Maximum Orders of Cyclic and Abelian Extendable Actions on Surfaces

Let $\Sigma_g (g>1)$ be a closed surface embedded in $S^3$. If a group $G$ can acts on the pair $(S^3, \Sigma_g)$, then we call such a group action on $\Sigma_g$ extendable over $S^3$. In this paper we show that the maximum order of extendable cyclic group actions is $4g+4$ when $g$ is even and $4g-4$ when $g$ is odd; the maximum order of extendable abelian group actions is $4g+4$. We also give results of similar questions about extendable group actions over handlebodies.Comment: 22pages, 10 figure

Alternating Heegaard diagrams and Williams solenoid attractors in 3--manifolds

We find all Heegaard diagrams with the property "alternating" or "weakly alternating" on a genus two orientable closed surface. Using these diagrams we give infinitely many genus two 3--manifolds, each admits an automorphism whose non-wondering set consists of two Williams solenoids, one attractor and one repeller. These manifolds contain half of Prism manifolds, Poincar\'e's homology 3--sphere and many other Seifert manifolds, all integer Dehn surgeries on the figure eight knot, also many connected sums. The result shows that many kinds of 3--manifolds admit a kind of "translation" with certain stability.Comment: 26 pages, 44 figure

Embedding surfaces into S3S^3 with maximum symmetry

We restrict our discussion to the orientable category. For $g > 1$, let $OE_g$ be the maximum order of a finite group $G$ acting on the closed surface $\Sigma_g$ of genus $g$ which extends over $(S^3, \Sigma_g)$, where the maximum is taken over all possible embeddings $\Sigma_g\hookrightarrow S^3$. We will determine $OE_g$ for each $g$, indeed the action realizing $OE_g$. In particular, with 23 exceptions, $OE_g$ is $4(g+1)$ if $g\ne k^2$ or $4(\sqrt{g}+1)^2$ if $g=k^2$, and moreover $OE_g$ can be realized by unknotted embeddings for all $g$ except for $g=21$ and $481$.Comment: 42 pages, 37 figures, 6 tables of figure

Graphs in the 3--sphere with maximum symmetry

We consider the orientation-preserving actions of finite groups $G$ on pairs $(S^3, \Gamma)$, where $\Gamma$ is a connected graph of genus $g>1$, embedded in $S^3$. For each $g$ we give the maximum order $m_g$ of such $G$ acting on $(S^3, \Gamma)$ for all such $\Gamma\subset S^3$. Indeed we will classify all graphs $\Gamma\subset S^3$ which realize these $m_g$ in different levels: as abstract graphs and as spatial graphs, as well as their group actions. Such maximum orders without the condition "orientation-preserving" are also addressed.Comment: 34 pages, to appear in Discrete Comput. Geo

Learning Emotion Representations from Verbal and Nonverbal Communication

Emotion understanding is an essential but highly challenging component of artificial general intelligence. The absence of extensively annotated datasets has significantly impeded advancements in this field. We present EmotionCLIP, the first pre-training paradigm to extract visual emotion representations from verbal and nonverbal communication using only uncurated data. Compared to numerical labels or descriptions used in previous methods, communication naturally contains emotion information. Furthermore, acquiring emotion representations from communication is more congruent with the human learning process. We guide EmotionCLIP to attend to nonverbal emotion cues through subject-aware context encoding and verbal emotion cues using sentiment-guided contrastive learning. Extensive experiments validate the effectiveness and transferability of EmotionCLIP. Using merely linear-probe evaluation protocol, EmotionCLIP outperforms the state-of-the-art supervised visual emotion recognition methods and rivals many multimodal approaches across various benchmarks. We anticipate that the advent of EmotionCLIP will address the prevailing issue of data scarcity in emotion understanding, thereby fostering progress in related domains. The code and pre-trained models are available at https://github.com/Xeaver/EmotionCLIP.Comment: CVPR 202

Investigating the Existence of "Secret Language'' in Language Models

In this paper, we study the problem of secret language in NLP, where current language models (LMs) seem to have a hidden vocabulary that allows them to interpret absurd inputs as meaningful concepts. We investigate two research questions: ``Does the secret language phenomenon exist in different language models?'' and ``Does secret language depend on specific context?'' To answer these questions, we introduce a novel method named \textit{SecretFinding}, a gradient-based approach that can automatically discover secret languages in LMs. We conduct experiments on five representative models (Electra, ALBERT, Roberta, DistillBERT, and CLIP) finetuned on four NLP benchmarks (SST-2, MRPC, SNLI, and SQuAD) and a language-grounding benchmark (MSCOCO). Our experimental results show that even when we replace the most important words with others that are semantically dissimilar to the original words in a sentence, LMs do not consider the new sentence semantically dissimilar to the original, as the output does not change with a high probability. This phenomenon holds true across the five models and five tasks and gives a positive answer to the first research question. As for the second research question, we find that the secret language discovered by \textit{SecretFinding} is quite general and could even be transferred to other models in the black-box settings, such as GPT-3 and ChatGPT. Finally, we discuss the causes of secret language, how to eliminate it, the potential connection to memorization, and ethical implications. Examples of secret language found by SecretFinding are available on https://huggingface.co/spaces/anonymousauthors/ACL23_SecretLanguage

Bordered surfaces in the 3-sphere with maximum symmetry

We consider finite group actions on the 3-sphere which leave invariant an embedded, compact, bounded surface of algebraic genus g > 1 (orientable or non-orientable), and determine for each g the maximum order of such an action. For example, the maximal possibility 12(g-1) is obtained for the finitely many values g = 2, 3, 4, 5, 9, 11, 25, 97, 121 and 241. For each g > 1, we classify the topological types of the surfaces and their embeddings into the 3-sphere

Embedding compact surfaces into the 3-dimensional Euclidean space with maximum symmetry

We give the maximum orders of finite group actions on Euclidean 3-space which leave invariant an embedded compact bordered surface (orientable or non-orientable), in terms of the algebraic genus of the surface. We also identify the topological types of the bordered surfaces realizing the maximum order, and find simple representative embeddings for such surfaces