It is well-known that deleting or contracting any element of a connected matroid always yields at least one connected minor. However, for a connected polymatroid, only two such elements can be guaranteed, proved by Hall in 2013. This note investigates the connectivity properties of slice-projections of connected polymatroids, which includes deletion and contraction. We establish that for any element of a connected polymatroid, at least one of its two consecutive slice-projections is connected. We also obtain that the $j$-th slice-projection from the top of a polymatroid and $j$-th slice-projection from the bottom of its dual have the same connectedness. These results both extend existing connectedness theorems for graphs and matroids.
We construct a class of Riemannian metrics in closed surfaces of genus greater than one, having Anosov geodesic flows, and some regions of positive curvature, such that for each such surface, there exists a smooth curve of conformal deformations that preserves the Anosov property and connects the surface with a Riemannian metric of negative curvature. The conformal deformation does not arise from geometric flows like the Ricci flow, since it is known that such flows might generate conjugate points in the presence of points of positive curvature in the surface.
You have a satellite spacecraft or asteroid that moves under the gravitational influence of a massive central body and follows a Keplerian orbit around it ellipse parabola or hyperbola Given measurements of two positions in its orbit what is the family of possible orbital paths that connects them I use the conic section orbits semilatus rectum directly related to orbital angular momentum to parameterise these orbits The solutions have applications to orbit determination ballistic missiles interplanetary interception and targeted reentry I also show how they can be applied to solve the Lambert problem of finding the unique transfer orbit that connects two points in a specified time interval These results are accessible to advanced undergraduate students in physics or aerospace engineering. Supplementary materials are provided online
This work establishes a Space-Time Connectivity Theorem for normal currents. In analogy to classical results by Federer and Fleming as well as a recent theorem for integral currents by the second author, this result allows one to witness the weak* convergence of a uniformly bounded sequence of boundaryless normal currents with a space-time normal current that connects the elements of the sequence with their limit. The space-time setting is distinguished from the classical case in that this connecting current has a time coordinate and thus constitutes a progressive-in-time way to deform an element of the sequence to the limit.
This paper considers a class of delay differential equations with unimodal feedback and describes the structure of certain unstable sets of stationary points and periodic orbits. These unstable sets consist of heteroclinic connections from stationary points and periodic orbits to stable stationary points, stable periodic orbits and some more complicated compact invariant sets. A prototype example is the Mackey--Glass type equation $y'(t)=-ay(t)+b \frac{y^2(t-1)}{1+y^n(t-1)}$ having three stationary solutions $0$, $ξ_{1,n}$ and $ξ_{2,n}$ with $0<ξ_{1,n}<ξ_{2,n}$, provided $b>a>0$, and $n$ is large. The 1-dimensional leading unstable set $W^u(\hatξ_{1,n})$ of the stationary point $\hatξ_{1,n}$ is decomposed into three disjoint orbits, $W^u(\hatξ_{1,n})=W^{u,-}(\hatξ_{1,n})\cup \{\hatξ_{1,n}\} \cup W^{u,+}(\hatξ_{1,n})$.. Here $\hatξ_{1,n}$ is a constant function in the phase space with value $ξ_{1,n}$. $W^{u,-}(\hatξ_{1,n})$ is a connecting orbit from $\hatξ_{1,n}$ to $\hat{0}$. There exists a threshold value $b^*=b^*(a)>a$ such that, in case $b\in (a,b^*)$, $W^{u,+}(\hatξ_{1,n})$ connects $\hatξ_{1,n}$ to $\hat{0}$; and in case $b>b^*$, $W^{u,+}(\hatξ_{1,n})$ conne
The rapid advancement of communication technologies has established cellular networks as the backbone for diverse applications, each with distinct quality of service requirements. Meeting these varying demands within a unified infrastructure presents a critical challenge that can be addressed through advanced techniques such as multi-connectivity. Multiconnectivity enables User equipments to connect to multiple BSs simultaneously, facilitating QoS differentiation and provisioning. This paper proposes a QoS-aware multi-connectivity framework leveraging machine learning to enhance network performance. The approach employs deep neural networks to estimate the achievable QoS metrics of BSs, including data rate, reliability, and latency. These predictions inform the selection of serving clusters and data rate allocation, ensuring that the User Equipment connects to the optimal BSs to meet its QoS needs. Performance evaluations demonstrate that the proposed algorithm significantly enhances Quality of Service (QoS) for applications where traditional and state-of-the-art methods are inadequate. Specifically, the algorithm achieves a QoS success rate of 98%. Furthermore, it improves spectru
Let $G = (V(G), E(G))$ be a simple connected graph and $Ω$ a subset of $ V(G)$ with $|Ω|\geq2$. An $Ω$-path in $G$ is a path that connects all vertices of $Ω$. Two $Ω$-paths $P_i$ and $P_j$ are said to be internally disjoint if $V(P_i)\cap V(P_j)=Ω$ and $E(P_i)\cap E(P_j)=\emptyset$. Denote $π_G(Ω)$ by the maximum number of internally disjoint $Ω$-paths in $G$. For an integer $k\geq2$, the $k$-path-connectivity $π_k(G)$ of $G$ is defined as $\min\{π_G(Ω)\midΩ\subseteq V(G)$ and $|Ω|=k\}$. Let $CW_n$ denote the Cayley graph generated by the $n$-vertex wheel graph. In this paper, we investigate the $3$-path-connectivity of $CW_n$ and prove that $π_3(CW_n)=\lfloor\frac{6n-9}4\rfloor$ for all $n\geq4$.
In this paper, we explore a taxonomy of connectivity for space-like structures. It is inspired by isolating posets of connected pieces of a space and examining its embedding in the ambient space. The taxonomy includes in its scope all standard notions of connectivity in point-set and point-free contexts, such as connectivity in graphs and hypergraphs (as well as k-connectivity in graphs), connectivity and path-connectivity in topology, and connectivity of elements in a frame.
We use a double-duality argument to give a new proof of Dieudonné's theorem on spaces of singular matrices. The argument connects the situation to the structure of spaces of operators with rank at most $1$, and works best over algebraically closed fields.
Residual connections are central to modern deep learning architectures, enabling the training of very deep networks by mitigating gradient vanishing. Hyper-Connections recently generalized residual connections by introducing multiple connection strengths at different depths, thereby addressing the seesaw effect between gradient vanishing and representation collapse. However, Hyper-Connections increase memory access costs by expanding the width of hidden states. In this paper, we propose Frac-Connections, a novel approach that divides hidden states into multiple parts rather than expanding their width. Frac-Connections retain partial benefits of Hyper-Connections while reducing memory consumption. To validate their effectiveness, we conduct large-scale experiments on language tasks, with the largest being a 7B MoE model trained on up to 3T tokens, demonstrating that Frac-Connections significantly outperform residual connections.
Non-sensory thalamic nuclei interact with the cortex through thalamocortical and cortico-basal ganglia-thalamocortical loops. Reciprocal connections between the mediodorsal thalamus (MD) and the prefrontal cortex are particularly important in cognition, while the reciprocal connections of the ventromedial (VM), ventral anterior (VA), and ventrolateral (VL) thalamus with the prefrontal and motor cortex are necessary for sensorimotor information processing. However, limited and often oversimplified understanding of the connectivity of the MD, VA, and VL nuclei in primates have hampered development of accurate models that explain their contribution to cognitive and sensorimotor functions. The current prevalent view suggests that the MD connects with the prefrontal cortex, while the VA and VL primarily connect with the premotor and motor cortices. However, past studies have also reported diverse connections that enable these nuclei to integrate information across a multitude of brain systems. In this review, we provide a comprehensive overview of the anatomical connectivity of the primate MD, VA, and VL with the cortex. By synthesizing recent findings, we aim to offer a valuable resour
One of the most intriguing findings in the structure of neural network landscape is the phenomenon of mode connectivity: For two typical global minima, there exists a path connecting them without barrier. This concept of mode connectivity has played a crucial role in understanding important phenomena in deep learning. In this paper, we conduct a fine-grained analysis of this connectivity phenomenon. First, we demonstrate that in the overparameterized case, the connecting path can be as simple as a two-piece linear path, and the path length can be nearly equal to the Euclidean distance. This finding suggests that the landscape should be nearly convex in a certain sense. Second, we uncover a surprising star-shaped connectivity: For a finite number of typical minima, there exists a center on minima manifold that connects all of them simultaneously via linear paths. These results are provably valid for linear networks and two-layer ReLU networks under a teacher-student setup, and are empirically supported by models trained on MNIST and CIFAR-10.
By the connection graph we mean an underlying weighted graph with a connection which associates edge set with an orthogonal group. This paper centers its investigation on the connection heat kernels on connection lattices and connection discrete torus. For one dimensional connection lattice, we derive the connection heat kernel expression by doing the Taylor expansion on the exponential function involving normalized connection Laplacian. We introduce a novel connection called product connection and prove that the connection heat kernel on arbitrary high dimensional lattice with product connection equals the Kronecker sum of one dimensional connection lattices' connection heat kernels. Furthermore, if the connection graph is consistent, we substantiate the interrelation between its connection heat kernel and its underlying graph's heat kernel. We define a connection called quotient connection such that discrete torus with quotient connection can be considered as a quotient graph of connection lattice, whose connection heat kernel is demonstrated to be the sum of connection lattices' connection heat kernels. In addition, we derive an alternative expression of connection heat kernel o
The introduction of connected and automated vehicles (CAV) is believed to reduce congestion, enhance safety, and improve traffic efficiency. Numerous research studies have focused on controlling pure CAV platoons in fully connected automated traffic, as well as single or multiple CAVs in mixed traffic with human-driven vehicles (HVs). CAV cruising control designs have been proposed to stabilize the car-following traffic dynamics, but few studies has considered their safety impact, particularly the trade-offs between stability and safety. In this paper, we study how cooperative control strategies for CAVs can be designed to enhance the safety and smoothness of mixed traffic under varying penetrations of connectivity and automation. Considering mixed traffic where a pair of CAVs travels amongst HVs, we design cooperative feedback controllers for the pair CAVs to stabilize traffic via cooperation and, possibly, by also leveraging connectivity with HVs. The real-time safety impact of the CAV controllers is investigated using control barrier functions (CBF). We construct CBF safety constraints, based on which we propose safety-critical control designs to guarantee CAV safety, HV safety
Schubert polynomials form a basis of the polynomial ring. This basis and its structure constants have received extensive study. Recently, Pan and Yu initiated the study of top Lascoux polynomials. These polynomials form a basis of a subalgebra of the polynomial ring where each graded piece has finite dimension. This paper connects Schubert polynomials and top Lascoux polynomials via a simple operator. We use this connection to show these two bases share the same structure constants. We also translate several results on Schubert polynomials to top Lascoux polynomials, including combinatorial formulas for their monomial expansions and supports.
In this paper, we present a novel interpretation of the so-called Weisfeiler-Lehman (WL) distance, introduced by Chen et al. (2022), using concepts from stochastic processes. The WL distance aims at comparing graphs with node features, has the same discriminative power as the classic Weisfeiler-Lehman graph isomorphism test and has deep connections to the Gromov-Wasserstein distance. This new interpretation connects the WL distance to the literature on distances for stochastic processes, which also makes the interpretation of the distance more accessible and intuitive. We further explore the connections between the WL distance and certain Message Passing Neural Networks, and discuss the implications of the WL distance for understanding the Lipschitz property and the universal approximation results for these networks.
A knot mosaic is a representation of a knot or link on a square grid using a collection of tiles that are either blank or contain a portion of the knot diagram. Traditionally, a piece of the knot on one tile connects to a piece of the knot on an adjacent tile at a connection point that is located at the midpoint of a tile edge. We introduce a new set of tiles in which the connection points are located at corners of the tile. By doing this, we can create more efficient knot mosaics for knots with small crossing number. In particular, when using these corner connection tiles, it is possible to create knot mosaic diagrams for all knots with crossing number 8 or less on a mosaic that is no larger and uses fewer non-blank tiles than is possible using the traditional tiles.
It is well-known that computing a Markov basis for a discrete loglinear model is very hard in general. Thus, we focus on connecting tables in a fiber via a subset of a Markov basis and in this paper, we consider connecting tables if we allow cell counts in each tale to be $-1$. In this paper we show that if a subset of a Markov basis connects all tables in the fiber which contains a table with all ones, then moves in this subset connect tables in the fiber if we allow cell counts to be $-1$. In addition, we show that in some cases under the no-three-way interaction model, we can connect tables by all basic moves of $2 \times 2 \times 2$ minors with allowing $X_{ijk} \geq -1$. We then apply this Markov Chain Monte Carlo (MCMC) scheme to an empirical data on Naval officer and enlisted population. Our computational experiments show it works well and we end with the conjecture on the no-three-way interaction model.
When the discriminants $Δ$ and $Δp^2$ are idoneal, Patane proved a theorem which connects the theta series associated to binary quadratic forms of each discriminant. This paper generalizes the main theorem of Patane by no longer requiring $Δ$ and $Δp^2$ to be idoneal. In particular, we state and prove an identity which connects the theta series associated to a single binary quadratic form of discriminant $Δ$ to a theta series associated to a subset of binary quadratic forms of discriminant $Δp^2$. Here and everywhere $p$ is a prime.