated by implementing leapfrog on two Riemannian manifolds: the unit 2-sphere and the manifold of all 2 × 2 symmetric positive definite matrices. [ABSTRACT FROM AUTHOR]

This paper concerns the time growth of the highest-order energy of the systems of incompressible isotropic Hookean elastodynamics in two space dimensions. The global well-posedness of smooth solutions near equilibrium was proved by Lei [Comm. Pure Appl. Math., 69 (2016), pp. 2072-2106] where the highest-order generalized energy may have certain growth in time. We develop novel energy estimate strategies based on the inherent structures and show that the highest-order generalized energy is uniformly bounded for all the time. [ABSTRACT FROM AUTHOR]

The small data global well-posedness of the 3D incompressible Navier--Stokes equations in R³ with only one-directional dissipation remains an outstanding open problem. The dissipation in just one direction, say, 2 1u is simply insufficient in controlling the nonlinearity in the whole space R³. The beautiful work of Paicu and Zhang [Sci. China Math., 62 (2019), pp. 1175-1204] solved the case when the spatial domain is bounded in the x1-direction by observing a crucial Poincaré-type inequality. Motivated by this Navier-Stokes open problem and by experimental observations on the stabilizing effects of background magnetic fields, this paper intends to understand the global well-posedness and stability of a special 3D magnetohydrodynamic (MHD) system near a background magnetic field. The spatial domain is R³, and the velocity in this MHD system obeys the 3D Navier-Stokes with only one-directional dissipation. With no Poincaré-type inequality, this problem appears to be impossible. By discovering the mathematical mechanism of the experimentally observed stabilizing effect and introducing several innovative techniques to deal with the derivative loss difficulties, we are able to bound the Navier-Stokes nonlinearity and solve the desired global well-posedness and stability problem. [ABSTRACT FROM AUTHOR]

We study the existence of nontrivial unbounded surfaces S ⊂ ℝ³ with the property that the constant charge distribution on S is an electrostatic equilibrium, i.e., the resulting electrostatic force is normal to the surface at each point on S . Among bounded regular surfaces S, only the round sphere has this property by a result of Reichel [Arch. Ration. Mech. Anal., 137 (1997), pp. 381--394] (see also Mendez and Reichel [Forum Math., 12 (2000), pp. 223--245]) confirming a conjecture of Gruber. In the present paper, we show the existence of nontrivial exceptional domains Ω ⊂ ℝ³ whose boundaries S=∂Ω enjoy the above property. [ABSTRACT FROM AUTHOR]

In this paper, we consider a mixed system of free boundary problem modeling multilayer tumor growth with time delay. The concept of a time delay goes back to Byrne [Math. Biosci., 144 (1997), pp. 83--117] and the formulation of a multilayer tumor model can be found in Zhou, Escher, and Cui [J. Math. Anal. Appl., 337 (2008), pp. 443--457] and the references therein. In a quasi-steady state approximation of the multilayer tumor growth model with a time delay, He, Xing, and Hu [Math. Methods Appl. Sci., to appear] established a critical value \mu \ast for which the system is linearly stable ifand linearly unstable i under nonflat perturbations. We consider the full system without quasi-steady state simplification in this paper. This is a free boundary system of mixed elliptic, parabolic, and hyperbolic equations. A key difference to our earlier work is the consideration of the full parabolic equation for the nutrient in the model, which makes the system more complex and more challenging. By applying series expansion and Laplace transform methods, we found a threshold 0 such that the unique flat stationary solution is linearly stable if \mu < \mu \ast and linearly unstable if \mu > \mu \ast under nonflat perturbations. This is accomplished with the inverse Laplace transform and a careful study of the spectrum of the system. Some of our results are presented in a more useful abstract form, which would be applicable to other systems. As far as we know, this is the first paper for the study of stability for multilayer tumor models with a time delay and with a parabolic equation for the nutrient. [ABSTRACT FROM AUTHOR]

Policy optimization, which learns the policy of interest by maximizing the value function via large-scale optimization techniques, lies at the heart of modern reinforcement learning (RL). In addition to value maximization, other practical considerations arise commonly as well, including the need of encouraging exploration, and that of ensuring certain structural properties of the learned policy due to safety, resource, and operational constraints. These considerations can often be accounted for by resorting to regularized RL, which augments the target value function with a structure-promoting regularization term. Focusing on an infinite-horizon discounted tabular Markov decision process, this paper proposes a generalized policy mirror descent (GPMD) algorithm for solving regularized RL. As a generalization of policy mirror descent [G. Lan, Math. Program., 198 (2023), pp. 1059-1106], the proposed algorithm accommodates a general class of convex regularizers as well as a broad family of Bregman divergence in cognizance of the regularizer in use. We demonstrate that our algorithm converges linearly to the global solution over an entire range of learning rates, in a dimension-free fashion, even when the regularizer lacks strong convexity and smoothness. In addition, this linear convergence feature is provably stable in the face of inexact policy evaluation and imperfect policy updates. Numerical experiments are provided to corroborate the applicability and appealing performance of GPMD. [ABSTRACT FROM AUTHOR]

The transversal number Τ (H) of a hypergraph H is the minimum number of vertices that intersect every edge of H. A 6-uniform hypergraph has all edges of size 6. On 10 November 2000 Tuza and Vestergaard [Discuss. Math. Graph Theory, 22 (2002), pp. 199-210] conjectured that if H is a 3-regular 6-uniform hypergraph of order n, then Τ(H) < 1/4n. In this paper we prove this conjecture, which has become known as the Tuza-Vestergaard conjecture. [ABSTRACT FROM AUTHOR]

While there have been many results on lower bounds for Max Cut in unweighted graphs, the only lower bound for noninteger weights is that by Poljak and Turzik [Discrete Math., 58 (1986), pp. 99--104]. In this paper, we launch an extensive study of lower bounds for Max Cut in weighted graphs. We introduce a new approach for obtaining lower bounds for Weighted Max Cut. Using it, the probabilistic method, Vizing's chromatic index theorem, and other tools, we obtain several lower bounds for arbitrary weighted graphs, weighted graphs of bounded girth, and triangle-free weighted graphs. We pose conjectures and open questions. [ABSTRACT FROM AUTHOR]

The global existence and convergence rates of solutions to the three-dimensional compressible Navier-Stokes equations without heat conductivity have been obtained since Duan and Ma [Indiana Univ. Math. J., 5 (2008), pp. 2299-2319]. However, as far as we know, how to construct the global small solutions to the compressible Navier-Stokes equations without heat conductivity in R² is still an open problem. In the present paper, we give a positive answer in Rn (n=2,3) with the initial data restricted in the Besov spaces. [ABSTRACT FROM AUTHOR]

For the Helmholtz equation posed in the exterior of a Dirichlet obstacle, we prove that if there exists a family of quasimodes (as is the case when the exterior of the obstacle has stable trapped rays), then there exist near-zero eigenvalues of the standard variational formulation of the exterior Dirichlet problem (recall that this formulation involves truncating the exterior domain and applying the exterior Dirichlet-to-Neumann map on the truncation boundary). Our motivation for proving this result is that (a) the finite-element method for computing approximations to solutions of the Helmholtz equation is based on the standard variational formulation, and (b) the location of eigenvalues, and especially near-zero ones, plays a key role in understanding how iterative solvers such as the generalized minimum residual method (GMRES) behave when used to solve linear systems, in particular those arising from the finite-element method. The result proved in this paper is thus the first step towards rigorously understanding how GMRES behaves when applied to discretizations of high-frequency Helmholtz problems under strong trapping (the subject of the companion paper [P. Marchand et al., Adv. Comput. Math., to appear]). [ABSTRACT FROM AUTHOR]

This paper describes the structure of solutions to Kolmogorov's equations for nonhomogeneous jump Markov processes and applications of these results to control of jump stochastic systems. These equations were studied by Feller [Trans. Amer. Math. Soc., 48 (1940), pp. 488--515], who clarified in 1945 in the errata to that paper that some of its results covered only nonexplosive Markov processes. In this work, which is largely of a survey nature, the case of explosive processes is also considered. This paper is based on the invited talk presented by the authors at the conference "P. L. Chebyshev -- 200," and it describes the results of their joint studies with Manasa Mandava (1984--2019). [ABSTRACT FROM AUTHOR]

In this paper, we investigate representations of links that are either centrally symmetric in R3 or antipodally symmetric in S3. By using the notions of antipodally self-dual and antipodally symmetric maps, introduced and studied by the authors in [L. Montejano, J. L. Ramírez Alfonsín, and I. Rasskin, SIAM J. Discrete Math., 36 (2022), pp. 1551-1566], we are able to present sufficient combinatorial conditions for a link L to admit such representations. The latter naturally provide sufficient conditions for L to be amphichiral. We also introduce another (closely related) method yielding again sufficient conditions for L to be amphichiral. We finally prove that a link L, associated to a map G, is amphichiral if the self-dual pairing of G is not one of 6 specific cases among the classification of the 24 self-dual pairing Cor(G)⊳Aut(G). [ABSTRACT FROM AUTHOR]

In this paper, we adopt the augmented Lagrangian method (ALM) to solve convex quadratic second-order cone programming problems (SOCPs). Fruitful results on the efficiency of the ALM have been established in the literature. Recently, it has been shown in [Cui, Sun, and Toh, Math. Program., 178 (2019), pp. 381-415] that if the quadratic growth condition holds at an optimal solution for the dual problem, then the KKT residual converges to zero R-superlinearly when the ALM is applied to the primal problem. Moreover, Cui, Ding, and Zhao [SIAM J. 0ptim., 27 (2017), pp. 2332-2355] provided sufficient conditions for the quadratic growth condition to hold under the metric subregularity and bounded linear regularity conditions for solving composite matrix optimization problems involving spectral functions. Here, we adopt these recent ideas to analyze the convergence properties of the ALM when applied to SOCPs. To the best of our knowledge, no similar work has been done for SOCPs so far. In our paper, we first provide sufficient conditions to ensure the quadratic growth condition for SOCPs. With these elegant theoretical guarantees, we then design an SOCP solver and apply it to solve various classes of SOCPs, such as minimal enclosing ball problems, classical trust-region subproblems, square-root Lasso problems, and DIMACS Challenge problems. Numerical results show that the proposed ALM based solver is efficient and robust compared to the existing highly developed solvers, such as Mosek and SDPT3. [ABSTRACT FROM AUTHOR]

In this paper, we develop, analyze and numerically test an invariant-preserving three-level linearized implicit difference scheme for a rotation-two-component Camassa--Holm system [L. Fan, H. Gao, and Y. Liu, Adv. Math. 291 (2016), pp. 59-89], which contains strongly nonlinear terms and high-order derivative terms. We prove that the numerical scheme is uniquely solvable and second-order convergent for both the spatial and temporal discretizations. Optimal error estimates for the velocity in the L∞-norm and for the surface elevation in the L2-norm are obtained under a suitable step-size ratio restriction based on energy analysis. In particular, the analysis for the strongly nonlinear term is carried out by splitting it into several recursive expressions. Moreover, the numerical scheme preserves at least two conservation invariants: mass and energy. Extensive numerical experiments including long time simulations for zero/nonzero rotation parameters demonstrate solution behavior and verify the convergence results as well as mass/energy conservation. [ABSTRACT FROM AUTHOR]

In this paper, it is shown that there does not exist a nontrivial Leray backward self-similar solution to the three-dimensional (3D) Navier-Stokes equations with profiles in Morrey spaces Ṁq,1 (ℝ³) provided 3/2 < q < 6, or in Ṁq,1 (ℝ³) provided 6 ≤ q < ∞ and 2 < l ≤ q. This generalizes the corresponding results obtained by Nečas, Råužička, and Šverák [ Acta. Math., 176 (1996), pp. 283--294] in L³ (ℝ³); Tsai [Arch. Ration. Mech. Anal., 143 (1998), pp. 29-51] in Lp(ℝ³) with p ≥ 3; Chae and Wolf [Arch. Ration. Mech. Anal., 225 (2017), pp. 549-572] in Lorentz spaces Lp,∞ (ℝ³) with p > 3/2; and Guevara and Phuc [SIAM J. Math. Anal., 50 (2018), pp. 541--556] in Ṁq,12-2q/3 (ℝ³) with 12/5 ≤ q < 3 and in Lq,∞ (ℝ³) with 12/5 ≤ q < 6. [ABSTRACT FROM AUTHOR]

In this paper, we propose an embedded low-regularity integrator (ELRI) under a new framework for solving the modified Korteweg-de Vries (mKdV) equation under rough data. Different from the previous work [Wu and Zhao, BIT, Number. Math., (2021)], the present ELRI scheme is constructed based on an approximation of a scaled Schrödinger operator and a new strategy of iterative regularizing through the inverse Miura transform. Moreover, the ELRI scheme is explicitly defined in the physical space, and it is efficient under the Fourier pseudospectral discretization. By rigorous error analysis, we show that ELRI achieves first-order accuracy by requiring the boundedness of one additional spatial derivative of the solution. Numerical results are presented to show the accuracy and efficiency of ELRI. [ABSTRACT FROM AUTHOR]

This paper presents the first family of conforming finite element div div complexes on tetrahedral grids in three dimensions. In these complexes, finite element spaces of H(div div, Ω; S) are from a recent article [L. Chen and X. Huang, Math. Comp., 91 (2022), pp. 1107-1142] while finite element spaces of both H(symcurl, Ω; T) and H¹( Ω; ℝ³) are newly constructed here. It is proved that these finite element complexes are exact. As a result, they can be used to discretize the linearized Einstein-Bianchi system within the dual formulation. [ABSTRACT FROM AUTHOR]

A modification of the Lindeberg and Rotar' conditions was considered in the papers by Presman and Formanov [Dokl. Math., 99 (2019), pp. 204-207] and [Dokl. Ross. Akad. Nauk Ser. Mat., 485 (2019), pp. 548-552 (in Russian)]. This modification was concerned with the sums of absolute (respectively, difference) moments of order 2 + α for the distributions of the summands truncated at the unit level. It was shown that, when checking the normal convergence, it is sufficient, instead of checking the convergence to zero of the Lindeberg or Rotar' characteristics for any ε > 0, to check that there exists an α > 0 such that a characteristic (introduced in these papers) corresponding to this α converges to zero. Moreover, from the existence of such α it follows that the characteristic corresponding to any α > 0 also tends to zero. We show that the moment functions can be changed to more general functions and describe the class of such functions. [ABSTRACT FROM AUTHOR]

This paper is devoted to the construction of generalized multiscale Young measures, which are the extension of Pedregal's multiscale Young measures [Trans. Amer. Math. Soc., 358 (2006), pp. 591-602] to the setting of generalized Young measures introduced by DiPerna and Majda [Comm. Math. Phys., 108 (1987), pp. 667-689]. As a tool for variational problems, these are well-suited objects for the study (at different length-scales) of oscillation and concentration effects of convergent sequences of measures. Important properties of multiscale Young measures such as compactness, representation of nonlinear compositions, localization principles, and differential constraints are extensively developed in the second part of this paper. As an application, we use this framework to address the Γ-limit characterization of the homogenized limit of convex integrals defined on spaces of measures satisfying a general linear PDE-constraint. [ABSTRACT FROM AUTHOR]

This paper provides a probabilistic proof of the comparison result for viscosity solutions of path-dependent semilinear PDEs. We consider the notion of viscosity solutions introduced in [I. Ekren, et al., Ann. Probab., 42 (2014), pp. 204-236], which considers as test functions all those smooth processes which are tangent in mean. When restricted to the Markovian case, this definition induces a larger set of test functions and reduces to the notion of stochastic viscosity solutions analyzed in [E. Bayraktar and M. Sirbu, Proc. Amer. Math. Soc., 140 (2012), pp. 3645-3654; SIAM J. Control Optim., 51 (2013), pp. 4274-4294]. Our main result takes advantage of this enlargement of the test functions and provides an easier proof of comparison. This is most remarkable in the context of the linear path-dependent heat equation. As a key ingredient for our methodology, we introduce a notion of punctual differentiation, similar to the corresponding concept in the standard viscosity solutions [L. A. Caffarelli and X. Cabre, Amer. Math. Soc. Colloq. Publ., 43, AMS, Providence, RI, 1995], and we prove that semimartingales are almost everywhere punctually differentiable. This smoothness result can be viewed as the counterpart of the Aleksandroff smoothness result for convex functions. A similar comparison result was established earlier in [I. Ekren et al., Ann. Probab., 42 (2014), pp. 204-236]. The result of this paper is more general and, more importantly, the arguments that we develop do not rely on any representation of the solution. [ABSTRACT FROM AUTHOR]

Most undergraduates have limited experience with mathematical modeling. In an effort to respond to various initiatives, such as the recommendations outlined in [S. Garfunkel and M. Montgomery, eds., GAIMME: Guidelines for Assessment & Instruction in Mathematical Modeling Education, SIAM, 2016], this paper describes a course on the mathematical models of cancer growth and treatment. Among its aims is to provide a template for a "bridge" course between the traditional calculus and differential equations sequence and more advanced courses in mathematics and statistics. Prerequisites include a course in ordinary differential equations. Linear algebra is a useful corequisite but no previous programming experience is required. The content includes classical models of tumor growth as well as models for the growth of specific cancer types. Relevant research articles are provided for further study. Material for student projects and effective communication is supplied, as well as suggestions for homework assignments and computer labs. This paper aims to assist instructors in developing their own "Mathematics + Cancer" course. [ABSTRACT FROM AUTHOR]

This paper provides a probabilistic proof of the comparison result for viscosity solutions of path-dependent semilinear PDEs. We consider the notion of viscosity solutions introduced in [I. Ekren, et al., Ann. Probab., 42 (2014), pp. 204-236], which considers as test functions all those smooth processes which are tangent in mean. When restricted to the Markovian case, this definition induces a larger set of test functions and reduces to the notion of stochastic viscosity solutions analyzed in [E. Bayraktar and M. Sirbu, Proc. Amer. Math. Soc., 140 (2012), pp. 3645-3654; SIAM J. Control Optim., 51 (2013), pp. 4274-4294]. Our main result takes advantage of this enlargement of the test functions and provides an easier proof of comparison. This is most remarkable in the context of the linear path-dependent heat equation. As a key ingredient for our methodology, we introduce a notion of punctual differentiation, similar to the corresponding concept in the standard viscosity solutions [L. A. Caffarelli and X. Cabre, Amer. Math. Soc. Colloq. Publ., 43, AMS, Providence, RI, 1995], and we prove that semimartingales are almost everywhere punctually differentiable. This smoothness result can be viewed as the counterpart of the Aleksandroff smoothness result for convex functions. A similar comparison result was established earlier in [I. Ekren et al., Ann. Probab., 42 (2014), pp. 204-236]. The result of this paper is more general and, more importantly, the arguments that we develop do not rely on any representation of the solution. [ABSTRACT FROM AUTHOR]

The Jones eigenvalue problem first described in [D. S. Jones, Quart. J. Mech. Appl. Math., 36 (1983), pp. 111--138] concerns unusual modes in bounded elastic bodies--time-harmonic displacements whose tractions and normal components are both identically zero on the boundary. This problem is usually associated with a lack of unique solvability for certain models of fluidstructure interaction. The boundary conditions in this problem appear, at first glance, to rule out any nontrivial modes unless the domain possesses significant geometric symmetries. Indeed, Jones modes were shown to not be possible in most C∞ domains in [T. Hargé, C. R. Acad. Sci. Paris Sér. I Math., 311 (1990), pp. 857--859]. However, in this paper we will see that while the existence of Jones modes sensitively depends on the domain geometry, such modes do exist in a broad class of domains. This paper presents the first detailed theoretical and computational investigation of this eigenvalue problem in Lipschitz domains. We also analytically demonstrate Jones modes on some simple geometries. [ABSTRACT FROM AUTHOR]

We prove global existence of weak solutions for a version of the one velocity Baer--Nunziato system with dissipation describing a mixture of two noninteracting viscous compressible fluids in a piecewise regular Lipschitz domain with general inflow/outflow boundary conditions. The geometrical setting is general enough to comply with most current domains important for applications, such as (curved) pipes of piecewise regular and axis-dependent cross-sections. For the existence proof, we adapt to the system the classical Lions--Feireisl approach to the compressible Navier--Stokes equations which is combined with a generalization of the theory of renormalized solutions to the transport equations in the spirit of Vasseur, Wen, and Yu [J. Math. Pures Appl. (9), 125 (2019), pp. 247-282]. The results related to the families of transport equations presented in this paper extend/improve some statements of the theory of renormalized solutions and are therefore of independent interest. [ABSTRACT FROM AUTHOR]

Let r \geq k \geq 2 and K(k) r,n denote the complete n-balanced r-partite k-uniform hypergraph, whose vertex set consists of r parts, each has n vertices, and whose edge set contains all the k-element subsets with no two vertices from one part. A decomposition of K(k) r,n is a partition of E(K(k) r,n). A perfect matching (resp., Hamilton tight cycle) decomposition of K(k) r,n is a decomposition of K(k) r,n into perfect matchings (resp., Hamilton tight cycles). In this paper, we prove that if k | n (resp., 2 - k and k | n), then K(k) k+1,n (resp., K(k) k+2,n) has a perfect matching decomposition. We also prove that for any integer k \geq 2, K(k) k+1,n has a Hamilton tight cycle decomposition. In all cases, we use constructive methods involving number theory. In fact, we confirm two conjectures proposed by Zhang, Lu, and Liu [Appl. Math. Comput., 386 (2020), 125492]. [ABSTRACT FROM AUTHOR]

Convergence of Dziuk's fully discrete linearly implicit scheme which was proposed in [G. Dziuk, Math. Models Methods Appl. Sci., 4 (1994), pp. 589--606] for curve shortening flow still remains open. In this paper, we prove that this scheme has an optimal error estimate in the H¹-norm. [ABSTRACT FROM AUTHOR]

In this paper, we complement the framework of bilevel unconstrained minimization (BLUM) [Y. Nesterov, Math. Program., to appear] by a new pth-order proximal-point method convergent as O(k-(3p+1)/2), where k is the iteration counter. As compared with [Y. Nesterov, Math. Program., to appear], we replace the auxiliary line search by a segment search. This allows us to bound its complexity by a logarithm of the desired accuracy. Each step in this search needs an approximate computation of a high-order proximal-point operator. Under the assumption on boundedness of (p+1)th derivative of the objective function, this can be done by one step of the pthorder augmented tensor method. In this way, for p = 2, we get a new second-order method with the rate of convergence O(k-7/2) and logarithmic complexity of the auxiliary search at each iteration. Another possibility is to compute the proximal-point operator by lower-order minimization methods. As an example, for p = 3, we consider the upper-level process convergent as O(k-5). Assuming the boundedness of fourth derivative, an appropriate approximation of the proximal-point operator can be computed by a second-order method in logarithmic number of iterations. This combination gives a second-order scheme with much better complexity than the existing theoretical limits. [ABSTRACT FROM AUTHOR]

In this paper we propose a method for computing the Faddeeva function w(z):... erfc(-iz) via truncated modified trapezoidal rule approximations to integrals on the real line. Our starting point is the method due to Matta and Reichel (Math. Comp. 25 (1971), pp. 339-344) and Hunter and Regan (Math. Comp. 26 (1972), pp. 339-541). Addressing shortcomings flagged by Weideman (SIAM. J. Numer. Anal. 31 (1994), pp. 1497-1518), we construct approximations which we prove are exponentially convergent as a function of N+1, the number of quadrature points, obtaining error bounds which show that accuracies of 2x10-15 in the computation of w(z) throughout the complex plane are achieved with N=11, this confirmed by computations. These approximations, moreover, provably achieve small relative errors throughout the upper complex half-plane where w(z) is non-zero. Numerical tests suggest that this new method is competitive, in accuracy and computation times, with existing methods for computing w(z) for complex z [ABSTRACT FROM AUTHOR]

In this paper, we investigate properties of entropy-penalized Wasserstein barycenters introduced in [J. Bigot, E. Cazelles, and N. Papadakis, SIAM J. Math. Anal., 51 (2019), pp. 2261-2285] as a regularization of Wasserstein barycenters [M. Agueh and G. Carlier, SIAM J. Math. Anal., 43 (2011), pp. 904-924]. After characterizing these barycenters in terms of a system of Monge--Ampère equations, we prove some global moment and Sobolev bounds as well as higher regularity properties. We finally establish a central limit theorem for entropic-Wasserstein barycenters. [ABSTRACT FROM AUTHOR]

Nonunique weak solutions in Leray--Hopf class are constructed for the threedimensional magneto-hydrodynamics with Hall effect. We adapt the widely appreciated convex integration framework developed in a recent work of Buckmaster and Vicol [Ann. of Math., 189 (2019), pp. 101-144] for the Navier--Stokes equation and with deep roots in a sequence of breakthrough papers for the Euler equation. [ABSTRACT FROM AUTHOR]

In this paper we introduce a general framework for time-inconsistent optimal control problems. We characterize the closed-loop equilibrium strategy in both the integral and pointwise forms with the newly developed methodology. We recover and improve the results of some wellknown models, including the classical optimal control, Bjork, Khapko, and Murgoci [Finance Stoch., 21 (2017), pp. 331-360], He and Jiang, preprint, ssrn:3308274, 2020, and Yong [Math. Control Related Fields, 2 (2012), pp. 271-329] models, and reveal some interesting aspects that appear for the first time in the literature. We illustrate the usefulness of the model and the results by a number of examples in dynamic portfolio selection, including mean-variance with state-dependent risk aversion, investment/consumption with nonexponential discounting, and utility-deviation-risk with coupled terminal state and expected terminal state. [ABSTRACT FROM AUTHOR]

In this paper, we consider a two-dimensional nonresistive magnetohydrodynamic model, taking the fluctuation of absolute temperature into account. Combining the method of weak convergence developed by Lions [Mathematical Topics in Fluid Mechanics: Volume 2: Compressible Models, Clarendon Press, Oxford, UK, 1998] and Feireisl et al. [E. Feireisl and A. Novotný, Singular Limits in Thermodynamics of Viscous Fluids, Birkhauser Verlag, Basel, 2009; E. Feireisl, A. Novotn ý, and H. Petzeltov á, J. Math. Fluid Mech., 3 (2001), pp. 358--392] from compressible Navier--Stokes(--Fourier) system and the new technique of variable reduction proposed by Vasseur, Wen, and Yu [J. Math. Pure. Appl., 125 (2019), pp. 247--282] and refined by Novotný and Pokorný [Arch. Ration. Mech. Anal., 235 (2020), pp. 355--403] from compressible two-fluid models, weak solutions are shown to exist globally in time with finite energy initial data. The result is the first one on global solvability to a full compressible, viscous, nonresistive magnetohydrodynamic system in multidimensions with large initial data. [ABSTRACT FROM AUTHOR]

In this paper we generalize a weak sequential result of [C. Fan, Int. Math. Res. Not. IMRN, 2021 (2021), pp. 4864--4906] to any nonscattering solutions in one dimension. No symmetry assumptions are required for the initial data. [ABSTRACT FROM AUTHOR]

The article discusses research featured in the Expository Research Papers section including one by Laurent Demanet and Lexing Ying on the efficient representations for functions of elliptic operators and another by Jeffrey Blanchard, Coralia Cartis and Jared Tanner on restricted isometry property.

In this paper, through an intuitive vanil la proximal method perspective, we derive a concise unified acceleration framework (UAF) for minimizing a convex function that has H\"older continuous derivatives with respect to general (non-Euclidean) norms. The UAF reconciles two different high-order acceleration approaches, one by Nesterov [Math. Program., 112 (2008), pp. 159-181] and one by Monteiro and Svaiter [SIAM J. Optim., 23 (2013), pp. 1092-1125]. As a result, the UAF unifies the high-order acceleration instances of the two approaches by only two problem-related parameters and two additional parameters for framework design. Meanwhile, the UAF (and its analysis) is the first approach to make high-order methods applicable for high-order smoothness conditions with respect to non-Euclidean norms. Furthermore, the UAF is the first approach that can match the existing lower bound of iteration complexity for minimizing a convex function with H\"older continuous derivatives. For practical implementation, we introduce a new and effective heuristic that significantly simplifies the binary search procedure required by the framework. We use experiments on logistic regression to verify the effectiveness of the heuristic. Finally, the UAF is proposed directly in the general composite convex setting and shows that the existing high-order algorithms can be naturally extended to the general composite convex setting. [ABSTRACT FROM AUTHOR]

For two graphs $F$ and $H$, the relative Turán number ${ex}(H,F)$ is the maximum number of edges in an $F$-free subgraph of $H$. Foucaud, Krivelevich, and Perarnau [SIAM J. Discrete Math., 29 (2015), pp. 65--78] and Perarnau and Reed [Combin. Probab. Comput., 26 (2017), pp. 448--467] studied these quantities as a function of the maximum degree of $H$. In this paper, we study a generalization for uniform hypergraphs. If $F$ is a complete $r$-partite $r$-uniform hypergraph with parts of sizes $s_1,s_2,\dots,s_r$ with each $s_{i + 1}$ sufficiently large relative to $s_i$, then with $1/\beta = \sum_{i = 2}^r \prod_{j = 1}^{i - 1} s_j$ we prove that for any $r$-uniform hypergraph $H$ with maximum degree $\Delta$, ${ex}(H,F)\ge \Delta^{-\beta - o(1)} \cdot e(H).$ This is tight as $\Delta \rightarrow \infty$ up to the $o(1)$ term in the exponent, since we show there exists a $\Delta$-regular $r$-graph $H$ such that ${ex}(H,F)=O(\Delta^{-\beta}) \cdot e(H)$. Similar tight results are obtained when $H$ is the random $n$-vertex $r$-graph $H_{n,p}^r$ with edge-probability $p$, extending results of Balogh and Samotij [J. Lond. Math. Soc., 83 (2011), pp. 1091--1094] and Morris and Saxton [Adv. Math., 298 (2016), pp. 534--580]. General lower bounds for a wider class of $F$ are also obtained. [ABSTRACT FROM AUTHOR]

This is a continuation and an extension of our recent work [J. Math. Pures Appl., 143 (2020), pp. 116-161] on the geometric structures of Laplacian eigenfunctions and their applications to inverse scattering problems. In that work, we studied the analytic behavior of the Laplacian eigenfunctions at a point where two nodal or generalized singular lines intersect. The results reveal an important and intriguing property that the vanishing order of the eigenfunction at the intersecting point is closely related to the rationality of the intersecting angle. In this paper, we continue this development in three dimensions and study the analytic behaviors of the Laplacian eigenfunctions at places where nodal or generalized singular planes intersect. Compared with the two-dimensional case, the geometric situation is much more complicated, and so is the corresponding analysis: the intersection of two planes generates an edge corner, whereas the intersection of more than three planes generates a vertex corner. We provide a systematic and comprehensive characterization of the relations between the analytic behaviors of an eigenfunction at a corner point and the geometric quantities of that corner for all these geometric cases. Moreover, we apply the spectral results to establish some novel unique identifiability results for the geometric inverse problems of recovering the shape as well as the (possible) surface impedance coefficient by the associated scattering far-field measurements. [ABSTRACT FROM AUTHOR]

The paper contains an elegant approach for eliminating the problems caused by idempotents in the arithmetics of probability distribution functions over Rd where the operation is the coordinate wise maximum of the corresponding random vectors. This method is, in fact, an application of the general approach in Ruzsa and Szekely. In the article the notion of \hair"=\maximal idempotent divisor" H(s) for any element s in abstract semigroups S was introduced and a new semigroup H(s) S was considered to settle several problems much easier than directly in S, e.g., the characterization of the class max-I0.

Let B(k) n be the book graph which consists of n copies of Kk+1 all sharing a common Kk, and let Cm be a cycle of length m. In this paper, we first determine the exact value of r(B(2) n,Cm) for 8 9n + 112 leq m leq lceil 3n 2 rceil + 1 and n geq 1000. This answers a question of Faudree, Rousseau, and Sheehan [Ars Combin., 31 (1991), pp. 239--248] in a stronger form when m and n are large. Building upon this exact result, we are able to determine the asymptotic value of r(B(k) n,Cn) for each k geq 3. Namely, we prove that for each k geq 3, r(B(k) n ,Cn) = (k + 1 + ok(1))n. This extends a result due to Rousseau and Sheehan [J. London Math. Soc., 18 (1978), pp. 392--396]. [ABSTRACT FROM AUTHOR]

M. A. Henning and A. Yeo conjectured in [SIAM J. Discrete Math., 26 (2012), pp. 687{694] that a bipartite digraph of minimum out-degree at least 3 contains two vertex disjoint directed cycles of different lengths. In this paper, we disprove this conjecture. Further, we classify strong tournaments and strong bipartite tournaments of minimum out-degree 3 without two vertex disjoint directed cycles of different lengths. [ABSTRACT FROM AUTHOR]

In this paper, we analyze a class of reaction-diffusion systems with delay and a Neumann condition from the dynamical behavior of the map that determines the equilibria. For scalar equations, a similar analysis was given by Yi and Zou in [Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), pp. 2955--2973]. There are strong differences between the scalar equations and the systems. We show with an example that the direct extension of the results by Yi and Zou in [Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), pp. 2955--2973] to systems is not possible. This forces us to introduce additional assumptions. However, our approach is sufficiently general to be applicable to a wide variety of models, including neural networks and population dynamics. In some cases, our technique gives the optimal conditions for the global attraction. [ABSTRACT FROM AUTHOR]

We consider a parabolic-parabolic Keller--Segel system in a ball of RN under the Neumann boundary condition. This was introduced as a model of aggregation of bacteria. The aggregation is mathematically defined as finite-time blowup. When N = 2, an optimal criterion for finite-time blowup was obtained in [N. Mizoguchi and M. Winkler, Boundedness of Global Solutions in the Two-Dimensional Parabolic Keller--Segel System, preprint]. On the other hand, there has been no criterion for finite-time blowup for N \geq 3 though existence of radial solutions blowing up in finite time was known due to [M. Winkler, J. Math. Pures Appl., 100 (2013), pp. 748--767]. In this paper, focusing on common nature in all dimensions, we give a criterion for finite-time blowup for N = 2, 3, 4. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of minimizing a smooth objective over multiple rank constraints on Hankel structured matrices. These kinds of problems arise in system identification, system theory, and signal processing, where the rank constraints are typically "hard constraints." To solve these problems, we propose a hybrid penalty method that combines a penalty method with a postprocessing scheme. Specifically, we solve the penalty subproblems until the penalty parameter reaches a given threshold, and then switch to a local alternating pseudoprojection"" method to further reduce constraint violation. Pseudoprojection is a generalization of the concept of projection. We show that a pseudoprojection onto a single low-rank Hankel structured matrix constraint can be computed efficiently by existing software such as SLRA [I. Markovsky and K. Usevich, J. Comput. Appl. Math., 256 (2014), pp. 278--292], under mild assumptions. We also demonstrate how the penalty subproblems in the hybrid penalty method can be solved by pseudoprojection-based optimization methods, and then present some convergence results for our hybrid penalty method. Finally, the efficiency of our method is illustrated by numerical examples. [ABSTRACT FROM AUTHOR]

A graph is said to be open-neighborhood conflict-free k-colorable if there exists an assignment of k different colors to some of the vertices such that, for every vertex v, there is a color that is assigned to exactly one vertex among the neighbors of v. The open-neighborhood conflict-free chromatic number χO(G) is the smallest k for which G is open-neighborhood conflict-free k-colorable. Z. Abel et al., [SIAM J. Discrete Math. 32 (2018), pp. 2675--2702] showed that χ O(G) ≤ 8 for every planar graph G and posed two open problems to ask whether χO(G) \leq 4 for every planar graph G and whether χO(G) ≤ 3 for every outerplanar graph G. We present in this paper positive answers for the above two open problems by establishing a stronger result which states that, for every integer k ≥ 2, every minor-k-colorable graph is open-neighborhood conflict-free k-colorable, where a graph G is said to be minor-k-colorable if every minor of G is k-colorable. [ABSTRACT FROM AUTHOR]

A classic theorem of Erdös and Pósa [Canad. J. Math., 17 (1965), pp. 347--352] states that every graph has either k vertex-disjoint cycles or a set of O(k log k) vertices meeting all its cycles. While the standard proof revolves around finding a large 'frame" in the graph (a subdivision of a large cubic graph), an alternative way of proving this theorem is to use a ball packing argument of Kühn and Osthus [Random Structures Algorithms, 22 (2003), pp. 213--225] and Diestel and Rempel [Combinatorica, 25 (2005), pp. 111--116]. In this paper, we argue that the latter approach is particularly well suited for studying edge variants of the Erdös-Pósa theorem. As an illustration, we give a short proof of a theorem of Bruhn, Heinlein, and Joos [Combinatorica, 39 (2019), pp. 1--36] that cycles of length at least l have the so-called edge-Erdös-Pósa property. More precisely, we show that every graph G contains either k edge-disjoint cycles of length at least l or an edge set F of size O(kl . log(kl)) such that G-F has no cycle of length at least l, For fixed l, this improves on the previously best known bound of O(k2 log k + kl). [ABSTRACT FROM AUTHOR]

This paper examines bounds on upper tails for cycle counts in Gn,p. For a fixed graph H define ξH =ξn,p H to be the number of copies of H in Gn,p. It is a much studied and surprisingly difficult problem to understand the upper tail of the distribution of ξH, for example, to estimate P(ξH > 2EξH). The best known result for general H and p is due to Janson, Oleszkiewicz, and Ruciński [Israel J. Math., 142 (2004), pp. 61--92], who proved that exp[ OH,η (MH(n, p) ln(1/p))] < P(ξH > (1 + η )EξH) < exp[-Ω H,η (MH(n, p))]. Thus they determined the upper tail up to a factor of ln(1/p) in the exponent. There has since been substantial work to improve these bounds for particular H and p. We close the ln(1/p) gap for cycles, up to a constant in the exponent. Here the lower bound stated above is accurate for l-cycles when p > ln¹/(l-2)n/n. [ABSTRACT FROM AUTHOR]