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Zhengfang Wang

Publications and source records attributed to Zhengfang Wang.

30 records · Page 2Linked to original sources
arXiv2021

GPRInvNet: Deep Learning-Based Ground Penetrating Radar Data Inversion for Tunnel Lining

A DNN architecture referred to as GPRInvNet was proposed to tackle the challenges of mapping the ground-penetrating radar (GPR) B-Scan data to complex permittivity maps of subsurface structures. The GPRInvNet consisted of a trace-to-trace encoder and a decoder. It was specially designed to take into account the characteristics of GPR inversion when faced with complex GPR B-Scan data, as well as addressing the spatial alignment issues between time-series B-Scan data and spatial permittivity maps. It displayed the ability to fuse features from several adjacent traces on the B-Scan data to enhance each trace, and then further condense the features of each trace separately. As a result, the sensitive zones on the permittivity maps spatially aligned to the enhanced trace could be reconstructed accurately. The GPRInvNet has been utilized to reconstruct the permittivity map of tunnel linings. A diverse range of dielectric models of tunnel linings containing complex defects has been reconstructed using GPRInvNet. The results have demonstrated that the GPRInvNet is capable of effectively reconstructing complex tunnel lining defects with clear boundaries. Comparative results with existing baseline methods also demonstrated the superiority of the GPRInvNet. For the purpose of generalizing the GPRInvNet to real GPR data, some background noise patches recorded from practical model testing were integrated into the synthetic GPR data to retrain the GPRInvNet. The model testing has been conducted for validation, and experimental results revealed that the GPRInvNet had also achieved satisfactory results with regard to the real data.

cs.CV↗
arXiv2021

Leavitt path algebras, $B_\infty$-algebras and Keller's conjecture for singular Hochschild cohomology

For a finite quiver without sinks, we establish an isomorphism in the homotopy category $\mathrm {Ho}(B_\infty)$ of $B_{\infty}$-algebras between the Hochschild cochain complex of the Leavitt path algebra $L$ and the singular Hochschild cochain complex of the corresponding radical square zero algebra $Λ$. Combining this isomorphism with a description of the dg singularity category of $Λ$ in terms of the dg perfect derived category of $L$, we verify Keller's conjecture for the singular Hochschild cohomology of $Λ$. More precisely, we prove that there is an isomorphism in $\mathrm{Ho}(B_\infty)$ between the singular Hochschild cochain complex of $Λ$ and the Hochschild cochain complex of the dg singularity category of $Λ$. One ingredient of the proof is the following duality theorem on $B_\infty$-algebras: for any $B_\infty$-algebra, there is a natural $B_\infty$-isomorphism between its opposite $B_\infty$-algebra and its transpose $B_\infty$-algebra. We prove that Keller's conjecture is invariant under one-point (co)extensions and singular equivalences with levels. Consequently, Keller's conjecture holds for those algebras obtained inductively from $Λ$ by one-point (co)extensions and singular equivalences with levels. These algebras include all finite dimensional gentle algebras.

math.RT↗
arXiv2020

Defect segmentation: Mapping tunnel lining internal defects with ground penetrating radar data using a convolutional neural network

This research proposes a Ground Penetrating Radar (GPR) data processing method for non-destructive detection of tunnel lining internal defects, called defect segmentation. To perform this critical step of automatic tunnel lining detection, the method uses a CNN called Segnet combined with the Lovász softmax loss function to map the internal defect structure with GPR synthetic data, which improves the accuracy, automation and efficiency of defects detection. The novel method we present overcomes several difficulties of traditional GPR data interpretation as demonstrated by an evaluation on both synthetic and real datas -- to verify the method on real data, a test model containing a known defect was designed and built and GPR data was obtained and analyzed.

cs.CV↗
arXiv2019

Invariance of the Gerstenhaber algebra structure on Tate-Hochschild cohomology

Keller proved in 1999 that the Gerstenhaber algebra structure on the Hochschild cohomology of an algebra is an invariant of the derived category. In this paper, we adapt his approach to show that the Gerstenhaber algebra structure on the Tate-Hochschild cohomology of an algebra is preserved under singular equivalences of Morita type with level, a notion introduced by the author in previous work.

math.RT↗
arXiv2019

The Batalin-Vilkovisky structure on the Tate-Hochschild cohomology ring of a group algebra

We determine the Batalin-Vilkovisky structure on the Tate-Hochschild cohomology of the group algebra $kG$ of a finite group $G$ in terms of the additive decomposition. In particular, we show that the Tate cohomology of $G$ is a Batalin-Vilkovisky subalgebra of the Tate-Hochschild cohomology of the group algebra $kG$, and that the Tate cochain complex of $G$ is a cyclic $A_{\infty}$-subalgebra of the Tate-Hochschild cochain complex of $kG$.

math.GR↗
arXiv2018

Singular Hochschild cohomology and algebraic string operations

Given a differential graded (dg) symmetric Frobenius algebra $A$ we construct an unbounded complex $\mathcal{D}^{*}(A,A)$, called the Tate-Hochschild complex, which arises as a totalization of a double complex having Hochschild chains as negative columns and Hochschild cochains as non-negative columns. We prove that the complex $\mathcal{D}^*(A,A)$ computes the singular Hochschild cohomology of $A$. We construct a cyclic (or Calabi-Yau) $A$-infinity algebra structure, which extends the classical Hochschild cup and cap products, and an $L$-infinity algebra structure extending the classical Gerstenhaber bracket, on $\mathcal{D}^*(A,A)$. Moreover, we prove that the cohomology algebra $H^*(\mathcal{D}^*(A,A))$ is a Batalin-Vilkovisky (BV) algebra with BV operator extending Connes' boundary operator. Finally, we show that if two Frobenius algebras are quasi-isomorphic as dg algebras then their Tate-Hochschild cohomologies are isomorphic and we use this invariance result to relate the Tate-Hochschild complex to string topology.

math.RT↗
arXiv2018

Gerstenhaber algebra and Deligne's conjecture on Tate-Hochschild cohomology

Using non-commutative differential forms, we construct a complex called singular Hochschild cochain complex for any associative algebra over a field. The cohomology of this complex is isomorphic to the Tate-Hochschild cohomology in the sense of Buchweitz. By a natural action of the cellular chain operad of the spineless cacti operad, introduced by R. Kaufmann, on the singular Hochschild cochain complex, we provide a proof of the Deligne's conjecture for this complex. More concretely, the complex is an algebra over the (dg) operad of chains of the little $2$-discs operad. By this action, we also obtain that the singular Hochschild cochain complex has a $B$-infinity algebra structure and its cohomology ring is a Gerstenhaber algebra. Inspired by the original definition of Tate cohomology for finite groups, we define a generalized Tate-Hochschild complex with the Hochschild chains in negative degrees and the Hochschild cochains in non-negative degrees. There is a natural embedding of this complex into the singular Hochschild cochain complex. In the case of a self-injective algebra, this embedding becomes a quasi-isomorphism. In particular, for a symmetric algebra, this allows us to show that the Tate-Hochschild cohomology ring, equipped with the Gerstenhaber algebra structure, is a Batalin-Vilkovisky algebra.

math.RT↗
arXiv2017

An explicit projective bimodule resolution of a Leavitt path algebra

We construct an explicit projective bimodule resolution for the Leavitt path algebra of a row-finite quiver. We prove that the Leavitt path algebra of a row-countable quiver has Hochschild cohomolgical dimension at most one, that is, it is quasi-free in the sense of Cuntz-Quillen. The construction of the resolution relies on an explicit derivation of the Leavitt path algebra.

math.RA↗
arXiv2015

Singular Hochschild Cohomology and Gerstenhaber Algebra Structure

In this paper, we define the singular Hochschild cohomology groups $HH_{sg}^i(A, A)$ of an associative $k$-algebra $A$ as morphisms from $A$ to $A[i]$ in the singular category $D_{sg}(A\otimes_k A^{op})$ for $i\in \mathbb{Z}$. We prove that $HH_{sg}^*(A, A)$ has a Gerstenhaber algebra structure and in the case of a symmetric algebra $A$, $HH_{sg}^*(A, A)$ is a Batalin-Vilkovisky (BV) algebra.

math.RT↗
arXiv2014

Singular Equivalence of Morita Type with Level

We generalize the notion of stable equivalence of Morita type and define what is called "singular equivalence of Morita type with level". Such an equivalence of induces an equivalence between singular categories. We will also prove that a derived equivalence of standard type induces a singular equivalence of Morita type with level.

math.RT↗