Archiv der Mathematik (ARCH MATH )

Publisher: Mathematisches Forschungsinstitut Oberwolfach, Springer Verlag


Archiv der Mathematik publishes original contributions preferably not longer than 10 printed pages from the entire field of mathematics and its applications which are of interest to a large readership. Short but comprehensive authors' summaries of longer works which are not yet published as well as limited number of survey articles are also published. The journal is well known for its emphasis on papers dealing with solutions of problems in classical and modern mathematics which are not overly technical in nature.

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    Archiv der Mathematik website
  • Other titles
    Archiv der Mathematik, Archives of mathematics, Archives mathématiques
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    Periodical, Internet resource
  • Document type
    Journal / Magazine / Newspaper, Internet Resource

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Springer Verlag

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Publications in this journal

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    ABSTRACT: Let $f(z)=\sum_{n=1}^\infty a(n)e^{2\pi i nz}\in S_k^{new}(\Gamma_0(N))$ be a newform of even weight $k\geq2$ that does not have complex multiplication. Then $a(n)\in\mathbb{R}$ for all $n$, so for any prime $p$, there exists $\theta_p\in[0,\pi]$ such that $a(p)=2p^{(k-1)/2}\cos(\theta_p)$. Let $\pi(x)=\#\{p\leq x\}$. For a given subinterval $I\subset[0,\pi]$, the now-proven Sato-Tate Conjecture tells us that as $x\to\infty$, \[ \#\{p\leq x:\theta_p\in I\}\sim \mu_{ST}(I)\pi(x),\quad \mu_{ST}(I)=\int_{I} \frac{2}{\pi}\sin^2(\theta)~d\theta. \] Let $\epsilon>0$. Assuming that the symmetric power $L$-functions of $f$ are automorphic, we prove that as $x\to\infty$, \[ \#\{p\leq x:\theta_p\in I\}=\mu_{ST}(I)\pi(x)+O\left(\frac{x}{(\log x)^{9/8-\epsilon}}\right), \] where the implied constant is effectively computable and depends only on $k,N,$ and $\epsilon$.
    Archiv der Mathematik 07/2014; 103(2).
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    ABSTRACT: We bound the order of a finite p-group in terms of its exponent and p-rank. Here the p-rank is the maximal rank of an abelian subgroup. These results are applied to defect groups of p-blocks of finite groups with given Loewy length. Doing so, we improve results in a recent paper by Koshitani, Külshammer, and Sambale. In particular, we determine possible defect groups for blocks with Loewy length 4.
    Archiv der Mathematik 07/2014; 103(1).
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    ABSTRACT: Let N≥5 and D 2,2 (ℝ N ) denote the closure of C 0 ∞ (ℝ N ) in the norm ∥u∥ D 2,2 (ℝ N ) 2 :=∫ ℝ N |Δu| 2 . Let K∈C 2 (ℝ N ). We consider the following problem for ε≥0: (P ε )Findu∈D 2,2 (ℝ N )solving:Δ 2 u=(1+εK(x))u N+4 N-4 inℝ N u>0inℝ N · We show an exact multiplicity result for (P ε ) for all small ε>0.
    Archiv der Mathematik 06/2014; 102(6).
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    ABSTRACT: The topological data of a group action on a compact Riemann surface can be encoded using a tuple (h;m 1 ,...,m s ) called its signature. There are two number theoretic conditions on a tuple necessary for it to be a signature: the Riemann-Hurwitz formula is satisfied and each m i equals the order of a non-trivial group element. We show on the genus spectrum of a group that asymptotically, satisfaction of these conditions is in fact sufficient. We also describe the order of growth for the number of tuples which satisfy these conditions but are not signatures in the case of cyclic groups.
    Archiv der Mathematik 06/2014; 102(6).
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    ABSTRACT: This paper deals with the dead-core rates problem for the fast diffusion equation with a spatially dependent strong absorption. where 0<p<m<1and−1<q<0. By using the self-similar transformation technique and the Zelenyak method, we proved that the temporal dead-core rate is non-self-similar.
    Archiv der Mathematik 05/2014; 102(2014):469–481.
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    ABSTRACT: We consider profinite groups in which all commutators are contained in a union of finitely many procyclic subgroups. It is shown that if G is a profinite group in which all commutators are covered by m procyclic subgroups, then G possesses a finite characteristic subgroup M contained in G' such that the order of M is m-bounded and G'/M is procyclic. If G is a pro-p group such that all commutators in G are covered by m procyclic subgroups, then G' is either finite of m-bounded order or procyclic.
    Archiv der Mathematik 05/2014; 103(2).
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    ABSTRACT: Let R be a polynomial ring over ${\mathbb{C}}$. In this paper we show that a polynomial in R [3] is a residual coordinate if and only if it is a stable coordinate.
    Archiv der Mathematik 05/2014; 100(1).
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    ABSTRACT: We give a non-probabilistic proof for the boundary integral representation of μ-harmonic functions on topological semigroups.
    Archiv der Mathematik 05/2014; 102(5).
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    ABSTRACT: For one-dimensional simple symmetric random walks, we prove that the irregular set associated with the rate of returns to the origin is residual.
    Archiv der Mathematik 05/2014; 102(5).
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    ABSTRACT: In an article in the Pure and Applied Mathematics Quarterly in 2008 Duke and Jenkins investigated a certain natural basis of the space of weakly holomorphic modular forms for the full modular group $SL_2({\bf Z})$. We show here that their results can be generalized to certain moonshine groups, also allowing characters that are real on the underlying subgroup $\Gamma_0(N)$.
    Archiv der Mathematik 04/2014; 103(1).
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    ABSTRACT: Let $R$ be a valuation ring with fraction field $K$ and $2\in R^\times$. We give an elementary proof of the following known result: Two unimodular quadratic forms over $R$ are isometric over $K$ if and only if they are isometric over $R$. Our proof does not use Witt's Cancelation Theorem and yields an explicit algorithm to construct an isometry over $R$ from a given isometry over $K$. The statement actually holds for hermitian forms over valuated involutary division rings, provided mild assumptions.
    Archiv der Mathematik 04/2014; 103(2).
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    ABSTRACT: We provide a new proof of Volberg's Theorem characterizing thin interpolating sequences as those for which the Gram matrix associated to the normalized reproducing kernels is a compact perturbation of the identity. In the same paper, Volberg characterized sequences for which the Gram matrix is a compact perturbation of a unitary as well as those for which the Gram matrix is a Schatten-$2$ class perturbation of a unitary operator. We extend this characterization from $2$ to $p$, where $2 \le p \le \infty$.
    Archiv der Mathematik 04/2014; 103(1).
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    ABSTRACT: We give necessary and sufficient conditions under which an HNN-extension with abelian base group or an amalgamated free product of abelian groups is a Howson group (that is the intersection of any two finitely generated subgroups is finitely generated). We describe HNN-extensions and amalgamated free products which are Howson groups without satisfying the Burns–Cohen statements.
    Archiv der Mathematik 04/2014; 98(2).
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    ABSTRACT: This paper explores the relationship between various notions related to interpolation by functions in the Paley-Wiener space PW τ p (1<p<∞), which consists of the entire functions of exponential type at most τ>0 such that ∥f∥ p =∫ ℝ |f(t)| p dt 1/p <∞· A sequence Λ={λ n } n≥1 ⊂ℂ is called “minimal” in PW τ p if there exists a sequence (f n ) n ⊂PW τ p such that f n (λ k )=1 if k=n and 0 if k≠n. A stronger notion is that of a “weak interpolation sequence” (WIS): Λ is a WIS. if there exists a sequence (f n ) n ⊂PW τ p such that f n (λ k )=∥k λ n ∥ PW τ q if k=n and 0 if k≠n. Here q is the conjugate exponent of p and k λ n (z)=sin[τ(z-λ ¯)] τ(z-λ ¯) is the reproducing kernel in PW τ p associated with λ n . In dual terms, Λ is minimal in PW τ p if and only if the family of reproducing kernels K(Λ)={k λ n } n≥1 is minimal in PW τ q , and it is a WIS if and only if K(Λ) is uniformly minimal in PW τ q . Finally Λ is an “interpolation sequence” for PW τ p if for every sequence of values a=(a n )∈l p there exists f∈PW τ p such that f(λ n )=a n ∥k λ n ∥ PW τ q for all n≥1. If, moreover, such f is unique then Λ is called “complete”. For a∈ℝ consider the upper and lower half-planes ℂ a ± ={z∈ℂ:ℑ(z)≷a}· The author showed in his thesis that if Λ is a WIS then for every a∈ℝ the sequences Λ∩ℂ a ± satisfy the Carleson condition: inf n≥1 ∏ k≠n λ n -λ k λ n -λ ¯ k -2ia>0·(1) It is well known that for the Hardy spaces in the half-plane H p (ℂ + ) (here a=0) weak interpolation sequences coincide with interpolation sequences, and that these are characterized by the Carleson condition [H. S. Shapiro and A. L. Shields, Am. J. Math. 83, 513–532 (1961; Zbl 0112.29701)]. The main result of the paper is the following. Theorem 1.1. Let τ>0, 1<p<∞, and Λ be a minimal sequence in PW τ p such that for every a∈ℝ the sequence Λ∩ℂ a ± satisfies (1). Then, for every ϵ>0, Λ is an interpolating sequence for PW τ+ϵ p . The main point here is that no uniform minimality is required. A consequence of this result is that a weak interpolation sequence for PW τ p is automatically an interpolation sequence for PW τ+ϵ p (but not for PW τ p , as shown by an example). Another consequence is that the upper density of weak interpolation sequences Λ contained in a horizontal band is bounded by τ/π: D Λ + =lim r→1 sup x∈ℝ #(ℜ(Λ)∩[x,x+r]) r≤τ π· This is the same upper bound given by K. Seip [J. Funct. Anal. 130, No. 1, 131–160 (1995; Zbl 0872.46006)] for interpolation sequences contained in a horizontal band. A final section is devoted to similar results on weighted interpolation in both the Paley-Wiener and the Hardy spaces.
    Archiv der Mathematik 04/2014; 102(4).