A criterion for the classification of Bott towers is presented, i.e., two ABott towers B_∗(A) and B_∗(A') are isomorphic if and only if the matrices A and A' are equivalent. The equivalence relation is defined by two operations on matrices. And it is based on the observation that any Bott tower B_∗(A) is uniquely determined by its structure matrix A, which is a strictly upper triangular integer matrix. The classification of Bott towers is closely related to the cohomological rigidity problem for both Bott towers and Bott manifolds.

We obtain that every irreducible quasifinite module with non-zero level over the twisted affine Nappi-Witten algebra is either a highest weight module or a lowest one.

Three families of low-density parity-check (LDPC) codes are constructed based on the totally isotropic subspaces of symplectic, unitary, and orthogonal spaces over finite fields, respectively. The minimum distances of the three families of LDPC codes in some special cases are settled.

This paper studies hypothesis testing in the Ornstein-Ulenbeck process with linear drift. With the help of large and moderate deviations for the log-likelihood ratio process, the decision regions and the corresponding decay rates of the error probabilities related to this testing problem are established.

We study the charged 3-body problem with the potential function being (-α)-homogeneous on the mutual distances of any two particles via the variational method and try to find the geometric characterizations of the minimizers. We prove that if the charged 3-body problem admits a triangular central configuration, then the variational minimizing solutions of the problem in the \frac{\pi }{2}-antiperiodic function space are exactly defined by the circular motions of this triangular central configuration.

We construct a class of modules for the twisted multi-loop algebra of type A1×A1 by applying Wakimoto free bosonic realization. We also discuss the structures and the irreducibility of the Fock space.

Boolean functions possessing multiple cryptographic criteria play an important role in the design of symmetric cryptosystems. The following criteria for cryptographic Boolean functions are often considered:high nonlinearity, balancedness, strict avalanche criterion, and global avalanche characteristics. The trade-off among these criteria is a difficult problem and has attracted many researchers. In this paper, two construction methods are provided to obtain balanced Boolean functions with high nonlinearity. Besides, the constructed functions satisfy strict avalanche criterion and have good global avalanche characteristics property. The algebraic immunity of the constructed functions is also considered.

For any module V over the two-dimensional non-abelian Lie algebra b and scalar α∈ℂ, we define a class of weight modules F^α(V) with zero central charge over the affine Lie algebra A⁽¹⁾₁. These weight modules have infinitedimensional weight spaces if and only if V is infinite dimensional. In this paper, we will determine necessary and sufficient conditions for these modules F^α(V) to be irreducible. In this way, we obtain a lot of irreducible weight A⁽¹⁾₁-modules with infinite-dimensional weight spaces.

We study the Grushin operators acting on {ℝ}_x^{d\mathrm{1}}\times {ℝ}_t^{d\mathrm{2}} and defined by the formula L=-{\sum }_{j=1}^{d_{\mathrm{1}}}{\partial }_{x_j}^{\mathrm{2}}-{\sum }_{j=1}^{d_{\mathrm{1}}}{\left|x_j\right|}^2{\sum }_{k=1}^{d_{\mathrm{2}}}{\partial }_{t_k}^{\mathrm{2}}. We establish a restriction theorem associated with the considered operators. Our result is an analogue of the restriction theorem on the Heisenberg group obtained by D. Müller [Ann. of Math., 1990, 131: 567–587].

We introduce the notions of IDS modules, IP modules, and Baer^* modules, which are new generalizations of von Neumann regular rings, PPrings, and Baer rings, respectively, in a general module theoretic setting. We obtain some characterizations and properties of IDS modules, IP modules and Baer∗ modules. Some important classes of rings are characterized in terms of IDS modules, IP modules, and Baer^*modules.

For an acyclic quiver Q and a finite-dimensional algebra A, we give a unified form of the indecomposable injective objects in the monomorphism category Mon(Q,A) and prove that Mon(Q,A) has enough injective objects.

We study quasi-stationarity for one-dimensional diffusions killed at 0, when 0 is a regular boundary and+∞ is an entrance boundary. We give a necessary and sufficient condition for the existence of exactly one quasistationary distribution, and we also show that this distribution attracts all initial distributions.

This paper is devoted to investigating the bounded behaviors of the oscillation and variation operators for Calderón-Zygmund singular integrals and the corresponding commutators on the weighted Morrey spaces. We establish several criterions of boundedness, which are applied to obtain the corresponding bounds for the oscillation and variation operators of Hilbert transform, Hermitian Riesz transform and their commutators with BMO functions, or Lipschitz functions on weighted Morrey spaces.

We prove that almost all positive even integers n can be written as n=p_2^2+p_3^3+p_4^4+p_5^5 with \left|p{\displaystyle \frac{k}{k}}-{\displaystyle \frac{N}{\mathrm{4}}}\right|\le N^{\frac{\mathrm{321}}{\mathrm{325}}+ϵ} for 2≤k≤5. Moreover, it is proved that each sufficiently large odd integer N can be represented as N=p_1+p_2^2+p_3^3+p_4^4+p_5^5 with \left|p{\displaystyle \frac{k}{k}}-{\displaystyle \frac{N}{\mathrm{5}}}\right|\le N^{\frac{\mathrm{321}}{\mathrm{325}}+ϵ}for 1≤k≤5.

Shift Harnack inequality and integration by parts formula are established for semilinear stochastic partial differential equations and stochastic functional partial differential equations by modifying the coupling used by F. -Y. Wang [Ann. Probab., 2012, 42(3): 994–1019]. Log-Harnack inequality is established for a class of stochastic evolution equations with non-Lipschitz coefficients which includes hyperdissipative Navier-Stokes/Burgers equations as examples. The integration by parts formula is extended to the path space of stochastic functional partial differential equations, then a Dirichlet form is defined and the log-Sobolev inequality is established.

We construct free Hom-semigroups when its unary operation is multiplicative and is an involution. Our method of construction is by bracketed words. As a consequence, we obtain free Hom-associative algebras generated by a set under the same conditions for the unary operation.