How persuaded are you? A typology of responses

Several recent studies have suggested that there are two different ways in which a person can proceed when assessing the persuasiveness of a mathematical argument: by evaluating whether it is personally convincing, or by evaluating whether it is publicly acceptable. In this paper, using Toulmin's (1958) argumentation scheme, we produce a more detailed theoretical classification of the ways in which participants can interpret a request to assess the persuasiveness of an argument. We suggest that there are (at least) five ways in which such a question can be interpreted. The classification is illustrated with data from a study that asked undergraduate students and research-active mathematicians to rate how persuasive they found a given argument. We conclude by arguing that researchers interested in mathematical conviction and proof validation need to be aware of the different ways in which participants can interpret questions about the persuasiveness of arguments, and that they must carefully control for these variations during their studies.     

How big are the increments of G-Brownian motion?

In this paper,we investigate the problem:How big are the increments of G-Brownian motion.We obtain the Csrg and R′ev′esz’s type theorem for the increments of G-Brownian motion.As applications of this result,we get the law of iterated logarithm and the Erds and R′enyi law of large numbers for G-Brownian motion.Furthermore,it turns out that our theorems are natural extensions of the classical results obtained by Csrg and R′ev′esz(1979).     

How fast are the particles of super-Brownian motion?

In this paper we investigate fast particles in the range and support ofsuper-Brownian motion in the historical setting. In this setting eachparticle of super-Brownian motion alive at time t is represented by apath w:[0,t]→ℝ ^d and the state of historical super-Brownian motionis a measure on the set of paths. Typical particles have Brownian paths,however in the uncountable collection of particles in the range of asuper-Brownian motion there are some which at exceptional times movefaster than Brownian motion. We determine the maximal speed of allparticles during a given time period E, which turns out to be afunction of the packing dimension of E. A path w in the support ofhistorical super-Brownian motion at time t is called a-fast if . Wecalculate the Hausdorff dimension of the set of a-fast paths in thesupport and the range of historical super-Brownian motion. A valuabletool in the proofs is a uniform dimension formula for the Browniansnake, which reduces dimension problems in the space of stopped paths to dimension problems on the line. Received: 27 January 2000 / Revised version: 28 August 2000 / Published online: 24 July 2001     

N-point Virasoro algebras are multipoint Krichever–Novikov-type algebras

We show how the recently again discussed N-point Witt, Virasoro, and a?ne Lie algebras are genus zero examples of the multipoint versions of Krichever–Novikov-type algebras as introduced and studied by Schlichenmaier. Using this more general point of view, useful structural insights and an easier access to calculations can be obtained. The concept of almost-grading will yield information about triangular decompositions which are of importance in the theory of representations. As examples, the algebra of functions, vector fields, differential operators, current algebras, a?ne Lie algebras, Lie superalgebras, and their central extensions are studied. Very detailed calculations for the three-point case are given.     

How small are the increments of a Wiener process?

Let $\text{γ}{}_{\mathrm{т}}\text{=(8(}\text{logTa}{}^{-1}{}_{\mathrm{T}}\text{+log log T)}\text{π}{}^2\text{a}{}_{\mathrm{T}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, 0<a_T?T<∞, and {W(t);0?t<∞} be a standard Wiener process. This exposition studies the almost sure behaviour of

, under varying conditions on a_T and T/a_T. The following analogue of Lévy's modulus of continuity of a Wiener Process is also given:

$\text{lim}{}_{\mathrm{h\to 0}}\text{inf}{}_{\mathrm{0?t?1}}\text{sup}{}_{\mathrm{0?s?h}}\text{(}\text{8 log h}{}^{-1}\text{π}{}^2\text{h}\text{)}\text{1}\text{2}\text{|W(t+s)?W(t)| =}{}^{\mathrm{a.s.}}\text{1.}$

and this may be viewed as the exact “modulus of non-differentiability” of a Wiener Process.     

How big are the increments of lp-valued Gaussian processes?

Let be a sequence of independent Gaussian processes with (h) Put . The large increments forY(·) with bounded σ(p, h) are investigated. As an example the large increments of infinite-dimensional fractional Ornstein-Uhlenbeck process in I^P are given. The method can also be applied to certain processes with stationary increments. Project supported by the National Natural Science Foundation of China and the Natural Science Foundation of Zhejiang Province.     

Homotopy theories in additive categories are homotopies of Δ-groups

In this paper we prove that theories obtained via a cone or an arc object in additive categories are homotopy theories of Δ-abelian groups.     

Jordan–Hölder theorems for derived module categories of piecewise hereditary algebras

A Jordan–Hölder theorem is established for derived module categories of piecewise hereditary algebras, in particular for representations of quivers and for hereditary abelian categories of a geometric nature. The resulting composition series of derived categories are shown to be independent of the choice of bounded or unbounded derived module categories, and also of the choice of finitely generated or arbitrary modules.     

How Topological are Topological Groups?

Abstract

A Characterization of the category of topological groups is provided which does not refer to the category of topological spaces at all, but only to the category of uniform spaces. Similarly, the category of SIN-groups is characterized in a purely uniform way.     

Induced modules of split Iwahori–Hecke algebras are reducible

We show that the modules of an Iwahori–Hecke algebra obtained via induction from a proper parabolic subalgebra are always reducible.     

How many graphs are unions of k‐cliques?

We study the number of n‐vertex graphs that can be written as the edge‐union of k‐vertex cliques. We obtain reasonably tight estimates for in the cases (i) k = n ? o(n) and (ii) k = o(n) but . We also show that exhibits a phase transition around . We leave open several potentially interesting cases, and raise some other questions of a similar nature. © 2006 Wiley Periodicals, Inc. J Graph Theory 52: 87–107, 2006     

How big are the increments of a multi-parameter Wiener process?

Summary Let _T = (2a _T(log Ta _T ^–1 +log log T))^–1/2, 0< a _T T< and let R^* be the set of sub-rectangles of the square [0, T ^1/2]x[0, T^1/2], having an area a _T . This paper studies the almost sure limiting behaviour of as T, where W is a two-time parameter Wiener process. With a _T =T, our results give the well-known law of iterated logarithm and a generalization of the latter is also attained. The multi-time parameter analogues of our twotime parameter Wiener process results are also stated in the text.Research partially supported by a Canadian NRC grant and a Canada Council Leave Fellowship     

How quadratic are the natural numbers?

Abstract. For natural numbers n we inspect all factorizations n = ab of n with a ￡ ba \le b in \Bbb N\Bbb N and denote by n=a_n b_nn=a_n b_n the most quadratic one, i.e. such that b_n - a_nb_n - a_n is minimal. Then the quotient k(n) : = a_n/b_n\kappa (n) := a_n/b_n is a measure for the quadraticity of n. The best general estimate for k(n)\kappa (n) is of course very poor: 1/n ￡ k(n) ￡ 11/n \le \kappa (n)\le 1. But a Theorem of Hall and Tenenbaum [1, p. 29], implies(logn)^-d-e ￡ k(n) ￡ (logn)^-d(\log n)^{-\delta -\varepsilon } \le \kappa (n) \le (\log n)^{-\delta } on average, with d = 1 - (1+log₂  2)/log2=0,08607 ?\delta = 1 - (1+\log _2 \,2)/\log 2=0,08607 \ldots and for every e > 0\varepsilon >0. Hence the natural numbers are fairly quadratic.¶k(n)\kappa (n) characterizes a specific optimal factorization of n. A quadraticity measure, which is more global with respect to the prime factorization of n, is k^*(n): = ?_{1 ￡ a ￡ b, ab=n} a/b\kappa ^*(n):= \textstyle\sum\limits \limits _{1\le a \le b, ab=n} a/b. We show k^*(n) ~ \frac 12\kappa ^*(n) \sim \frac {1}{2} on average, and k^*(n)=W(2^{\frac 12(1-e) log n/log ₂n})\kappa ^*(n)=\Omega (2^{\frac {1}{2}(1-\varepsilon ) {\log}\, n/{\log} _2n})for every e > 0\varepsilon>0.