Existence and Asymptotic Behavior of Solutions for the Unstirred Chemostat Model with Ratio-Dependent Function

A system of reaction-diffusion equations arising from the unstirred chemostat model with ratio-dependent function is considered. The asymptotic behavior of solutions is given and all positive steady-state solutions to this model lie on a single smooth solution curve. It turns out that the ratio-dependence effect will not affect the dynamics, compared with (Hsu and Waltman in SIAM J. Appl. Math. 53(4):1026–1044, 1993) and (Nie and Wu in Sci. China Math. 56(10):2035–2050, 2013).

     

An Infinite Antichain of Planar Tanglegrams

Contrary to the expectation arising from the tanglegram Kuratowski theorem of Czabarka et al. (SIAM J. Discrete Math. 31(3), 1732–1750, 2017), we construct an infinite antichain of planar tanglegrams with respect to the induced subtanglegram partial order. R.E. Tarjan, R. Laver, D.A. Spielman and M. Bóna, and possibly others, showed that the partially ordered set of finite permutations ordered by deletion of entries contains an infinite antichain, i.e., there exists an infinite collection of permutations, such that none of them contains another as a pattern. Our construction adds a twist to the construction of Spielman and Bóna (Electr. J. Comb. 7, N2, 2000).

     

New subspace minimization conjugate gradient methods based on regularization model for unconstrained optimization

In this paper, two new subspace minimization conjugate gradient methods based on p-regularization models are proposed, where a special scaled norm in p-regularization model is analyzed. Different choices of special scaled norm lead to different solutions to the p-regularized subproblem. Based on the analyses of the solutions in a two-dimensional subspace, we derive new directions satisfying the sufficient descent condition. With a modified nonmonotone line search, we establish the global convergence of the proposed methods under mild assumptions. R-linear convergence of the proposed methods is also analyzed. Numerical results show that, for the CUTEr library, the proposed methods are superior to four conjugate gradient methods, which were proposed by Hager and Zhang (SIAM J. Optim. 16(1):170–192, 2005), Dai and Kou (SIAM J. Optim. 23(1):296–320, 2013), Liu and Liu (J. Optim. Theory. Appl. 180(3):879–906, 2019) and Li et al. (Comput. Appl. Math. 38(1):2019), respectively.

     

Cohen positive strongly p-summing and p-convex multilinear operators

The aim of this work is to give and study the notion of Cohen positive p-summing multilinear operators. We prove a natural analog of the Pietsch domination theorem for these classes and characterize their conjugates. As an application, we generalize a result due to Bu and Shi (J. Math. Anal. Appl. 401:174–181, 2013), and we compare this class with the class of multiple p-convex m-linear operators.

     

Two Tilts of Higher Spherical Algebras

We introduce and study two exotic families of finite-dimensional algebras over an algebraically closed field. We prove that every such an algebra is derived equivalent to a higher spherical algebra studied by Erdmann and Skowroński (Arch. Math. 114, 25–39, 2020), and hence that it is a tame symmetric periodic algebra of period 4. This together with the results of Erdmann and Skowroński (Algebr. Represent. Theor. 22, 387–406, 2019; Arch. Math. 114, 25–39, 2020) shows that every trivial extension algebra of a tubular algebra of type (2,2,2,2) admits a family of periodic symmetric higher deformations which are tame of non-polynomial growth and have the same Gabriel quiver, answering the question recently raised by Skowroński.

     

Some characterizations of magnetic Sobolev spaces

ABSTRACT

The aim of this note is to survey recent results contained in Nguyen H-M, Squassina M. [On anisotropic Sobolev spaces. Commun Contemp Math, to appear. DOI:10.1142/S0219199718500177]; Nguyen H-M, Pinamonti A, Squassina M, et al. [New characterizations of magnetic Sobolev spaces. Adv Nonlinear Anal. 2018;7(2):227–245]; Pinamonti A, Squassina M, Vecchi E. [Magnetic BV functions and the Bourgain-Brezis-Mironescu formula. Adv Calc Var, to appear. DOI:10.1515/acv-2017-0019]; Pinamonti A, Squassina M, Vecchi E. [The Maz'ya-Shaposhnikova limit in the magnetic setting. J Math Anal Appl. 2017;449:1152–1159] and Squassina M, Volzone B. [Bourgain-Brezis-Mironescu formula for magnetic operators. C R Math Acad Sci Paris. 2016;354:825–831], where the authors extended to the magnetic setting several characterizations of Sobolev and BV functions.     

A wide neighborhood interior-point algorithm for linear optimization based on a specific kernel function

This paper presents an interior point algorithm for solving linear optimization problems in a wide neighborhood of the central path introduced by Ai and Zhang (SIAM J Optim 16:400–417, 2005). In each iteration, the algorithm computes the new search directions by using a specific kernel function. The convergence of the algorithm is shown and it is proved that the algorithm has the same iteration bound as the best short-step algorithms. We demonstrate the computational efficiency of the proposed algorithm by testing some Netlib problems in standard form. To best our knowledge, this is the first wide neighborhood path-following interior-point method with the same complexity as the best small neighborhood path-following interior-point methods that uses the kernel function.

     

Competitive Exclusion for Chemostat Equations with Variable Yields

In this paper, we study the global dynamics of a chemostat model with a single nutrient and several competing species. Growth rates are not required to be proportional to food uptakes. Our approach is based on the construction of Lyapunov functions. The Lyapunov functions extend those used by Hsu (SIAM J. Appl. Math. 34:760–763, 1978) and by Wolkowicz and Lu (SIAM J. Appl. Math. 57:1019–1043, 1997) in the case when growth rates are proportional to food uptakes. Our result generalizes a large variety of previous results obtained by Lyapunov techniques.     

On inexact generalized proximal methods with a weakened error tolerance criterion

Two inexact versions of a Bregman-function-based proximal method for finding a zero of a maximal monotone operator, suggested in [J. Eckstein (1998). Approximate iterations in Bregman-function-based proximal algorithms. Math. Programming, 83, 113–123; P. da Silva, J. Eckstein and C. Humes (2001). Rescaling and stepsize selection in proximal methods using separable generalized distances. SIAM J. Optim., 12, 238–261], are considered. For a wide class of Bregman functions, including the standard entropy kernel and all strongly convex Bregman functions, convergence of these methods is proved under an essentially weaker accuracy condition on the iterates than in the original papers.

Also the error criterion of a logarithmic–quadratic proximal method, developed in [A. Auslender, M. Teboulle and S. Ben-Tiba (1999). A logarithmic-quadratic proximal method for variational inequalities. Computational Optimization and Applications, 12, 31–40], is relaxed, and convergence results for the inexact version of the proximal method with entropy-like distance functions are described.

For the methods mentioned, like in [R.T. Rockafellar (1976). Monotone operators and the proximal point algorithm. SIAM J. Control Optim., 14, 877–898] for the classical proximal point algorithm, only summability of the sequence of error vector norms is required.     

Characterizing bounded orthogonally additive polynomials on vector lattices

We derive formulas for characterizing bounded orthogonally additive polynomials in two ways. Firstly, we prove that certain formulas for orthogonally additive polynomials derived in Kusraeva (Vladikavkaz Math J 16(4):49–53, 2014) actually characterize them. Secondly, by employing complexifications of the unique symmetric multilinear maps associated with orthogonally additive polynomials, we derive new characterizing formulas.

     

Multifactor Heston's stochastic volatility model for European option pricing

In this study, we extend the multiscale stochastic volatility model of [Fouque J‐P, Lorig MJ, SIAM J Financial Math. 2011;2(1):221‐254] by incorporating a slow varying factor of volatility. The resulting model can be viewed as a multifactor extension of the Heston model with two additional factors driving the volatility levels. An asymptotic analysis consisting of singular and regular perturbation expansions is developed to obtain an approximation to European option prices. We also find explicit expressions for some essential functions that are available only in integral formulas in the work of [Fouque J‐P, Lorig MJ, SIAM J Financial Math. 2011;2(1):221‐254]. This finding basically leads to considerable reduction in computational time for numerical calculation as well as calibration problems. An accuracy result of the asymptotic approximation is also provided. For numerical illustration, the multifactor Heston model is calibrated to index options on the market, and we find that the resulting implied volatility surfaces fit the market data better than those produced by the multiscale stochastic volatility model of [Fouque J‐P, Lorig MJ, SIAM J Financial Math. 2011;2(1):221‐254], particularly for long‐maturity call options.     

Enumerating typical abelian prime-fold coverings of a circulant graph

Enumerating the isomorphism classes of several types of graph coverings is one of the central research topics in enumerative topological graph theory (see [R. Feng, J.H. Kwak, J. Kim, J. Lee, Isomorphism classes of concrete graph coverings, SIAM J. Discrete Math. 11 (1998) 265-272; R. Feng, J.H. Kwak, Typical circulant double coverings of a circulant graph, Discrete Math. 277 (2004) 73-85; R. Feng, J.H. Kwak, Y.S. Kwon, Enumerating typical circulant covering projections onto a circulant graph, SIAM J. Discrete Math. 19 (2005) 196-207; SIAM J. Discrete Math. 21 (2007) 548-550 (erratum); M. Hofmeister, Graph covering projections arising from finite vector spaces over finite fields, Discrete Math. 143 (1995) 87-97; M. Hofmeister, Enumeration of concrete regular covering projections, SIAM J. Discrete Math. 8 (1995) 51-61; M. Hofmeister, A note on counting connected graph covering projections, SIAM J. Discrete Math. 11 (1998) 286-292; J.H. Kwak, J. Chun, J. Lee, Enumeration of regular graph coverings having finite abelian covering transformation groups, SIAM J. Discrete Math. 11 (1998) 273-285; J.H. Kwak, J. Lee, Isomorphism classes of graph bundles, Canad. J. Math. XLII (1990) 747-761]). A covering is called abelian (or circulant, respectively) if its covering graph is a Cayley graph on an abelian (or a cyclic, respectively) group. A covering p from a Cayley graph onto another Cay (Q,Y) is called typical if the map p:A→Q on the vertex sets is a group epimorphism. Recently, the isomorphism classes of connected typical circulant r-fold coverings of a circulant graph are enumerated in [R. Feng, J.H. Kwak, Typical circulant double coverings of a circulant graph, Discrete Math. 277 (2004) 73-85] for r=2 and in [R. Feng, J.H. Kwak, Y.S. Kwon, Enumerating typical circulant covering projections onto a circulant graph, SIAM J. Discrete Math. 19 (2005) 196-207; SIAM J. Discrete Math. 21 (2007) 548-550 (erratum)] for any r. As a continuation of these works, we enumerate in this paper the isomorphism classes of typical abelian prime-fold coverings of a circulant graph.     

Global well-posedness for the 2D dispersive SQG equation and inviscid Boussinesq equations

In this paper, we obtain global well-posedness for the 2D dispersive SQG equation and inviscid Boussinesq equations. Our works are consistent with the corresponding works by Elgindi–Widmayer (SIAM J Math Anal 47:4672–4684, 2015) in the special case \({A=\kappa=1}\). In addition, our result concerning the SQG equation can be regarded as the borderline case of the work by Cannone et al. (Proc Lond Math Soc 106:650–674, 2013).     

Common fixed points and invariant approximation for Gregus type contraction mappings

Some new common fixed point theorems for Gregus type contraction mappings have been obtained in convex metric spaces. As applications, invariant approximation results for these types of mappings are obtained. The proved results generalize, unify and extend some of the known results of M.A. Al-Thagafi (Int. J. Math. Sci. 18:613–616, 1995; J. Approx. Theory 85:318–323, 1996), M.A. Al-Thagafi and N. Shahzad (Nonlinear Anal. 64:2778–2786, 2006), L. ?iri? (Publ. Inst. Math. 49:174–178, 1991; Arch. Math. (BRNO) 29:145–152, 1993), M.L. Diviccaro, B. Fisher, S. Sessa (Publ. Math. (Debr.) 34:83–89, 1987), B. Fisher and S. Sessa (Int. J. Math. Math. 9:23–28, 1986), M. Gregus (Boll. Un. Mat. Ital. (5) 7-A:193–198, 1980), L. Habiniak (J. Approx. Theory 56:241–244, 1989), N. Hussain, B.E. Rhoades and G. Jungck (Numer. Func. Anal. Optim. 28:1139–1151, 2007), G. Jungck (Int. J. Math. Math. Sci. 13:497–500, 1990), G. Jungck and S. Sessa (Math. Jpn. 42:249–252, 1995), R.N. Mukherjee and V. Verma (Math. Jpn. 33:745–749, 1988), T.D. Narang and S. Chandok (Ukr. Math. J. 62:1367–1376, 2010), S.A. Sahab, M.S. Khan and S. Sessa (J. Approx. Theory 55:349–351, 1988), N. Shahzad (J. Math. Anal. Appl. 257:39–45, 2001; Rad. Math. 10:77–83, 2001; Int. J. Math. Game Theory Algebra 13:157–159, 2003), S.P. Singh (J. Approx. Theory 25:89–90, 1979), A. Smoluk (Mat. Stosow. 17:17–22, 1981), P.V. Subrahmanyam (J. Approx. Theory 20:165–172, 1977) and of few others.     

Coherent and convex loss-based risk measures for portfolio vectors

In this paper, we introduce two new classes of risk measures, named coherent and convex loss-based risk measures for portfolio vectors. These new risk measures can be considered as a multivariate extension of univariate loss-based risk measures introduced by Cont et al. (Stat Risk Model 30:133–167, 2013). Representation results for these new introduced risk measures are provided. The links between convex loss-based risk measures for portfolios and convex risk measures for portfolios introduced by Burgert and Rüschendorf (Insur Math Econ 38:289–297, 2006) or Wei and Hu (Stat Probab Lett 90:114–120, 2014) are stated. Finally, applications to the multi-period coherent and convex loss-based risk measures are addressed.

     

Existence,Nonexistence and Multiplicity Results of a Chern-Simons-Schrödinger System

We study the existence, nonexistence and multiplicity of solutions to Chern-Simons-Schrödinger system

$$\begin{aligned} \left \{ \textstyle\begin{array}{l@{\quad }l} -\Delta u+u+\lambda (\frac{h^{2}(|x|)}{|x|^{2}}+\int _{|x|}^{+ \infty }\frac{h(s)}{s}u^{2}(s)ds )u=|u|^{p-2}u,\quad x\in \mathbb{R}^{2}, \\ u\in H^{1}_{r}(\mathbb{R}^{2}), \end{array}\displaystyle \right . \end{aligned}$$

where \(\lambda >0\) is a parameter, \(p\in (2,4)\) and

$$ h(s)=\frac{1}{2} \int _{0}^{s}ru^{2}(r)dr. $$

We prove that the system has no solutions for \(\lambda \) large and has two radial solutions for \(\lambda \) small by studying the decomposition of the Nehari manifold and adapting the fibering method. We also give the qualitative properties about the energy of the solutions and a variational characterization of these extremals values of \(\lambda \). Our results improve some results in Pomponio and Ruiz (J. Eur. Math. Soc. 17:1463–1486, 2015).

     

Accounting for risk aversion in derivatives purchase timing

We study the problem of optimal timing to buy/sell derivatives by a risk-averse agent in incomplete markets. Adopting the exponential utility indifference valuation, we investigate this timing flexibility and the associated delayed purchase premium. This leads to a stochastic control and optimal stopping problem that combines the observed market price dynamics and the agent??s risk preferences. Our results extend recent work on indifference valuation of American options, as well as the authors?? first paper (Leung and Ludkovski, SIAM J Finan Math 2(1) 768?C793, 2011). In the case of Markovian models of contracts on non-traded assets, we provide analytical characterizations and numerical studies of the optimal purchase strategies, with applications to both equity and credit derivatives.     

On the Gauss–Newton method

We provide a new semilocal convergence analysis of the Gauss–Newton method (GNM) for solving nonlinear equation in the Euclidean space. Using a combination of center-Lipschitz, Lipschitz conditions, and our new idea of recurrent functions, we provide under the same or weaker hypotheses than before (Ben-Israel, J. Math. Anal. Appl. 15:243–252, 1966; Chen and Nashed, Numer. Math. 66:235–257, 1993; Deuflhard and Heindl, SIAM J. Numer. Anal. 16:1–10, 1979; Guo, J. Comput. Math. 25:231–242, 2007; Häußler, Numer. Math. 48:119–125, 1986; Hu et al., J. Comput. Appl. Math. 219:110–122, 2008; Kantorovich and Akilov, Functional Analysis in Normed Spaces, Pergamon, Oxford, 1982), a finer convergence analysis. The results can be extended in case outer or generalized inverses are used. Numerical examples are also provided to show that our results apply, where others fail (Ben-Israel, J. Math. Anal. Appl. 15:243–252, 1966; Chen and Nashed, Numer. Math. 66:235–257, 1993; Deuflhard and Heindl, SIAM J. Numer. Anal. 16:1–10, 1979; Guo, J. Comput. Math. 25:231–242, 2007; Häußler, Numer. Math. 48:119–125, 1986; Hu et al., J. Comput. Appl. Math. 219:110–122, 2008; Kantorovich and Akilov, Functional Analysis in Normed Spaces, Pergamon, Oxford, 1982).     

Optimal Transport, Convection, Magnetic Relaxation and Generalized Boussinesq Equations

We establish a connection between optimal transport theory (see Villani in Topics in optimal transportation. Graduate studies in mathematics, vol. 58, AMS, Providence, 2003, for instance) and classical convection theory for geophysical flows (Pedlosky, in Geophysical fluid dynamics, Springer, New York, 1979). Our starting point is the model designed few years ago by Angenent, Haker, and Tannenbaum (SIAM J. Math. Anal. 35:61–97, 2003) to solve some optimal transport problems. This model can be seen as a generalization of the Darcy–Boussinesq equations, which is a degenerate version of the Navier–Stokes–Boussinesq (NSB) equations. In a unified framework, we relate different variants of the NSB equations (in particular what we call the generalized hydrostatic-Boussinesq equations) to various models involving optimal transport (and the related Monge–Ampère equation, Brenier in Commun. Pure Appl. Math. 64:375–417, 1991; Caffarelli in Commun. Pure Appl. Math. 45:1141–1151, 1992). This includes the 2D semi-geostrophic equations (Hoskins in Annual review of fluid mechanics, vol. 14, pp. 131–151, Palo Alto, 1982; Cullen et al. in SIAM J. Appl. Math. 51:20–31, 1991, Arch. Ration. Mech. Anal. 185:341–363, 2007; Benamou and Brenier in SIAM J. Appl. Math. 58:1450–1461, 1998; Loeper in SIAM J. Math. Anal. 38:795–823, 2006) and some fully nonlinear versions of the so-called high-field limit of the Vlasov–Poisson system (Nieto et al. in Arch. Ration. Mech. Anal. 158:29–59, 2001) and of the Keller–Segel for Chemotaxis (Keller and Segel in J. Theor. Biol. 30:225–234, 1971; Jäger and Luckhaus in Trans. Am. Math. Soc. 329:819–824, 1992; Chalub et al. in Mon. Math. 142:123–141, 2004). Mathematically speaking, we establish some existence theorems for local smooth, global smooth or global weak solutions of the different models. We also justify that the inertia terms can be rigorously neglected under appropriate scaling assumptions in the generalized Navier–Stokes–Boussinesq equations. Finally, we show how a “stringy” generalization of the AHT model can be related to the magnetic relaxation model studied by Arnold and Moffatt to obtain stationary solutions of the Euler equations with prescribed topology (see Arnold and Khesin in Topological methods in hydrodynamics. Applied mathematical sciences, vol. 125, Springer, Berlin, 1998; Moffatt in J. Fluid Mech. 159:359–378, 1985, Topological aspects of the dynamics of fluids and plasmas. NATO adv. sci. inst. ser. E, appl. sci., vol. 218, Kluwer, Dordrecht, 1992; Schonbek in Theory of the Navier–Stokes equations, Ser. adv. math. appl. sci., vol. 47, pp. 179–184, World Sci., Singapore, 1998; Vladimirov et al. in J. Fluid Mech. 390:127–150, 1999; Nishiyama in Bull. Inst. Math. Acad. Sin. (N.S.) 2:139–154, 2007).