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Multiobjective optimization problems (MOPs) are prevalent in machine learning, with applications in multi-task learning, learning under fairness or robustness constraints, etc. Instead of reducing multiple objective functions into a scalar objective, MOPs aim to optimize for the so-called Pareto optimality or Pareto set learning, which involves optimizing more than one objective function simultaneously, over models with millions of parameters. Existing benchmark libraries for MOPs mainly focus on evolutionary algorithms, most of which are zeroth-order methods that do not effectively utilize higher-order information from objectives and cannot scale to large-scale models with millions of parameters. In light of the above gap, this paper introduces LibMOON, the first multiobjective optimization library that supports state-of-the-art gradient-based methods, provides a fair benchmark, and is open-sourced for the community.

In this paper, we present a novel 3D head avatar creation approach capable of generalizing from few-shot in-the-wild data with high-fidelity and animatable robustness. Given the underconstrained nature of this problem, incorporating prior knowledge is essential. Therefore, we propose a framework comprising prior learning and avatar creation phases. The prior learning phase leverages 3D head priors derived from a large-scale multi-view dynamic dataset, and the avatar creation phase applies these priors for few-shot personalization. Our approach effectively captures these priors by utilizing a Gaussian Splatting-based auto-decoder network with part-based dynamic modeling. Our method employs identity-shared encoding with personalized latent codes for individual identities to learn the attributes of Gaussian primitives. During the avatar creation phase, we achieve fast head avatar personalization by leveraging inversion and fine-tuning strategies. Extensive experiments demonstrate that our model effectively exploits head priors and successfully generalizes them to few-shot personalization, achieving photo-realistic rendering quality, multi-view consistency, and stable animation., Comment: Project page: https://headgap.github.io/

Multi-objective optimization can be found in many real-world applications where some conflicting objectives can not be optimized by a single solution. Existing optimization methods often focus on finding a set of Pareto solutions with different optimal trade-offs among the objectives. However, the required number of solutions to well approximate the whole Pareto optimal set could be exponentially large with respect to the number of objectives, which makes these methods unsuitable for handling many optimization objectives. In this work, instead of finding a dense set of Pareto solutions, we propose a novel Tchebycheff set scalarization method to find a few representative solutions (e.g., 5) to cover a large number of objectives (e.g., $>100$) in a collaborative and complementary manner. In this way, each objective can be well addressed by at least one solution in the small solution set. In addition, we further develop a smooth Tchebycheff set scalarization approach for efficient optimization with good theoretical guarantees. Experimental studies on different problems with many optimization objectives demonstrate the effectiveness of our proposed method.

Multi-objective optimization problems can be found in many real-world applications, where the objectives often conflict each other and cannot be optimized by a single solution. In the past few decades, numerous methods have been proposed to find Pareto solutions that represent optimal trade-offs among the objectives for a given problem. However, these existing methods could have high computational complexity or may not have good theoretical properties for solving a general differentiable multi-objective optimization problem. In this work, by leveraging the smooth optimization technique, we propose a lightweight and efficient smooth Tchebycheff scalarization approach for gradient-based multi-objective optimization. It has good theoretical properties for finding all Pareto solutions with valid trade-off preferences, while enjoying significantly lower computational complexity compared to other methods. Experimental results on various real-world application problems fully demonstrate the effectiveness of our proposed method., Comment: Accepted by the 41st International Conference on Machine Learning (ICML 2024)

It is desirable in many multi-objective machine learning applications, such as multi-task learning with conflicting objectives and multi-objective reinforcement learning, to find a Pareto solution that can match a given preference of a decision maker. These problems are often large-scale with available gradient information but cannot be handled very well by the existing algorithms. To tackle this critical issue, this paper proposes a novel predict-and-correct framework for locating a Pareto solution that fits the preference of a decision maker. In the proposed framework, a constraint function is introduced in the search progress to align the solution with a user-specific preference, which can be optimized simultaneously with multiple objective functions. Experimental results show that our proposed method can efficiently find a particular Pareto solution under the demand of a decision maker for standard multiobjective benchmark, multi-task learning, and multi-objective reinforcement learning problems with more than thousands of decision variables. Code is available at: https://github.com/xzhang2523/pmgda. Our code is current provided in the pgmda.rar attached file and will be open-sourced after publication.}

Multiobjective optimization (MOO) is prevalent in numerous applications, in which a Pareto front (PF) is constructed to display optima under various preferences. Previous methods commonly utilize the set of Pareto objectives (particles on the PF) to represent the entire PF. However, the empirical distribution of the Pareto objectives on the PF is rarely studied, which implicitly impedes the generation of diverse and representative Pareto objectives in previous methods. To bridge the gap, we suggest in this paper constructing \emph{uniformly distributed} Pareto objectives on the PF, so as to alleviate the limited diversity found in previous MOO approaches. We are the first to formally define the concept of ``uniformity" for an MOO problem. We optimize the maximal minimal distances on the Pareto front using a neural network, resulting in both asymptotically and non-asymptotically uniform Pareto objectives. Our proposed method is validated through experiments on real-world and synthetic problems, which demonstrates the efficacy in generating high-quality uniform Pareto objectives and the encouraging performance exceeding existing state-of-the-art methods. The detailed model implementation and the code are scheduled to be open-sourced upon publication.

We aim to participate in the calibration of the X-ray photon count rate to halo mass scaling relation of galaxy clusters selected in the first eROSITA All-Sky Survey on the Western Galactic Hemisphere (eRASS1) using KiDS-1000 weak-lensing (WL) data. We measure the radial shear profiles around eRASS1 galaxy clusters using background galaxies in KiDS-1000, as well as the cluster member contamination. Furthermore we provide consistency checks with the other stage-III WL surveys who take part in the eRASS1 mass calibration, DES Y3 and HSC-Y3. We determine the cluster member contamination of eRASS1 clusters present in KiDS-1000 based on source number density profiles, where we account for the obscuration caused by cluster galaxies. The extracted shear profiles, together with the contamination model and the lens sample selection, are then analysed through a Bayesian population model. We calibrate the WL mass bias parameter by analysing realistic synthetic shear profiles from mock cluster catalogues. Our consistency checks between KiDS-1000 and DES Y3 & HSC-Y3 include the comparison of contamination-corrected density contrast profiles employing the union of background sources around common clusters, as well as the individual scaling relation results. We present a global contamination model for eRASS1 clusters in KiDS-1000 and the calibration results of the X-ray photon count rate to halo mass relation. The results of the WL mass bias parameter show that the uncertainty of the multiplicative shear bias dominates the systematic error budget at low clusters redshifts while the uncertainty of our contamination model does at high ones. The cross-checks between the three WL surveys show that they are statistically consistent with each other. This enables for the first time cosmological constraints from clusters calibrated by three state-of-the-art WL surveys. (abridged), Comment: 22 pages, 23 figures

We present the analytic calculation of the leading order three-point energy correlator (EEEC) in hadronic Higgs decays, including both gluon-initiated channel $H\rightarrow g g+X$ and quark-initiated channel $H\rightarrow q\bar q+X$. The phase space integration is evaluated directly using Mandelstam variables $s_{ij}=(p_i+p_j)^2$, and the appearing square roots can be rationalized by either conformal ratios or celestial coordinate variables. Throughout the calculation, we observe the same transcendental function space as in $\mathcal{N}=4$ super Yang-Mills (SYM) theory and $e^+e^-\rightarrow \text{ hadrons}$. Different infrared limits are also explored using the full analytic result, offering the fixed-order data for EEEC factorization and resummation. Given its non-trivial shape dependence, the EEEC presents an excellent opportunity to explore the dynamics of gluon jets originating from the $H \to gg$ decay channel at future lepton colliders., Comment: 18 pages, 9 figures, ancillary files with results

Current methods for large language model alignment typically use scalar human preference labels. However, this convention tends to oversimplify the multi-dimensional and heterogeneous nature of human preferences, leading to reduced expressivity and even misalignment. This paper presents Panacea, an innovative approach that reframes alignment as a multi-dimensional preference optimization problem. Panacea trains a single model capable of adapting online and Pareto-optimally to diverse sets of preferences without the need for further tuning. A major challenge here is using a low-dimensional preference vector to guide the model's behavior, despite it being governed by an overwhelmingly large number of parameters. To address this, Panacea is designed to use singular value decomposition (SVD)-based low-rank adaptation, which allows the preference vector to be simply injected online as singular values. Theoretically, we prove that Panacea recovers the entire Pareto front with common loss aggregation methods under mild conditions. Moreover, our experiments demonstrate, for the first time, the feasibility of aligning a single LLM to represent an exponentially vast spectrum of human preferences through various optimization methods. Our work marks a step forward in effectively and efficiently aligning models to diverse and intricate human preferences in a controllable and Pareto-optimal manner.

Understanding how the properties of galaxies relate to the properties of the hot circum-galactic medium (CGM) around them can constrain galaxy evolution models. We measured the X-ray luminosity of the hot CGM based on the surface brightness profiles of central galaxy samples measured from Spectrum Roentgen Gamma (SRG)/eROSITA all-sky survey data. We related the X-ray luminosity to the galaxies' stellar and halo mass, and we compared the observed relations to the self-similar model and intrinsic (i.e., not forward-modeled) output of the IllustrisTNG, EAGLE, and SIMBA simulations. The average hot CGM X-ray luminosity ($L_{\rm X,CGM}$) correlates with the galaxy's stellar mass ($M_*$). It increases from $(1.6 \pm 2.1)\times10^{39} \rm erg\,s^{-1}$ to $(3.4 \pm 0.3)\times10^{41} \rm erg\,s^{-1}$, when $\log(M_*)$ increases from 10.0 to 11.5. A power law describes the correlation as $\log(L_{\rm X,CGM})= (2.4\pm 0.1)\times \log(M_*)+(14.6\pm1.5)$. The hot CGM X-ray luminosity as a function of halo mass is measured within $\log(M_{\rm 500c})=11.3-13.7$, extending our knowledge of the scaling relation by more than two orders of magnitude. $L_{\rm X,CGM}$ increases with $M_{\rm 500c}$ from $(3.0 \pm 1.6)\times10^{39}\ \rm erg\,s^{-1}$ at $\log(M_{\rm 500c})=11.3$ to $(1.3 \pm 0.1)\times10^{42}\ \rm erg\,s^{-1}$ at $\log(M_{\rm 500c})=13.7$. The relation follows a power law of $\log(L_{\rm X,CGM})= (1.32\pm 0.05)\times \log(M_{\rm 500c})+(24.1\pm0.7)$. Our observations highlight the necessity of non-gravitational processes at the galaxy group scale while suggesting these processes are sub-dominant at the galaxy scale. We show that the outputs of current cosmological galaxy simulations generally align with the observational results uncovered here but with possibly important deviations in selected mass ranges., Comment: Abstract abridged for arXiv submission, accepted by A&A

The circumgalactic medium (CGM) provides the material needed for galaxy formation and influences galaxy evolution. The hot ($T>10^6K$) CGM is poorly detected around galaxies with stellar masses ($M_*$) lower than $3\times10^{11}M_\odot$ due to the low surface brightness. We used the X-ray data from the first four SRG/eROSITA All-Sky Surveys (eRASS:4). Based on the SDSS spectroscopic survey and halo-based group finder algorithm, we selected central galaxies with spectroscopic redshifts of $z_{\rm spec}<0.2$ and stellar masses of $10.0<\log(M_*/M_\odot)<11.5$ (85,222 galaxies) -- or halo masses of $11.5<\log(M_{\rm 200m}/M_\odot)<14.0$ (125,512 galaxies). By stacking the X-ray emission around galaxies, masking the detected X-ray point sources and carefully modeling the X-ray emission from the unresolved active galactic nuclei (AGN) and X-ray binaries (XRB), we obtain the X-ray emission from the hot CGM. We detected the X-ray emission around MW-mass and more massive central galaxies extending up to the virial radius ($R_{\rm vir}$). We used a $\beta$ model to describe the X-ray surface brightness profile and found $\beta =0.43^{+0.10}_{-0.06}\,(0.37^{+0.04}_{-0.02})$ for MW-mass (M31-mass) galaxies.We estimated the baryon budget of the hot CGM and obtained a value that is lower than the prediction of $\Lambda$CDM cosmology, indicating significant gas depletion in these halos. We extrapolated the hot CGM profile measured within $R_{\rm vir}$ to larger radii and found that within $\approx 3 R_{\rm vir}$, the baryon budget is close to the $\Lambda$CDM cosmology prediction. Our results set a firm footing for the presence of the hot CGM around such galaxies. These measurements constitute a new benchmark for galaxy evolution models and possible implementations of feedback processes therein., Comment: Abstract abridged for arXiv submission, accepted by A&A

As the closest galaxy cluster, the Virgo Cluster is an exemplary environment for the study of the large-scale filamentary structure and physical effects that are present in cluster outskirts but absent from the more easily studied inner regions. Here, we present an analysis of the SRG/eROSITA data from five all-sky surveys. eROSITA allows us to resolve the entire Virgo cluster and its outskirts on scales between 1 kpc and 3 Mpc, covering a total area on the sky of about 25$^\circ$ by 25$^\circ$. We utilized image manipulation techniques and surface brightness profiles to search for extended emission, surface brightness edges, and features in the outskirts. We employed a method of comparing mean and median profiles to measure gas clumping out to and beyond the virial radius. Surface brightness analysis of the cluster and individual sectors of the cluster reveal the full extent of previously identified cold fronts to the north and south. The emissivity bias due to gas clumping, which we quantify over three orders of magnitude in the radial range, is found to be mild, consistent with previous findings. We find uniform clumping measurements in all directions, with no enhancements along candidate filaments. We find an estimated virial gas mass of $M_{\mathrm{gas},r

The generalization of decision-making agents encompasses two fundamental elements: learning from past experiences and reasoning in novel contexts. However, the predominant emphasis in most interactive environments is on learning, often at the expense of complexity in reasoning. In this paper, we introduce CivRealm, an environment inspired by the Civilization game. Civilization's profound alignment with human history and society necessitates sophisticated learning, while its ever-changing situations demand strong reasoning to generalize. Particularly, CivRealm sets up an imperfect-information general-sum game with a changing number of players; it presents a plethora of complex features, challenging the agent to deal with open-ended stochastic environments that require diplomacy and negotiation skills. Within CivRealm, we provide interfaces for two typical agent types: tensor-based agents that focus on learning, and language-based agents that emphasize reasoning. To catalyze further research, we present initial results for both paradigms. The canonical RL-based agents exhibit reasonable performance in mini-games, whereas both RL- and LLM-based agents struggle to make substantial progress in the full game. Overall, CivRealm stands as a unique learning and reasoning challenge for decision-making agents. The code is available at https://github.com/bigai-ai/civrealm.

In the past few decades, many multiobjective evolutionary optimization algorithms (MOEAs) have been proposed to find a finite set of approximate Pareto solutions for a given problem in a single run, each with its own structure. However, in many real-world applications, it could be desirable to have structure constraints on the entire optimal solution set, which define the patterns shared among all solutions. The current population-based MOEAs cannot properly handle such requirements. In this work, we make the first attempt to incorporate the structure constraints into the whole solution set by a single Pareto set model, which can be efficiently learned by a simple evolutionary stochastic optimization method. With our proposed method, the decision-makers can flexibly trade off the Pareto optimality with preferred structures among all solutions, which is not supported by previous MOEAs. A set of experiments on benchmark test suites and real-world application problems fully demonstrates the efficiency of our proposed method.

Homotopy optimization is a traditional method to deal with a complicated optimization problem by solving a sequence of easy-to-hard surrogate subproblems. However, this method can be very sensitive to the continuation schedule design and might lead to a suboptimal solution to the original problem. In addition, the intermediate solutions, often ignored by classic homotopy optimization, could be useful for many real-world applications. In this work, we propose a novel model-based approach to learn the whole continuation path for homotopy optimization, which contains infinite intermediate solutions for any surrogate subproblems. Rather than the classic unidirectional easy-to-hard optimization, our method can simultaneously optimize the original problem and all surrogate subproblems in a collaborative manner. The proposed model also supports real-time generation of any intermediate solution, which could be desirable for many applications. Experimental studies on different problems show that our proposed method can significantly improve the performance of homotopy optimization and provide extra helpful information to support better decision-making., Comment: Accepted by the 40th International Conference on Machine Learning (ICML 2023)

The projected energy correlator measures the energy deposited in multiple detectors as a function of the largest angular distance $x_L = (1 - \cos\chi_L)/2$ between detectors. The collinear limit $x_L\to 0$ of the projected energy correlator is particularly interesting for understanding the jet-substructures, while the large logarithms of $x_L$ could potentially spoil the perturbation theory and must be resummed. As a necessary ingredient for its resummation at next-to-next-to-leading logarithmic (NNLL) accuracy, we calculate the two-loop jet functions for the projected three-point energy correlator (E3C), using direct integration method and the parameter space Integration-by-Part (IBP) method. We then present the NNLL resummation for $e^+e^-$ annihilation and an approximate NNLL resummation for $pp\rightarrow jj$ process, where the two-loop hard constant is estimated in the latter case. The convergence is improved and the hadronization effect in the collinear limit is suppressed when considering the ratio of E3C distribution to two-point energy-energy correlator (EEC). Our results show potential in precision determination of strong coupling constant using energy correlators from both $e^+e^-$ data and $pp$ data., Comment: 28 pages + appendix, 13 figures

The heavy jet mass event shape has large perturbative logarithms near the leading order kinematic threshold at $\rho = \frac{1}{3}$. Catani and Webber named these logarithms Sudakov shoulders and resummed them at double-logarithmic level. A resummation to next-to-leading logarithmic level was achieved recently. Here, we extend the resummation using an effective field theory framework to next-to-next-to-leading logarithmic order and show how to combine it with the resummation of dijet logarithms. We also solve the open problem of an unphysical singularity in the resummed momentum space distribution, in a way similar to how it is resolved in the Drell-Yan $q_T$ spectrum: through a careful analysis of the kinematics and scale-setting in position space. The heavy jet mass Sudakov shoulder is the first observable that does not involve transverse momentum for which position space resummation is critical. These advances may lead to a more precise extraction of the strong coupling constant from $e^+ e^-$ data., Comment: 33 pages + appendices, 10 figures

We develop a neural network based pipeline to estimate masses of galaxy clusters with a known redshift directly from photon information in X-rays. Our neural networks are trained using supervised learning on simulations of eROSITA observations, focusing in this paper on the Final Equatorial Depth Survey (eFEDS). We use convolutional neural networks which are modified to include additional information of the cluster, in particular its redshift. In contrast to existing work, we utilize simulations including background and point sources to develop a tool which is usable directly on observational eROSITA data for an extended mass range from group size halos to massive clusters with masses in between $10^{13}M_\odot

We present the serendipitous discovery of (1) a large double radio relic associated with the galaxy cluster PSZ2 G277.93+12.34 and (2) a new odd radio circle, ORC J1027-4422, both found in the same deep MeerKAT 1.3 GHz wide-band radio continuum image. The angular separation of the two arc-shaped cluster relics is ~16 arcmin or ~2.6 Mpc for a cluster redshift of z ~ 0.158. The thin southern relic, which shows several ridges/shocks including one possibly moving inwards, has a linear extent of ~1.64 Mpc. In contrast, the northern relic is about twice as wide, twice as bright, but only has a largest linear size of ~0.66 Mpc. Complementary SRG/eROSITA X-ray images reveal extended emission from hot intracluster gas between the two relics and around the narrow-angle tail (NAT) radio galaxy PMN J1033-4335 (z ~ 0.153) located just east of the northern relic. The radio morphologies of the NAT galaxy and the northern relic, which are also detected with the Australian Square Kilometer Array Pathfinder (ASKAP) at 888 MHz, suggest both are moving in the same outward direction. The discovery of ORC J1027-4422 in a different part of the same MeerKAT image makes it the 4th known single ORC. It has a diameter of ~90 arcsec corresponding to 400 kpc at a tentative redshift of z ~ 0.3 and remains undetected in X-ray emission. Supported by simulations, we discuss similarities between outward moving galaxy and cluster merger shocks as the formation mechanisms for ORCs and radio relics, respectively., Comment: 16 pages, 11 figures, MNRAS, in press

Expensive multi-objective optimization problems can be found in many real-world applications, where their objective function evaluations involve expensive computations or physical experiments. It is desirable to obtain an approximate Pareto front with a limited evaluation budget. Multi-objective Bayesian optimization (MOBO) has been widely used for finding a finite set of Pareto optimal solutions. However, it is well-known that the whole Pareto set is on a continuous manifold and can contain infinite solutions. The structural properties of the Pareto set are not well exploited in existing MOBO methods, and the finite-set approximation may not contain the most preferred solution(s) for decision-makers. This paper develops a novel learning-based method to approximate the whole Pareto set for MOBO, which generalizes the decomposition-based multi-objective optimization algorithm (MOEA/D) from finite populations to models. We design a simple and powerful acquisition search method based on the learned Pareto set, which naturally supports batch evaluation. In addition, with our proposed model, decision-makers can readily explore any trade-off area in the approximate Pareto set for flexible decision-making. This work represents the first attempt to model the Pareto set for expensive multi-objective optimization. Experimental results on different synthetic and real-world problems demonstrate the effectiveness of our proposed method., Comment: To appear in 36th Conference on Neural Information Processing Systems (NeurIPS 2022)

The energy correlator measures the energy deposited in multiple detectors as a function of the angles among them. In this paper, an analytic formula is given for the three-point energy correlator with full angle dependence at leading order in electron-positron annihilation. This is the first analytic computation of trijet event shape observables in QCD, which provides valuable data for phenomenological studies. The result is computed with direct integration, where appropriate parameterizations of both phase space and kinematic space are adopted to simplify the calculation. With full shape dependence, our result provides the expansions in various kinematic regions such as equilateral, triple collinear and squeezed limits, which benefit studies on both factorization and large logarithm resummation., Comment: 42 pages, 8 figures

Mergers of galaxy clusters are the most energetic events in the universe, driving shock and cold fronts, generating turbulence, and accelerating particles that create radio halos and relics. The galaxy cluster CL 0217+70 is a remarkable late stage merger, with a double peripheral radio relic and a giant radio halo. A Chandra study detects surface brightness edges that correspond to radio features within the halo. In this work, we present a study of this cluster with NuSTAR and Chandra data using spectro-imaging methods. The global temperature is found to be kT = 9.1 keV. We set an upper limit for the IC flux of ~2.7x10^(-12) erg s^(-1) cm^(-2), and a lower limit to the magnetic field of 0.08 microG. Our local IC search revealed a possibility that IC emission may have a significant contribution at the outskirts of a radio halo emission and on/near shock regions within ~0.6 r500 of clusters. We detected a "hot spot" feature in our temperature map coincident a surface brightness edge, but our investigation on its origin is inconclusive. If the "hot spot" is the downstream of a shock, we set a lower limit of kT > 21 keV to the plasma, that corresponds to M~2. We found three shock fronts within 0.5 r500. Multiple weak shocks within the cluster center hint at an ongoing merger activity and continued feeding of the giant radio halo. CL 0217+70 is the only example hosting these secondary shocks in multiple form., Comment: 21 pages, 14 figures, 7 tables, submitted

Electricity consumption forecasting has vital importance for the energy planning of a country. Of the enabling machine learning models, support vector regression (SVR) has been widely used to set up forecasting models due to its superior generalization for unseen data. However, one key procedure for the predictive modeling is feature selection, which might hurt the prediction accuracy if improper features were selected. In this regard, a modified discrete particle swarm optimization (MDPSO) was employed for feature selection in this study, and then MDPSO-SVR hybrid mode was built to predict future electricity consumption. Compared with other well-established counterparts, MDPSO-SVR model consistently performs best in two real-world electricity consumption datasets, which indicates that MDPSO for feature selection can improve the prediction accuracy and the SVR equipped with the MDPSO can be a promised alternative for electricity consumption forecasting.

Polylogrithmic functions, such as the logarithm or dilogarithm, satisfy a number of algebraic identities. For the logarithm, all the identities follow from the product rule. For the dilogarithm and higher-weight classical polylogarithms, the identities can involve five functions or more. In many calculations relevant to particle physics, complicated combinations of polylogarithms often arise from Feynman integrals. Although the initial expressions resulting from the integration usually simplify, it is often difficult to know which identities to apply and in what order. To address this bottleneck, we explore to what extent machine learning methods can help. We consider both a reinforcement learning approach, where the identities are analogous to moves in a game, and a transformer network approach, where the problem is viewed analogously to a language-translation task. While both methods are effective, the transformer network appears more powerful and holds promise for practical use in symbolic manipulation tasks in mathematical physics., Comment: 41 pages, 10 figures

When the allowed range of an observable grows order-by-order in perturbation theory, its perturbative expansion can have discontinuities (as in the $C$ parameter) or discontinuities in its derivatives (as in thrust or heavy jet mass) called Sudakov shoulders. We explore the origin of these logarithms using both perturbation theory and effective field theory. We show that for thrust and heavy jet mass, the logarithms arise from kinematic configurations with narrow jets and deduce the next-to-leading logarithmic series. The left-shoulder logarithms in heavy jet mass ($\rho$) of the form $r\ \alpha_s^n \ln^{2n}r $ with $r=\frac{1}{3}-\rho$ are particularly dangerous, because they invalidate fixed order perturbation theory in regions traditionally used to extract $\alpha_s$. Although the factorization formula shows there are no non-global logarithms, we find Landau-pole like singularities in the resummed distribution associated with the cusp anomalous dimension, and that power corrections are exceptionally important., Comment: 49 pages, 8 figures. Published in Physical Review D

In this paper, we mainly study the function spaces related to H-sober spaces. For an irreducible subset system H and $T_{0}$ spaces $X$ and $Y$, it is proved that $Y$ is H-sober iff the function space $\mathbb{C}(X, Y)$ of all continuous functions $f : X\longrightarrow Y$ equipped with the topology of pointwise convergence is H-sober iff the function space $\mathbb{C}(X, Y)$ equipped with the Isbell topology is H-sober. One immediate corollary is that for a $T_{0}$ space $X$, $Y$ is a sober space (resp., $d$-space, well-filtered space) iff the function space $\mathbb{C}(X, Y)$ equipped with the topology of pointwise convergence is a sober space (resp., $d$-space, well-filtered space) iff the function space $\mathbb{C}(X, Y)$ equipped with the the Isbell topology is a sober space (resp., $d$-space, well-filtered space). It is shown that $T_{0}$ spaces $X$ and $Y$, if the function space $\mathbb{C}(X, Y)$ equipped with the compact-open topology is H-sober, then $Y$ is H-sober. The function space $\mathbb{C}(X, Y)$ equipped with the Scott topology is also discussed., Comment: 11 pages

An analytic formula is given for the three-point energy correlator (EEEC) at leading order (LO) in maximally supersymmetric Yang-Mills theory ($\mathcal{N}=4$ sYM). This is the first analytic calculation of a three-parameter event shape observable, which provides valuable data for various studies ranging from conformal field theories to jet substructure. The associated class of functions define a new type of single-valued polylogarithms characterized by 16 alphabet letters, which manifest a $D_6 \times Z_2$ dihedral symmetry of the event shape. With the unexplored simplicity in the perturbative structure of EEEC, all kinematic regions including collinear, squeezed and coplanar limits are now available., Comment: 9 pages, 4 figures; typo in eq(4) corrected