Construction of entire modular forms of weights 5 and 6 for the congruence group Γ0(4N)

Two classes of entire modular forms of weight 5 and two of weight 6 are constructed for the congruence subgroup ₀(4N). The constructed modular forms as well as the modular forms from [1] will be helpful in the theory of representation of numbers by the quadratic forms in 10 and 12 variables.     

Lattice-theoretic characterizations of the group classes {\mathfrak{N}^{k-1}\mathfrak{A}} and {\mathfrak{N}^k} for k?�� 2

Let ${2\leq k\in \mathbb{N}}$ . Recently, Costantini and Zacher obtained a lattice-theoretic characterization of the classes ${\mathfrak{N}^k}$ of finite soluble groups with nilpotent length at most k. It is the aim of this paper to give a lattice-theoretic characterization of the classes ${\mathfrak{N}^{k-1}\mathfrak{A}}$ of finite groups with commutator subgroup in ${\mathfrak{N}^{k-1}}$ ; in addition, our method also yields a new characterization of the classes ${\mathfrak{N}^k}$ . The main idea of our approach is to use two well-known theorems of Gaschütz on the Frattini and Fitting subgroups of finite groups.     

Extremal Self-Dual Codes over \mathbb{F} 2 × \mathbb{F} 2

In this paper, it is shown that extremal (Hermitian) self-dual codes over ₂ × ₂ exist only for lengths 1, 2, 3, 4, 5, 8 and 10. All extremal self-dual codes over ₂ × ₂ are found. In particular, it is shown that there is a unique extremal self-dual code up to equivalence for lengths 8 and 10. Optimal self-dual codes are also investigated. A classification is given for binary [12, 7, 4] codes with dual distance 4, binary [13, 7, 4] codes with dual distance 4 and binary [13, 8, 4] codes with dual distance 4.     

On Some Entire Modular Forms of an Integral Weight for the Congruence subgroup Γ0(4N)

Four types of entire modular forms of weight are constructed for the congruence subgroup ₀(4N) when s is even. One can find these forms helpful in revealing the arithmetical meaning of additional terms in the formulas for the number of representations of positive integers by positive quadratic forms with integral coefficients in an even number of variables.     

第二大特征值小于$\frac{\sqrt{5}-1}{2}$的平面图

若能将图$G$画在一个平面上,使得任何两条边仅在顶点处相交,则称$G$是平面图.本文刻画了第二大特征值小于$\frac{\sqrt{5}-1}{2}$的所有无孤立点的平面图.     

A necessary condition for the completeness of the system \left\{ {e^{ - \lambda _n t} |Re\lambda _n > 0} \right\} in the spaces C 0(ℝ+) and L p (ℝ+), p > 2

We obtain a necessary condition for the completeness of the system $$ e(\Lambda ) = \left\{ {e^{ - \lambda _n t} |Re\lambda _n > 0,n \in \mathbb{Z}} \right\} $$ in the spaces C ₀ and L ^p (?₊), p > 2, for the case in which the set of limit points of the sequence {λ _n } is countable and separable.     

Poincaré polynomial of the space $$\overline {{M_{0,n}}} \left( \mathbb{C} \right)$$ and the number of points of the space $$\overline {{M_{0,n}}} \left( {{F_q}} \right)$$M0,n¯(Fq)

A combinatorial proof that the number of points of the space \(\overline {{M_{0,n}}} \left( {{F_q}} \right)\) satisfies the recurrent formula for the Poincaré polynomials of the space \(\overline {{M_{0,n}}} \left( \mathbb{C} \right)\) is obtained.     

Hybrid subconvexity bounds for $$L(1/2,\hbox {sym}^{2} f \otimes \chi )$$ L ( 1 / 2 , sym 2 f ⊗ χ )

The Ramanujan Journal - Let $$k\ge 2$$ be an even integer, and let M be a positive integer. We prove a hybrid subconvexity bound for $$L(1/2,\text {sym}^2 f\otimes \chi )$$ with the associated...     

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 \((x,\ y,\ z)\)=\((1,\ 1,\ 2)\).     

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     

Covers ℘ for Abstract Regular Polytopes \mathcal {Q} such that \mathcal{Q}=\mathcal{P}/\mathbf{Z}_{p}^{k}

One key problem in the theory of abstract polytopes is the so-called amalgamation problem. In its most general form, this is the problem of characterising the polytopes with given facets  $\mathcal {K}$ and vertex figures ?. The first step in solving it for particular  $\mathcal{K}$ and ? is to find the universal such polytope, which covers all the others. This article explains a construction that may be attempted on an arbitrary polytope ?, which often yields an infinite family of finite polytopes covering ? and sharing its facets and vertex figures. The existence of such an infinite family proves that the universal polytope is infinite; alternatively, the construction can produce an explicit example of an infinite polytope of the desired type. An algorithm for attempting the construction is explained, along with sufficient conditions for it to work. The construction is applied to a few  $\mathcal{K}$ and ? for which it was previously not known whether or not the universal polytope was infinite, or for which only a finite number of finite polytopes was previously known. It is conjectured that the construction is quite broadly applicable.     

A $\mathbb{P}_N × \mathbb{P}_N$ Spectral Element Projection Method for the Unsteady Incompressible Navier-Stokes Equations

In this paper, we present a $\mathbb{P}_N × \mathbb{P}_N$ spectral element method and a detailed comparison with existing methods for the unsteady incompressible Navier-Stokes equations. The main purpose of this work consists of: (i) detailed comparison and discussion of some recent developments of the temporal discretizations in the frame of spectral element approaches in space; (ii) construction of a stable $\mathbb{P}_N × \mathbb{P}_N$ method together with a $\mathbb{P}_N → \mathbb{P}_{N-2}$post-filtering. The link of different methods will be clarified. The key feature of our method lies in that only one grid is needed for both velocity and pressure variables, which differs from most well-known solvers for the Navier-Stokes equations. Although not yet proven by rigorous theoretical analysis, the stability and accuracy of this one-grid spectral method are demonstrated by a series of numerical experiments.     

Hecke operators for the principal congruence subgroup of the group Sp(n,ℤ)

One computes the action of the Hecke operators on the generalized theta series with respect to progressions and one obtains the Euler decomposition of the theta transformations of the modular Siegel forms relative to the principal congruence subgroups of an integral symplectic group. The results are based on the technique of the decomposition of symplectic polynomials.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 154, pp. 130–135, 1986.     

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\mathop \Sigma \limits_{p \leqq N} e(\alpha p)

Ohne Zusammenfassung     

Метод многопараметрической интерполяции и теоремы вложения пространств БесоваB_{\vec p}^{\vec \alpha } \left[ {0,2\pi } \right)

In this paper we determine the method of multi-parameter interpolation and the scales of Lebesgue spaces $B_{\vec p} \left[ {0,2\pi } \right)$ and Besov spaces $B_{\vec p}^{\vec \alpha } \left[ {0,2\pi } \right)$ , which are generalizations of the Lorentz spacesL _pq [0, 2π) and Besov spacesB _pq ^α [0, 2π). We also prove imbedding theorems.     

The behavior of \sum\nolimits_{n = 1}^\infty {{{\zeta ^{ \llcorner n\theta \lrcorner } } \mathord{\left/ {\vphantom {{\zeta ^{ \llcorner n\theta \lrcorner } } n}} \right. \kern-\nulldelimiterspace} n}} for particular values of θ

Let ζ be a primitive q′-root of unity. We prove that the series $ \sum\nolimits_{n = 1}^\infty {{{\zeta ^{ \llcorner n\theta \lrcorner } } \mathord{\left/ {\vphantom {{\zeta ^{ \llcorner n\theta \lrcorner } } n}} \right. \kern-0em} n}} $ for θ ∈ Q converges if and only if θ = p/q with (p,q) = 1 and q′ ? p, and that there exists an uncountable set S of Liouville’s numbers such that the series does not converge when θ ∈ S.     

System Representations for the Paley–Wiener Space $$\mathcal {PW}_{\pi }^2$$

In this paper we study the approximation of stable linear time-invariant systems for the Paley–Wiener space \(\mathcal {PW}_{\pi }^2\), i.e., the set of bandlimited functions with finite \(L^2\)-norm, by convolution sums. It is possible to use either, the convolution sum where the time variable is in the argument of the bandlimited impulse response, or the convolution sum where the time variable is in the argument of the function, as an approximation process. In addition to the pointwise and uniform convergence behavior, the convergence behavior in the norm of the considered function space, i.e. the \(L^2\)-norm in our case, is important. While it is well-known that both convolution sums converge uniformly on the whole real axis, the \(L^2\)-norm of the second convolution sum can be divergent for certain functions and systems. We show that the there exist an infinite dimensional closed subspace of functions and an infinite dimensional closed subspace of systems, such that for any pair of function and system from these two sets, we have norm divergence.     

The Aronszajn–Donoghue Theory for Rank One Perturbations of the $$\mathcal{H}_{-2} {\text{-Class}}$$

A singular rank one perturbation of a self-adjoint operator A in a Hilbert space is considered, where and but with the usual A–scale of Hilbert spaces. A modified version of the Aronszajn-Krein formula is given. It has the form where F denotes the regularized Borel transform of the scalar spectral measure of A associated with . Using this formula we develop a variant of the well known Aronszajn–Donoghue spectral theory for a general rank one perturbation of the class.Submitted: March 14, 2002 Revised: December 15, 2002