The Sharp Log-Sobolev Inequality on a Compact Interval

We provide a proof of the sharp log-Sobolev inequality on a compact interval

Perimeter-minimizing Tilings by Convex and Non-convex Pentagons

We study the presumably unnecessary convexity hypothesis in a theorem of Chung et al. on perimeter-minimizing planar tilings by convex pentagons. We prove that the theorem holds without the convexity hypothesis in certain special cases, and we offer direction for further research

Surface-area-minimizing n-hedral Tiles

We provide a list of conjectured surface-area-minimizing n-hedral tiles of space for n from 4 to 14, previously known only for n equal to 5 and 6. We find the optimal orientation-preserving tetrahedral tile (n=4), and we give a nice new proof for the optimal 5-hedron (a triangular prism)

General Bootstrapping Approach for RLWE-based Homomorphic Encryption

We propose a new bootstrapping approach that works for all three Brakerski-Gentry-Vaikuntanathan (BGV), Brakerski/Fan-Vercauteren (BFV), and Cheon-Kim-Kim-Song (CKKS) schemes. This approach adopts a blind rotation technique from FHEW-type schemes. For BGV and BFV, our bootstrapping does not have any restrictions on plaintext modulus unlike typical cases of the previous methods. For CKKS, our approach introduces an error comparable to a rescaling error which enables more than 70 bits of precision after bootstrapping while consuming only 1-2 levels. Due to the high precision of the proposed bootstrapping algorithm, it is the first bootstrapping resistant to the security vulnerability of CKKS found by Li and Micciancio (Eurocrypt 2021). In addition, we introduce methods to reduce the size of public keys required for blind rotations generated by a secret key holder

Indirect Reciprocity with Optional Interactions and Private Information

We consider indirect reciprocity with optional interactions and private information. A game is offered between two players and accepted unless it is known that the other person is a defector. Whenever a defector manages to exploit a cooperator, his or her reputation is revealed to others in the population with some probability. Therefore, people have different private information about the reputation of others, which is a setting that is difficult to analyze in the theory of indirect reciprocity. Since a defector loses a fraction of his social ties each time he exploits a cooperator, he is less efficient at exploiting cooperators in subsequent rounds. We analytically calculate the critical benefit-to-cost ratio above which cooperation is successful in various settings. We demonstrate quantitative agreement with simulation results of a corresponding Wright–Fisher process with optional interactions and private information. We also deduce a simple necessary condition for the critical benefit-to-cost ratio

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