Simple proofs of open problems about the structure of involutions in the Riordan group

AbstractWe prove that if D=(g(x),f(x)) is an element of order 2 in the Riordan group then g(x)=±exp[Φ(x,xf(x)] for some antisymmetric function Φ(x,z). Also we prove that every element of order 2 in the Riordan group can be written as BMB-1 for some element B and M=(1,-1) in the Riordan group. These proofs provide solutions to two open problems presented by L. Shapiro [L.W. Shapiro, Some open questions about random walks, involutions, limiting distributions and generating functions, Adv. Appl. Math. 27 (2001) 585–596]

PROGrasp: Pragmatic Human-Robot Communication for Object Grasping

Interactive Object Grasping (IOG) is the task of identifying and grasping the desired object via human-robot natural language interaction. Current IOG systems assume that a human user initially specifies the target object's category (e.g., bottle). Inspired by pragmatics, where humans often convey their intentions by relying on context to achieve goals, we introduce a new IOG task, Pragmatic-IOG, and the corresponding dataset, Intention-oriented Multi-modal Dialogue (IM-Dial). In our proposed task scenario, an intention-oriented utterance (e.g., "I am thirsty") is initially given to the robot. The robot should then identify the target object by interacting with a human user. Based on the task setup, we propose a new robotic system that can interpret the user's intention and pick up the target object, Pragmatic Object Grasping (PROGrasp). PROGrasp performs Pragmatic-IOG by incorporating modules for visual grounding, question asking, object grasping, and most importantly, answer interpretation for pragmatic inference. Experimental results show that PROGrasp is effective in offline (i.e., target object discovery) and online (i.e., IOG with a physical robot arm) settings.Comment: 7 pages, 6 figure

Matrix periods and competition periods of Boolean Toeplitz matrices

In this paper, we study the matrix period and the competition period of Toeplitz matrices over a binary Boolean ring $\mathbb{B} = \{0,1\}$. Given subsets $S$ and $T$ of $\{1,\ldots,n-1\}$, an $n\times n$ Toeplitz matrix $A=T_n\langle S ; T \rangle$ is defined to have $1$ as the $(i,j)$-entry if and only if $j-i \in S$ or $i-j \in T$. We show that if $\max S+\min T \le n$ and $\min S+\max T \le n$, then $A$ has the matrix period $d/d'$ and the competition period $1$ where $d = \gcd (s+t \mid s \in S, t \in T)$ and $d' = \gcd(d, \min S)$. Moreover, it is shown that the limit of the matrix sequence $\{A^m(A^T)^m\}_{m=1}^\infty$ is a directed sum of matrices of all ones except zero diagonal. In many literatures we see that graph theoretic method can be used to prove strong structural properties about matrices. Likewise, we develop our work from a graph theoretic point of view

PGA: Personalizing Grasping Agents with Single Human-Robot Interaction

Language-Conditioned Robotic Grasping (LCRG) aims to develop robots that ground and grasp objects based on natural language instructions. While robots capable of recognizing personal objects like "my wallet" can interact more naturally with non-expert users, current LCRG systems primarily limit robots to understanding only generic expressions. To this end, we introduce a task scenario GraspMine with a novel dataset that aims to locate and grasp personal objects given personal indicators via learning from a single human-robot interaction. To address GraspMine, we propose Personalized Grasping Agent (PGA), that learns personal objects by propagating user-given information through a Reminiscence-a collection of raw images from the user's environment. Specifically, PGA acquires personal object information by a user presenting a personal object with its associated indicator, followed by PGA inspecting the object by rotating it. Based on the acquired information, PGA pseudo-labels objects in the Reminiscence by our proposed label propagation algorithm. Harnessing the information acquired from the interactions and the pseudo-labeled objects in the Reminiscence, PGA adapts the object grounding model to grasp personal objects. Experiments on GraspMine show that PGA significantly outperforms baseline methods both in offline and online settings, signifying its effectiveness and personalization applicability on real-world scenarios. Finally, qualitative analysis shows the effectiveness of PGA through a detailed investigation of results in each phase.Comment: 7 pages, under revie

On k-11-representable graphs

Distinct letters x and y alternate in a word w if after deleting in w all letters but the copies of x and y we either obtain a word of the form xyxy... (of even or odd length) or a word of the form yxyx... (of even or odd length). A simple graph G=(V,E) is word-representable if there exists a word w over the alphabet V such that letters x and y alternate in w if and only if xy is an edge in E. Thus, edges of G are defined by avoiding the consecutive pattern 11 in a word representing G, that is, by avoiding xx and yy. In 2017, Jeff Remmel introduced the notion of a k-11-representable graph for a non-negative integer k, which generalizes the notion of a word-representable graph. Under this representation, edges of G are defined by containing at most k occurrences of the consecutive pattern 11 in a word representing G. Thus, word-representable graphs are precisely 0-11-representable graphs. Our key result in this paper is showing that every graph is 2-11-representable by a concatenation of permutations, which is rather surprising taking into account that concatenation of permutations has limited power in the case of 0-11-representation. Also, we show that the class of word-representable graphs, studied intensively in the literature, is contained strictly in the class of 1-11-representable graphs. Another result that we prove is the fact that the class of interval graphs is precisely the class of 1-11-representable graphs that can be represented by uniform words containing two copies of each letter. This result can be compared with the known fact that the class of circle graphs is precisely the class of 0-11-representable graphs that can be represented by uniform words containing two copies of each letter