Through a geometric capacitary analysis based on space dualities, this paper addresses several fundamental aspects of functional analysis and potential theory for the Morrey spaces in harmonic analysis over the Euclidean spaces.

The Orlicz–John ellipsoids, which are in the framework of the booming Orlicz Brunn–Minkowski theory, are introduced for the first time. It turns out that they are generalizations of the classical John ellipsoid and the evolved Lp John ellipsoids. The analog of Ball’s volume-ratio inequality is established for the new Orlicz–John ellipsoids. The connection between the isotropy of measures and the characterization of Orlicz–John ellipsoids is demonstrated.

We show that the category of finite-length generalized modules for the singlet vertex algebra M(p), p∈Z_+, is equal to the category O_M(p) of C_1-cofinite M(p)-modules, and that this category admits the vertex algebraic braided tensor category structure of Huang-Lepowsky-Zhang. Since O_M(p) includes the uncountably many typical M(p)-modules, which are simple M(p)-module structures on Heisenberg Fock modules, our results substantially extend our previous work on tensor categories of atypical M(p)-modules. We also introduce a tensor subcategory O_M(p)^T, graded by an algebraic torus T, which has enough projectives and is conjecturally tensor equivalent to the category of finite-dimensional weight modules for the unrolled restricted quantum group of sl_2 at a 2pth root of unity. We compute all tensor products involving simple and projective M(p)-modules, and we prove that both tensor categories O_M(p) and O_M(p)^T are rigid and thus also ribbon. As an application, we use vertex operator algebra extension theory to show that the representation categories of all finite cyclic orbifolds of the triplet vertex algebras W(p) are non-semisimple modular tensor categories, and we confirm a conjecture of Adamović-Lin-Milas on the classification of simple modules for these finite cyclic orbifolds.

Lattice problem such as NTRU problem and LWE problem is widely used as the security base of post-quantum cryptosystems. And currently doing lattice reduction by BKZ algorithm is the most efficient way to solve it. In this paper, we give several further improvements on BKZ algorithm, which can be used for different SVP subroutines base on both enumeration and sieving. These improvements in combination provide a speed up of 2^{3~4} in total. It is significant in concrete attacks. Using these new techniques, we solved the 656 dimensional ideal lattice challenge in only 380 thread hours (also with a enumeration based SVP subroutine), much less than the previous records (which costs 4600 thread hours in total). With these improvements enabled, we can still simulate the new BKZ algorithm easily. One can also use this simulator to find the blocksize strategy (and the corresponding cost) to make Pot of the basis (defined in section 5.2) decrease as fast as possible, which means the length of the first basis vector decrease the fastest if we accept the GSA assumption. It is useful for analyzing concrete attacks on lattice-based cryptography.

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