On the equivalence of Lagrange’s axiom to the Lotschnittaxiom

We prove that, in Hilbert’s plane absolute geometry, an axiom used by Lagrange in a proof of the Euclidean parallel postulate in a paper read on 3 February 1806 at the Institut de France, which states that “If a and b are two parallels from P to g, then the reflection of a in b is parallel to g as well”, is equivalent to F. Bachmann’s Lotschnittaxiom, which states that “The perpendiculars on the sides a right angle always intersect.”     

Heat Kernels, Smoothness Estimates, and Exponential Decay

In this article, we characterize functions whose Fourier transforms have exponential decay. We characterize such functions by showing that they satisfy a family of estimates that we call quantitative smoothness estimates (QSE). Using the QSE, we establish Gaussian decay in the “bad direction” for the □ _b -heat kernel on polynomial models in ? ⁿ⁺¹. On the transform side, the problem becomes establishing QSE on a heat kernel associated to the weighted $\bar{\partial}$ -operator on L ²(?). The bounds are established with Duhamel’s formula and careful estimation. In ?², we can prove both on and off-diagonal decay for the □ _b -heat kernel on polynomial models.     

Approximation of almost Sahoo–Riedel’s points by Sahoo–Riedel’s points

In this paper, we present an extention of Hyers–Ulam stability of Sahoo–Riedel’s points for real-valued differentiable functions on [a, b] and then we obtain stability results of Flett’s points for functions in the class of continuously differentiable functions on [a, b] with f′(a) = f′(b).     

Proof of the conjectures of Bernstein and Erdös concerning the optimal nodes for polynomial interpolation

Convergence properties of sequences of continuous functions, with kth order divided differences bounded from above or below, are studied. It is found that for such sequences, convergence in a “monotone norm” (e.g., L_p) on [a, b] to a continuous function implies uniform convergence of the sequence and its derivatives up to order k ? 1 (whenever they exist), in any closed subinterval of [a, b]. Uniform convergence in the closed interval [a, b] follows from the boundedness from below and above of the kth order divided differences. These results are applied to the estimation of the degree of approximation in Monotone and Restricted Derivative approximation, via bounds for the same problems with only one restricted derivative.     

A reconsideration of Legendre-Jacobi symbols

The classical definition of the Jacobi symbol (a:b) was badly conceived for negative values of b. Alternative useful definitions of (a:?1) are proposed here. This is an elaboration of a point in the article “Spinor genera of binary quadratic forms” in this issue.     

Erratum to: The generalized strong recurrence for non-zero rational parameters

In the present paper, we prove that self-approximation of \({\log \zeta (s)}\) with d = 0 is equivalent to the Riemann Hypothesis. Next, we show self-approximation of \({\log \zeta (s)}\) with respect to all nonzero real numbers d. Moreover, we partially filled a gap existing in “The strong recurrence for non-zero rational parameters” and prove self-approximation of \({\zeta(s)}\) for 0 ≠ d = a/b with |a?b| ≠ 1 and gcd(a,b) = 1.     

Rational Parking Functions and Catalan Numbers

The “classical” parking functions, counted by the Cayley number (n+1) ^n?1, carry a natural permutation representation of the symmetric group S _n in which the number of orbits is the Catalan number \({\frac{1}{n+1} \left( \begin{array}{ll} 2n \\ n \end{array} \right)}\). In this paper, we will generalize this setup to “rational” parking functions indexed by a pair (a, b) of coprime positive integers. These parking functions, which are counted by b ^a?1, carry a permutation representation of S _a in which the number of orbits is the “rational” Catalan number \({\frac{1}{a+b} \left( \begin{array}{ll} a+b \\ a \end{array} \right)}\). First, we compute the Frobenius characteristic of the S _a -module of (a, b)-parking functions, giving explicit expansions of this symmetric function in the complete homogeneous basis, the power-sum basis, and the Schur basis. Second, we study q-analogues of the rational Catalan numbers, conjecturing new combinatorial formulas for the rational q-Catalan numbers \({\frac{1}{[a+b]_{q}} {{\left[ \begin{array}{ll} a+b \\ a \end{array} \right]}_{q}}}\) and for the q-binomial coefficients \({{{\left[ \begin{array}{ll} n \\ k \end{array} \right]}_{q}}}\). We give a bijective explanation of the division by [a+b] _q that proves the equivalence of these two conjectures. Third, we present combinatorial definitions for q, t-analogues of rational Catalan numbers and parking functions, generalizing the Shuffle Conjecture for the classical case. We present several conjectures regarding the joint symmetry and t = 1/q specializations of these polynomials. An appendix computes these polynomials explicitly for small values of a and b.     

A “strange” vector-valued quantum modular form

Since their definition in 2010 by Zagier, quantum modular forms have been connected to numerous topics such as strongly unimodal sequences, ranks, cranks, and asymptotics for mock theta functions near roots of unity. These are functions that are not necessarily defined on the upper half plane but a priori are defined only on a subset of ${\mathbb{Q}}$ Q , and whose obstruction to modularity is some analytically “nice” function. Motivated by Zagier’s example of the quantum modularity of Kontsevich’s “strange” function F(q), we revisit work of Andrews, Jiménez-Urroz, and Ono to construct a natural vector-valued quantum modular form whose components are similarly “strange”.     

Embedding edge-colored complete graphs in binary affine spaces

Under what conditions on an edge-coloring of a complete graph can it be embedded in an affine space over GF(2) in such a way that two edges have the same color if and only if they are parallel? Necessary conditions are that each vertex lies on at most one edge of each color, and the “parallelogram property” (if {a, b} ～ {c, d} then {a, c} ～ {b, d}); these are not sufficient. In this paper, several sufficient conditions are given, including one depending on the classification of graphs with the “triangle property” by Shult and Seidel. If a coloring is embeddable, its automorphisms are all induced by affine transformations; this is used to study colorings with doubly transitive automorphism groups. Connections with coding theory, elliptic geometry, and difference sets are mentioned.     

On joint conditional complexity (Entropy)

The conditional Kolmogorov complexity of a word a relative to a word b is the minimum length of a program that prints a given b as an input. We generalize this notion to quadruples of strings a, b, c, d: their joint conditional complexity K((a → c)∧(b → d)) is defined as the minimum length of a program that transforms a into c and transforms b into d. In this paper, we prove that the joint conditional complexity cannot be expressed in terms of the usual conditional (and unconditional) Kolmogorov complexity. This result provides a negative answer to the following question asked by A. Shen on a session of the Kolmogorov seminar at Moscow State University in 1994: Is there a problem of information processing whose complexity is not expressible in terms of the conditional (and unconditional) Kolmogorov complexity? We show that a similar result holds for the classical Shannon entropy. We provide two proofs of both results, an effective one and a “quasi-effective” one. Finally, we present a quasi-effective proof of a strong version of the following statement: there are two strings whose mutual information cannot be extracted. Previously, only a noneffective proof of that statement has been known.     

Notes on the Feynman integral,II

In the setting of Cameron and Storvick's recent theory, our main result establishes the existence of the analytic Feynman integral for functions on v-dimensional Wiener space of the form F(X) = exp {? ∝_a^b (A(s) X(s), X(s)) ds}. Here X is a $\text{R}$^v-valued continuous function on [a, b] such that X(a) = 0 and {A(s): a ? s ? b} is a commutative family of real, symmetric, positive definite matrices such that the square roots of the eigenvalues are functions of bounded variation on [a, b]. We obtain the existence theorem just referred to without having to construct special spaces, quadratic forms, etc., to fit the particular problem of interest.     

Selection principle for pointwise bounded sequences of functions

For a number ? > 0 and a real function f on an interval [a, b], denote by N(?, f, [a, b]) the least upper bound of the set of indices n for which there is a family of disjoint intervals [a _i , b _i ], i = 1, …, n, on [a, b] such that |f(a _i ) ? f(b _i )| > ? for any i = 1, …, n (sup Ø = 0). The following theorem is proved: if {f _j } is a pointwise bounded sequence of real functions on the interval [a, b] such that n(?) ≡ lim sup _j→∞ N(?, f _j , [a, b]) < ∞ for any ? > 0, then the sequence {f _j } contains a subsequence which converges, everywhere on [a, b], to some function f such that N(?, f, [a, b]) ≤ n(?) for any ? > 0. It is proved that the main condition in this theorem related to the upper limit is necessary for any uniformly convergent sequence {f _j } and is “almost” necessary for any everywhere convergent sequence of measurable functions, and many pointwise selection principles generalizing Helly’s classical theorem are consequences of our theorem. Examples are presented which illustrate the sharpness of the theorem.     

Compactly supported multiwindow dual Gabor frames of rational sampling density

We consider multiwindow Gabor systems (G _N ; a, b) with N compactly supported windows and rational sampling density N/ab. We give another set of necessary and sufficient conditions for two multiwindow Gabor systems to form a pair of dual frames in addition to the Zibulski–Zeevi and Janssen conditions. Our conditions come from the back transform of Zibulski–Zeevi condition to the time domain but are more informative to construct window functions. For example, the masks satisfying unitary extension principle (UEP) condition generate a tight Gabor system when restricted on [0, 2] with a?=?1 and b?=?1. As another application, we show that a multiwindow Gabor system (G _N ; 1, 1) forms an orthonormal basis if and only if it has only one window (N?=?1) which is a sum of characteristic functions whose supports ‘essentially’ form a Lebesgue measurable partition of the unit interval. Our criteria also provide a rich family of multiwindow dual Gabor frames and multiwindow tight Gabor frames for the particular choices of lattice parameters, number and support of the windows. (Section 4)     

Jensen, multi-Jensen and polynomial functions on arbitrary abelian groups

In contrast to Cauchy’s functional equation, the consideration of Jensen’s equation makes no sense for functions f : G → H if G, H are arbitrary abelian groups. This difficulty may be circumvented by implementing a third variable into the equation in such a way that there is no interference with the desired result that the solutions should be of the form “additive function plus constant”. Proceeding analogously with the multi-Jensen equation we can add certain possibilities to define polynomial functions on abelian groups to possibilities given earlier by Laczkovich.     

Faulhaber’s theorem on power sums

We observe that the classical Faulhaber’s theorem on sums of odd powers also holds for an arbitrary arithmetic progression, namely, the odd power sums of any arithmetic progression a+b,a+2b,…,a+nb is a polynomial in na+n(n+1)b/2. While this assertion can be deduced from the original Fauhalber’s theorem, we give an alternative formula in terms of the Bernoulli polynomials. Moreover, by utilizing the central factorial numbers as in the approach of Knuth, we derive formulas for r-fold sums of powers without resorting to the notion of r-reflective functions. We also provide formulas for the r-fold alternating sums of powers in terms of Euler polynomials.     

Normal families of meromorphic functions concerning shared values

In this paper we study the problem of normal families of meromorphic functions concerning shared values and prove that a family F of meromorphic functions in a domain D is normal if for each pair of functions f and g in F, f^′−afn and g^′−agn share a value b in D where n is a positive integer and a,b are two finite constants such that n?4 and a≠0. This result is not true when n?3.     

A new expansion of the confluent hypergeometric function in terms of modified Bessel functions

An asymptotic expansion of the confluent hypergeometric function U(a,b,x) for large positive 2a − b is given in terms of modified Bessel functions multiplied by Buchholz polynomials, a family of double polynomials in the variables b and x with rational coefficients.     

On the Complexity of Self-Validating Numerical Integration and Approximation of Functions with Singularities

We consider the complexity of numerical integration and piecewise polynomial at approximation of bounded functions from a subclass ofC^k([a, b]\Z), whereZis a finite subset of [a, b]. Using only function values or values of derivatives, we usually cannot guarantee that the costs for obtaining an error less thanεare bounded byO(ε^−1/k) and we may have much higher costs. The situation changes if we also allow “realistic” estimates of ranges of functions or derivatives on intervals as observations. A very simple algorithm now yields an error less thanεwithO(ε^−1/k)-costs and an analogous result is also obtained for uniform approximation with piecewise polynomials. In a practical implementation, estimation of ranges may be done efficiently with interval arithmetic and automatic differentiation. The cost for each such evaluation (also of ranges of derivatives) is bounded by a constant times the cost for a function evaluation. The mentioned techniques reduce the class of integrands, but still allow numerical integration of functions from a wide class withO(ε^−1/k) arithmetical operations and guaranteed precisionε.     

Polynomial testing of the query “IS ab ≥ cd?” with application to finding a minimal cost reliability ratio spanning tree

Given integers a, b, c, d, we present a polynomial algorithm for the query “is a^b ≥ c^d?”. The result is applied to yield a polynomial algorithm for the minimal cost reliability ratio spanning tree problem.