Particular and Universal: Hypothesis in Plato's Divided Line

The present study aims to examine high school mathematics teachers’ content knowledge of the logical structure of proof in geometry. For this purpose, three scenarios were established based on errors in proof deriving from the figural-concept interactions. Based on these scenarios, clinical interviews were held with five mathematics teachers, who were working at different high schools, who used proof actively in their classrooms, and who had varying experiences. The obtained data from the interviews were analyzed descriptively under three categories. The results of the study revealed that the teachers did not have sufficient content knowledge to cope with proof errors deriving from figural-concept interactions in geometry. The teachers generally failed to base the information they obtained from figures on an axiom or theorem, to avoid sweeping generalizations based on geometric figures, and making accurate changes in figures when contradictions emerged as a result of valid reasoning processes. It is recommended that discussions be held on the justifications underlying the propositions constituting the proof and mistakes in geometric proof at the undergraduate level of education.

We provide a fairly self-contained account of the localisation and cofinality theorems for the algebraic $\operatorname K$-theory of stable $\infty$-categories. It is based on a general formula for the evaluation of an additive functor on a Verdier quotient closely following work of Waldhausen. We also include a new proof of the additivity theorem of $\operatorname K$-theory, strongly inspired by Ranicki's algebraic Thom construction, a short proof of the universality theorem of Blumberg, Gepner and Tabuada, and a second proof of the cofinality theorem which is based on the universal property of $\operatorname K$-theory.

Publication [1] contains a number of inaccuracies. Particularly point 9 below justifies a correction, since the concerned Lemma 20 forms an essential part in our work [2]: 1. Page 408, first paragraph of Sect. 2: It should be “d := limt→−∞ d(t)/t” and “d := limt→∞ d(t)/t”. I want to thank an anonymous referee for indicating this typo. 2. Page 411, Lemma 1: “w = 0” and “idR− y ∈ I” should be added to statement 2, and “q = 0” and “idR − y ∈ I” to statement 3. 3. Page 413, Corollary 3: w = 0 and idR − y ∈ I is missing in statement 2, and q = 0 and idR − y ∈ I in statement 3. 4. Page 413, paragraph after Corollary 3: It should be “x = Xs,a,r(y)”. 5. Page 414, proof of Theorem 4: In the display “u′t < c(t) < d(t) < v′t” the second “<” should be “ ”. The sentence between page 414 and 415 should be “Since y(−2) is Lipschitz continuous with constant one and y(+2) non-decreasing, . . . ”. 6. Page 417, Theorem 7: The proof contains a couple of typos and gaps. Instead of a detailed correction we refer to the proof of the more general Theorem 2 in [2]. 7. Page 419, fourth line in Sect. 6: It should be x < s. 8. Page 423: In line 5 and in the first line of the proof of Lemma 16 it should be “an absolute vector norm” instead of “a vector norm,” since the monotonicity of the vector norm is implicitly used in the proof of Lemma 16. Moreover, in the second display in the proof both terms “diag(s−σ)” should be “diag(s−σ)−1”. 9. Page 427, Lemma 20: In assumption 2, the words “random variable” should be dropped and the prerequisite a(0)= 0 and r(0)= 0 added.

In this chapter we prove Theorem 1.4 and part (ii) of Theorem 1.3. This chapter is the heart of the subject. General existence theorems for Feller semigroups are formulated in terms of elliptic boundary value problems with spectral parameter (Theorem 9.12). First, we study Feller semigroups with reflecting barrier (Theorem 9.14) and then, by using these Feller semigroups we construct Feller semigroups corresponding to such a diffusion phenomenon that either absorption or reflection phenomenon occurs at each point of the boundary (Theorem 9.18). Our proof is based on the generation theorems of Feller semigroups discussed in Section 2.2.

Statements and proofs of theorems 4 and 6 in Kendall (1988) (the paper referred to above) require correction, which is supplied by this short note. In particular it gives a full and correct proof of the probabilistic approach to a general Liouville theorem for harmonic maps (theorem 6 of Kendall, 1988). The note is concluded by a summary of progress made on the open questions posed in Kendall (1988), two of which have now been resolved

SummaryIn 1693, Gottfried Wilhelm Leibniz published in the Acta Eruditorum a geometrical proof of the fundamental theorem of the calculus. It is shown that this proof closely resembles Isaac Barrow's proof in Proposition 11, Lecture 10, of his Lectiones Geometricae, published in 1670. This comparison provides evidence that Leibniz gained substantial help from Barrow's book in formulating and presenting his geometrical formulation of this theorem. The analysis herein also supports the work of J. M. Child, who in 1920 studied the early manuscripts of Leibniz and concluded that he had frequently copied his diagrams from Barrow's book, but without acknowledgement. It is also shown that the diagram of Leibniz associated with his 1693 proof has often been reproduced with errors that make some aspects of his text difficult to comprehend.

A Geometric Proof of the Gap Theorem

We extend Witten’s proof [“A new proof of the positive energy theorem,” Commun. Math. Phys. 80, 381–402 (1981)] of the positive mass theorem to a rigorous proof for Lorentzian manifolds of any dimension. This includes the original higher-dimensional positive mass theorem proved by spinors [Parker, T. and Taubes, C., “On Witten’s proof of the positive energy theorem,” Commun. Math. Phys. 84, 223–238 (1982)] for dimension 4 and [Zhang, X., “Positive mass conjecture for five-dimensional Lorentzian manifolds,” J. Math. Phys. 40, 3540–3552 (1999)] for dimension 5. Stimulated by the article of Zhang [“Positive mass theorem for hypersurface in 5-dimensional Lorentzian manifolds,” Commun. Anal. Geom. 8, 635–652 (2000)], we weaken the spin condition on the spacelike hypersurface to require only a spinc structure and give a modified positive mass theorem for Lorentzian manifolds for dimensions 4, 5, and 6.

In a paper published in 1927, Landau gave a new proof of Hadamard's theorem on the factorization of an integral function of finite order. His proof arose out of the study of the problem of factorization of (s-I)ζ(s), which had been for the first time solved by Hadamard and for which Ananda Rau gave an alternative proof by the use of Gauchy's theorem of residues. The object of this paper is to combine the methods of Landau and Ananda Rau to obtain another proof of Hadamard's general theorem of factorization.

Geometric proofs of some theorems of Schur-Horn type

The analysis of efficiency of proof of general theorem is in-process executed about impossibility of presentation of sum of any degrees of independent integers by one number of that degree by the method of the use of the indefinite mutually conditioned coefficients of multipleness.

We give a simple proof of a general theorem of Kotake-Narasimhan for elliptic operators in the setting of ultradifferentiable functions in the sense of Braun, Meise and Taylor. We follow the ideas of Komatsu. Based on an example of M\'etivier, we also show that the ellipticity is a necessary condition for the theorem to be true. The present new version of the paper modifies the proof of Theorem 1.4 for an observation by Hoepfner and Rampazo who pointed out that an induction hypothesis depends on a constant $C_q$ that changes in the induction process, and hence the argument might not work as it was written. However, the statement of the result was originally correct and modifying $C_q$ with a more concrete expression in the induction hypothesis, the induction procedure is easily clarified with almost the same proof. Moreover, we eliminate the condition that the weight is identically zero in the interval [0,1], showing that the statements hold true with very similar arguments.

A joint degree matrix (JDM) specifies the number of connections between nodes of given degrees in a graph, for all degree pairs and uniquely determines the degree sequence of the graph. We consider the space of all balanced realizations of an arbitrary JDM, realizations in which the links between any two degree groups are placed as uniformly as possible. We prove that a swap Markov Chain Monte Carlo (MCMC) algorithm in the space of all balanced realizations of an {\em arbitrary} graphical JDM mixes rapidly, i.e., the relaxation time of the chain is bounded from above by a polynomial in the number of nodes $n$. To prove fast mixing, we first prove a general factorization theorem similar to the Martin-Randall method for disjoint decompositions (partitions). This theorem can be used to bound from below the spectral gap with the help of fast mixing subchains within every partition and a bound on an auxiliary Markov chain between the partitions. Our proof of the general factorization theorem is direct and uses conductance based methods (Cheeger inequality).

A generalization of ordinary renewal processes has been treated byHinderer and the author: for the stochastic process with values in ℝ+ the aim is (a,a+β) with 0≦a<∞, 0<β≦∞ instead of (a, ∞), and a possible passing over the aim is replaced by a phase of stagnation. In the present paper for the open case of an infinite mean length of the undisturbed steps the relationV(a)/a→0(a→∞), with β fixed, for the mean waiting timeV(a) until reaching the aim is obtained. The proof uses the theory of inversepositive operators ofCollatz andSchroder. This concept also yields an elementary proof of the so-called elementary renewal theorem. Finally the Tauberian theorem ofIkehara inAgmon's version is generalized; it yields the so-called general renewal theorem in the case of finite mean length of the steps, which is also a corollary of a general Tauberian theorem ofBenes.

On Peleg's theorem for stability of convex effectivity functions: An alternative proof and an application to authority structures