Linking (n − 2)-dimensional panels inn-space I: (k − 1,k)-graphs and (k − 1,k)-frames

A (k – 1,k)-graph is a multi-graph satisfyinge (k – 1)v – k for every non-empty subset ofe edges onv vertices, with equality whene = |E(G)|. A (k – 1,k)-frame is a structure generalizing an (n – 2, 2)-framework inn-space, a structure consisting of a set of (n – 2)-dimensional bodies inn-space and a set of rigid bars each joining a pair of bodies using ball joints. We prove that a graph is the graph of a minimally rigid (with respect to edges) (k – 1,k)-frame if and only if it is a (k – 1,k)-graph. Rigidity here means infinitesimal rigidity or equivalently statical rigidity.     

Anisotropic weak Hardy spaces and interpolation theorems

In this paper, the authors establish the anisotropic weak Hardy spaces associated with very general discrete groups of dilations. Moreover, the atomic decomposition theorem of the anisotropic weak Hardy spaces is also given. As some applications of the above results, the authors prove some interpolation theorems and obtain the boundedness of the singular integral operators on these Hardy spaces.     

Euler polynomials,Bernoulli polynomials,and Lévyʼs stochastic area formula

In 1951, P. Lévy represented the Euler and Bernoulli numbers in terms of the moments of Lévy?s stochastic area. Recently the authors extended his result to the case of Eulerian polynomials of types A and B. In this paper, we continue to apply the same method to the Euler and Bernoulli polynomials, and will express these polynomials with the use of Lévy?s stochastic area. Moreover, a natural problem, arising from such representations, to calculate the expectations of polynomials of the stochastic area and the norm of the Brownian motion will be solved.     

De l'Hôpital,Bernoulli, and the genesis of Analyse des infiniment petits

The Marquis de l'Hôpital's Analyse des infiniment petits (1696) was the first ever calculus textbook. Its introductory chapters were based on Johann Bernoulli's Lectiones de calculo differentialium, a series of lessons written for the Marquis in 1691–92. Bernoulli also assisted with the later chapters, while the Marquis supported him with a stipend during 1694–96. We examine the text and its source material in this paper, which was presented at the research symposium ‘Editing historical mathematics: techniques and traditions since 1900’ at All Soul's College, Oxford, in December 2011.

With this in mind he published his Analyse des infiniment petits, a book that was both good and well-written?…?We may only find fault in that Mr. de l'Hôpital did not make well enough known the debt he owed to Mr. Bernoulli?…

(Montucla 1798 Montucla, J F. 1798. Histoire des mathématiques, 2nd Vol. 2, Paris [Google Scholar], 397)     

Atiyah’s L 2-index theorem,Kan-Thurston construction,and weak Bass conjecture

We conjecture that for a group G of type FP, the L ²-Euler characteristic of a group G is the same as the ordinary Euler characteristic of G, and show that this conjecture is closely related with the weak Bass conjecture. We also present a class of groups satisfying this conjecture. Ourmethod combines the Kan-Thurston construction, Atiyah’s L ²-index theorem, and a result of Berrick, Chatterji, and Mislin.     

On the Diophantine Equations (2 n − 1)(6 n − 1) = x 2 and (a n − 1)(a kn − 1) = x 2

In this paper we prove that the equation (2 ⁿ – 1)(6 ⁿ – 1) = x ² has no solutions in positive integers n and x. Furthermore, the equation (a ⁿ – 1) (a ^kn – 1) = x ² in positive integers a > 1, n, k > 1 (kn > 2) and x is also considered. We show that this equation has the only solutions (a,n,k,x) = (2,3,2,21), (3,1,5,22) and (7,1,4,120).     

On eigenvalues of symmetric (+1, −1) matrices

To every symmetric matrixA with entries ±1, we associate a graph G(A), and ask (for two different definitions of distance) for the distance ofG(A) to the nearest complete bipartite graph (cbg). Letλ ₁(A),λ ¹ (A) be respectively the algebraically largest and least eigenvalues ofA. The Frobenius distance (see Section 4) to the nearest cbg is bounded above and below by functions ofn −λ ₁ (A), wheren=ord A. The ordinary distance (see Section 1) to the nearest cbg is shown to be bounded above and below by functions ofλ ¹ (A). A curious corollary is: there exists a functionf (independent ofn, and given by (1.1)), such that |λ _i (A) | ≦f(λ ¹(A), whereλ _i (A) is any eigenvalue ofA other thanλ _i (A). This work was supported (in part) by the U.S. Army under contract #DAHC04-C-0023.     

Propertyf −1(S) =g −1(S) for entire and meromorphicP-adic functions

LetW be an algebraically closed filed of characteristic zero, letK be an algebraically closed field of characteristic zero, complete for an ultrametric absolute value, and letA(K) (resp. ℳ(K)) be the set of entire (resp. meromorphic) functions inK. For everyn≥7, we show that the setS _n(b) of zeros of the polynomialx ⁿ−b (b≠0) is such that, iff, g ∈W[x] or iff, g ∈A(K), satisfyf ⁻¹(S _n(b))=g ⁻¹(S _n(b)), thenf ⁿ=g ⁿ. For everyn≥14, we show thatS _n(b) is such that iff, g ∈W({tx}) or iff, g ∈ ℳ(K) satisfyf ⁻¹(S _n(b))=g ⁻¹(S _n(b)), then eitherf ⁿ=g ⁿ, orfg is a constant. Analogous properties are true for complex entire and meromorphic functions withn≥8 andn≥15, respectively. For everyn≥9, we show that the setY _n(c) of zeros of the polynomial , (withc≠0 and 1) is an ursim ofn points forW[x], and forA(K). For everyn≥16, we show thatY _n(c) is an ursim ofn points forW(x), and for ℳ(K). We follow a method based on thep-adic Nevanlinna Theory and use certain improvement of a lemma obtained by Frank and Reinders.     

Group Laws [x,y −1] ≡ u(x,y) and Varietal Properties

Let F = ?x, y? be a free group. It is known that the commutator [x, y ^?1] cannot be expressed in terms of basic commutators, in particular in terms of Engel commutators. We show that the laws imposing such an expression define specific varietal properties. For a property 𝒫 we consider a subset U(𝒫) ? F such that every law of the form [x, y ^?1] ≡ u, u ∈ U(𝒫) provides the varietal property 𝒫. For example, we show that each subnormal subgroup is normal in every group of a variety 𝔙 if and only if 𝔙 satisfies a law of the form [x, y ^?1] ≡ u, where u ∈ [F′, ?x?].     

P and Q polynomial schemes with q = −1

A theorem of D. Leonard, as it appears in Bannai and Ito (Benjamin-Cummings Lecture Note Series, No. 58, Menlo Park, Ca., 1984), says the intersection numbers of distance-regular graphs with the Q-polynomial property take seven possible forms, called 1, 1A, 2, 2A, 2B, 2C, and 3. In this paper we show the known list of examples of type 3 is complete.     

Weak martingale Hardy spaces and weak atomic decompositions

In this paper we define some weak martingale Hardy spaces and three kinds of weak atoms. They are the counterparts of martingale Hardy spaces and atoms in the classical martingale Hp-theory. And then three atomic decomposition theorems for martingales in weak martingale Hardy spaces are proved. With the help of the weak atomic decompositions of martingale, a sufficient condition for a sublinear operator defined on the weak martingale Hardy spaces to be bounded is given. Using the sufficient condition, we obtain a series of martingale inequalities with respect to the weak Lp-norm, the inequalities of weak (p ,p)-type and some continuous imbedding relationships between various weak martingale Hardy spaces. These inequalities are the weak versions of the basic inequalities in the classical martingale Hp-theory.     

Maschke-type theorem and Morita context over weak Hopf algebras

This paper gives a Maschke-type theorem over semisimple weak Hopf algebras, extends the well-known Maschke-type theorem given by Cohen and Fishman and constructs a Morita context over weak Hopf algebras.     

Generic Vopěnka’s Principle,remarkable cardinals,and the weak Proper Forcing Axiom

We introduce and study the first-order Generic Vopěnka’s Principle, which states that for every definable proper class of structures \(\mathcal {C}\) of the same type, there exist \(B\ne A\) in \(\mathcal {C}\) such that B elementarily embeds into A in some set-forcing extension. We show that, for \(n\ge 1\), the Generic Vopěnka’s Principle fragment for \(\Pi _n\)-definable classes is equiconsistent with a proper class of n-remarkable cardinals. The n-remarkable cardinals hierarchy for \(n\in \omega \), which we introduce here, is a natural generic analogue for the \(C^{(n)}\)-extendible cardinals that Bagaria used to calibrate the strength of the first-order Vopěnka’s Principle in Bagaria (Arch Math Logic 51(3–4):213–240, 2012). Expanding on the theme of studying set theoretic properties which assert the existence of elementary embeddings in some set-forcing extension, we introduce and study the weak Proper Forcing Axiom, \(\mathrm{wPFA}\). The axiom \(\mathrm{wPFA}\) states that for every transitive model \(\mathcal M\) in the language of set theory with some \(\omega _1\)-many additional relations, if it is forced by a proper forcing \(\mathbb P\) that \(\mathcal M\) satisfies some \(\Sigma _1\)-property, then V has a transitive model \(\bar{\mathcal M}\), satisfying the same \(\Sigma _1\)-property, and in some set-forcing extension there is an elementary embedding from \(\bar{\mathcal M}\) into \(\mathcal M\). This is a weakening of a formulation of \(\mathrm{PFA}\) due to Claverie and Schindler (J Symb Logic 77(2):475–498, 2012), which asserts that the embedding from \(\bar{\mathcal M}\) to \(\mathcal M\) exists in V. We show that \(\mathrm{wPFA}\) is equiconsistent with a remarkable cardinal. Furthermore, the axiom \(\mathrm{wPFA}\) implies \(\mathrm{PFA}_{\aleph _2}\), the Proper Forcing Axiom for antichains of size at most \(\omega _2\), but it is consistent with \(\square _\kappa \) for all \(\kappa \ge \omega _2\), and therefore does not imply \(\mathrm{PFA}_{\aleph _3}\).     

Über die Anzahl der Lösungen von [p−1,q−1]≤x

Ohne Zusammenfassung