Abschätzungen für\sum\limits_{n \leqslant N} {B_1 (n\alpha )}

Estimates for . Let and letB ₁(x)={x}–1/2. In this paper we shall give best possible estimates for . On the importance of this sum see the papers ofBehnke [2],Hardy andLittlewood [5],Hecke [6] andOstrowski [9].

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Meinem verehrten Lehrer, Herrn Professor E. Hlawka zum siebzigsten Geburtstag gewidmet     

Finite Simple Groups with Hall {2, r}-Subgroups,r �� ��(G)\{2, t}, t �� ��(G)

Ukrainian Mathematical Journal - We describe finite simple groups with Hall biprimary subgroups of even order that contain Sylow subgroups of odd order of the G-group, with the exception of one...     

Asymptotic behavior of sums of the type\sum\limits_{\kappa = m}^{n - 1} {\exp } (i\omega \sqrt {n\kappa } ) as n,m → ∞

The asymptotic behavior asn, m → ∞ of the sum $$\sum\limits_{\kappa ,\ell = m}^{n - 1} {\exp \left[ {i\omega \sqrt n \left( {\sqrt \kappa + \sqrt \ell } \right)} \right]} \Phi \left( {1 - \frac{{\left| {\sqrt \kappa - \sqrt \ell } \right|}}{\Delta }} \right)$$ is studied where π(t)=0 for t?0 and φ(t)=t for t > 0.     

Über die Summe\sum\limits_{n \leqslant x} {\varphi (f)(n)} und verwandte Probleme

Ohne Zusammenfassung     

不存在秩为r(G)+2c(G)-1或r(G)-2c(G)+1的符号图

设Γ=(G,σ)是一个符号图,其中G是Γ的基图.设r(G,σ)是Γ的秩.[Linear Algebra Appl.,2018,538:166-186]和[Linear Multilinear Algebra,2019,67:2520-2539]分别证明了r(G)-2c(G)≤r(G,σ)≤r(G)+2c(G),其中,r(G)和c(G)分别是G的秩和圈空间维数.本文主要证明没有符号图的秩能够达到r(G)+2c(G)-1和r(G)-2c(G)+1,并且证明了存在无穷多个符号图的秩r(G,σ)=r(G)+2c(G)-s,其中s∈[0,4c(G)]且s≠1及4c(G)-1.     

A proof of the Anderson–Badawi $$ \mathbf{rad}\varvec{(I)}^{\varvec{n}} \varvec{\subseteq } {\varvec{I}}$$ formula for $$\varvec{n}$$n-absorbing ideals

In [1], Anderson and Badawi conjectured that \(\mathrm{rad}(I)^n \subseteq I\) for every n-absorbing ideal I of a commutative ring. In this article, we prove their conjecture. We also prove related conjectures for radical ideals.     

Some Arithmetical Properties of the Generating Power Series for the Sequence {ζ(2k+1)}k=1 ∞

Let f_odd(z):= ∑^∞ _k=1ζ(2k + 1)z^2k be the power series with the values of the Riemann ζ function at odd integers as coefficients. This function can be analytically continued to a meromorphic function over C. We prove that 1 and the values of f_odd at rational points with relatively prime denominators are linearly independent over ―Q. Some arithmetical properties of the sequence {ζ(2k+1)} _k=1 ^∞ are deduced. This revised version was published online in June 2006 with corrections to the Cover Date.     

Extracting BB′IW Inhabitants of Simple Types From Proofs in the Sequent Calculus {LT_\to^{t}} for Implicational Ticket Entailment

The decidability of the logic of pure ticket entailment means that the problem of inhabitation of simple types by combinators over the base { B, B′, I, W } is decidable too. Type-assignment systems are often formulated as natural deduction systems. However, our decision procedure for this logic, which we presented in earlier papers, relies on two sequent calculi and it does not yield directly a combinator for a theorem of ${T_\to}$ . Here we describe an algorithm to extract an inhabitant from a sequent calculus proof—without translating the proof into another proof system.     

On the exponential diophantine equation $$\left( am^{2}+1\right) ^{x}+\left( bm^{2}-1\right) ^{y}=(cm)^{z}$$ with $$ c\mid m $$c∣m

Let \(a,\ b,\ c,\ m\) be positive integers such that \(a+b=c^2, 2\mid a, 2\not \mid c\) and \(m>1\). In this paper we prove that if \(c\mid m \) and \(m>36c^3 \log c\), then the equation \((am^2+1)^x+(bm^2-1)^y=(cm)^z\) has only the positive integer solution \((x,\ y,\ z)\)=\((1,\ 1,\ 2)\).     

Über die Differentialgleichung\frac{{dx}}{{dy}} = kf\left( {xy} \right), deren Lösungskurve durch zwei gegebene Punkte hindurchgehen soll

Ohne Zusammenfassung     

On certain degenerate Whittaker Models for cuspidal representations of $${\mathrm{GL}_{k \cdot n}(\mathbb {F}_q)}$$ GL k · n ( F q )

Mathematische Zeitschrift - Let $$\pi $$ be an irreducible cuspidal representation of $$\mathrm {GL}_{kn}(\mathbb {F}_q)$$ . Assume that $$\pi = \pi _{\theta }$$ , corresponds to a regular...     

Beweis der Existenz einer Lösung der Differentialgleichung\frac{{dy}}{{dx}} = f\left( {x,y} \right) ohne Hinzunahme der Cauchy-Lipschitz'schen Bedingung

Ohne Zusammenfassung     

Über die Funktionalgleichungf(x) = d^{m - 1} \sum\limits_{i = 0}^{d - 1} {f(\frac{{x + i}}{d})} in den Körpern ℤp

Functional equations of the type (1) $$f(x) = d^{m - 1} \sum\limits_{i = 0}^{d - 1} {f(\frac{{x + i}}{d})}$$ for functions f: ?_n→?_n have first been considered in [1] and [2] in the cases m ∈ {0,1} (?_n=Z/n? means the ring of integers modulo n). In this note we give the complete solution of (1) in the case f: ?_p→?_p (p a prime) for each m ∈ ?∪{0}.