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ZusammenfassungDiagnostische Kompetenz ist eine zentrale Komponente der professionellen Kompetenzen von Lehrkräften, die die Qualität von Unterricht und somit das Lernen von Schülerinnen und Schülern beeinflusst. Aufgrund dieser zentralen Bedeutung rückt die systematische Schulung der diagnostischen Kompetenz zunehmend in den Forschungsfokus. Diese Arbeit zielt auf die Erforschung der Förderung diagnostischer Kompetenz im Bereich der Mathematik ab. Wir stellen eine quasi-experimentelle Studie mit Treatment- und Kontrollgruppe vor, die den Effekt einer Intervention bei Studierenden des Grundschullehramts (n = 74) untersucht. Die Intervention erfolgt im Rahmen eines Seminars, das die als effektiv bekannten Bestandteile der Förderung diagnostischer Kompetenz aufgreift. Diagnostische Kompetenz modellieren wir als Fähigkeit, Lernprodukte von Schülerinnen adäquat und multiperspektivisch beurteilen zu können und messen deren Entwicklung in einem Pre-Post-Design. Die Ergebnisse zeigen, dass unsere Schulung insbesondere auf einen Teil sogenannter epistemischen Aktivitäten wirkt: die Entwicklung und Stützung von Hypothesen zu Fähigkeiten von Schülerinnen und Schülern. Hier unterscheidet sich die Treatmentgruppe im Post-Test signifikant von der Kontrollgruppe. Die Arbeit leistet insgesamt einen Beitrag zur Entwicklung und mehrperspektivischen Messung diagnostischer Kompetenz bezogen auf epistemische Aktivitäten im diagnostischen Prozess und der multiperspektivischen Beurteilung der Lernprodukte von Schülerinnen und Schülern zu offenen Lernangebote zur Arithmetik.

A strong inspiration for studying perturbation theory for fractional evolution equations comes from the fact that they have proven to be useful tools in modeling many physical processes. In this paper, we study fractional evolution equations of order $\alpha\in (1,2]$ associated with the infinitesimal generator of an operator fractional cosine function generated by bounded time-dependent perturbations in a Banach space. We show that the abstract fractional Cauchy problem associated with the infinitesimal generator $A$ of a strongly continuous fractional cosine function remains uniformly well-posed under bounded time-dependent perturbation of $A$. We also provide some necessary special cases.

We study the upper bound of the colength of trace of the canonical module in one-dimensional Cohen-Macaulay rings. We answer the two questions posed by Herzog-Hibi-Stamate and Kobayashi., 9 pages

Lockdowns imposed by many countries on their populations at the beginning of the COVID-19 crisis forced teachers to adapt quickly and without adequate preparation to distance teaching. In this paper, we focus on one of the most formidable challenges that teachers faced during the lockdowns and even in the post-lockdown emergency period, namely, developing assessment that maintains the pedagogical continuity that educational institutions typically require. Based on the results of a previous study, focused on the analysis of answers to an open-ended questionnaire administered to a population of 700 teachers from France, Germany, Israel and Italy, a semi-structured interview series was designed and implemented by the authors of this paper with a small group of teachers. The transcripts of these interviews were analysed according to the interpretative phenomenological analysis methodology, with the aim of investigating teachers’ own perspectives on the following: (a) the difficulties with which they had to contend, with respect to the question of assessment; (b) the techniques adopted to deal with these difficulties; and (c) the ways in which the lockdown experience could affect the future evolution of teachers’ assessment practices. This analysis supported us in formulating hypotheses concerning the possible long-term effects of lockdown on modes of assessment in mathematics.

In this work, we consider optimal stopping problems with model uncertainty incorporated into the formulation of the underlying objective function. Typically, the robust, efficient hedging of American options in incomplete markets may be described as optimal stopping of such kind. Based on a generalisation of the additive dual representation of Rogers (Math. Financ. 12:271–286, 2002) to the case of optimal stopping under model uncertainty, we develop a novel regression-based Monte Carlo algorithm for the approximation of the corresponding value function. The algorithm involves optimising a penalised empirical dual objective functional over a class of martingales. This formulation allows us to construct upper bounds for the optimal value with reduced complexity. Finally, we carry out a convergence analysis of the proposed algorithm and illustrate its performance by several numerical examples.

We consider weak solutions $(u,\pi ):{\Omega }\to \mathbb {R}^{n}\times \mathbb {R}$ ( u , π ) : Ω → ℝ n × ℝ to stationary ϕ-Navier-Stokes systems of the type $ \left \{ \begin {array}{ll} -\mathrm {div~} a(x,\mathcal {E} u)+\nabla \pi +[Du]u=f \\ \mathrm {div~} u=0 \end {array} \right . $ − div a ( x , E u ) + ∇ π + [ D u ] u = f div u = 0 in ${\Omega }\subset \mathbb {R}^{n}$ Ω ⊂ ℝ n , and to the corresponding ϕ-Stokes systems, in which the convective term [Du]u does not appear. In the above system, the function a(x,ξ) depends Hölder continuously on x and satisfies growth conditions with respect to the second variable expressed through a Young function ϕ. The notation $\mathcal {E} u$ E u is used for the symmetric part of the gradient Du. We prove results on the fractional higher differentiability of both the symmetric part of the gradient $\mathcal {E} u$ E u and of the pressure π.

The paper deals with t-designs that can be partitioned into s-designs, each missing a point of the underlying set, called point-missing s-resolvable t-designs, with emphasis on their applications in constructing t-designs. The problem considered may be viewed as a generalization of overlarge sets which are defined as a partition of all the $$\left( {\begin{array}{c}v +1\\ k\end{array}}\right) $$ v + 1 k k-sets chosen from a $$(v+1)$$ ( v + 1 ) -set X into $$(v+1)$$ ( v + 1 ) mutually disjoint s-$$(v,k,\delta )$$ ( v , k , δ ) designs, each missing a different point of X. Among others, it is shown that the existence of a point-missing $$(t-1)$$ ( t - 1 ) -resolvable t-$$(v,k,\lambda )$$ ( v , k , λ ) design leads to the existence of a t-$$(v,k+1,\lambda ')$$ ( v , k + 1 , λ ′ ) design. As a result, we derive various infinite series of 4-designs with constant index using overlarge sets of disjoint Steiner quadruple systems. These have parameters 4-$$(3^n,5,5)$$ ( 3 n , 5 , 5 ) , 4-$$(3^n+2,5,5)$$ ( 3 n + 2 , 5 , 5 ) and 4-$$(2^n+1,5,5)$$ ( 2 n + 1 , 5 , 5 ) , for $$n \ge 2$$ n ≥ 2 , and were unknown until now. We also include a recursive construction of point-missing s-resolvable t-designs and its application.

The paper introduces a method for constructing 2-resolvable t-designs for $$t=3,4$$ t = 3 , 4 . The main idea is based on the assumption that there exists a partition of a t-design into Steiner 2-designs. A remarkable property of the method is that it enables the construction of 2-resolvable t-designs with a large variety of block sizes. For $$t=4$$ t = 4 , it is required that the Steiner 2-designs of the partition are projective planes and this case would also lead to a construction of 3-resolvable 5-designs. For instance, we show the existence of an infinite series of 3-resolvable 5-designs having $$N=5$$ N = 5 resolution classes with parameters 5-$$(14+8m,7, 10(9+8m)(1+m))$$ ( 14 + 8 m , 7 , 10 ( 9 + 8 m ) ( 1 + m ) ) for any $$m \ge 0$$ m ≥ 0 as a byproduct. Moreover, it turns out that the method is very effective, as it yields infinitely many 2-resolvable 3-designs. However, the question of simplicity of the constructed designs has not been yet investigated.

We revisit a construction principle of Fredholm operators using Hilbert complexes of densely defined, closed linear operators and apply this to particular choices of differential operators. The resulting index is then computed using an explicit description of the cohomology groups of generalised (‘harmonic’) Dirichlet and Neumann tensor fields. The main results of this contribution are the computation of the indices of Dirac type operators associated to the elasticity complex and the newly found biharmonic complex, relevant for the biharmonic equation, elasticity, and for the theory of general relativity. The differential operators are of mixed order and cannot be seen as leading order type with relatively compact perturbation. As a by-product we present a comprehensive description of the underlying generalised Dirichlet–Neumann vector and tensor fields defining the respective cohomology groups, including an explicit construction of bases in terms of topological invariants, which are of both analytical and numerical interest. Though being defined by certain projection mechanisms, we shall present a way of computing these basis functions by solving certain PDEs given in variational form. For all of this we rephrase core arguments in the work of Rainer Picard [42] applied to the de Rham complex and use them as a blueprint for the more involved cases presented here. In passing, we also provide new vector-analytical estimates of generalised Poincaré–Friedrichs type useful for elasticity or the theory of general relativity.

Let G be a simple finite graph. A famous theorem of Dirac says that G is chordal if and only if G admits a perfect elimination order. It is known by Fr ̈oberg that the edge ideal I(G) of G has a linear resolution if and only if the complementary graph Gc of G is chordal. In this article, we discuss some algebraic consequences of Dirac’s theorem in the theory of homological shift ideals of edge ideals. Recall that if I is a monomial ideal, HSk(I) is the monomial ideal generated by the kth multigraded shifts of I. We prove that HS1(I) has linear quotients, for any monomial ideal I with linear quotients generated in a single degree. For and edge ideal I(G) with linear quotients, it is not true that HSk(I(G)) has linear quotients for all k ≥ 0. On the other hand, if Gc is a proper interval graph or a forest, we prove that this is the case. Finally, we discuss a conjecture of Bandari, Bayati and Herzog that predicts that if I is polymatroidal, HSk(I) is polymatroidal too, for all k ≥ 0. We are able to prove that this conjecture holds for all polymatroidal ideals generated in degree two.

A family of irreducible holomorphic symplectic (ihs) manifolds over the complex projective line has unobstructed deformations if its period map is an embedding. This applies in particular to twistor spaces of ihs manifolds. Moreover, a family of ihs manifolds over a subspace of the period domain extends to a universal family over an open neighborhood in the period domain.

We analyse an optimal trade execution problem in a financial market with stochastic liquidity. To this end, we set up a limit order book model in continuous time. Both order book depth and resilience are allowed to evolve randomly in time. We allow trading in both directions and for càdlàg semimartingales as execution strategies. We derive a quadratic BSDE that under appropriate assumptions characterises minimal execution costs, and we identify conditions under which an optimal execution strategy exists. We also investigate qualitative aspects of optimal strategies such as e.g. appearance of strategies with infinite variation or existence of block trades, and we discuss connections with the discrete-time formulation of the problem. Our findings are illustrated in several examples.

When implementing educational innovations, teachers’ approaches to integrating new teaching demands are a crucial factor in their professional development. This becomes especially important in inclusive mathematics education, where teachers are demanded to integrate two jobs: (a) create joint learning experiences for all students and (b) provide focused learning opportunities for individuals. These jobs and the orientations in which they are pursued are perceived as conflicting demands by some teachers, while others do not. Within the model of content-related teacher expertise, this qualitative study investigates teachers’ practices for integrating these demands by disentangling the interplay between teachers’ self-reported practices and their underlying orientations about inclusive education. The analysis reveals that only some teachers explicate conflicts between inclusive teaching demands. It also shows that teachers command a variety of different practices for dealing with the new teaching demands and that some complex practices can indeed integrate different demands simultaneously. Implications include the insight that professional development programs on inclusive education should pay careful attention to teachers’ articulated jobs and orientations in order to promote the integration of teaching demands. Beyond this specific professional development content, the search for practices for integrating demands arising from an interplay of jobs and orientations might be a promising professional development research approach for increasing the scope of professional development.

in press We study monomial cut ideals associated to a graph G, which are a monomial analogue of toric cut ideals as introduced by Sturmfels and Sullivant. Primary decompositions, projective dimensions, and Castelnuovo–Mumford regularities are investigated if the graph can be decomposed as 0-clique sums and disjoint union of subgraphs. The total Betti numbers of a cycle are computed. Moreover, we classify all Freiman ideals among monomial cut ideals.

Partial differential equations (PDEs) are a fundamental tool in the modeling of many real-world phenomena. In a number of such real-world phenomena the PDEs under consideration contain gradient-dependent nonlinearities and are high-dimensional. Such high-dimensional nonlinear PDEs can in nearly all cases not be solved explicitly, and it is one of the most challenging tasks in applied mathematics to solve high-dimensional nonlinear PDEs approximately. It is especially very challenging to design approximation algorithms for nonlinear PDEs for which one can rigorously prove that they do overcome the so-called curse of dimensionality in the sense that the number of computational operations of the approximation algorithm needed to achieve an approximation precision of sizee > 0 grows at most polynomially in both the PDE dimension d is an element of N and the reciprocal of the prescribed approximation accuracy epsilon. In particular, to the best of our knowledge there exists no approximation algorithm in the scientific literature which has been proven to overcome the curse of dimensionality in the case of a class of nonlinear PDEs with general time horizons and gradient-dependent non-linearities. It is the key contribution of this article to overcome this difficulty. More specifically, it is the key contribution of this article (i) to propose a new full-history recursive multilevel Picard approximation algorithm for high-dimensional nonlinear heat equations with general time horizons and gradient-dependent nonlinearities and (ii) to rigorously prove that this full-history recursive multilevel Picard approximation algorithm does indeed overcome the curse of dimensionality in the case of such nonlinear heat equations with gradient-dependent nonlinearities., Foundations of Computational Mathematics, 2 (4), ISSN:1615-3375, ISSN:1615-3383

We classify all rotational symmetric solutions of the singular minimal surface equation in both cases $$\alpha α < 0 and $$\alpha >0$$ α > 0 . In addition, we discuss further geometric and analytic properties of the solutions, in particular stability, minimizing properties and Bernstein-type results.

This article proves a case of the p-adic Birch and Swinnerton–Dyer conjecture for Garrett p-adic L-functions of (Bertolini et al. in On p-adic analogues of the Birch and Swinnerton–Dyer conjecture for Garrett L-functions, 2021), in the imaginary dihedral exceptional zero setting of extended analytic rank 4.

We recall numerical criteria for Cohen–Macaulayness related to system of parameters and introduce monomial ideals of Konig type which include the edge ideals of Konig graphs. We show that a monomial ideal is of Konig type if and only if its corresponding residue class ring admits a system of parameters whose elements are of the form $$x_i-x_j$$ . This provides an algebraic characterization of Konig graphs. We use this special parameter systems for the study of the edge ideal of Konig graphs and the study of the order complex of a certain family of posets. Finally, for any simplicial complex $$\Delta $$ we introduce a system of parameters for $$K[\Delta ]$$ with a universal construction principle, independent of the base field and only dependent on the faces of $$\Delta $$ . This system of parameters is an efficient tool to test Cohen–Macaulayness of the Stanley–Reisner ring of a simplicial complex.

We consider motion by anisotropic curvature of a network of three curves immersed in the plane meeting at a triple junction and with the other ends fixed. We show existence, uniqueness and regularity of a maximal geometric solution and we prove that, if the maximal time is finite, then either the length of one of the curves goes to zero or the $L^2$ norm of the anisotropic curvature blows up., 39 pages, 1 figure

Starting from a Cosserat-type model for curved rods, we derive analytical expressions for the effective stiffness coefficients of multilayered composite beams with an arbitrary number of layers. For this purpose, we employ the comparison with analytical solutions of some bending, torsion, and extension problems for three-dimensional beams and rods. The layers of the composite beam consist of different orthotropic or isotropic non-homogeneous elastic materials. We apply the obtained general formulas to calculate exact analytical solutions of some beam problems and compare them with corresponding results of numerical simulations. The numerical study shows a wide range of validity and applicability of the obtained formulas.

Functional error estimates are well-established tools for a posteriori error estimation and related adaptive mesh-refinement for the finite element method (FEM). The present work proposes a first functional error estimate for the boundary element method (BEM). One key feature is that the derived error estimates are independent of the BEM discretization and provide guaranteed lower and upper bounds for the unknown error. In particular, our analysis covers Galerkin BEM and the collocation method, what makes the approach of particular interest for scientific computations and engineering applications. Numerical experiments for the Laplace problem confirm the theoretical results., Comment: Key words and phrases. boundary element method, functional a posteriori error estimate, adaptive mesh-refinement

We consider the cylindrical bending problem for an infinite plate as modeled with a family of generalized continuum models, including the micromorphic approach. The models allow to describe length scale effects in the sense that thinner specimens are comparatively stiffer. We provide the analytical solution for each case and exhibits the predicted bending stiffness. The relaxed micromorphic continuum shows bounded bending stiffness for arbitrary thin specimens, while classical micromorphic continuum or gradient elasticity as well as Cosserat models (Neff et al. in Acta Mechanica 211(3–4):237–249, 2010) exhibit unphysical unbounded bending stiffness for arbitrary thin specimens. This finding highlights the advantage of using the relaxed micromorphic model, which has a definite limit stiffness for small samples and which aids in identifying the relevant material parameters.

In this paper we study graded Bourbaki ideals. It is a well-known fact that for torsionfree modules over Noetherian normal domains, Bourbaki sequences exist. We give criteria in terms of certain attached matrices for a homomorphism of modules to induce a Bourbaki sequence. Special attention is given to graded Bourbaki sequences. In the second part of the paper, we apply these results to the Koszul cycles of the residue class field and determine particular Bourbaki ideals explicitly. We also obtain in a special case the relationship between the structure of the Rees algebra of a Koszul cycle and the Rees algebra of its Bourbaki ideal., Comment: 29 pages

We consider the problem of designing a cloak for waves described by the Helmholtz equation from an integer programming point of view. The problem can be modeled as a PDE-constrained optimization problem with integer-valued control inputs that are distributed in the computational domain. A first-discretize-then-optimize approach results in a large-scale mixed-integer nonlinear program that is in general intractable because of the large number of integer variables that arise from the discretization of the domain. Instead, we propose an efficient algorithm that is able to approximate the local infima of the underlying nonconvex infinite-dimensional problem arbitrarily close without the need to solve the discretized finite-dimensional integer programs to optimality. We optimize only the continuous relaxations of the approximations for local minima and then apply the sum-up rounding methodology to obtain integer-valued controls. If the solutions of the discretized continuous relaxations converge to a local minimizer of the continuous relaxation, then the resulting discrete-valued control sequence converges weakly $$^*$$ in $$L^\infty$$ to the same local minimizer. These approximation properties follow under suitable refinements of the involved discretization grids. Our results use familiar concepts arising from the analytical properties of the underlying PDE and complement previous results, derived from a topology optimization point of view.

International audience; In this paper we propose a novel variance reduction approach for additive functionals of Markov chains based on minimization of an estimate for the asymptotic variance of these functionals over suitable classes of control variates. A distinctive feature of the proposed approach is its ability to significantly reduce the overall finite sample variance. This feature is theoretically demonstrated by means of a deep non asymptotic analysis of a variance reduced functional as well as by a thorough simulation study. In particular we apply our method to various MCMC Bayesian estimation problems where it favourably compares to the existing variance reduction approaches.

This article is mainly motivated by the urge to answer two kinds of questions regarding the Bundesliga, which is Germany’s primary football (soccer) division having the highest average stadium attendance worldwide: “At any point in the season, what is the lowest final rank a certain team can achieve?” and “At any point in the season, what is the highest final rank a certain team can achieve?”. Although we focus on the Bundesliga in particular, the integer programming formulations we introduce to answer these questions can easily be adapted to a variety of other league systems and tournaments.

In this paper, we explore the use of micromorphic-type interface conditions for the modeling of a finite-sized metamaterial. We show how finite-domain boundary value problems can be approached in the framework of enriched continuum mechanics (relaxed micromorphic model) by imposing continuity of macroscopic displacement and of generalized tractions, as well as additional conditions on the micro-distortion tensor and on the double-traction. The case of a metamaterial slab of finite width is presented, its scattering properties are studied via a semi-analytical solution of the relaxed micromorphic model and compared to a direct finite-element simulation encoding all details of the selected microstructure. The reflection and transmission coefficients obtained via the two methods are presented as a function of the frequency and of the direction of propagation of the incident wave. We find excellent agreement for a large range of frequencies going from the long-wave limit to frequencies beyond the first band-gap and for angles of incidence ranging from normal to near-parallel incidence. The present paper sets the basis for a new viewpoint on finite-size metamaterial modeling enabling the exploration of meta-structures at large scales.

International audience

Let $E/K$ be a finite Galois extension of totally real number fields with Galois group $G$. Let $p$ be an odd prime and let $r>1$ be an odd integer. The $p$-adic Beilinson conjecture relates the values at $s=r$ of $p$-adic Artin $L$-functions attached to the irreducible characters of $G$ to those of corresponding complex Artin $L$-functions. We show that this conjecture, the equivariant Iwasawa main conjecture and a conjecture of Schneider imply the `$p$-part' of the equivariant Tamagawa number conjecture for the pair $(h^0(\mathrm{Spec}(E))(r), \mathbb Z[G])$. If $r>1$ is even we obtain a similar result for Galois CM-extensions after restriction to `minus parts'., 32 pages, v2 contains considerable improvements (the vanishing of $\mu$ is no longer required); v3 minor changes and corrections, Remark 3.22 added; v4 further minor changes. To appear in Sel. Math. New Ser

In this contribution, we study a class of doubly nonlinear elliptic equations with bounded, merely integrable right-hand side on the whole space $$\mathbb {R}^N$$ R N . The equation is driven by the fractional Laplacian $$(-\varDelta )^{\frac{s}{2}}$$ ( - Δ ) s 2 for $$s\in (0,1]$$ s ∈ ( 0 , 1 ] and a strongly continuous nonlinear perturbation of first order. It is well known that weak solutions are in genreral not unique in this setting. We are able to prove an $$L^1$$ L 1 -contraction and comparison principle and to show existence and uniqueness of entropy solutions.

Following Rivière’s study of conservation laws for second order quasilinear systems with critical nonlinearity and Lamm/Rivière’s generalization to fourth order, we consider similar systems of order 2m. Typical examples are m-polyharmonic maps. Under natural conditions, we find a conservation law for weak solutions on 2m-dimensional domains. This implies continuity of weak solutions.

We study the dynamics of many charges interacting with the Maxwell field. The particles are modeled by means of nonnegative distribution functions $$f^+$$ and $$f^-$$ representing two species of charged matter with positive and negative charge, respectively. If their initial velocities are small compared to the speed of light, $$\mathrm{c}$$, then in lowest order, the Newtonian or classical limit, their motion is governed by the Vlasov–Poisson system. We investigate higher-order corrections with an explicit control on the error terms. The Darwin order correction, order $$|\bar{\mathrm{v}}/\mathrm{c}|^2$$, has been proved previously. In this contribution, we obtain the dissipative corrections due to radiation damping, which are of order $$|\bar{\mathrm{v}}/\mathrm{c}|^3$$ relative to the Newtonian limit. If all particles have the same charge-to-mass ratio, the dissipation would vanish at that order.

In this paper we investigate simple t-designs having s-resolutions for $$t \ge 3$$ and $$1 \le s

In this paper we propose a canonical variational framework for rate-independent phenomenological geometrically linear gradient plasticity with plastic spin. The model combines the additive decomposition of the total distortion into non-symmetric elastic and plastic distortions, with a defect energy contribution taking account of the Burgers vector through a dependence only on the dislocation density tensor Curl(p) giving rise to a non-symmetric nonlocal backstress, and isotropic hardening response only depending on the accumulated equivalent plastic strain. The model is fully isotropic and satisfies linearized gauge-invariance conditions, i.e., only true state-variables appear. The model satisfies also the principle of maximum dissipation which allows to show existence for the weak formulation. For this result, a recently introduced Korn's inequality for incompatible tensor fields is necessary. Uniqueness is shown in the class of strong solutions. For vanishing energetic length scale, the model reduces to classical elasto-plasticity with symmetric plastic strain sym(p) and standard isotropic hardening., 5 figures

In this paper we will show that the pull-back of any regular differential form defined on the smooth locus of a GIT quotient of dimension at most four to any resolution yields a regular differential form.

We are interested in nontrivial conditions on the nonnegative masks that guarantee the convergence of the correspondent subdivision schemes. Roughly speaking, a certain convexity of the support of the given mask implies the convergence of the subdivision scheme. Moreover, those conditions are computable. The key of proving our main theorem is to find out an irreducible or primitive mapping on some multi-integer set and to show the uniqueness of this mapping.

Freiman’s theorem gives a lower bound for the cardinality of the doubling of a finite set in $${\mathbb R}^n$$ . In this paper we give an interpretation of his theorem for monomial ideals and their fiber cones. We call a quasi-equigenerated monomial ideal a Freiman ideal, if the set of its exponent vectors achieves Freiman’s lower bound for its doubling. Algebraic characterizations of Freiman ideals are given, and finite simple graphs are classified whose edge ideals or matroidal ideals of its cycle matroids are Freiman ideals.

In this paper, we give sufficient conditions guaranteeing the validity of the well-known minimax theorem for the lower Snell envelope. Such minimax results play an important role in the characterisation of arbitrage-free prices of American contingent claims in incomplete markets. Our conditions do not rely on the notions of stability under pasting or time-consistency and reveal some unexpected connection between the minimax result and path properties of the corresponding process of densities. We exemplify our general results in the case of families of measures corresponding to diffusion exponential martingales.

We consider a doubly nonlinear evolution equation with multiplicative noise and show existence and pathwise uniqueness of a strong solution. Using a semi-implicit time discretization we get approximate solutions with monotonicity arguments. We establish a-priori estimates for the approximate solutions and show tightness of the sequence of image measures induced by the sequence of approximate solutions. As a consequence of the theorems of Prokhorov and Skorokhod we get a.s. convergence of a subsequence on a new probability space which allows to show the existence of martingale solutions. Pathwise uniqueness is obtained by an $$L^1$$ -method. Using this result, we are able to show existence and uniqueness of strong solutions.

We show that for an odd prime p, the p-primary parts of refinements of the (imprimitive) non-abelian Brumer and Brumer–Stark conjectures are implied by the equivariant Iwasawa main conjecture (EIMC) for totally real fields. Crucially, this result does not depend on the vanishing of the relevant Iwasawa $$\mu $$ -invariant. In combination with the authors’ previous work on the EIMC, this leads to unconditional proofs of the non-abelian Brumer and Brumer–Stark conjectures in many new cases.

Broadly neutralizing antibodies (bNAbs) are very promising agents for HIV-1 prophylaxis and AIDS treatment. However, the neutralization susceptibility of circulating recombinants such as CRF01_AE, which is becoming increasingly prevalent, has not been studied in detail until now. Here, we focused on CRF01_AE in China and aimed to find bNAbs that can be used for neutralization of CRF01_AE. Full-length env clones were obtained from the plasma samples of 22 HIV-1-infected individuals sampled in 2009 and 2015. An env-pseudovirus-based neutralization assay was conducted using five categories of bNAbs: VRC01, NIH45-46G54W, and 3BNC117 (targeting the CD4 binding site); PG9 and PG16 (targeting the V1V2 loop); 2G12 (glycan specific), PGT121 and 10-1074 (targeting the V3 glycan); 2F5, 4E10, and 10E8 (targeting the membrane-proximal external region (MPER)). The neutralizing efficiency was compared, and features of the escape pseudoviruses were analyzed. The CRF01_AE pseudoviruses exhibited different susceptibility to these bNAbs. Overall, 4E10, 10E8, and 3BNC117 neutralized all 22 env-pseudotyped viruses, followed by NIH45-46G54W and VRC01, which neutralized more than 90% of the viruses. 2F5, PG9, and PG16 showed only moderate breadth, while the other three bNAbs neutralized none of these pseudoviruses. Specifically, 10E8, NIH45-46G54Wand 3BNC117 showed the highest efficiency, combining neutralization potency and breadth. Mutations at position 160, 169, 171 were associated with resistance to PG9 and PG16, while loss of a potential glycan at position 332 conferred insensitivity to V3-glycan-targeting bNAbs. Our results may help for choosing bNAbs that can be used preferentially for prophylactic or therapeutic approaches in China.

Results of Bernstein type are proven for supersolutions of the singular minimal surface equation when $$\alpha

We reconsider anti-plane shear deformations based on prior work of Knowles and relate the existence of anti-plane shear deformations to fundamental constitutive concepts of elasticity theory like polyconvexity, rank-one convexity and tension–compression symmetry. In addition, we provide finite element simulations to visualize our theoretical findings.

We give sufficient conditions for the expected excess and the mean-upper-semideviation of recourse functions to be strongly convex. This is done in the setting of two-stage stochastic programs with complete linear recourse and random right-hand side. This work extends results on strong convexity of risk-neutral models.

Let E be a vector bundle over an irreducible projective variety X defined over an algebraically closed field. We give a necessary and sufficient condition for E to be a direct image of a vector bundle on an etale cover, of degree more than one, of X. In fact, we describe all possible ways E can be realized as a direct image. Given a connection D on E, a criterion is given for D to be induced by a connection on a vector bundle whose direct image, by an etale covering map of degree more than one, is E.

The purpose of this article is to show how the philosophical theory of inferentialism can be used to understand students’ conceptual development in the field of mathematics. Based on the works of philosophers such as Robert Brandom, an epistemological theory in mathematics education is presented that offers the opportunity to trace students’ conceptual development in both its individual and social facets through analyzing patterns of reasoning. A design experiment on decimal numbers serves as a paradigmatic example. The overall goal is to illustrate the relationship between mathematical standard and individual ways of reasoning in conceptual development processes.

We address the single-machine batch scheduling problem with the objective of minimizing the total setup cost. This problem arises when there are n jobs that are partitioned into F families and when setup operations are required whenever the machine switches from processing a job of one family to processing a job of another family. We assume that setups do not require time but are associated with a fixed cost which is identical for all setup operations. Each job has a processing time and an associated deadline. The objective is to schedule all jobs such that they are on time with respect to their deadlines and the total setup cost is minimized. We show that the decision version of this problem is NP-complete in the strong sense. Furthermore, we present properties of optimal solutions and an $$O(n\log n+nF)$$ algorithm that approximates the cost of an optimal schedule by a factor of F. The algorithm is analyzed in computational tests.

Recently, the scaling limit of cluster sizes for critical inhomogeneous random graphs of rank-1 type having finite variance but infinite third moment degrees was obtained (see previous work by Bhamidi, van der Hofstad and van Leeuwaarden). It was proved that when the degrees obey a power law with exponent in the interval (3,4), the sequence of clusters ordered in decreasing size and scaled appropriately converges as n goes to infinity to a sequence of decreasing non-degenerate random variables. Here, we study the tails of the limit of the rescaled largest cluster, i.e., the probability that the scaling limit of the largest cluster takes a large value u, as a function of u. This extends a related result of Pittel for the Erd\H{o}s-R\'enyi random graph to the setting of rank-1 inhomogeneous random graphs with infinite third moment degrees. We make use of delicate large deviations and weak convergence arguments., Comment: corrected and updated first reference

In this paper we prove the existence of solutions to doubly nonlinear equations whose prototype is given by $$\partial_t u^m- {\rm div}\, D_{\xi}\, f(x,Du) =0,$$ with $${m\in (0,\infty )}$$ , or more generally with an increasing and piecewise C1 nonlinearity b and a function f depending on u $$\partial_{t}b(u) - {\rm div}\, D_{\xi}\, f(x,u,Du)= -D_u f(x,u,Du).$$ For the function f we merely assume convexity and coercivity, so that, for instance, $${f(x,u,\xi)=\alpha(x)|\xi|^p + \beta(x)|\xi|^q}$$ with 1 0}$$ , and $${f(\xi)=\exp(\tfrac12|\xi|^2)}$$ are covered. Thus, for functions $${f(x,u,\xi )}$$ satisfying only a coercivity assumption from below but very general growth conditions from above, we prove the existence of variational solutions; mean while, if $${f(x,u,\xi )}$$ grows naturally when $${\left\vert \xi \right\vert \rightarrow +\infty }$$ as a polynomial of order p (for instance in the case of the p-Laplacian operator), then we obtain the existence of solutions in the sense of distributions as well as the existence of weak solutions. Our technique is purely variational and we treat both the cases of bounded and unbounded domains. We introduce a nonlinear version of the minimizing movement approach that could also be useful for the numerics of doubly nonlinear equations.