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Abstract: Abstract The focus of this work is to provide an extensive numerical study of the parallel efficiency and robustness of a staggered dual-primal Newton–Krylov deluxe solver for implicit time discretizations of the Bidomain model. This model describes the propagation of the electrical impulse in the cardiac tissue, by means of a system of parabolic reaction-diffusion partial differential equations. This system is coupled to a system of ordinary differential equations, modeling the ionic currents dynamics. A staggered approach is employed for the solution of a fully implicit time discretization of the problem, where the two systems are solved successively. The arising nonlinear algebraic system is solved with a Newton–Krylov approach, preconditioned by a dual-primal Domain Decomposition algorithm in order to improve convergence. The theoretical analysis and numerical validation of this strategy has been carried out in Huynh et al. (SIAM J. Sci. Comput. 44, B224–B249, 2022) considering only simple ionic models. This paper extends this study to include more complex biophysical ionic models, as well as the presence of ischemic regions, described mathematically by heterogeneous diffusion coefficients with possible discontinuities between subregions. The results of several numerical experiments show robustness and scalability of the proposed parallel solver. PubDate: 2022-09-19

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Abstract: Abstract For α,γ ≥ 0, 0 ≤ β < 1 and k ≥ 1, we introduce the functions class \(\mathcal {R}_{H}^{k}(\alpha ,\gamma ,\beta )\) , which is the harmonic analogue to the family \(\mathcal R(\alpha ,\gamma ,\beta )\) due to Ali et al. (Complex Var. Elliptic Equ. 58, 1569–1590, 2013). It is observed that the family \({\mathcal {R}}^{k}_{H}(\alpha ,\gamma ,\beta )\) is close-to-convex in \({\mathbb {D}}\) . We first determine the sharp bounds on harmonic Bieberbach-type conjecture, sufficient condition and growth theorems of functions in the family \({\mathcal {R}}_{H}^{k}(\alpha ,\gamma ,\beta )\) . It is proved that the family is invariant under the convex combination and convolution. Various radii related problems on the partial sums of f in the class \(\mathcal {R}_{H}^{k}(\alpha ,\gamma ,\beta )\) are also presented. Other results include radii of convexity and starlikeness of the functions in the class \({\mathcal {W}}_{H}^{0}(\alpha ,\beta )\) . PubDate: 2022-09-19

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Abstract: Abstract In this work we study convergence properties of sparse polynomial approximations for a class of affine parametric saddle point problems, including the Stokes equations for viscous incompressible flow, mixed formulation of diffusion equations for groundwater flow, time-harmonic Maxwell equations for electromagnetics, etc. Due to the lack of knowledge or intrinsic randomness, the (viscosity, diffusivity, permeability, permittivity, etc.) coefficients of such problems are uncertain and can often be represented or approximated by high- or countably infinite-dimensional random parameters equipped with suitable probability distributions, and the coefficients affinely depend on a series of either globally or locally supported basis functions, e.g., Karhunen–Loève expansion, piecewise polynomials, or adaptive wavelet approximations. We consider sparse polynomial approximations of the parametric solutions, in particular sparse Taylor approximations, and study their convergence properties for these parametric problems. Under suitable sparsity assumptions on the parametrization of the random coefficients, we show the algebraic convergence rates O(N−r) for the sparse polynomial approximations of the parametric solutions based on the results for affine parametric elliptic PDEs (Cohen, A. et al.: Anal. Appl. 9, 11–47, 2011), (Bachmayr, M., et al.: ESAIM Math. Model. Numer. Anal. 51, 321–339, 2017), (Cohen, A., DeVore, R.: Acta Numer. 24, 1–159, 2015), (Chkifa, A., et al.: J. Math. Pures Appl. 103, 400–428, 2015), (Chkifa, A., et al.: ESAIM Math. Model. Numer. Anal. 47, 253–280, 2013), (Cohen, A., Migliorati, G.: Contemp. Comput. Math., 233–282, 2018), with the rate r depending only on a sparsity parameter in the parametrization, not on the number of active parameter dimensions or the number of polynomial terms N. We note that parametric saddle point problems were considered in (Cohen, A., DeVore, R.: Acta Numer. 24, 1–159, 2015, Section 2.2) with the anticipation that the same results on the approximation of the solution map obtained for elliptic PDEs can be extended to more general saddle point problems. In this paper, we consider a general formulation of saddle point problems, different from that presented in (Cohen, A., DeVore, R.: Acta Numer. 24, 1–159, 2015, Section 2.2), and obtain convergence rates for the two variables, e.g., velocity and pressure in Stokes equations, which are different for the case of locally supported basis functions. PubDate: 2022-09-19

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Abstract: Abstract In this work we introduce some representations of isometric transformations on the sphere of 2-adic units, then give some necessary and sufficient conditions for their ergodicty. PubDate: 2022-09-15

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Abstract: Abstract A degenerate Zakharov system arises as a model for the description of laser-plasma interactions. It is a coupled system of a Schrödinger and a wave equation with a non-dispersive direction. In this paper, a new local well-posedness result for rough initial data is established. The proof is based on an efficient use of local smoothing and maximal function norms. PubDate: 2022-09-15

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Abstract: Abstract Let S be a semigroup, and let \(\mathbb {F}\) be a quadratically closed field of characteristic ≠ 2 with identity element 1. We describe, in terms of multiplicative functions of S, the solutions \(f:S\rightarrow \mathbb {F}\) of the new functional equation $$ f(x\phi (y))+\mu (y)f(\varphi (y)x)=2f(x)f(y),\quad x,y\in S, $$ where \(\phi ,\varphi :S\rightarrow S\) are two endomorphisms that need not be involutive and \(\mu :S\rightarrow \mathbb {F}\) is a multiplicative map such that μ(xφ(x)) = 1 for all x ∈ S. Significant consequences of this result are presented. PubDate: 2022-09-15

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Abstract: Abstract The aim of this paper is to define a new class of surfaces in Euclidean space using the concept of osculating circle. Given a regular curve C, the surface of osculating circles generated by C is the set of all osculating circles at all points of C. It is proved that these surfaces contain a one-parametric family of planar lines of curvature. A classification of surfaces of osculating circles is given in the family of canal surfaces, Weingarten surfaces, surfaces with constant Gauss curvature and surfaces with constant mean curvature. PubDate: 2022-09-15

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Abstract: Abstract In embryogenesis, epithelial cells acting as individual entities or as coordinated aggregates in a tissue, exhibit strong coupling between mechanical responses to internally or externally applied stresses and chemical signalling. One of the most important chemical signals in this process is calcium. This mechanochemical coupling and intercellular communication drive the coordination of morphogenetic movements which are characterised by drastic changes in the concentration of calcium in the tissue. In this paper we extend the recent mechanochemical model in Kaouri et al. (J. Math. Biol. 78, 2059–2092, 2019), for an epithelial continuum in one dimension, to a more realistic multi-dimensional case. The resulting parametrised governing equations consist of an advection-diffusion-reaction system for calcium signalling coupled with active-stress linear viscoelasticity and equipped with pure Neumann boundary conditions. We implement a finite element method in perturbed saddle-point form for the simulation of this complex multiphysics problem. Special care is taken in the treatment of the stress-free boundary conditions for the viscoelasticity in order to eliminate rigid motions from the space of admissible displacements. The stability and solvability of the continuous weak formulation is shown using fixed-point theory. Guided by the bifurcation analysis of the one-dimensional model, we analyse the behaviour of the system as two bifurcation parameters vary: the level of IP3 concentration and the strength of the mechanochemical coupling. We identify the parameter regions giving rise to solitary waves and periodic wavetrains of calcium. Furthermore, we demonstrate the nucleation of calcium sparks into synchronous calcium waves coupled with deformation. This model can be employed to gain insights into recent experimental observations in the context of embryogenesis, but also in other biological systems such as cancer cells, wound healing, keratinocytes, or white blood cells. PubDate: 2022-09-14

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Abstract: Abstract In this work we review discontinuous Galerkin finite element methods on polytopal grids (PolydG) for the numerical simulation of multiphysics wave propagation phenomena in heterogeneous media. In particular, we address wave phenomena in elastic, poro-elastic, and poro-elasto-acoustic materials. Wave propagation is modeled by using either the elastodynamics equation in the elastic domain, the acoustics equations in the acoustic domain and the low-frequency Biot’s equations in the poro-elastic one. The coupling between different models is realized by means of (physically consistent) transmission conditions, weakly imposed at the interface between the subdomains. For all models configuration, we introduce and analyse the PolydG semi-discrete formulation, which is then coupled with suitable time marching schemes. For the semi-discrete problem, we present the stability analysis and derive a-priori error estimates in a suitable energy norm. A wide set of two-dimensional verification tests with manufactured solutions are presented in order to validate the error analysis. Examples of physical interest are also shown to demonstrate the capability of the proposed methods. PubDate: 2022-07-16 DOI: 10.1007/s10013-022-00566-3

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Abstract: Abstract We consider the vector partition problem, where n agents, each with a d-dimensional attribute vector, are to be partitioned into p parts so as to minimize cost which is a given function on the sums of attribute vectors in each part. The problem has applications in a variety of areas including clustering, logistics and health care. We consider the complexity and parameterized complexity of the problem under various assumptions on the natural parameters p,d,a,t of the problem where a is the maximum absolute value of any attribute and t is the number of agent types, and raise some of the many remaining open problems. PubDate: 2022-07-01 DOI: 10.1007/s10013-021-00523-6

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Abstract: Abstract Let q be a nondegenerate quadratic form on V. Let X ⊂ V be invariant for the action of a Lie group G contained in SO(V,q). For any f ∈ V consider the function df from X to \(\mathbb C\) defined by df(x) = q(f − x). We show that the critical points of df lie in the subspace orthogonal to \({\mathfrak g}\cdot f\) , that we call critical space. In particular any closest point to f in X lie in the critical space. This construction applies to singular t-ples for tensors and to flag varieties and generalizes a previous result of Draisma, Tocino and the author. As an application, we compute the Euclidean Distance degree of a complete flag variety. PubDate: 2022-07-01 DOI: 10.1007/s10013-021-00547-y

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Abstract: Abstract A reaction system exhibits “absolute concentration robustness” (ACR) in some species if the positive steady-state value of that species does not depend on initial conditions. Mathematically, this means that the positive part of the variety of the steady-state ideal lies entirely in a hyperplane of the form xi = c, for some c > 0. Deciding whether a given reaction system – or those arising from some reaction network – exhibits ACR is difficult in general, but here we show that for many simple networks, assessing ACR is straightforward. Indeed, our criteria for ACR can be performed by simply inspecting a network or its standard embedding into Euclidean space. Our main results pertain to networks with many conservation laws, so that all reactions are parallel to one other. Such “one-dimensional” networks include those networks having only one species. We also consider networks with only two reactions, and show that ACR is characterized by a well-known criterion of Shinar and Feinberg. Finally, up to some natural ACR-preserving operations – relabeling species, lengthening a reaction, and so on – only three families of networks with two reactions and two species have ACR. Our results are proven using algebraic and combinatorial techniques. PubDate: 2022-07-01 DOI: 10.1007/s10013-021-00524-5

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Abstract: Abstract Sparse polynomials that vanish on algebraic sets are preferred in many computations since they are easy to evaluate and often arise from underlying structure. For example, a monomial vanishes on an algebraic set if and only if the algebraic set is contained in the union of the coordinate hyperplanes. Eisenbud and Sturmfels initiated a detailed study of binomial ideals roughly 25 years ago and showed that they had many special properties including that each component is rational. Given a general point on a component and its tangent space, this paper exploits rationality to develop a local approach that decides if the component is defined by binomials or not. When a component is not defined by binomials, one often is interested in computing sparse polynomials that vanish on the component. Thus, this paper also develops an approach for computing sparse polynomials using a witness set for the component. Our approach relies on using numerical homotopy methods to sample points on the algebraic set along with incorporating multiplicity information using Macaulay dual spaces. If the algebraic set is defined by polynomials with rational coefficients, exactness recovery such as lattice based methods can be used to find exact representations of the sparse polynomials. Several examples are presented demonstrating the new methods. PubDate: 2022-07-01 DOI: 10.1007/s10013-021-00543-2

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Abstract: Abstract Motivated by Gröbner basis theory for finite point configurations, we define and study the class of standard complexes associated to a matroid. Standard complexes are certain subcomplexes of the independence complex that are invariant under matroid duality. For the lexicographic term order, the standard complexes satisfy a deletion-contraction-type recurrence. We explicitly determine the lexicographic standard complexes for lattice path matroids using classical bijective combinatorics. PubDate: 2022-07-01 DOI: 10.1007/s10013-021-00546-z

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Abstract: Abstract We introduce the class of “smooth rough paths” and study their main properties. Working in a smooth setting allows us to discard sewing arguments and focus on algebraic and geometric aspects. Specifically, a Maurer–Cartan perspective is the key to a purely algebraic form of Lyons’ extension theorem, the renormalization of rough paths following up on [Bruned et al.: A rough path perspective on renormalization, J. Funct. Anal. 277(11), 2019], as well as a related notion of “sum of rough paths”. We first develop our ideas in a geometric rough path setting, as this best resonates with recent works on signature varieties, as well as with the renormalization of geometric rough paths. We then explore extensions to the quasi-geometric and the more general Hopf algebraic setting. PubDate: 2022-06-23 DOI: 10.1007/s10013-022-00570-7

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Abstract: Abstract We show, in this first part, that the maximal number of singular points of a normal quartic surface \(X \subset \mathbb {P}^{3}_{K}\) defined over an algebraically closed field K of characteristic 2 is at most 16. We produce examples with 14, respectively 12, singular points and show that, under several geometric assumptions ( \(\mathfrak S_{4}\) -symmetry, or behaviour of the Gauss map, or structure of tangent cone at one of the singular points P, separability/inseparability of the projection with centre P), we can obtain smaller upper bounds for the number of singular points of X. PubDate: 2022-06-09 DOI: 10.1007/s10013-022-00556-5

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Abstract: Abstract Results of Koebe (Ber. Sächs. Akad. Wiss. Leipzig, Math.-phys. Kl. 88, 141–164, 1936), Schramm (Invent. Math. 107(3), 543560, 1992), and Springborn (Math. Z. 249, 513–517, 2005) yield realizations of 3-polytopes with edges tangent to the unit sphere. Here we study the algebraic degrees of such realizations. This initiates the research on constrained realization spaces of polytopes. PubDate: 2022-05-06 DOI: 10.1007/s10013-022-00559-2

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Abstract: Abstract We explore four approaches to the question of defectivity for a complex projective toric variety XA associated with an integral configuration A. The explicit tropicalization of the dual variety \(X_{A}^{\vee }\) due to Dickenstein, Feichtner, and Sturmfels allows for the computation of the defect in terms of an affine combinatorial invariant ρ(A). We express ρ(A) in terms of affine invariants ι(A) associated to Esterov’s iterated circuits and λ(A), an invariant defined by Curran and Cattani in terms of a Gale dual of A. Thus we obtain formulae for the dual defect in terms of iterated circuits and Gale duals. An alternative expression for the dual defect of XA is given by Furukawa–Ito in terms of Cayley decompositions of A. We give a Gale dual interpretation of these decompositions and apply it to the study of defective configurations. PubDate: 2022-04-27 DOI: 10.1007/s10013-022-00554-7

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Abstract: Abstract In this paper, we survey the theory of Cartwright–Sturmfels ideals. These are \(\mathbb {Z}^{n}\) -graded ideals, whose multigraded generic initial ideal is radical. Cartwright–Sturmfels ideals have surprising properties, mostly stemming from the fact that their Hilbert scheme only contains one Borel-fixed point. This has consequences, e.g., on their universal Gröbner bases and on the family of their initial ideals. In this paper, we discuss several known classes of Cartwright–Sturmfels ideals and we find a new one. Among determinantal ideals of same-size minors of a matrix of variables and Schubert determinantal ideals, we are able to characterize those that are Cartwright–Sturmfels. PubDate: 2022-04-20 DOI: 10.1007/s10013-022-00551-w

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Abstract: Abstract In this paper we study the set of tensors that admit a special type of decomposition called an orthogonal tensor train decomposition. Finding equations defining varieties of low-rank tensors is generally a hard problem, however, the set of orthogonally decomposable tensors is defined by appealing quadratic equations. The tensors we consider are an extension of orthogonally decomposable tensors. We show that they are defined by similar quadratic equations, as well as an interesting higher-degree additional equation. PubDate: 2022-04-08 DOI: 10.1007/s10013-022-00558-3