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Automating architectural floorplan design is vital for housing and interior design, offering a faster, cost-effective alternative to manual sketches by architects. However, existing methods, including rule-based and learning-based approaches, face challenges in design complexity and constrained generation with extensive post-processing, and tend to obvious geometric inconsistencies such as misalignment, overlap, and gaps. In this work, we propose a novel generative framework for vector floorplan design via structural graph generation, called GSDiff, focusing on wall junction generation and wall segment prediction to capture both geometric and semantic aspects of structural graphs. To improve the geometric rationality of generated structural graphs, we propose two innovative geometry enhancement methods. In wall junction generation, we propose a novel alignment loss function to improve geometric consistency. In wall segment prediction, we propose a random self-supervision method to enhance the model's perception of the overall geometric structure, thereby promoting the generation of reasonable geometric structures. Employing the diffusion model and the Transformer model, as well as the geometry enhancement strategies, our framework can generate wall junctions, wall segments and room polygons with structural and semantic information, resulting in structural graphs that accurately represent floorplans. Extensive experiments show that the proposed method surpasses existing techniques, enabling free generation and constrained generation, marking a shift towards structure generation in architectural design.

We propose a novel method to construct large-scale boundary-identical microstructure datasets with high attribute coverage for highly efficient multiscale design. Central to our technique is using a deep generative model to generate microstructures under the two conditions, including the specified boundary and homogenized elastic tensor. We achieve the desired dataset by alternately adding microstructures with diverse attributes constructed by the deep generative model into the dataset and retraining the deep generative model. Sixteen compatible metamaterial datasets with high attribute coverage are constructed by our method. We demonstrate the effectiveness and practicability of the obtained datasets over various multiscale design examples. Specifically, in the design of a mechanical cloak, we utilize macrostructures with $30 \times 30$ elements and microstructures filled with $256 \times 256$ elements. The entire reverse design process is completed within one minute, significantly enhancing the efficiency of the multiscale optimization.

Mechanical metamaterial is a synthetic material that can possess extraordinary physical characteristics, such as abnormal elasticity, stiffness, and stability, by carefully designing its internal structure. To make metamaterials contain delicate local structures with unique mechanical properties, it is a potential method to represent them through high-resolution voxels. However, it brings a substantial computational burden. To this end, this paper proposes a fast inverse design method, whose core is an advanced deep generative AI algorithm, to generate voxel-based mechanical metamaterials. Specifically, we use the self-conditioned diffusion model, capable of generating a microstructure with a resolution of $128^3$ to approach the specified homogenized tensor matrix in just 3 seconds. Accordingly, this rapid reverse design tool facilitates the exploration of extreme metamaterials, the sequence interpolation in metamaterials, and the generation of diverse microstructures for multi-scale design. This flexible and adaptive generative tool is of great value in structural engineering or other mechanical systems and can stimulate more subsequent research., Comment: 13 pages, 6 figures

Let $\mathcal H$ be a finite dimensional complex Hilbert space with dimension $n \ge 3$ and $\mathcal P(\mathcal H)$ the set of projections on $\mathcal H$. Let $\varphi: \mathcal P(\mathcal H) \to \mathcal P(\mathcal H)$ be a surjective map. We show that $\varphi$ shrinks the joint spectrum of any two projections if and only if it is joint spectrum preserving for any two projections and thus is induced by a ring automorphism on $\mathbb C$ in a particular way. In addition, for an arbitrary $k \ge 3$, $\varphi$ shrinks the joint spectrum of any $k$ projections if and only if it is induced by a unitary or an anti-unitary. Assume that $\phi$ is a surjective map on the Grassmann space of rank one projections. We show that $\phi$ is joint spectrum preserving for any $n$ rank one projections if and only if it can be extended to a surjective map on $\mathcal P(\mathcal{H})$ which is spectrum preserving for any two projections. Moreover, for any $k >n$, $\phi$ is joint spectrum shrinking for any $k$ rank one projections if and only if it is induced by a unitary or an anti-unitary., Comment: 14 pages

Based on the characterization of surjective $L^p$-isometries of unitary groups in finite factors, we describe all surjective $L^p$-isometries between Grassmann spaces of projections with the same trace value in semifinite factors., Comment: 14 pages

A unital ring is called clean (resp. strongly clean) if every element can be written as the sum of an invertible element and an idempotent (resp. an invertible element and an idempotent that commutes). T.Y. Lam proposed a question: which von Neumann algebras are clean as rings? In this paper, we characterize strongly clean von Neumann algebras and prove that all finite von Neumann algebras and all separable infinite factors are clean., Comment: 19 pages(A citation mistake is corrected)

Sparse representation classification achieves good results by addressing recognition problem with sufficient training samples per subject. However, SRC performs not very well for small sample data. In this paper, an inverse-projection group sparse representation model is presented for breast tumor classification, which is based on constructing low-rank variation dictionary. The proposed low-rank variation dictionary tackles tumor recognition problem from the viewpoint of detecting and using variations in gene expression profiles of normal and patients, rather than directly using these samples. The inverse projection group sparsity representation model is constructed based on taking full using of exist samples and group effect of microarray gene data. Extensive experiments on public breast tumor microarray gene expression datasets demonstrate the proposed technique is competitive with state-of-the-art methods. The results of Breast-1, Breast-2 and Breast-3 databases are 80.81%, 89.10% and 100% respectively, which are better than the latest literature., Comment: 31 pages, 14 figures, 12 tables. arXiv admin note: text overlap with arXiv:1803.03562

Microarray gene expression data-based tumor classification is an active and challenging issue. In this paper, an integrated tumor classification framework is presented, which aims to exploit information in existing available samples, and focuses on the small sample problem and unbalanced classification problem. Firstly, an inverse space sparse representation based classification (ISSRC) model is proposed by considering the characteristics of gene-based tumor data, such as sparsity and a small number of training samples. A decision information factors (DIF)-based gene selection method is constructed to enhance the representation ability of the ISSRC. It is worth noting that the DIF is established from reducing clinical misdiagnosis rate and dimension of small sample data. For further improving the representation ability and classification stability of the ISSRC, feature learning is conducted on the selected gene subset. The feature learning method is constructed by complementing the advantages of non-negative matrix factorization (NMF) and deep learning. Without confusion, the ISSRC combined with gene selection and feature learning is called the integrated ISSRC, whose stability, optimization and the corresponding convergence are analyzed. Extensive experiments on six public microarray gene expression datasets show the integrated ISSRC-based tumor classification framework is superior to classical and state-of-the-art methods. There are significant improvements in classification accuracy, specificity and sensitivity, whether there is a tumor in the early diagnosis, what kind of tumor, or whether metastasis occurs after tumor surgery., Comment: 40 pages, 18 figures, 13 tables

The Wigner's theorem, which is one of the cornerstones of the mathematical formulation of quantum mechanics, asserts that every symmetry of quantum system is unitary or anti-unitary. This classical result was first given by Wigner in 1931. Thereafter it has been proved and generalized in various ways by many authors. Recently, G. P. Geh\'{e}r extended Wigner's and Moln\'{a}r's theorems and characterized the transformations on the Grassmann space of all rank-$n$ projections which preserve the transition probability. The aim of this paper is to provide a new approach to describe the general form of the transition probability preserving (not necessarily bijective) maps between Grassmann spaces. As a byproduct, we are able to generalize the results of Moln\'{a}r and G. P. Geh\'{e}r.