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.     

A trace formula for hecke operators on L2(S2) and modular forms on Γ0(4)

To any finite symmetric subsetR ⊂ O₃ corresponds a Hecke operatorT _R on L²(S ²) which leaves the eigenspaces ℋ_n (n ≥ 0) of the Laplacian invariant. We compute the trace ofT _R | ℋ_n and prove that the sum of the positive eigenvalues ofT _R on ⊕_k=0 ^n-1ℋ_k prevails over the modulus of the sum of the negative eigenvalues. For anym ∈ ℕ the integral quaternions of normm define such a Hecke operator , and renormalizing the traces ofT _{R m} | ℋ _n slightly, we obtain sequences of Fourier coefficients of modular forms on Γ₀(4). Dedicated to R. Remmert on the occasion of his seventieth birthday     

Dimensions of cusp forms for Γ0(p) in degree two and small weights

We investigate degree two Siegel cusp forms of small weight for Γ₀(p). Using the Restriction Technique we compute some dimensions and verify the conjectures ofHashimoto in some examples of weights three and four. For weight two we determine the dimension for primesp ≤ 41 and find only lifts. We explain in general how to compute spaces of Siegel cusp forms for subgroups of finite index in Γ _n .     

Differential automorphisms for modular forms on Γ<Subscript>0</Subscript>(4)

If , then let M _k (Γ₀(4)) be the usual space of half integral weight modular forms. Ono constructed differential endomorphisms of M _k (Γ₀(4)) by using the usual differential operator. Here we construct a similar set of differential endomorphisms using a linear combination of the differential operator and the quasi-modular forms E ₂, E ₂|V ₂, and E ₂|V ₄. We compute a full set of eigenforms with eigenvalues, and we prove that these endomorphisms are in fact automorphisms. The author would like to thank the NSF for its support through the REGS program.     

On the cauchy problem for some non-Kowalewskian equations inD {σ} L2 (*)

A pseudo-differential operator is considered, which generalizes some peculiar non-Kowalewskian operators of 2-evolution type. A result is proved about the well-posedness of the Cauchy problem inD _{} ^L2 , where 1 is Gevrey index.     

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.     

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.     

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.     

On the description of the spacesL_p^{\overrightarrow \alpha } (Ω)

We consider the problem of describing the spaces \(L_p^{\overrightarrow \alpha } \) (Ω), Ω ≠E _n of Bessel potentials in terms of the “modified” fractional derivatives of Marchaud.     

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

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.     

Global solvability of the Cauchy problem for the Navier–Stokes equation in \mathbb {R}^3 for some class of initial data

In this paper we prove the existence of regular solutions to the Navier–Stokes equations if the initial data v ₀ have some finite weighted norm and supp v ₀ belongs to , is a ball with radius R ₀, where R ₀ is sufficiently large. The proof follows from appropriate estimates in weighted Sobolev spaces. K. Pileckas was supported by EC FP6 MC-ToK programme SPADE 2, MTKD-CT-2004-014508.     

Chain of interacting SU(2)4 anyons and quantum SU(2)k×$\overline { SU(2)_k } $ doubles

We consider a chain of SU(2) ₄ anyons with transitions to a topologically ordered phase state. For half-integer and integer indices of the type of strongly correlated excitations, we find an effective low-energy Hamiltonian that is an analogue of the standard Heisenberg Hamiltonian for quantum magnets. We describe the properties of the Hilbert spaces of the system eigenstates. For the Drinfeld quantum SU(2)_k×[`(SU(2)_k )]\overline {SU(2)_k } doubles, we use numerical computations to show that the largest eigenvalues of the adjacency matrix for graphs that are extended Dynkin diagrams coincide with the total quantum dimensions for the levels k = 2, 3, 4, 5. We also formulate a hypothesis about the reason for the universal behavior of the system in the long-wave limit.     

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 number of partitions into parts not congruent to 0, $$\pm 3 \pmod {12}$$ ± 3 ( mod 12 )

Periodica Mathematica Hungarica - We consider the partitions of n into parts not congruent to 0, $$\pm 3\pmod {12}$$ and provide some new results related to the number of these partitions. In this...     

On q-analogs of weight multiplicities for the Lie superalgebras \mathfrak{gl}(n,m) and \mathfrak{spo(}2n,M)

The paper is devoted to the generalization of Lusztig’s q-analog of weight multiplicities to the Lie superalgebras $\mathfrak{gl}(n,m)$ and $\mathfrak{spo(}2n,M).$ We define such q-analogs K _λ,μ (q) for the typical modules and for the irreducible covariant tensor $\mathfrak{gl}(n,m)$ -modules of highest weight λ. For $\mathfrak{gl}(n,m),$ the defined polynomials have nonnegative integer coefficients if the weight μ is dominant. For $\mathfrak{spo(}2n,M)$ , we show that the positivity property holds when μ is dominant and sufficiently far from a specific wall of the fundamental chamber. We also establish that the q-analog associated to an irreducible covariant tensor $\mathfrak{gl}(n,m)$ -module of highest weight λ and a dominant weight μ is the generating series of a simple statistic on the set of semistandard hook-tableaux of shape λ and weight μ. This statistic can be regarded as a super analog of the charge statistic defined by Lascoux and Schützenberger.     

A necessary and sufficient condition for the biorthogonality of {ρ 1,ρ 2}

In this paper, we present a necessary and sufficient condition for the biorthogonality of a class of special functionsρ ₁ andρ ₂. The functions are useful in the theory of biorthogonal wavelet.     

Schiffer’s Problem and an Isoperimetric Inequality for the First Buckling Eigenvalue of Domains on $$\mathbb{S}^{2}$$

We consider the overdetermined eigenvalue problem on a sufficiently regular connected open domain Ω on the 2-sphere :