Boundary maps and maximal representations on infinite-dimensional Hermitian symmetric spaces

We define a Toledo number for actions of surface groups and complex hyperbolic lattices on infinite-dimensional Hermitian symmetric spaces, which allows us to define maximal representations. When the target is not of tube type, we show that there cannot be Zariski-dense maximal representations, and whenever the existence of a boundary map can be guaranteed, the representation preserves a finite-dimensional totally geodesic subspace on which the action is maximal. In the opposite direction, we construct examples of geometrically dense maximal representation in the infinite-dimensional Hermitian symmetric space of tube type and finite rank. Our approach is based on the study of boundary maps, which we are able to construct in low ranks or under some suitable Zariski density assumption, circumventing the lack of local compactness in the infinite-dimensional setting.

Analytical modeling of tool failure boundary map in milling titanium alloy

As a typical aerospace difficult-to-machine material, tool failure in milling titanium alloy Ti6Al4V will reduce the stability of the milling process and affect the surface quality of the workpiece. Aiming at the fact that cemented carbide tools are prone to wear failure and breakage failure in milling titanium alloy, a safe tool failure boundary map is provided to ensure that the tools will not occur failure with the cutting parameters selected in the safe area during the prediction time. Based on the processing characteristics of Ti6Al4V, the failure boundary map mainly considers three forms of tool failure: flank wear, rake wear, and cutting edge breakage. By revealing the three failure mechanisms, the failure analytical model is established and the failure boundary map is obtained. Compared with the experimental results, it has good consistency, and the research results can provide a reference for the field of titanium alloy cutting process.

BGM-Net: Boundary-Guided Multiscale Network for Breast Lesion Segmentation in Ultrasound

Automatic and accurate segmentation of breast lesion regions from ultrasonography is an essential step for ultrasound-guided diagnosis and treatment. However, developing a desirable segmentation method is very difficult due to strong imaging artifacts e.g., speckle noise, low contrast and intensity inhomogeneity, in breast ultrasound images. To solve this problem, this paper proposes a novel boundary-guided multiscale network (BGM-Net) to boost the performance of breast lesion segmentation from ultrasound images based on the feature pyramid network (FPN). First, we develop a boundary-guided feature enhancement (BGFE) module to enhance the feature map for each FPN layer by learning a boundary map of breast lesion regions. The BGFE module improves the boundary detection capability of the FPN framework so that weak boundaries in ambiguous regions can be correctly identified. Second, we design a multiscale scheme to leverage the information from different image scales in order to tackle ultrasound artifacts. Specifically, we downsample each testing image into a coarse counterpart, and both the testing image and its coarse counterpart are input into BGM-Net to predict a fine and a coarse segmentation maps, respectively. The segmentation result is then produced by fusing the fine and the coarse segmentation maps so that breast lesion regions are accurately segmented from ultrasound images and false detections are effectively removed attributing to boundary feature enhancement and multiscale image information. We validate the performance of the proposed approach on two challenging breast ultrasound datasets, and experimental results demonstrate that our approach outperforms state-of-the-art methods.

Superrigidity of maximal measurable cocycles of complex hyperbolic lattices

Let $$\Gamma $$ Γ be a torsion-free lattice of $$PU (p,1)$$ P U ( p , 1 ) with $$p \ge 2$$ p ≥ 2 and let $$(X,\mu _X)$$ ( X , μ X ) be an ergodic standard Borel probability $$\Gamma $$ Γ -space. We prove that any maximal Zariski dense measurable cocycle $$\sigma : \Gamma \times X \longrightarrow SU (m,n)$$ σ : Γ × X ⟶ S U ( m , n ) is cohomologous to a cocycle associated to a representation of $$PU (p,1)$$ P U ( p , 1 ) into $$SU (m,n)$$ S U ( m , n ) , with $$1 \le m \le n$$ 1 ≤ m ≤ n . The proof follows the line of Zimmer’ Superrigidity Theorem and requires the existence of a boundary map, that we prove in a much more general setting. As a consequence of our result, there cannot exist maximal measurable cocycles with the above properties when $$1< m < n$$ 1 < m < n .

Flexibility of measure-theoretic entropy of boundary maps associated to Fuchsian groups

Given a closed, orientable, compact surface S of constant negative curvature and genus $g \geq 2$ , we study the measure-theoretic entropy of the Bowen–Series boundary map with respect to its smooth invariant measure. We obtain an explicit formula for the entropy that only depends on the perimeter of the $(8g-4)$ -sided fundamental polygon of the surface S and its genus. Using this, we analyze how the entropy changes in the Teichmüller space of S and prove the following flexibility result: the measure-theoretic entropy takes all values between 0 and a maximum that is achieved on the surface that admits a regular $(8g-4)$ -sided fundamental polygon. We also compare the measure-theoretic entropy to the topological entropy of these maps and show that the smooth invariant measure is not a measure of maximal entropy.

Overconvergent modular forms are highest-weight vectors in the Hodge-Tate weight zero part of completed cohomology

We construct a $(\mathfrak {gl}_2, B(\mathbb {Q}_p))$ and Hecke-equivariant cup product pairing between overconvergent modular forms and the local cohomology at $0$ of a sheaf on $\mathbb {P}^1$ , landing in the compactly supported completed $\mathbb {C}_p$ -cohomology of the modular curve. The local cohomology group is a highest-weight Verma module, and the cup product is non-trivial on a highest-weight vector for any overconvergent modular form of infinitesimal weight not equal to $1$ . For classical weight $k\geq 2$ , the Verma has an algebraic quotient $H^1(\mathbb {P}^1, \mathcal {O}(-k))$ , and on classical forms, the pairing factors through this quotient, giving a geometric description of ‘half’ of the locally algebraic vectors in completed cohomology; the other half is described by a pairing with the roles of $H^1$ and $H^0$ reversed between the modular curve and $\mathbb {P}^1$ . Under minor assumptions, we deduce a conjecture of Gouvea on the Hodge-Tate-Sen weights of Galois representations attached to overconvergent modular forms. Our main results are essentially a strict subset of those obtained independently by Lue Pan, but the perspective here is different, and the proofs are short and use simple tools: a Mayer-Vietoris cover, a cup product, and a boundary map in group cohomology.

Pelatihan Pendampingan Pembuatan Peta Tematik Menggunakan GPS dan Aplikasi CAD

One of the problems in the development of human resource professionalism so far is related to land issues (boundaries). The purpose of this activity is to provide training on the use of GPS and processing data using Autodesk map software. The training was attended by the village and sub-district staff of Samarinda City, as many as 50 people. The method used is 1) giving theoretical and practical knowledge about data collection and processing 2) assignment of making land boundaries per area, 3) presentation of results in the field and 4) output (boundary map). The results of this activity were: First, all participants of this training activity were very enthusiastic, enthusiastic, and had high motivation to take part in the implementation of the second boundary land making training activity, the participants who succeeded in making a land boundary map of 25 people or 50% of 50 participants. Third, the quality of the maps is quite good. They found differences in boundary map shape between version Samarinda City Bappeda and field data collection. Based on field data from the Bugis Kelurahan staff, the shape of the land boundary map in the Bugis sub-district in Samarinda almost resembles the land boundary map of Samarinda City of 2.5 ha, because Samarinda City Bappeda does not include the Segiri GOR Stadium as an area of Bugis sub-district that should be included as a map of the border. The map results from Samarinda Kota sub-district staff were recombined by sub-district and sub-district lurah to be determined as a map of sub-district Kota Samarinda and displayed in the A0 printout in each of the kelurahan offices located in sub-district Samarinda Kota.

Algebraic hull of maximal measurable cocycles of surface groups into Hermitian Lie groups

Following the work of Burger, Iozzi and Wienhard for representations, in this paper we introduce the notion of maximal measurable cocycles of a surface group. More precisely, let $$\mathbf {G}$$ G be a semisimple algebraic $${\mathbb {R}}$$ R -group such that $$G=\mathbf {G}({\mathbb {R}})^{\circ }$$ G = G ( R ) ∘ is of Hermitian type. If $$\Gamma \le L$$ Γ ≤ L is a torsion-free lattice of a finite connected covering of $$\mathrm{PU}(1,1)$$ PU ( 1 , 1 ) , given a standard Borel probability $$\Gamma $$ Γ -space $$(\Omega ,\mu _\Omega )$$ ( Ω , μ Ω ) , we introduce the notion of Toledo invariant for a measurable cocycle $$\sigma :\Gamma \times \Omega \rightarrow G$$ σ : Γ × Ω → G . The Toledo invariant remains unchanged along G-cohomology classes and its absolute value is bounded by the rank of G. This allows to define maximal measurable cocycles. We show that the algebraic hull $$\mathbf {H}$$ H of a maximal cocycle $$\sigma $$ σ is reductive and the centralizer of $$H=\mathbf {H}({\mathbb {R}})^{\circ }$$ H = H ( R ) ∘ is compact. If additionally $$\sigma $$ σ admits a boundary map, then H is of tube type and $$\sigma $$ σ is cohomologous to a cocycle stabilizing a unique maximal tube type subdomain. This result is analogous to the one obtained for representations. In the particular case $$G=\mathrm{PU}(n,1)$$ G = PU ( n , 1 ) maximality is sufficient to prove that $$\sigma $$ σ is cohomologous to a cocycle preserving a complex geodesic. We conclude with some remarks about boundary maps of maximal Zariski dense cocycles.

Potential Use of UAV-Based Mapping System to Accelerate the Production of Parcel Boundary Map in Indonesia

Recently, the Ministry of Agrarian Affairs announces a decree to accelerate a completion of the registration of land parcels of cadaster maps up to a scale of 1:1000. It is known that the use of Unmanned Aerial Vehicle based mapping system (UAV-Map) can produce orthophoto image with spatial resolution less than 10 cm, but it is not yet known whether UAV-Map implementation is able to identify boundary of land parcel in any condition. Therefore, this paper would analyze the planimetric accuracy that is conformed to the regulation of State Minister of Agrarian Affairs/Head of National Land Agency No.3, 1997 both for urban and rural areas. Two testing areas are established which located on urban area and rural area respectively. Flight missions are conducted using a fixed-wing aircraft equipped with a consumer grade camera and a navigational grade GPS-INS system. Orthophoto maps are produced by using Agisoft Photoscan software. Digitizing of parcel boundaries are followed both on an existing map and on the orthophoto maps. Deviations in areas are expressed in terms of the RMSE figures. Planimetric accuracies as indicated by the RMSE value are of 0,044 m for urban areas and 0,122 m for rural area. It is showed that all discrepancies of the parcels area are still below the recommended threshold values of the regulation. It is can be concluded that the orthophoto maps obtained by using a low cost UAV-Map system can be used to identify land parcels boundaries and to determine the parcel area.