命题逻辑公式的压缩表示及其相应的形式系统

提出一种将命题逻辑公式压缩表示的方法--公式的压缩图,给出相应的形式系统,并证明该系统的证明效率比传统相继式演算系统Gentzen\{cut}有指数级的提高,从而为命题逻辑提供了一种新的有效的推理系统.     

Exploring the Link Between Preservice Teachers' Conception of Proof and the Use of Dynamic Geometry Software

This case study investigated how secondary preservice mathematics teachers perceive the need for and the benefits of formal proof when given geometric tasks in the context of dynamic geometry software. Results indicate that preservice teachers are concerned that after using dynamic software high school students will not see the need for proofs. The participants stated that multiple examples are not equivalent to a proof but, nonetheless, questioned the value of formal proof for high school students. Finally, preservice teachers found the greatest value of geometric software to be in helping students understand key relationships within a problem or theorem. Participants also tended to study a problem more deeply with the software than without it.     

The phase diagram of a directed polymer in random media with <Emphasis Type="Italic">p</Emphasis>-spin ferromagnetic interactions

We consider a directed polymer model with an additive p-spin (p>2) ferromagnetic term in the Hamiltonian. We give a rigorous proof for the specific free energy and derive the phase diagram. This model was proposed previously, and a detailed proof was given in the case p = 2, while the main result was only stated for p > 2. We give a detailed proof of the main result and show the behavior of the model as p → ∞ by constructing the phase diagram also in this case. These results are important in many applications, for instance, in telecommunication and immunology. Our major finding is that in the phase diagram for p > 2, a new transition curve (absent for p = 2) emerges between the paramagnetic region and the so-called mixed region and that the ferromagnetic region diminishes as p → ∞.     

基于Coq的第三代微积分机器证明系统

本文基于证明辅助工具Coq,完整实现林群院士和张景中院士等倡导的第三代微积分|没有极限的微积分|理论构架的形式化验证,包括对张景中等发表的题为\微积分基础的新视角"论文中全部定义和定理的Coq描述.进而,对定理无例外地给出Coq的机器证明代码,所有形式化过程已被Coq验证,并在计算机上运行通过,体现了基于Coq的数学定理机器证明具有可读性和交互性的特点,其证明过程规范、严谨、可靠.本文是实践研究人员利用计算机学习、理解、构建乃至教育现代数学理论的一个尝试.     

On hooks of Young diagrams

The well-known fact that there is always one more addable than removable box for a Young diagram is generalized to arbitrary hooks. As an application, this immediately implies a simple proof of a conjecture of Regev and Vershik [3] for which inductive proofs have recently been given by Regev and Zeilberger [4] and Janson [1].     

Macdonald identities and multidimensional theta-functions

A new proof of the combinatorial Macdonald identities is presented. It is shown that one may regard these identities as a decomposition of certain multidimensional theta-functions into infinite products. The proof is based on some analytical properties of theta-functions. It is briefly discussed how one can modify the proof in order to replace analytical arguments by formal ones involving only operations with formal series.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 240, 1997, pp. 67–77.The author is grateful to A. M. Vershik and Yu. M. Bazlov for their interest and helpful comments.This research was supported by ISSEP (grant No. A96-1965).     

Topos中的Galois联络

In this paper, we investigate the Galois connections between two partially ordered objects in an arbitrary elementary topos. Some characterizations of Galois adjunctions which is similar to the classical case are obtained by means of the diagram proof. This shows that the diagram method can be used to reconstruct the classical order theory in an arbitrary elementary topos.     

Mathematics Majors' Perceptions of Conviction,Validity, and Proof

In this paper, 28 mathematics majors who completed a transition-to-proof course were given 10 mathematical arguments. For each argument, they were asked to judge how convincing they found the argument and whether they thought the argument constituted a mathematical proof. The key findings from this data were (a) most participants did not find the empirical argument in the study to be convincing or to meet the standards of proof, (b) the majority of participants found a diagrammatic argument to be both convincing and a proof, (c) participants evaluated deductive arguments not by their form but by their content, but (d) participants often judged invalid deductive arguments to be convincing proofs because they did not recognize their logical flaws. These findings suggest improving undergraduates' comprehension of mathematical arguments does not depend on making undergraduates aware of the limitations of empirical arguments but instead on improving the ways in which they process the arguments that they read.     

Le rang du systeme lineaire des racines d'une algebre de lie rigide resoluble complexe

Adams and Conway have stated without proof a result which says, roughly speaking, that the representation ring R(G) of a compact, connected Lie group G is generated as a λ-ring by elements in 1-to-1 correspondence with the branches of the Dynkin diagram. In this note we present an elementary proof of this.     

Counting trees with random walks

We give a simple proof of Tutte’s matrix-tree theorem, a well-known result providing a closed-form expression for the number of rooted spanning trees in a directed graph. Our proof stems from placing a random walk on a directed graph and then applying the Markov chain tree theorem to count trees. The connection between the two theorems is not new, but it appears that only one direction of the formal equivalence between them is readily available in the literature. The proof we now provide establishes the other direction. More generally, our approach is another example showing that random walks can serve as a powerful glue between graph theory and Markov chain theory, allowing formal statements from one side to be carried over to the other.     

Cognitive trajectory of proof by contradiction for transition-to-proof students

History and research on proof by contradiction suggests proof by contradiction is difficult for students in a number of ways. Students’ comprehension of already-written proofs by contradiction is one such aspect that has received relatively little attention. Applying the cognitive lens of Action-Process-Object-Schema (APOS) Theory to proof by contradiction, we constructed and tested a cognitive model that describes how a student might construct the concept ‘proof by contradiction’ in an introduction to proof course. Data for this study was collected from students in a series of five teaching interventions focused on proof by contradiction. This paper will report on two participants as case studies to illustrate that our cognitive trajectory for proof by contradiction is a useful model for describing how students may come to understand the proof method.     

A Counting Formula for Labeled, Rooted Forests

Given a power series, the coefficients of the formal inverse may be expressed as polynomials in the coefficients of the original series. Further, these polynomials may be parameterized by certain ordered, labeled forests. There is a known formula for the formal inverse, which indirectly counts these classes of forests, developed in a non-direct manner. Here, we provide a constructive proof for this counting formula that explains why it gives the correct count. Specifically, we develop algorithms for building the forests, enabling us to count them in a direct manner.     

The shapley transfer value without zero weights

Shapley's wellknown existence proof for the Shapley transfer value must allow for rather troublesome zero utility weights. In order to avoid these, a different existence proof is given in this paper. As an application of this approach it is shown that such a Shapley transfer value of a convex non-sidepayment game is an element of its strong core, and furthermore, an axiomatization of the Shapley transfer value is presented, basing upon the formal specification of a notion of independence of irrelevant alternatives.     

An intuitionistic formula hierarchy based on high‐school identities

We revisit the notion of intuitionistic equivalence and formal proof representations by adopting the view of formulas as exponential polynomials. After observing that most of the invertible proof rules of intuitionistic (minimal) propositional sequent calculi are formula (i.e., sequent) isomorphisms corresponding to the high‐school identities, we show that one can obtain a more compact variant of a proof system, consisting of non‐invertible proof rules only, and where the invertible proof rules have been replaced by a formula normalization procedure. Moreover, for certain proof systems such as the G4ip sequent calculus of Vorob'ev, Hudelmaier, and Dyckhoff, it is even possible to see all of the non‐invertible proof rules as strict inequalities between exponential polynomials; a careful combinatorial treatment is given in order to establish this fact. Finally, we extend the exponential polynomial analogy to the first‐order quantifiers, showing that it gives rise to an intuitionistic hierarchy of formulas, resembling the classical arithmetical hierarchy, and the first one that classifies formulas while preserving isomorphism.     

A higher-dimensional Kurzweil theorem for formal Laurent series over finite fields

In a recent paper, Kim and Nakada proved an analogue of Kurzweil?s theorem for inhomogeneous Diophantine approximation of formal Laurent series over finite fields. Their proof used continued fraction theory and thus cannot be easily extended to simultaneous Diophantine approximation. In this note, we give another proof which works for simultaneous Diophantine approximation as well.     

A higher-dimensional Kurzweil theorem for formal Laurent series over finite fields

In a recent paper, Kim and Nakada proved an analogue of Kurzweilʼs theorem for inhomogeneous Diophantine approximation of formal Laurent series over finite fields. Their proof used continued fraction theory and thus cannot be easily extended to simultaneous Diophantine approximation. In this note, we give another proof which works for simultaneous Diophantine approximation as well.     

K-theory and intersection theory revisited

Another proof that the product structure on K-theory may be used to define the product structure on the Chow ring of a smooth variety over a field is presented. The virtue of this proof is that it is essentially a formal argument using natural properties of Quillen's spectral sequence, the K-theory product, cycle classes, and the classical intersection product.