On some hyperbolic systems of temple class

The aim of this paper is the statement of a general class of Temple systems of conservation laws that includes both the chromatography/electrophoresis like systems and the 2×2 LeRoux system that we generalize to any dimension. We show that this class actually belongs to the Temple type, and compute a complete set of strict Riemann invariants in a generic situation. As the property “the integral curve of this eigenvector is a straight line”is essentially a linear algebra property, we aim to deduce the results with simple (linear algebra) hypothesis and arguments, from the structure of the jacobian matrix.     

Are You Convinced? Middle‐Grade Students' Evaluations of Mathematical Arguments

Students learn norms of proving by observing teachers generating proofs, engaging in proving, and generalizing features of proofs deemed convincing by an authority, such as a textbook. Students at all grade levels have difficulties generating valid proof; however, little research exists on students' understandings about what makes a mathematical argument convincing prior to more formal instruction in methods of proof. This study investigated middle‐school students' (ages 12–14) evaluations of arguments for a statement in number theory. Students evaluated both an empirical and a general argument in an interview setting. The results show that students tend to prefer empirical arguments because examples enhance an argument's power to show that the statement is true. However, interview responses also reveal that a significant number of students find arguments to be most convincing when examples are supported with an explanation that “tells why” the statement is true. The analysis also examined the alignment of students' reasons for choosing arguments as more convincing along with the strategies they employ to make arguments more convincing. Overall, the findings show middle‐school students' conceptions about what makes arguments convincing are more sophisticated than their performance in generating arguments suggests.     

Une approche séquentielle de l'hypothèse de Riemann généralisée

A generalisation of the Beurling and Nyman criterion implies that the Riemann hypothesis for a function F in the Selberg class is equivalent to the statement that χ, the characteristic function of (0,1) belongs to the adherence in L²(0,+∞) of a subspace BF. In this article, the author generalises a result of Báez-Duarte to the Selberg class and gives a construction of some sequence of BF converging to χ in L²(0,+∞) under Riemann hypothesis for F.     

Diophantine equations and the generalized Riemann hypothesis

We show that, for a listable set P of polynomials with integer coefficients, the statement “for all roots θ of all polynomials in P, the generalized Riemann hypothesis for Q(θ) holds” is Diophantine. That is, the statement is equivalent to the unsolvability of a particular Diophantine equation. This is achieved by finding a decidable property P such that the aforementioned statement may be written in the form “P holds for all natural numbers”.     

Remarks on DiPerna's Paper ``Convergence of the Viscosity Method for Isentropic Gas Dynamics'

Concerns have been voiced about the correctness of certain technical points in DiPerna's paper (Comm. Math. Phys.91 (1983), 1-30) related to the vacuum state. In this note, we provide clarifications. Our conclusion is that these concerns mainly arise from the statement of a lemma for constructing the viscous approximate solutions and some typos; however, the gap can be either fixed by correcting the statement of the lemma and the typos or bypassed by employing the finite difference methods.

     

A compelling argument for the gravity p-median model

The p-median model is used to locate P facilities to serve a geographically distributed population. Conventionally, it is assumed that the population always travels to the nearest facility.  and  re-estate three arguments on why this assumption might be incorrect, and they introduce the gravity p-median model to relax the assumption. We favor the gravity p-median model, but we note that in an applied setting, the three arguments are incomplete. In this communication, we point at the existence of a fourth compelling argument for the gravity p-median model.     

Corrigendum to “The Conley conjecture for Hamiltonian systems on the cotangent bundle and its analogue for Lagrangian systems” [J. Funct. Anal. 256 (9) (2009) 2967-3034]

In lines 8-11 of Lu (2009) [18, p. 2977] we wrote: “For integer m?3, if M is Cm-smooth and Cm−1-smooth L:R×TM→R satisfies the assumptions (L1)-(L3), then the functional Lτ is C²-smooth, bounded below, satisfies the Palais-Smale condition, and all critical points of it have finite Morse indexes and nullities (see [1, Prop. 4.1, 4.2] and [4])”. However, as proved in Abbondandolo and Schwarz (2009) [2] the claim that Lτ is C²-smooth is true if and only if for every (t,q) the function v?L(t,q,v) is a polynomial of degree at most 2. So the arguments in Lu (2009) [18] are only valid for the physical Hamiltonian in (1.2) and corresponding Lagrangian therein. In this note we shall correct our arguments in Lu (2009) [18] with a new splitting lemma obtained in Lu (2011) [20].     

Generalizations of ?wierczkowski’s lemma and the arity gap of finite functions

?wierczkowski’s lemma-as it is usually formulated-asserts that if f:Aⁿ→A is an operation on a finite set A, n≥4, and every operation obtained from f by identifying a pair of variables is a projection, then f is a semiprojection. We generalize this lemma in various ways. First, it is extended to B-valued functions on A instead of operations on A and to essentially at most unary functions instead of projections. Then we characterize the arity gap of functions of small arities in terms of quasi-arity, which in turn provides a further generalization of ?wierczkowski’s lemma. Moreover, we explicitly classify all pseudo-Boolean functions according to their arity gap. Finally, we present a general characterization of the arity gaps of B-valued functions on arbitrary finite sets A.     

The opaque nature of generic examples: The structure of student teachers’ arguments in multiplicative reasoning

The study aims to explore the structural aspects of generic examples, to get better insight into what makes them potentially opaque for learners. We have analyzed 27 written arguments, for which student teachers (grades 1–10) were asked to use a generic example to prove a given statement in multiplication. Using Toulmin’s framework, we developed five categories of arguments based on their structure: examples, empirical arguments, leap arguments, embedded arguments, and other arguments. Also, we conclude that none of the student teachers provided arguments that we recognize as complete generic examples. The results bring us to a discussion about features of generic examples making them difficult to come to grips with, having implications for how teacher educators can support student teachers’ learning to prove. From this, we propose a definition of generic examples that attends to the criteria suggested in previous research, yet, emphasizing their structural nature.     

Relative critical exponents,non-vanishing and metrics with minimal singularities

In this article we establish a very general non-vanishing statement, as well as several properties of metrics with minimal singularities of adjoint bundles. Our arguments combine the proof of the classical non-vanishing theorem due to V. Shokurov, as well as many ideas from Y.-T. Siu’s analytic proof of the finite generation of the canonical ring. An important tool is the notion of relative critical exponent of two closed positive currents with respect to a measure.     

Leveraging interdiscursivity to support elementary students in bridging the empirical-deductive gap: the case of parity

Research has recognized deductive reasoning as challenging but not impossible for young mathematics learners. In this paper, we present a learning environment developed to assist elementary-school students to bridge the empirical-deductive gap in the context of parity of numbers. Using the commognitive framework, we construe the empirical-deductive gap as part of a broader divide between two discourses that abide by different rules of a “mathematical game”: a discourse on specific numbers and a discourse on numeric patterns. Interdiscursivity is leveraged as a mechanism for instructional design, where students’ familiar routines with specific numbers are teased out and advanced to make sense in the new discourse. We mobilize this mechanism to create opportunities for students to play an active role in recognizing issues with empirical reasoning and generating deductive arguments to establish the validity of universal statements. The environment is illustrated with a small group of 8-year-olds who learned to justify deductively that “odd + odd = even”.     

An equivalence to the Gleason problem

In this article we study the Gleason problem locally. A new method for solving the Gleason A problem is presented. This is done by showing an equivalent statement to the Gleason A problem. In order to prove this statement, necessary and a sufficient conditions for a bounded domain to have the Gleason A property are found. Also an example of a bounded but not smoothly-bounded domain in Cn is given, which satisfies the sufficient condition at the origin, and hence has the Gleason A property there.     

Monotonicity properties of the Neumann heat kernel in the ball

A well-known conjecture of R. Laugesen and C. Morpurgo asserts that the diagonal of the Neumann heat kernel of the unit ball U⊂Rn is a strictly increasing radial function. In this paper we use probabilistic arguments to settle this conjecture and to prove some inequalities for the Neumann heat kernel in the ball.     

Weierstrass points and their impact in the study of algebraic curves: a historical account from the “Lückensatz” to the 1970s

In this note we give a historical account of the origin and the development of the concept of Weierstrass point. We also explain how Weierstrass points have contributed to the study of compact Riemann surfaces and algebraic curves in the century from Weierstrass’ statement of the gap theorem to the 1970s. In particular, we focus on the seminal work of Hürwitz that raised questions which are at the center of contemporary research on this topic.      

Approximation diophantienne sur les variétés semi-abéliennes

Let A be a semi-abelian variety over , Γ a subgroup of of finite rank and X a subvariety of A which is not a translate of a semi-abelian subvariety of A. Work by P. Vojta and M. McQuillan shows that is not dense in X. B. Poonen has then conjectured that the same remains true if Γ is replaced by a fattening for a certain ε>0 where h is a canonical height. B. Poonen and S. Zhang have shown independently this to hold when A is almost split. On the other hand, the statement contains the Bogomolov property (with Γ=0) now proven by S. David and P. Philippon. In this paper, we prove Poonen's conjecture for any A. We also consider the slightly more general sets instead of Γ_ε. We use the case Γ=0 as well as a generalized Vojta inequality.     

An <Emphasis Type="Italic">s</Emphasis>-Hamiltonian Line Graph Problem

For an integer k > 0, a graph G is k-triangular if every edge of G lies in at least k distinct 3-cycles of G. In (J Graph Theory 11:399–407 (1987)), Broersma and Veldman proposed an open problem: for a given positive integer k, determine the value s for which the statement “Let G be a k-triangular graph. Then L(G), the line graph of G, is s-hamiltonian if and only L(G) is (s + 2)-connected” is valid. Broersma and Veldman proved in 1987 that the statement above holds for 0 ≤ s ≤ k and asked, specifically, if the statement holds when s = 2k. In this paper, we prove that the statement above holds for 0 ≤ s ≤ max{2k, 6k − 16}.     

Sequences of 0’s and 1’s: Special sequence spaces with the separable Hahn property

As pointed out in [4] the paper [2], authored by G. Bennett, J. Boos and T. Leiger, contains a nontrivial gap in the argumentation of the proof of Theorem 5.2 which is one of main results of that paper and has been applied three times. Till now neither the gap is closed nor a counterexample has been stated. That is why the authors have examined in [4] the situation around the ‘gap’ aiming to a better understanding for the gap. The aim of this paper is to prove the mentioned applications of the theorem in doubt by using gliding hump arguments (quite similar to the classical proofs of the Theorems of Schur and Hahn in the first case (cf. [3]) and a very technical and artful construction, being of independent mathematical interest, in the second case). Research of T. Leiger supported by Estonian Science Foundation Grant 5376.     

A study of the lex plus powers conjecture

An {a₁,…,a_n}-lex plus powers ideal is a monomial ideal in I⊂k[x₁,…,x_n] which minimally contains the regular sequence x₁^a₁,…,x_n^a_n and such that whenever m∈R_t is a minimal generator of I and m′∈R_t is greater than m in lex order, then m′∈I. Conjectures of Eisenbud et al. and Charalambous and Evans predict that after restricting to ideals containing a regular sequence in degrees {a₁,…,a_n}, then {a₁,…,a_n}-lex plus powers ideals have extremal properties similar to those of the lex ideal. That is, it is proposed that a lex plus powers ideal should give maximum possible Hilbert function growth (Eisenbud et al.), and, after fixing a Hilbert function, that the Betti numbers of a lex plus powers ideal should be uniquely largest (Charalambous, Evans). The first of these assertions would extend Macaulay's theorem on Hilbert function growth, while the second improves the Bigatti, Hulett, Pardue theorem that lex ideals have largest graded Betti numbers. In this paper we explore these two conjectures. First we give several equivalent forms of each statement. For example, we demonstrate that the conjecture for Hilbert functions is equivalent to the statement that for a given Hilbert function, lex plus powers ideals have the most minimal generators in each degree. We use this result to prove that it is enough to show that lex plus powers ideals have the most minimal generators in the highest possible degree. A similar result holds for the stronger conjecture. In this paper we also prove that if the weaker conjecture holds, then lex plus powers ideals are guaranteed to have largest socles. This suffices to show that the two conjectures are equivalent in dimension ?3, which proves the monomial case of the conjecture for Betti numbers in those degrees. In dimension 2, we prove both conjectures outright.     

Algorithms for generating arguments and counterarguments in propositional logic

A common assumption for logic-based argumentation is that an argument is a pair 〈Φ,α〉 where Φ is minimal subset of the knowledgebase such that Φ is consistent and Φ entails the claim α. Different logics provide different definitions for consistency and entailment and hence give us different options for formalising arguments and counterarguments. The expressivity of classical propositional logic allows for complicated knowledge to be represented but its computational cost is an issue. In previous work we have proposed addressing this problem using connection graphs and resolution in order to generate arguments for claims that are literals. Here we propose a development of this work to generate arguments for claims that are disjunctive clauses of more than one disjunct, and also to generate counteraguments in the form of canonical undercuts (i.e. arguments that with a claim that is the negation of the conjunction of the support of the argument being undercut).