• Volume 123, Issue 3

August 2013,   pages  303-454

• On the 𝑢-Invariant of Hermitian Forms

Let 𝐾 be a field of characteristic not 2 and 𝐴 a central simple algebra with an involution 𝜎. A result of Mahmoudi provides an upper bound for the 𝑢-invariants of hermitian forms and skew-hermitian forms over $(A,\sigma)$ in terms of the 𝑢-invariant of 𝐾. In this paper we give a different upper bound when 𝐴 is a tensor product of quaternion algebras and 𝜎 is a the tensor product of canonical involutions. We also show that our bounds are sharper than those of Mahmoudi.

• A Short Note on Sign Changes

In this paper, we present a quantitative result for the number of sign changes for the sequences $\{a(n^j)\}_{n\geq 1},j=2,3,4$ of the Fourier coefficients of normalized Hecke eigencusp forms for the full modular group $SL_2(\mathbb{Z})$. We also prove a similar kind of quantitative result for the number of sign changes of the 𝑞-exponents $c(p)$ (𝑝 vary over primes) of certain generalized modular functions for the congruence subgroup $\Gamma_0(N)$, where 𝑁 is square-free.

• A Note on a Kind of Character Sums over the Short Interval

Let 𝑝 be a prime, 𝜒 denote the Dirichlet character modulo 𝑝 and $L(p)=\{a\in\mathbb{Z}^+|(a,p)=1,aā\equiv 1(\mathrm{mod} p),|a - ā|\leq H\}$. We study the distribution of elements in the set $L(p)$ in character over the short interval. In this paper, we use the analytic method and show the distribution property of

\begin{equation*}\sum_{\substack{n\leq N}\\{n\in L(p)}}\chi(n),\end{equation*}

and give a non-trivial estimate.

• Maps into Projective Spaces

We compute the cohomology of the Picard bundle on the desingularization $\overline{J}^d (Y)$ of the compactified Jacobian of an irreducible nodal curve 𝑌. We use it to compute the cohomology classes of the Brill–Noether loci in $\overline{J}^d(Y)$.

We show that the moduli space 𝑀 of morphisms of a fixed degree from 𝑌 to a projective space has a smooth compactification. As another application of the cohomology of the Picard bundle, we compute a top intersection number for the moduli space 𝑀 confirming the Vafa–Intriligator formulae in the nodal case.

• On the Number of Isomorphism Classes of Transversals

In this paper we prove that there does not exist a subgroup 𝐻 of a finite group 𝐺 such that the number of isomorphism classes of normalized right transversals of 𝐻 in 𝐺 is four.

• Harder-Narasimhan Filtrations which are not Split by the Frobenius Maps

We will produce a smooth projective scheme 𝑋 over $\mathbb{Z}$, a rank 2 vector bundle 𝑉 on 𝑋 with a line subbundle 𝐿 having the following property. For a prime 𝑝, let $F_p$ be the absolute Fobenius of $X_p$, and let $L_p\subset V_p$ be the restriction of $L\subset V$. Then for almost all primes 𝑝, and forall $t\geq 0,(F^∗_p)^t L_P\subset (F^∗_p)^t V_p$ is a non-split Harder-Narasimhan filtration. In particular, $(F^∗_p)^t V_p$ is not a direct sum of strongly semistable bundles for any 𝑡. This construction works for any full flag veriety $G/B$, with semisimple rank of $G\geq 2$. For the construction, we will use Borel–Weil–Bott theorem in characteristic 0, and Frobenius splitting in characteristic 𝑝.

• Blowing-up Semilinear Wave Equation with Exponential Nonlinearity in Two Space Dimensions

We investigate the initial value problem for some semi-linear wave equation in two space dimensions with exponential nonlinearity growth.

• Porosity of Free Boundaries in the Obstacle Problem for Quasilinear Elliptic Equations

In this paper, we establish growth rate of solutions near free boundaries in the identical zero obstacle problem for quasilinear elliptic equations. As a result, we obtain porosity of free boundaries, which is naturally an extension of the previous works by Karp et al. (J. Diff. Equ. 164 (2000) 110–117) for 𝑝-Laplacian equations, and by Zheng and Zhang (J. Shaanxi Normal Univ. 40(2) (2012) 11–13, 18) for 𝑝-Laplacian type equations.

• The Boundedness of Multilinear Calderón-Zygmund Operators on Hardy Spaces

In this paper, we study the boundedness of the multilinear Calderón–Zygmund operators on products of Hardy spaces.

• On a Class of Smooth Frechet Subalgebras of $C^\ast$-Algebras

The paper contributes to understanding the differential structure in a $C^∗$-algebra. Refining the Banach $(D^∗_p)$-algebras investigated by Kissin and Shulman as noncommutative analogues of the algebra $C^p[a,b]$ of 𝑝-times continuously differentiable functions, we investigate a Frechet $(D^∗\infty)$-subalgebra $\mathcal{B}$ of a $C^∗$-algebra as a noncommutative analogue of the algebra $C^\infty[a,b]$ of smooth functions. Regularity properties like spectral invariance, closure under functional calculi and domain invariance of homomorphisms are derived expressing $\mathcal{B}$ as an inverse limit over 𝑛 of Banach $(D^∗_n)$-algebras. Several examples of such smooth algebras are exhibited.

• Direct and Reverse Inclusions for Strongly Multiple Summing Operators

We prove some direct and reverse inclusion results for strongly summing and strongly multiple summing operators under the assumption that the range has finite cotype.

• Outgoing Cuntz Scattering System for a Coisometric Lifting and Transfer Function

We study a coisometry that intertwines Popescu’s presentations of minimal isometric dilations of a given operator tuple and of a coisometric lifting of the tuple. Using this we develop an outgoing Cuntz scattering system which gives rise to an input–output formalism. A transfer function is introduced for the system. We also compare the transfer function and the characteristic function for the associated lifting.

• A Class of Degenerate Stochastic Differential Equations with Non-Lipschitz Coefficients

We obtain sufficient condition for SDEs to evolve in the positive orthant. We use arguments based on comparison theorems for SDEs to achieve this. As an application we prove the existence of a unique strong solution for a class of multidimensional degenerate SDEs with non-Lipschitz diffusion coefficients.

• # Proceedings – Mathematical Sciences

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November 2019

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019