Bijective Map Research Articles

Jordan isomorphisms of nest algebras

Let N and M be two nontrivial nests on a real or complex Hilbert space H. Let alg N and alg M be two nest algebras. Suppose that ϕ is a linear bijective mapping from alg N onto alg M such that ϕ ( AB + BA ) = ϕ ( A ) ϕ ( B ) + ϕ ( B ) ϕ ( A ) for any A , B ∈ alg N with AB = G , where G is an arbitrary but fixed operator in alg N and ϕ ( I ) = I . Then ϕ is either an isomorphism or an anti-isomorphism.

Nonlinear maps preserving the mixed Jordan type products on *-algebras

Suppose R and C represent two unital *-algebras, with R containing a nontrivial projection. In the present article, we demonstrate, under certain restrictions that if a bijective map Δ : R → C satisfies Δ ( K ° N • W ) = Δ ( K ) ° Δ ( N ) • Δ ( W ) for all K , N , W ∈ R , then Δ is a *-preserving ring isomorphism. Also, this result is investigated in factor von Neumann algebras.

A Complete Characterization of Magic Constants Arising from Distance Magic Graphs

A positive integer \(k\) is called a magic constant if there is a graph \(G\) along with a bijective function \(f\) from \(V(G)\) to the first \(|V(G)|\) natural numbers such that the weight of the vertex \(w(v) = \sum_{uv \in E} f(u) = k\) for all \(v \in V\). It is known that all odd positive integers greater than or equal to \(3\) and the integer powers of \(2\), \(2^{t}\), \(t \geq 6\), are magic constants. In this paper, we characterize all positive integers that are magic constants and generate all distance magic graphs, up to isomorphism, of order up to \(10\).

Bi-Skew Lie (Jordan) Product on Factor Von Neumann Algebras

In this manuscript, we characterize the bijective mapping between two von Neumann algebras and with dim ⩾ 1. The mapping Λ is to satisfy the following condition on the mixed bi-skew Lie (Jordan) product , for all , where the bi-skew Lie (Jordan) product is defined as , and . Specifically, we elaborate on the properties of the mapping Λ, establishing that it is additive, a∗-somorphism, and linear.

Linear maps preserving the inclusion of fixed subsets into the local spectrum at some fixed vector

For a natural number n≥2, denote by Mn the space of all n×n matrices over the complex field C. Let x0∈Cn be a fixed nonzero vector, and fix also two nonempty subsets K1,K2⊆C, each having at most n distinct elements. Under the assumption that |K1|≤|K2|, we characterize linear bijective maps φ on Mn having the property that, for each matrix T, we have that K2 is a subset of the local spectrum of φ(T) at x0 whenever K1 is a subset of the local spectrum of T at x0. As a corollary, we also characterize linear maps φ on Mn having the property that, for each matrix T, we have that K1 is a subset of the local spectrum of T at x0 if and only if K2 is a subset of the local spectrum of φ(T) at x0, without the bijectivity assumption on the map φ and with no assumption made regarding the number of elements of K1 and K2.

Reordered Computable Numbers

A real number is called left-computable if there exists a computable increasing sequence of rational numbers converging to it. In this article we are investigating a proper subset of the left-computable numbers. We say that a real number x is reordered computable if there exist a computable function f:N→N with ∑k=0∞2-f(k)=x and a bijective function σ:N→N such that the rearranged series ∑k=0∞2-f(σ(k)) converges computably. In this article we will give some examples and counterexamples for reordered computable numbers and we will show that these numbers are closed under addition, multiplication and the Solovay reduction. Finally, we will also present a density theorem for reordered computable numbers.

Uni-to-Multi Modal Knowledge Distillation for Bidirectional LiDAR-Camera Semantic Segmentation.

Combining LiDAR points and images for robust semantic segmentation has shown great potential. However, the heterogeneity between the two modalities (e.g. the density, the field of view) poses challenges in establishing a bijective mapping between each point and pixel. This modality alignment problem introduces new challenges in network design and data processing for cross-modal methods. Specifically, 1) points that are projected outside the image planes; 2) the complexity of maintaining geometric consistency limits the deployment of many data augmentation techniques. To address these challenges, we propose a cross-modal knowledge imputation and transition approach. First, we introduce a bidirectional feature fusion strategy that imputes missing image features and performs cross-modal fusion simultaneously. This allows us to generate reliable predictions even when images are missing. Second, we propose a Uni-to-Multi modal Knowledge Distillation (U2MKD) framework, leveraging the transfer of informative features from a single-modality teacher to a cross-modality student. This overcomes the issues of augmentation misalignment and enables us to train the student effectively. Extensive experiments on the nuScenes, Waymo, and SemanticKITTI datasets demonstrate the effectiveness of our approach. Notably, our method achieves an 8.3 mIoU gain over the LiDAR-only baseline on the nuScenes validation set and achieves state-of-the-art performance on the three datasets.

A result on nonlinear maps preserving mixed Jordan triple η-*-product on factor von Neumann algebras

In this paper, it is shown that every nonlinear bijective map ψ preserving mixed Jordan triple η-products with η ≠ − 1 from a factor von Neumann algebra M with dim M > 1 into another factor von Neumann algebra N with dim N > 1 is additive. Also, for α ∈ { − 1 , 1 } , we establish that α ψ is a linear *-isomorphism, and is a conjugate linear *-isomorphism only when η ∈ R .

<i>D</i>-Continuity

We call a map f : X → Y D-continuous if its restriction to any set of points that do not possess compact neighborhoods is continuous. We investigate this weaker version of continuity and provide examples to compare D-continuity with other types of continuity. Let f : X → Y be a D-continuous bijective map such that f(A) is locally finite, where A is the set of all points that do not possess compact neighborhoods in X. Then, we show that f|A is a homeomorphism. We also show that if X is a countably generated topological space, then any D-continuous f : X → Y is continuous. We discuss C-normality, illustrating the relationship between this property and D-continuity. Finally, we investigate the space of all real-valued D-continuous maps of an arbitrary topological space X and obtain some results.

A Progressive Embedding Approach to Bijective Tetrahedral Maps driven by Cluster Mesh Topology

We present a novel algorithm to map ball-topology tetrahedral meshes onto star-shaped domains with guarantees regarding bijectivity. Our algorithm is based on the recently introduced idea of Shrink-and-Expand, where images of interior vertices are initially clustered at one point (Shrink-), before being sequentially moved to non-degenerate positions yielding a bijective map (-and-Expand). In this context, we introduce the concept of the cluster mesh , i.e. the unexpanded interior mesh consisting of geometrically degenerate simplices. Using local, per-vertex connectivity information solely from the cluster mesh, we show that a viable expansion sequence guaranteed to produce a bijective map can always be found as long as the mesh is shellable. In addition to robustness guarantees for this ubiquitous class of inputs, other practically relevant benefits include improved parsimony and reduced algorithmic complexity. While inheriting some of the worst-case high run time requirements of the state of the art, significant acceleration for the average case is experimentally demonstrated.

Bijective Volumetric Mapping via Star Decomposition

A method for the construction of bijective volumetric maps between 3D shapes is presented. Arbitrary shapes of ball-topology are supported, overcoming restrictions of previous methods to convex or star-shaped targets. In essence, the mapping problem is decomposed into a set of simpler mapping problems, each of which can be solved with previous methods for discrete star-shaped mapping problems. Addressing the key challenges in this endeavor, algorithms are described to reliably construct structurally compatible partitions of two shapes with constraints regarding star-shapedness and to compute a parsimonious common refinement of two triangulations.

Hom-actions and class equation for Hom-groups

The notion of Hom-groups is defined as a generalization of a non-associative group. They can be obtained by twisting the associative operation with a compatible bijection mapping. In this article, we provide some constructions by twisting and also discuss properties related to Hom-groups. We introduce different notions of actions concerning Hom-groups. We then present a theorem for a class equation, which is proven. Following that, we illustrate some applications for p-Hom groups.

Robust Shared Control With Stable Contact Servoing for Enhanced Object Transportation by Telerobotic Bimanual Mobile Manipulators

ABSTRACTBimanual mobile manipulation significantly enhances the capabilities of field robots that interact with diverse dynamic environments and expands the range of possible manipulation actions. This paper introduces a robust shared control architecture for remote manipulation of the Collaborative Dual‐arm Robot Manipulator (CURI), aimed at stable object transportation. Given that stable contact is pivotal for successful object transportation, our focus is on stabilizing the contact between the robot end‐effectors and the object using a contact servoing strategy. To maintain stable contact, it is essential that the contact wrenches consistently stay within the predefined friction cones throughout the task. We propose a novel shared control strategy to stabilize the contact using the contact parameterization model. This model formulates the contact stability manifold using a set of unconstrained reparameterized variables through a bijective mapping function, ensuring that the controller derived from these unconstrained reparameterized variables consistently maintains contact stability. The efficacy of the proposed approach is convincingly demonstrated through a series of experiments involving representative bimanual transportation tasks within a logistics context. Both theoretical analysis and experimental validations confirm that our proposed control strategy not only ensures the stable transportation of objects by CURI but also substantially enhances the overall robustness of the shared control system.

C2-MILDLY NORMAL TOPOLOGICAL SPACES

This work studies a new topological property called C2-mild normality. A space X is called a C2-mildly normal space if there exist a Hausdorff mildly normal space Y and a bijective function f : X → Y such that the restriction function f|A : A → f(A) is a homeomorphism for each compact subspace A ⊆ X. We investigate this property and present some examples to illustrate the relationships between C2-mild normality and other kinds of topological properties.

Arithmetic Sequential Graceful Labeling For Arrow Graphs

Let G be a simple, finite, connected, undirected, non-trivial graph with 𝑝 vertices and 𝑞 edges. 𝑉(𝐺) be the vertex set and 𝐸(𝐺) be the edge set of 𝐺. Let 𝑓: 𝑉(𝐺) → {𝑎, 𝑎 + 𝑑, 𝑎 + 2𝑑, 𝑎 + 3𝑑, … , 𝑎 + 2𝑞𝑑} where a ≥ 0 and 𝑑 ≥ 1 is an injective function. If for each edge 𝑢𝑣 ∈ 𝐸(𝐺) , 𝑓 ∗ : 𝐸(𝐺) → {𝑑, 2𝑑, 3𝑑, 4𝑑, … , 𝑞𝑑} defined by 𝑓 ∗ (𝑢𝑣) = |𝑓(𝑢)− 𝑓(𝑣)| is a bijective function then the function 𝑓 is called arithmetic sequential graceful labeling. The graph with arithmetic sequential graceful labeling is called arithmetic sequential graceful graph. In this paper, we prove that one side arrow graphs 𝐴𝑅𝜂 2 , 𝐴𝑅𝜂 3 , 𝐴𝑅𝜂 5 and double-sided arrow graphs 𝐷(𝐴𝑅𝜂 2 ),𝐷(𝐴𝑅𝜂 3 ) are arithmetic sequential graceful graph

Results on super edge magic deficiency of somewell-known classes of finite graphs

A graph Ω(Λ, Γ) is considered super edge magic if there exists a bijective function ϕ:Λ(Ω)∪Γ(Ω)→ {1, 2, 3, …,|Λ(Ω)|+|Γ(Ω)|} such that ϕ(τ1)+(τ1τ2)+ ϕ(τ2) is a constant for every edge τ1τ2 ∈ Γ(Ω), and ϕ(Λ(Ω))= {1, 2, 3, …,|Λ(Ω)|}. Furthermore, the super edge magic deficiency of a graph Ω, denoted as μs (Ω), is either the minimum non-negative integer η such that Ω∪ηK1 is a super edge magic graph or +∞ if such an integer η does not exist. In this paper, we investigate the super edge magic deficiency of certain families of graphs.

Novel non‐involutive solutions of the Yang–Baxter equation from (skew) braces

AbstractWe produce novel non‐involutive solutions of the Yang–Baxter equation coming from (skew) braces. These solutions are generalisations of the known ones coming from braces and skew braces, and surprisingly in the case of braces, they are not necessarily involutive. In the case of two‐sided (skew) braces, one can assign such solutions to every element of the set. Novel bijective maps associated to the inverse solutions are also introduced. Moreover, we show that the recently derived Drinfeld twists of the involutive case are still admissible in the non‐involutive frame, and we identify the twisted ‐matrices and twisted coproducts. We observe, as in the involutive case, that the underlying quantum algebra is not a quasi‐triangular bialgebra, as one would expect, but a quasi‐triangular quasi‐bialgebra. The same applies to the quantum algebra of the twisted ‐matrices, contrary to the involutive case.

Closing duality gaps of SDPs completely through perturbation when singularity degree is one

Let ( P , D ) be a primal-dual pair of SDPs with a nonzero finite duality gap. Under such circumstances, P and D are weakly feasible and if we perturb the problem data to recover strong feasibility, the (common) optimal value function v as a function of the perturbation is not well-defined at zero (i.e. at the unperturbed data). Nevertheless, under the assumption that both P and D have singularity degree one, we show that a limiting version v a of v is a well-defined monotone decreasing continuous bijective function with domain [ 0 , π / 2 ] and image [ v ( P ) , v ( D ) ] , where v ( P ) and v ( D ) are the optimal values of P and D , respectively. The domain [ 0 , π / 2 ] corresponds to directions of perturbation defined in a certain manner. Thus, v a ‘completely fills’ the nonzero duality gap under a mild regularity condition. Our result is tight in that there exists an instance with singularity degree two for which v a is not continuous.

On Computing the k -Shortcut Fréchet Distance

The Fréchet distance is a popular measure of dissimilarity for polygonal curves. It is defined as a min–max formulation that considers all orientation-preserving bijective mappings between the two curves. Because of its susceptibility to noise, Driemel and Har-Peled introduced the shortcut Fréchet distance in 2012, where one is allowed to take shortcuts along one of the curves, similar to the edit distance for sequences. We analyse the parameterised version of this problem, where the number of shortcuts is bounded by a parameter \(k\) . The corresponding decision problem can be stated as follows: Given two polygonal curves \(T\) and \(B\) of at most \(n\) vertices, a parameter \(k\) and a distance threshold \(\delta\) , is it possible to introduce \(k\) shortcuts along \(B\) such that the Fréchet distance of the resulting curve and the curve \(T\) is at most \(\delta\) ? We study this problem for polygonal curves in the plane. We provide a complexity analysis for this problem with the following results: (1) there exists a decision algorithm with running time in \(\mathcal{O}(kn^{2k+2}\log n)\) ; (2) assuming the exponential-time hypothesis (ETH), there exists no algorithm with running time bounded by \(n^{o(k)}\) . In contrast, we also show that efficient approximate decider algorithms are possible, even when \(k\) is large. We present a \((3+\varepsilon)\) -approximate decider algorithm with running time in \(\mathcal{O}(kn^{2}\log^{2}n)\) for fixed \(\varepsilon\) . In addition, we can show that, if \(k\) is a constant and the two curves are \(c\) -packed for some constant \(c\) , then the approximate decider algorithm runs in near-linear time.

Hierarchical disentangled representation for image denoising and beyond

Image denoising is a typical ill-posed problem due to complex degradation. Leading methods based on normalizing flows have tried to solve this problem with an invertible transformation instead of a deterministic mapping. However, it is difficult to construct feasible bijective mapping to remove spatial-variant noise while recovering fine texture and structure details due to latent ambiguity in inverse problems. Inspired by a common observation that noise tends to appear in the high-frequency part of the image, we propose a fully invertible denoising method that injects the idea of disentangled learning into a general invertible architecture to split noise from the high-frequency part. More specifically, we decompose the noisy image into clean low-frequency and hybrid high-frequency parts with an invertible transformation and then disentangle case-specific noise and high-frequency components in the latent space. In this way, denoising is made tractable by inversely merging noiseless low and high-frequency parts. Furthermore, we construct a flexible hierarchical disentangling framework, which aims to decompose most of the low-frequency image information while disentangling noise from the high-frequency part in a coarse-to-fine manner. Extensive experiments on real image denoising, JPEG compressed artifact removal, and medical low-dose CT image restoration have demonstrated that the proposed method achieves competitive performance on both quantitative metrics and visual quality, with significantly less computational cost.