On the coefficients of the asymptotic expansion of the kernel of Berezin–Toeplitz quantization

We give new methods for computing the coefficients of the asymptotic expansions of the kernel of Berezin?CToeplitz quantization obtained recently by Ma?CMarinescu, and of the composition of two Berezin?CToeplitz quantizations. Our main tool is the stationary phase formula of Melin?CSj?strand.     

The Lefschetz–Lunts formula for deformation quantization modules

We adapt to the case of deformation quantization modules a formula of Lunts (Lefschetz fixed point theorems for Fourier–Mukai functors and DG algebras. arXiv:1102.2884. ArXiv e-prints, 2011) who calculates the trace of a kernel acting on the Hochschild homology of a DQ-algebroid.     

A^p（φ）上的Toeplitz算子与Berezin变换

讨论了加权Bergman空间A^p(φ）（1＜p＜∞）上的Toeplitz算子的有限乘积的有限和是紧算子当且仅当其Berezin变换在边界趋向于零。     

Stiefel–Whitney invariants for bilinear forms

We examine potential extensions of the Stiefel–Whitney invariants from quadratic forms to bilinear forms which are not necessarily symmetric. We show that as long as the symbolic nature of the invariants is maintained, some natural extensions carry only low dimensional information. In particular, the generic invariant on upper triangular matrices is equivalent to the dimension and determinant. Along the process, we show that every non-alternating matrix is congruent to an upper triangular matrix, and prove a version of Witt?s Chain Lemma for upper-triangular bases. (The classical lemma holds for orthogonal bases.)     

Density lower bound estimates for local minimizers of the 2d Mumford–Shah energy

We prove, using direct variational arguments, an explicit energy-treshold criterion for regular points of 2-dimensional Mumford-Shah energy minimizers. From this we infer an explicit constant for the density lower bound of De Giorgi, Carriero and Leaci.     

$$L^{\infty }$$-norm and energy quantization for the planar Lane–Emden problem with large exponent

For any smooth bounded domain \(\Omega \subset {\mathbb {R}}^2\), we consider positive solutions to

$$\begin{aligned} \left\{ \begin{array}{lr}-\Delta u= u^p &{} \text{ in } \Omega \\ u=0 &{} \text{ on } \partial \Omega \end{array}\right. \end{aligned}$$

which satisfy the uniform energy bound

$$\begin{aligned}p\Vert \nabla u\Vert _{\infty }\le C\end{aligned}$$

for \(p>1\). We prove convergence to \(\sqrt{e}\) as \(p\rightarrow +\infty \) of the \(L^{\infty }\)-norm of any solution. We further deduce quantization of the energy to multiples of \(8\pi e\), thus completing the analysis performed in De Marchis et al. (J Fixed Point Theory Appl 19:889–916, 2017).

     

A~p()上的Toeplitz算子与Berezin变换

讨论了加权Bergman空间AP（）（1＜p＜∞）上的Toeplitz算子证明了Toeplitz算子的有限乘积的有限和是紧算子当区仅当其Berezin变换在边界趋向于零．     

Eichler–Shimura theory for mock modular forms

We use mock modular forms to compute generating functions for the critical values of modular $L$ -functions, and we answer a generalized form of a question of Kohnen and Zagier by deriving the “extra relation” that is satisfied by even periods of weakly holomorphic cusp forms. To obtain these results we derive an Eichler–Shimura theory for weakly holomorphic modular forms and mock modular forms. This includes two “Eichler–Shimura isomorphisms”, a “multiplicity two” Hecke theory, a correspondence between mock modular periods and classical periods, and a “Haberland-type” formula which expresses Petersson’s inner product and a related antisymmetric inner product on $M_{k}^{!}$ in terms of periods.     

A Radon–Nikodym type theorem for forms

The Lebesgue decomposition theorem and the Radon–Nikodym theorem are the cornerstones of the classical measure theory. These theorems were generalized in several settings and several ways. Hassi, Sebestyén, and de Snoo recently proved a Lebesgue type decomposition theorem for nonnegative sesquilinear forms defined on complex linear spaces. The main purpose of this paper is to formulate and prove also a Radon–Nikodym type result in this setting. As an application, we present a Lebesgue type decomposition theorem and solve a special case of the infimum problem for densely defined (not necessarily bounded) positive operators.     

Alternative dual frames for digital-to-analog conversion in sigma–delta quantization

We design alternative dual frames for linearly reconstructing signals from sigma–delta (ΣΔ) quantized finite frame coefficients. In the setting of sampling expansions for bandlimited functions, it is known that a stable rth order sigma–delta quantizer produces approximations where the approximation error is at most of order 1 / λ ^r , and λ > 1 is the oversampling ratio. We show that the counterpart of this result is not true for several families of redundant finite frames for \mathbbR^d\mathbb{R}^d when the canonical dual frame is used in linear reconstruction. As a remedy, we construct alternative dual frame sequences which enable an rth order sigma–delta quantizer to achieve approximation error of order 1/N ^r for certain sequences of frames where N is the frame size. We also present several numerical examples regarding the constructions.     

Polynomial forms for quantum elliptic Calogero–Moser Hamiltonians

We hypothesize the form of a transformation reducing the elliptic A _N Calogero–Moser operator to a differential operator with polynomial coefficients. We verify this hypothesis for N ≤ 3 and, moreover, give the corresponding polynomial operators explicitly.     

The Eichler–Shimura cohomology theorem for Jacobi forms

Let \(\Gamma \) be a subgroup of finite index in \(\mathrm {SL}(2,\mathbb {Z})\). Eichler defined the first cohomology group of \(\Gamma \) with coefficients in a certain module of polynomials. Eichler and Shimura established that this group is isomorphic to the direct sum of two spaces of cusp forms on \(\Gamma \) with the same integral weight. These results were generalized by Knopp to cusp forms of real weights. In this paper, we define the first parabolic cohomology groups of Jacobi groups \(\Gamma ^{(1,j)}\) and prove that these are isomorphic to the spaces of (skew-holomorphic) Jacobi cusp forms of real weights. We also show that if \(j=1\) and the weights of Jacobi cusp forms are in \(\frac{1}{2}\mathbb {Z}-\mathbb {Z}\), then these isomorphisms can be written in terms of special values of partial L-functions of Jacobi cusp forms.     

Two closed forms for the Apostol–Bernoulli polynomials

In this note, we shall obtain two closed forms for the Apostol–Bernoulli polynomials.     

On generalized Bohr–Sommerfeld quantization rules for operators with PT symmetry

We give Bohr–Sommerfeld rules corresponding to quasi-eigenvalues in the pseudospectrum for a one-dimensional h-pseudodifferential operator verifying PT symmetry.     

Nonvanishing of Rankin–Selberg L-functions for Hilbert modular forms

Let F be a totally real number field of degree n over $\mathbb{Q}$ with ring of integers $\mathcal{O}$ and narrow class number one. Let S _2k (Γ) be the vector space of cuspidal Hilbert modular forms of parallel weight 2k for $\varGamma=\mathrm{SL}_{2}(\mathcal{O})$ , and let B _2k be an orthogonal Hecke eigenbasis for this space. For any fixed Hecke eigenform f∈S _2k (Γ) and any ε>0, we prove that $$\# \biggl\{ g \in B_{2k}: L \biggl(f \times g, \frac{1}{2} \biggr) \ne0 \biggr\} \gg k^{n- \varepsilon}, $$ where L(f×g,s) is the Rankin–Selberg L–function of f and g.     

Dirichlet空间上Toeplitz算子与Berezin型变换相关的一些性质

本文研究了单位圆盘D 的Dirichlet 空间上Toeplitz 算子和小Hankel 算子. 利用Berezin 型变换讨论了Toeplitz 算子的不变子空间问题, 具有Berezin 型符号的Toeplitz 算子的渐进可乘性以及Toeplitz 算子的Riccati 方程的可解性. 应用Berezin 变换得到了Toeplitz 算子和小Hankel 算子可逆的充分条件. 此外, 还利用Hankel 算子和Berezin 变换刻画了算子2T_uv-T_uT_v-T_vT_u 的紧性, 其中函数u,v ∈ L^2,1.     

Bohr–Sommerfeld–Heisenberg theory in geometric quantization

In the framework of geometric quantization we extend the Bohr–Sommerfeld rules to a full quantum theory which resembles the Heisenberg matrix theory. This extension is possible because Bohr–Sommerfeld rules not only provide an orthogonal basis in the space of quantum states, but also give a lattice structure to this basis. This permits the definition of appropriate shifting operators. As examples, we discuss the 1–dimensional harmonic oscillator and the coadjoint orbits of the rotation group.     

Conservation of energy for the Euler–Korteweg equations

In this article we study the principle of energy conservation for the Euler–Korteweg system. We formulate an Onsager-type sufficient regularity condition for weak solutions of the Euler–Korteweg system to conserve the total energy. The result applies to the system of Quantum Hydrodynamics.