scholarly journals SOME REFINED RESULTS ON THE MIXED LITTLEWOOD CONJECTURE FOR PSEUDO-ABSOLUTE VALUES

2018 ◽  
Vol 107 (1) ◽  
pp. 91-109
Author(s):  
WENCAI LIU
Keyword(s):  
Parameter Family ◽  
General Sequence ◽  
Absolute Value ◽  
Littlewood Conjecture ◽  
Minor Condition ◽  
Image Position ◽  
A Minor

In this paper, we study the mixed Littlewood conjecture with pseudo-absolute values. For any pseudo-absolute-value sequence ${\mathcal{D}}$, we obtain a sharp criterion such that for almost every $\unicode[STIX]{x1D6FC}$ the inequality $$\begin{eqnarray}|n|_{{\mathcal{D}}}|n\unicode[STIX]{x1D6FC}-p|\leq \unicode[STIX]{x1D713}(n)\end{eqnarray}$$ has infinitely many coprime solutions $(n,p)\in \mathbb{N}\times \mathbb{Z}$ for a certain one-parameter family of $\unicode[STIX]{x1D713}$. Also, under a minor condition on pseudo-absolute-value sequences ${\mathcal{D}}_{1},{\mathcal{D}}_{2},\ldots ,{\mathcal{D}}_{k}$, we obtain a sharp criterion on a general sequence $\unicode[STIX]{x1D713}(n)$ such that for almost every $\unicode[STIX]{x1D6FC}$ the inequality $$\begin{eqnarray}|n|_{{\mathcal{D}}_{1}}|n|_{{\mathcal{D}}_{2}}\cdots |n|_{{\mathcal{D}}_{k}}|n\unicode[STIX]{x1D6FC}-p|\leq \unicode[STIX]{x1D713}(n)\end{eqnarray}$$ has infinitely many coprime solutions $(n,p)\in \mathbb{N}\times \mathbb{Z}$.

Analogue of a theorem of Khintchine in fields of formal power series

1966 ◽  
Vol 62 (4) ◽  
pp. 637-642 ◽  
Author(s):  
T. W. Cusick
Keyword(s):  
Power Series ◽  
Positive Constant ◽  
Classical Result ◽  
Real Numbers ◽  
Linear Forms ◽  
Absolute Value ◽  
The Difference ◽  
Image Position ◽  
Formal Power ◽  
Positive Integers

For a real number λ, ‖λ‖ is the absolute value of the difference between λ and the nearest integer. Let X represent the m-tuple (x1, x2, … xm) and letbe any n linear forms in m variables, where the Θij are real numbers. The following is a classical result of Khintchine (1):For all pairs of positive integers m, n there is a positive constant Г(m, n) with the property that for any forms Lj(X) there exist real numbers α1, α2, …, αn such thatfor all integers x1, x2, …, xm not all zero.

scholarly journals Pointwise characteristic factors for Wiener–Wintner double recurrence theorem

2015 ◽  
Vol 36 (4) ◽  
pp. 1037-1066 ◽  
Author(s):  
IDRIS ASSANI ◽  
DAVID DUNCAN ◽  
RYO MOORE
Keyword(s):  
Full Measure ◽  
Orthogonal Complement ◽  
Ergodic System ◽  
Absolute Value ◽  
Recurrence Theorem ◽  
The Absolute ◽  
Image Position ◽  
Null Set

In this paper we extend Bourgain’s double recurrence result to the Wiener–Wintner averages. Let $(X,{\mathcal{F}},{\it\mu},T)$ be a standard ergodic system. We will show that for any $f_{1},f_{2}\in L^{\infty }(X)$, the double recurrence Wiener–Wintner average $$\begin{eqnarray}\frac{1}{N}\mathop{\sum }_{n=1}^{N}f_{1}(T^{an}x)f_{2}(T^{bn}x)e^{2{\it\pi}int}\end{eqnarray}$$ converges off a single null set of $X$ independent of $t$ as $N\rightarrow \infty$. Furthermore, we will show a uniform Wiener–Wintner double recurrence result: if either $f_{1}$ or $f_{2}$ belongs to the orthogonal complement of the Conze–Lesigne factor, then there exists a set of full measure such that the supremum on $t$ of the absolute value of the averages above converges to $0$.

Topological Hopf bifurcation in the plane

1983 ◽  
Vol 93 (1) ◽  
pp. 113-119
Author(s):  
Dieter Erle
Keyword(s):  
Hopf Bifurcation ◽  
Dynamical Systems ◽  
Asymptotic Stability ◽  
Stationary Point ◽  
Parameter Family ◽  
Stability Behaviour ◽  
Closed Orbits ◽  
The Stability ◽  
Image Position ◽  
Negative System

Classical bifurcation theorems for a 1 -parameter family of plane dynamical systemsassert the presence of closed orbits clustering at some distinguished parameter value (∈ = 0, say). Here, for any ∈, the origin is the only stationary point. The topological content of the mostly analytic hypotheses imposed is some change in the stability behaviour of the origin at ∈ = 0, roughly the passing of a kind of stability to a kind of instability. Topologically speaking, e.g. some of the conditions demanded are asymptotic stability of the origin for the negative system at ∈ > 0 and asymptotic stability of the origin for at ∈ < 0 (Hopf (8), Ruelle and Takens(11)) or ∈ = 0 (Chafee(2)).

Optical properties of silica glasses having O2 and Cl2 molecules

1991 ◽  
Vol 244 ◽  
Author(s):  
Koichi Awazu ◽  
Hiroshi Kawazoe ◽  
Ken-ichi Muta
Keyword(s):  
Optical Properties ◽  
Absorption Band ◽  
Gas Phase ◽  
Excimer Laser ◽  
Silica Glasses ◽  
Minor Fraction ◽  
Image Position ◽  
A Minor

ABSTRACTOxygen or chlorine was incorporated into silica glasses by sintering porous soot rods under 02/He or C12/He atmosphere. In case of oxygen, the Schumann-Runge band of 02 molecule was found in VUV region. An absorption band at 4. 8eV having 1. 9eV emission was generated with irradiation of excimer laser. We proposed that such optical properties were due to 02 molecule trapped in glass because they were similar to photochemistry of 02 molecule in the gas phase which could be shown as:When silica glasses were fabricated under chlorine ambient, a minor fraction was found to be present as Cl2 molecule trapped in the glass, which gives the absorption band at 3.8eV.

On the Product of Three Homogeneous Linear Forms. IV

1943 ◽  
Vol 39 (1) ◽  
pp. 1-21 ◽  
Author(s):  
H. Davenport
Keyword(s):  
Lower Bound ◽  
Real Number ◽  
Linear Transformation ◽  
Linear Forms ◽  
Special Linear ◽  
Absolute Value ◽  
Cubic Equation ◽  
Image Position ◽  
Homogeneous Linear

Let L1, L2, L3 be three homogeneous linear forms in u, v, w with real coefficients and determinant 1. Let M denote the lower bound offor integral values of u, v, w, not all zero. I proved a few years ago (1) thatmore precisely, thatexcept when L1, L2, L3 are of a special type, in which case If we denote by θ, ø, ψ the roots of the cubic equation t3+t2-2t-1 = 0, the special linear forms are equivalent, by an integral unimodular linear transformation, to(in any order), where λ1,λ2,λ3 are real number whose product is In this case, L1L2L3|λ1λ2λ3 is a non-zero integer, and the minimum of its absolute value is 1, giving

On the K-theory of the extended power construction

1982 ◽  
Vol 92 (2) ◽  
pp. 263-274 ◽  
Author(s):  
J. E. McClure ◽  
V. P. Snaith
Keyword(s):  
Minor Error ◽  
Ring Spectra ◽  
Power Construction ◽  
Complete Treatment ◽  
K Theory ◽  
Image Position ◽  
A Minor ◽  
Extended Power

The construction of Dyer-Lashof operations in K-theory outlined in (6) and refined in (12) depends in an essential way on the descriptions of the mod-p K-theory of EZp, ×ZpXp and EΣ ×σ p Xp given there. Unfortunately, these descriptions are incorrect when p is odd except in the case where the Bockstein β is identically zero in K*(X; Zp), and even in this case the methods of proof used in (6) and (12) are not strong enough to show that the answer given there is correct. In this paper we repair this difficulty, obtaining a complete corrected description of K*(EZp ×ZpXp; Zp) and K*(EΣp) (theorem 3·1 below, which should be compared with ((12); theorems 3·8 and 3·9) and ((6); theorem 3)). Because of the error, the method used in (6) and (12) to construct Dyer-Lashof operations fails to go through for odd primes when non-zero Bocksteins occur, and it is not clear that this method can be repaired. We shall not deal with the construction of Dyer-Lashof operations in this paper. Instead, the first author will give a complete treatment of these operations in (5), using our present results and the theory of H∞-ring spectra to obtain strengthened versions of the results originally claimed in ((12); theorem 5·1). There is also a minor error in the mod-2 results of (12) (namely, the second formula in (12), theorem 3·8 (a) (ii)) should readwhere B2 is the second mod-2 Bockstein, and a similar change is necessary in the second formula of ((12), theorem 3·8(b) (ii)). The correction of this error requires the methods of (5) and will not be dealt with here; fortunately, the mod-2 calculations of ((12), §6–9), (10) and (11) are unaffected and remain true as stated.

scholarly journals ON SOME PEXIDER-TYPE FUNCTIONAL EQUATIONS CONNECTED WITH THE ABSOLUTE VALUE OF ADDITIVE FUNCTIONS. PART II

2012 ◽  
Vol 85 (2) ◽  
pp. 202-216 ◽  
Author(s):  
BARBARA PRZEBIERACZ
Keyword(s):  
Functional Equation ◽  
Abelian Group ◽  
Functional Equations ◽  
Additive Functions ◽  
Absolute Value ◽  
Real Functions ◽  
The Absolute ◽  
Image Position

AbstractWe investigate the Pexider-type functional equation where f, g, h are real functions defined on an abelian group G. We solve this equation under the assumptions G=ℝ and f is continuous.

scholarly journals Fatou-Riesz-theorems in general sequence spaces

1980 ◽  
Vol 23 (2) ◽  
pp. 199-205
Author(s):  
Gerhard Otto Müller ◽  
Rolf Trautner
Keyword(s):  
Power Series ◽  
Formal Series ◽  
Sequence Spaces ◽  
Partial Sums ◽  
Classical Theorem ◽  
General Sequence ◽  
Image Position

Consider a formal serieswith partial sumsand the corresponding power series. Throughout we will assume thatfis analytic for |z| <1, i.e. thatA classical theorem of Fatou-Riesz (see (1,4)) states that ifandthenis convergent to 0.

A rational invariant for certain infinite discrete groups

1993 ◽  
Vol 113 (3) ◽  
pp. 473-478
Author(s):  
F. E. A. Johnson
Keyword(s):  
Euler Characteristic ◽  
Discrete Groups ◽  
Fundamental Groups ◽  
Intersection Form ◽  
Absolute Value ◽  
Rational Invariant ◽  
The Absolute ◽  
Image Position ◽  
Aspherical Manifolds

We introduce a rational-valued invariant which is capable of distinguishing between the commensurability classes of certain discrete groups, namely, the fundamental groups of smooth closed orientable aspherical manifolds of dimensional 4k(k ≥ 1) whose Euler characteristic χ(Λ) is non-zero. The invariant in question is the quotientwhere Sign (Λ) is the absolute value of the signature of the intersection formand [Λ] is a generator of H4k(Λ; ℝ).

scholarly journals ON SOME PEXIDER-TYPE FUNCTIONAL EQUATIONS CONNECTED WITH THE ABSOLUTE VALUE OF ADDITIVE FUNCTIONS. PART I

2012 ◽  
Vol 85 (2) ◽  
pp. 191-201 ◽  
Author(s):  
BARBARA PRZEBIERACZ
Keyword(s):  
Functional Equation ◽  
Abelian Group ◽  
Functional Equations ◽  
Additive Functions ◽  
Absolute Value ◽  
Real Functions ◽  
The Absolute ◽  
Image Position

AbstractWe investigate the Pexider-type functional equation where f,g,h are real functions defined on an abelian group G.

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