The Gauss–Wilson theorem for quarter-intervals

We define a Gauss factorial N _n ! to be the product of all positive integers up to N that are relatively prime to n. It is the purpose of this paper to study the multiplicative orders of the Gauss factorials $\left\lfloor\frac{n-1}{4}\right\rfloor_{n}!$ for odd positive integers n. The case where n has exactly one prime factor of the form p≡1(mod4) is of particular interest, as will be explained in the introduction. A fundamental role is played by p with the property that the order of  $\frac{p-1}{4}!$ modulo p is a power of 2; because of their connection to two different results of Gauss we call them Gauss primes. Our main result is a complete characterization in terms of Gauss primes of those n of the above form that satisfy $\left\lfloor\frac{n-1}{4}\right\rfloor_{n}!\equiv 1\pmod{n}$ . We also report on computations that were required in the process.     

The structure relation for Askey–Wilson polynomials

An explicit structure relation for Askey–Wilson polynomials is given. This involves a divided q-difference operator which is skew symmetric with respect to the Askey–Wilson inner product and which sends polynomials of degree n   to polynomials of degree n+1$n+1$. By specialization of parameters and by taking limits, similar structure relations, as well as lowering and raising relations, can be obtained for other families in the q-Askey scheme and the Askey scheme. This is explicitly discussed for Jacobi polynomials, continuous q-Jacobi polynomials, continuous q-ultraspherical polynomials, and for big q-Jacobi polynomials. An already known structure relation for this last family can be obtained from the new structure relation by using the three-term recurrence relation and the second order q-difference formula. The results are also put in the framework of a more general theory. Their relationship with earlier work by Zhedanov and Bangerezako is discussed. There is also a connection with the string equation in discrete matrix models and with the Sklyanin algebra.     

The factorization method for the Askey–Wilson polynomials

A special Infeld–Hull factorization is given for the Askey–Wilson second order q-difference operator. It is then shown how to deduce a generalization of the corresponding Askey–Wilson polynomials.     

The Morse–Sard theorem for Clarke critical values

The Morse–Sard theorem states that the set of critical values of a C^k$C^k$ smooth function defined on a Euclidean space R^d${\mathbb{R}}^d$ has Lebesgue measure zero, provided k≥d$k\ge d$. This result is hereby extended for (generalized) critical values of continuous selections over a compactly indexed countable family of C^k$C^k$ functions: it is shown that these functions are Lipschitz continuous and the set of their Clarke critical values is null.     

The Bott–Samelson theorem for positive Legendrian isotopies

The classical Bott–Samelson theorem states that if on a Riemannian manifold all geodesics issuing from a certain point return to this point, then the universal cover of the manifold has the cohomology ring of a compact rank one symmetric space. This result on geodesic flows has been generalized to Reeb flows and partially to positive Legendrian isotopies by Frauenfelder–Labrousse–Schlenk. We prove the full theorem for positive Legendrian isotopies.     

The Eichler–Shimura cohomology theorem for Jacobi forms

Let \(\Gamma \) be a subgroup of finite index in \(\mathrm {SL}(2,\mathbb {Z})\). Eichler defined the first cohomology group of \(\Gamma \) with coefficients in a certain module of polynomials. Eichler and Shimura established that this group is isomorphic to the direct sum of two spaces of cusp forms on \(\Gamma \) with the same integral weight. These results were generalized by Knopp to cusp forms of real weights. In this paper, we define the first parabolic cohomology groups of Jacobi groups \(\Gamma ^{(1,j)}\) and prove that these are isomorphic to the spaces of (skew-holomorphic) Jacobi cusp forms of real weights. We also show that if \(j=1\) and the weights of Jacobi cusp forms are in \(\frac{1}{2}\mathbb {Z}-\mathbb {Z}\), then these isomorphisms can be written in terms of special values of partial L-functions of Jacobi cusp forms.     

The Bowman–Bradley theorem for multiple zeta-star values

TextThe Bowman–Bradley theorem asserts that the multiple zeta values at the sequences obtained by inserting a fixed number of twos between $3,1,\dots ,3,1$ add up to a rational multiple of a power of π. We establish its counterpart for multiple zeta-star values by showing an identity in a non-commutative polynomial algebra introduced by Hoffman.VideoFor a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=LpqA2OJ6vP8.     

The de Bruijn–Erd?s theorem for hypergraphs

Fix integers n ≥ r ≥ 2. A clique partition of ${{[n] \choose r}}$ is a collection of proper subsets ${A_1, A_2, \ldots, A_t \subset [n]}$ such that ${\bigcup_i{A_i \choose r}}$ is a partition of ${{[n]\choose r}}$ . Let cp(n, r) denote the minimum size of a clique partition of ${{[n] \choose r}}$ . A classical theorem of de Bruijn and Erd?s states that cp(n, 2) =?n. In this paper we study cp(n, r), and show in general that for each fixed r ≥ 3, $${\rm cp}(n, r) \geq (1 + o(1))n^{r/2} \quad \quad {\rm as} \, \, n \rightarrow \infty.$$ We conjecture cp(n, r) =?(1 +?o(1))n ^r/2. This conjecture has already been verified (in a very strong sense) for r = 3 by Hartman–Mullin–Stinson. We give further evidence of this conjecture by constructing, for each r ≥ 4, a family of (1?+?o(1))n ^r/2 subsets of [n] with the following property: no two r-sets of [n] are covered more than once and all but o(n ^r ) of the r-sets of [n] are covered. We also give an absolute lower bound ${{\rm cp}(n, r) \geq {n \choose r}/{q + r - 1 \choose r}}$ when n =?q ² + q +?r ? 1, and for each r characterize the finitely many configurations achieving equality with the lower bound. Finally we note the connection of cp(n, r) to extremal graph theory, and determine some new asymptotically sharp bounds for the Zarankiewicz problem.     

The Larson–Sweedler theorem for weak multiplier Hopf algebras

The Larson–Sweedler theorem says that a finite-dimensional bialgebra with a faithful integral is a Hopf algebra [15 Larson, R. G., Sweedler, M. E. (1969). An associative orthogonal bilinear form for Hopf algebras. Amer. J. Math. 91:75–93.[Crossref], [Web of Science ®] , [Google Scholar]]. The result has been generalized to finite-dimensional weak Hopf algebras by Vecsernyés [44 Vecsernyés, P. (2003). Larson–Sweedler theorem and the role of grouplike elements in weak Hopf algebras. J. Algebra 270:471–520. See also arXiv: 0111045v3 [math.QA] for an extended version.[Crossref], [Web of Science ®] , [Google Scholar]]. In this paper, we show that the result is still true for weak multiplier Hopf algebras. The notion of a weak multiplier bialgebra was introduced by Böhm et al. in [4 Böhm, G., Gómez-Torecillas, J., López-Centella, E. (2015). Weak multiplier bialgebras. Weak multiplier bialgebras. 367(12):8681–8872. See also arXiv: 1306.1466 [math.QA]. [Google Scholar]]. In this note it is shown that a weak multiplier bialgebra with a regular and full coproduct is a regular weak multiplier Hopf algebra if there is a faithful set of integrals. Weak multiplier Hopf algebras are introduced and studied in [40 Van Daele, A., Wang, S. (2015). Weak multiplier Hopf algebras I. The main theory. J. Ange. Math. (Crelles J.) 705:155–209, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: 10.1515/crelle-2013-0053, July 2013. See also arXiv:1210.4395v1 [math.RA].[Web of Science ®] , [Google Scholar]]. Integrals on (regular) weak multiplier Hopf algebras are treated in [43 Van Daele, A., Wang, S. (2016). Weak multiplier Hopf algebras III. Integrals and duality. Preprint University of Leuven (Belgium) and Southeast University of Nanjing (China), See arXiv: 1701.04951.v3 [math.RA]. [Google Scholar]]. This result is important for the development of the theory of locally compact quantum groupoids in the operator algebra setting, see [13 Kahng, B.-J., Van Daele, A. A class of C*-algebraic locally compact quantum groupoids I. Preprint Canisius College Buffalo (USA) and University of Leuven (Belgium). [Google Scholar]] and [14 Kahng, B.-J., Van Daele, A. A class of C*-algebraic locally compact quantum groupoids II. Preprint Canisius College Buffalo (USA) and University of Leuven (Belgium). [Google Scholar]]. Our treatment of this material is motivated by the prospect of such a theory.     

The Paley–Wiener theorem for certain nilpotent Lie groupsJean

We generalize the classical Paley–Wiener theorem to special types of connected, simply connected, nilpotent Lie groups: First we consider nilpotent Lie groups whose Lie algebra admits an ideal which is a polarization for a dense subset of generic linear forms on the Lie algebra. Then we consider nilpotent Lie groups such that the co-adjoint orbits of all the elements of a dense subset of the dual of the Lie algebra 𝔤^* are flat (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Delandtsheer–Doyen parameters for block-transitive point-imprimitive 2-designs

Designs, Codes and Cryptography - Delandtsheer and Doyen bounded, in terms of the block size, the number of points of a point-imprimitive, block-transitive 2-design. To do this they introduced two...     

The generalized Morse–Sard theorem

The Morse–Sard theorem gives conditions under which the set of critical values of a function between Euclidean spaces has Lebesgue measure zero. Over the years the result has been extended and strengthened in various ways. We present a result, along with a simple proof, that subsumes many of these generalizations. We also review methods of constructing examples showing that differentiability hypotheses cannot be weakened, and we construct a complete set of examples for our result.     

The Riemann–Roch theorem for the Dynnikov–Novikov discrete complex analysis

We prove an analog of the Riemann–Roch Theorem for the Dynnikov–Novikov discrete complex analysis.     

The Kolmogorov–Riesz compactness theorem

We show that the Arzelà–Ascoli theorem and Kolmogorov compactness theorem both are consequences of a simple lemma on compactness in metric spaces. Their relation to Helly's theorem is discussed. The paper contains a detailed discussion on the historical background of the Kolmogorov compactness theorem.     

The Blok–Ferreirim theorem for normal GBL-algebras and its application

Generalized basic logic algebras (GBL-algebras for short) have been introduced in [JT02] as a generalization of Hájek’s BL-algebras, and constitute a bridge between algebraic logic and ℓ-groups. In this paper we investigate normal GBL-algebras, that is, integral GBL-algebras in which every filter is normal. For these structures we prove an analogue of Blok and Ferreirim’s [BF00] ordinal sum decomposition theorem. This result allows us to derive many interesting consequences, such as the decidability of the universal theory of commutative GBL-algebras, the fact that n-potent GBL-algebras are commutative, and a representation theorem for finite GBL-algebras as poset sums of GMV-algebras, a result which generalizes Di Nola and Lettieri’s [DL03] representation of finite BL-algebras. Presented by J. G. Raftery. Received May 23, 2007; accepted in final form February 20, 2008.