Formality Theorems for Hochschild Complexes and Their Applications

We give a popular introduction to formality theorems for Hochschild complexes and their applications. We review some of the recent results and prove that the truncated Hochschild cochain complex of a polynomial algebra is non-formal.     

A Formality Theorem for Poisson Manifolds

Letters in Mathematical Physics - Let M be a differential manifold. Using different methods, Kontsevich and Tamarkin have proved a formality theorem, which states the existence of a Lie...     

Formality Theorem for Quantizations of Lie Bialgebras

Using the theory of props we prove a formality theorem associated with universal quantizations of Lie bialgebras.     

Formality for Lie Algebroids

Using Dolgushev’s generalization of Fedosov’s method for deformation quantization, we give a positive answer to a question of P. Xu: can one prove a formality theorem for Lie algebroids ? As a direct application of this result, we obtain that any triangular Lie bialgebroid is quantizable.     

A Nonlinear Transfer Operator Theorem

In recent papers, Kenyon et al. (Ergod Theory Dyn Syst 32:1567–1584 2012), and Fan et al. (C R Math Acad Sci Paris 349:961–964 2011, Adv Math 295:271–333 2016) introduced a form of non-linear thermodynamic formalism based on solutions to a non-linear equation using matrices. In this note we consider the more general setting of Hölder continuous functions.     

Poisson Action and Formality

Using a formality on a Poisson manifold, we construct a star product and for each Poisson vector field a derivation of this star product. Starting with a Poisson action of a Lie group, we are able under a natural cohomological assumption to define a representation of its Lie algebra in the space of derivations of the star product. Finally, we use these results to define some generically tangential star products on duals of Lie algebra as in [1] but in a more realistic context. This work was supported by the CMCU contract 00 F 15 02.     

Formality Conjectures and Deformation

Let M be a Poisson manifold. Kontsevich proved that star products exist on M and he gave a classification. To relate his classification with other classifications, one could try to extend the Connes–Flato–Sternheimer invariant to a general Poisson manifold. We show how generalization of this invariant is related to the formality conjecture for chains. Finally, we show how to prove those conjectures step by step. Our approach, different from Tamarkin's, will give explicit formulas but doesn't yet solve the general conjecture.     

Theorem of Levinson via the Spectral Density

We deduce Levinson’s theorem in non-relativistic quantum mechanics in one dimension as a sum rule for the spectral density constructed from asymptotic data. We assume a self-adjoint Hamiltonian which guarantees completeness; the potential needs not to be isotropic and a zero-energy resonance is automatically taken into account. Peculiarities of this one-dimension case are explained because of the “critical” character of the free case u(x) = 0, in the sense that any attractive potential forms at least a bound state. We believe this method is more general and direct than the usual one in which one proves the theorem first for single wave modes and performs analytical continuation.     

Formality of Chain Operad of Little Discs

We prove that the chain operad of little disks is formal in characteristic zero, and discuss briefly the relation with Kontsevich formality in deformation quantization.     

A New Pooling Approach Based on Zeckendorf’s Theorem for Texture Transfer Information

The pooling layer is at the heart of every convolutional neural network (CNN) contributing to the invariance of data variation. This paper proposes a pooling method based on Zeckendorf’s number series. The maximum pooling layers are replaced with Z pooling layer, which capture texels from input images, convolution layers, etc. It is shown that Z pooling properties are better adapted to segmentation tasks than other pooling functions. The method was evaluated on a traditional image segmentation task and on a dense labeling task carried out with a series of deep learning architectures in which the usual maximum pooling layers were altered to use the proposed pooling mechanism. Not only does it arbitrarily increase the receptive field in a parameterless fashion but it can better tolerate rotations since the pooling layers are independent of the geometric arrangement or sizes of the image regions. Different combinations of pooling operations produce images capable of emphasizing low/high frequencies, extract ultrametric contours, etc.     

Quantum State Transfer via Adiabatic Passage

我们采用周期极化KTP晶体为非线性介质，通过光学参量振荡器运转于阈值以下的简并参量振荡过程，产生了单模正交压缩真空态光场，在泵浦功率为123mW，Local光功率为842uW，晶体温度为32．1摄氏度时我们使用平衡零拍探测法测得输出场噪声功率低于散粒噪声基准3．41dB。     

Quantum State Transfer via Parity Measurement

We propose two schemes for quantum state transfer using parity measurement in a cavity-waveguide system, and the two schemes can be generalized to multidipole’s case. An important advantage is that quantum state transfer can be completed by single-qubit rotations and the measurement of parity. Therefore, our scheme can be realized in the scope of current experimental technology.     

Generalized Deformations and Hochschild Cohomology

We present a generalization of Gerstenhaber's theory of deformations. We no longer assume that the deformation parameter t acts in its usual free and symmetric way on the elements of the original algebra A, but in the following manner: t · a = (a)t and a · t = (a)t, where and are endomorphisms of A. We develop the cohomological framework adapted to these deformations.     

Hastings’s Additivity Counterexample via Dvoretzky’s Theorem

The goal of this note is to show that Hastings’s counterexample to the additivity of minimal output von Neumann entropy can be readily deduced from a sharp version of Dvoretzky’s theorem.     

利用绝热过程实现量子信息转移

基于绝热过程,仅需一个真空腔即可分别实现未知单原子态、双原子纠缠态及GHZ态的转移.在这些方案中,量子信息都存储在原子的基态,且系统仅在暗态空间中演化,原子激发态上无布居,这使得原子的自发辐射效应大大受到抑制.相比之前的方案,这些方案所需资源更少,操作更简单.另外,还讨论了其在实验上实现的可行性.     

On Continuous and Differential Hochschild Cohomology

We show that there are canonical isomorphisms between Hochschild cohomology spaces , where is the algebra of smooth functions on a manifold M and the space of skew multivector fields over M. This implies that continuous and differential deformation theories of coincide.     

Formality of the Chain Operad of Framed Little Disks

We extend Tamarkin’s formality of the little disk operad to the framed little disk operad.     

Transfer information remotely via noise entangled coherent channels

A simple scheme of quantum teleportation is introduced to investigate the possibility of remotely transfer unknown tripartite state by using a multipartite entangled coherent state. A theoretical technique is introduced to generate maximum entangled coherent states and a generalized teleportation protocol is introduced. The fidelity of the teleported state increases significantly when increasing the number of modes in the case of large noise, whereas it shows the opposite behavior in the case of small noise.     

Evaluate More General Integrals Involving Universal Associated Legendre Polynomials via Taylor's Theorem

A few important integrals involving the product of two universal associated Legendre polynomials P_(l′)~(m′)(x),P_k~n′~′(x)and x~(2a)(1-x~2)~(-p-1),x~b(1±x)~(-p-1)and x~c(1-x~2)~(-p-1)(1±x)are evaluated using the operator form of Taylor’s theorem and an integral over a single universal associated Legendre polynomial.These integrals are more general since the quantum numbers are unequal,i.e.l~′≠k~′and m~′≠n~′.Their selection rules are also given.We also verify the correctness of those integral formulas numerically.     

用腔场QED技术实现量子信息转移

提出了一种量子信息转移方案,它是基于两个耦合的二能级原子同时与单模腔场发生大失谐相互作用实现的.通过控制腔场与双原子的相互作用时间,量子信息可以从一个原子转移到另一个原子,或从单模腔场转移到双原子纠缠态上,而包含在欲转移量子态上的信息被完全檫除. 关键词： 大失谐Jaynes Cummings模型 量子信息转移 偶极 偶极相互作用