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The third law of thermodynamics has been verified experimentally, but how to perfectly express such a law in theory has become a cross-century problematic issue. It is found that by introducing an innovative method, the Nernst equation can be obtained directly from the experimental data of chemical reactions at low temperatures without the need for artificial additional assumptions appearing in textbooks, so that the Nernst theorem should be replaced by the Nernst statement. It is also found that the heat capacity statement can be obtained from the experimental data of the heat capacity at low temperatures. The heat capacity statement and the Nernst statement are proved to be mutually derivable and the two are equivalent. The unattainability principle of absolute zero temperature is only a corollary of the Nernst statement or the heat capacity statement. Simultaneously, the defects and deficiencies related to the contents of the third law of thermodynamics appearing in textbooks are pointed out and corrected. The results obtained show clearly that the Nernst theorem and the unattainability principle of absolute zero temperature should be withdrawn from the statements of the third law of thermodynamics. It is important to find that the Nernst statement and the heat capacity statement are two equivalent statements of the third law of thermodynamics, which can solve the centennial debate problems of the third law of thermodynamics and supply the perfect statements for the third law of thermodynamics.

Personalized image completion aims to restore occluded regions in personal photos while preserving identity and appearance. Existing methods either rely on generic inpainting models that often fail to maintain identity consistency, or assume that suitable reference images are explicitly provided. In practice, suitable references are often not explicitly provided, requiring the system to search for identity-consistent images within personal photo collections. We present AlbumFill, a training-free framework that retrieves identity-consistent references from personal albums for personalized completion. Given an occluded image and a personal album, a vision-language model infers missing semantic cues to guide composed image retrieval, and the retrieved references are used by reference-based completion models. To facilitate this task, we introduce a dataset containing 54K human-centric samples with associated album images. Experiments across multiple baselines demonstrate the difficulty of personalized completion and highlight the importance of identity-consistent reference retrieval. Project Page: https://liagm.github.io/AlbumFill/

We study Collins azimuthal asymmetries in pion-in-jet production in polarized lepton-proton collisions, extending previous analyses of polarized $pp$ scattering to a complementary and theoretically simpler process. We keep adopting a simplified transverse momentum dependent (TMD) approach, with a collinear configuration for the initial state, and employ the transversity and Collins fragmentation functions as extracted from semi-inclusive deep inelastic scattering and $e^+ e^-$ annihilation processes. We then compute azimuthal asymmetries for the Electron-Ion Collider (EIC) kinematics, both within a leading order (LO) approach and by including quasireal photon exchange in the Weizsäcker-Williams approximation. Although this contribution is relevant in the whole kinematical range explored, it does not spoil the dominance of quark-initiated channels, leaving only a marginal role to their gluon counterparts. In this respect, Collins asymmetries in lepton-proton processes allow for a much clearer access to the transversity distribution, including its sea-quark component. As we will argue, a comparison with future EIC data could represent a further step in testing the hypothesis of the universality of the Collins function as well as of the TMD factorization for this class of processes.

This paper introduces gluing diagrams a combinatorial tool to construct homomorphisms between the shift pseudogroups of directed graphs and thus also their full groups of shifts. We will establish which of these diagrams produce isomorphisms. As an application, using the interpretation of Higman-Thompson groups as full groups of shifts of specific graphs, we will describe a procedure that constructs gluing diagrams that explicitly describe the isomorphisms between Higman-Thompson groups, conjectured by Higman and whose existence was proven by Pardo arXiv:1006.1759.

We show that for $d\geq 2$ every finite $d$-dimensional simplicial complex is a deformation retract of a $(2d-1)$-dimensional pseudomanifold with boundary. Moreover, it embeds as a retract in a closed $(2d-1)$-dimensional pseudomanifold.