A Mildly Relativistic Outflow Launched Two Years after Disruption in the Tidal Disruption Event AT2018hyz

Abstract: We present late-time radio/millimeter (as well as optical/UV and X-ray) detections of the tidal disruption event (TDE) AT2018hyz, spanning $970 - 1300$ d after optical discovery. In conjunction with earlier deeper limits, including at $\approx 700$ d, our observations reveal rapidly rising emission at $0.8-240$ GHz, steeper than $F_\nu\propto t^5$ relative to the time of optical discovery. Such a steep rise cannot be explained in any reasonable scenario of an outflow launched at the time of disruption (e.g., off-axis jet, sudden increase in the ambient density), and instead points to a delayed launch. Our multi-frequency data allow us to directly determine the radius and energy of the radio-emitting outflow, showing that it was launched $\approx 750$ d after optical discovery. The outflow velocity is mildly relativistic, with $\beta\approx 0.25$ and $\approx 0.6$ for a spherical and a $10^\circ$ jet geometry, respectively, and the minimum kinetic energy is $E_K\approx 5.8\times 10^{49}$ and $\approx 6.3\times 10^{49}$ erg, respectively. This is the first definitive evidence for the production of a delayed mildly-relativistic outflow in a TDE; a comparison to the recently-published radio light curve of ASASSN-15oi suggests that the final re-brightening observed in that event (at a single frequency and time) may be due to a similar outflow with a comparable velocity and energy. Finally, we note that the energy and velocity of the delayed outflow in AT2018hyz are intermediate between those of past non-relativistic TDEs (e.g., ASASSN-14li, AT2019dsg) and the relativistic TDE Sw\,J1644+57. We suggest that such delayed outflows may be common in TDEs.

• The paper provides a detailed catalog of mathematical symbols along with precise LaTeX commands for standardized academic typesetting.
• It organizes symbols into clear categories, including binary operators, accents, and Greek and Hebrew letters, to enhance usability.
• The comprehensive classification supports efficient document formatting and offers a foundation for improved AI-driven mathematical processing.

A Comprehensive Overview of Symbol Usage in Mathematical Communications

The presented document serves as a detailed cataloging of mathematical symbols used in various contexts within academic and professional settings. The primary focus of the paper is on providing a comprehensive reference for the American Astronomical Society (AAS) and American Mathematical Society (AMS) symbols, textual and mathematical accents, Greek and Hebrew letters, binary operators, miscellaneous symbols, arrows, relations, and function names. This encapsulation of symbols is particularly useful for ensuring consistency and standardization in typesetting mathematical documents, especially when utilizing LaTeX, a widely adopted typesetting system.

Contribution and Scope

The document's contribution lies in its detailed tabulation of symbols, including precise LaTeX commands for their deployment. By categorizing symbols into specific tables based on their function and mode of usage, the document enhances usability for researchers and academicians who require quick reference to these symbols. The organization helps facilitate efficient document formatting and encourages uniformity across mathematical and scientific publications.

Symbol Categorization

• Binary Operators and Relations: Tables delineate the numerous binary operators, both standard and AMS, spanning operations that are foundational in mathematical expressions. The paper also identifies the corresponding LaTeX syntax to simplify document typesetting. Relations and their negated forms are also covered extensively, illustrating the breadth of comparison symbols required in formal proofs and scholarly writing.
• Accents and Special Characters: The text-mode and math-mode accents, alongside national symbols, provide essential tools for correctly representing accented characters and other language-specific symbols in professional documents, maintaining linguistic accuracy and cultural representation.
• Greek and Hebrew Letters: The inclusion of Greek and Hebrew letters underscores their significance in representing constants, variables, and various mathematical constructs. By documenting their respective LaTeX commands, the paper aids in their seamless inclusion in texts.
• Miscellaneous and Geometric Symbols: Miscellaneous symbols, including those specific to set theory, logic, and basic geometric constructs, are encapsulated, thereby offering a broad spectrum of notations critical for expressing complex mathematical ideas.
• Arrows and Delimiters: Arrows are fundamental in depicting mappings and transformations, especially in discussions involving vector spaces and functional analysis. The table extends to AMS arrows, reflecting the importance of comprehensive arrow representation. Delimiters further add to this by setting bounds and comprehensively bracketing mathematical expressions.
• Function Names: Listing common function names wraps up the document's thorough symbol coverage, ensuring that fundamental and advanced functions are easily representable in LaTeX.

Implications and Future Developments

The detailed tabulation found in the document implies significant utility in enhancing the accuracy and efficiency of mathematical typesetting in academic papers, teaching materials, and professional communications. As these symbols constitute the visual language of mathematics and related fields, their standardized use encourages clearer communication of abstract concepts.

In the future development of AI and machine learning tools, such standardized references can be instrumental in developing systems capable of understanding and generating complex mathematical content. Integrating such standardized symbols into neural network training can lead to more robust natural language processing systems tailored to mathematical discourse.

The paper does not imply any particular novel theoretical advancement but rather facilitates the ongoing development of mathematical communication. As LaTeX evolves and possibly integrates with more AI-driven processing tools, maintaining and expanding such symbol catalogs will remain necessary to support the growing complexity of mathematical and scientific writing.