A positive proportion of locally soluble quartic Thue equations are globally insoluble

For any fixed nonzero integer h, we show that a positive proportion of integral binary quartic forms F do locally everywhere represent h, but do not globally represent h. We order classes of integral binary quartic forms by the two generators of their ring of ${\rm GL}_{2}({\mathbb Z})$ -invariants, classically denoted by I and J.

Automorphisms of a generalized extraspecial ℚ-group and its extension

Let [Formula: see text] be a set of primes and [Formula: see text] be a ring consisting of all rational numbers as [Formula: see text], where [Formula: see text] and [Formula: see text] are coprime, [Formula: see text] is a [Formula: see text]-number. The additive group of [Formula: see text] is denoted by [Formula: see text]. Suppose that [Formula: see text] and [Formula: see text] are two sets of primes and [Formula: see text] is a nonzero integer. Let [Formula: see text] be a generalized extraspecial [Formula: see text]-group as follows: [Formula: see text] Suppose that [Formula: see text] is a direct product of [Formula: see text] and [Formula: see text] copies of [Formula: see text]. Let [Formula: see text] be the normal subgroup of [Formula: see text] consisting of all elements of [Formula: see text] which act trivially on the derived subgroup [Formula: see text] of [Formula: see text], and [Formula: see text] be the normal subgroup of [Formula: see text] consisting of all central automorphisms of [Formula: see text] which also act trivially on the center [Formula: see text] of [Formula: see text]. Then, (i) The extension [Formula: see text] is split; (ii) [Formula: see text]; (iii) If [Formula: see text], then [Formula: see text] and [Formula: see text]; If [Formula: see text], then [Formula: see text] and [Formula: see text].

ON VALUES TAKEN BY THE LARGEST PRIME FACTOR OF SHIFTED PRIMES

Denote by$\mathbb{P}$the set of all prime numbers and by$P(n)$the largest prime factor of positive integer$n\geq 1$with the convention$P(1)=1$. In this paper, we prove that, for each$\unicode[STIX]{x1D702}\in (\frac{32}{17},2.1426\cdots \,)$, there is a constant$c(\unicode[STIX]{x1D702})>1$such that, for every fixed nonzero integer$a\in \mathbb{Z}^{\ast }$, the set$$\begin{eqnarray}\{p\in \mathbb{P}:p=P(q-a)\text{ for some prime }q\text{ with }p^{\unicode[STIX]{x1D702}}<q\leq c(\unicode[STIX]{x1D702})p^{\unicode[STIX]{x1D702}}\}\end{eqnarray}$$has relative asymptotic density one in$\mathbb{P}$. This improves a similar result due to Banks and Shparlinski [‘On values taken by the largest prime factor of shifted primes’,J. Aust. Math. Soc.82(2015), 133–147], Theorem 1.1, which requires$\unicode[STIX]{x1D702}\in (\frac{32}{17},2.0606\cdots \,)$in place of$\unicode[STIX]{x1D702}\in (\frac{32}{17},2.1426\cdots \,)$.

Hyers-Ulam Stability of Jensen Functional Inequality inp-Banach Spaces

We prove the Hyers-Ulam stability of the following Jensen functional inequality∥f((x-y)/n+z)+f((y-z)/n+x)+f((z-x)/n+y)∥≤∥f((x+y+z)∥inp-Banach spaces for any fixed nonzero integern.

On the size of Diophantine m-tuples

Let n be a nonzero integer. A set of m positive integers {a1, a2, …, am} is said to have the property D(n) if aiaj+n is a perfect square for all 1 [les ] i [les ] j [les ] m. Such a set is called a Diophantine m-tuple (with the property D(n)), or Pn-set of size m.Diophantus found the quadruple {1, 33, 68, 105} with the property D(256). The first Diophantine quadruple with the property D(1), the set {1, 3, 8, 120}, was found by Fermat (see [8, 9]). Baker and Davenport [3] proved that this Fermat’s set cannot be extended to the Diophantine quintuple, and a famous conjecture is that there does not exist a Diophantine quintuple with the property D(1). The theorem of Baker and Davenport has been recently generalized to several parametric families of quadruples [12, 14, 16], but the conjecture is still unproved.On the other hand, there are examples of Diophantine quintuples and sextuples like {1, 33, 105, 320, 18240} with the property D(256) [11] and {99, 315, 9920, 32768, 44460, 19534284} with the property D(2985984) [19]].

Vibration and Stability of a Spinning Disk Under Stationary Distributed Edge Loads

The vibration and stability of a spinning disk under conservative distributed edge tractions are studied both numerically and analytically. The edge traction is circumferentially stationary in the space. When the compressive traction is uniform, it is found that no modal interaction occurs and the natural frequencies of all nonreflected waves decrease, while the natural frequencies of the reflected waves increase. When the spinning disk is under distributed traction in the form of cos kθ, where k is a nonzero integer, it is found that the eigenvalue only changes slightly under the edge traction if the natural frequency of interest is well separated from others. When two modes are almost degenerate, however, modal interaction may or may not occur. It is observed that when the difference between the number of nodal diameters of these two modes is equal to ±k, frequency veering occurs when both modes are nonreflected, and merging occurs when one of these two modes is a reflected wave. In applying this rule, the number of nodal diameters of the forward and the reflected wave is considered as negative.

Quadratic forms in normal open induction

Models of normal open induction (NOI) are those discretely ordered rings, integrally closed in their fraction field whose nonnegative part satisfy Peano's induction axioms for open formulas in the language of ordered semirings.Here we study the problem of representability of an element a of a model M of NOI (in some extension of M) by a quadratic form of the type X2 + b Y2 where b is a nonzero integer. Using either a trigonometric or a hyperbolic parametrization we prove that except in some trivial cases, M[x, y] with x2 + by2 = a can be embedded in a model of NOI.We also study quadratic extensions of a model M of NOI; we first prove some properties of the ring of Gaussian integers of M. Then we study the group of solutions of a Pell equation in NOI; we construct a model in which the quotient group by the squares has size continuum.