We investigate and concentrate on new infinitesimal generators of Lie symmetries for an extended (2+ 1)-dimensional Calogero-Bogoyavlenskii-Schif (eCBS) equation using the commutator table which results in a system of nonlinear ordinary differential equations (ODEs) which can be manually solved. Through two stages of Lie symmetry reductions, the eCBS equation is reduced to non-solvable nonlinear ODEs using different combinations of optimal Lie vectors. Using the integration method and the Riccati and Bernoulli equation methods, we investigate new analytical solutions to those ODEs. Back substituting to the original variables generates new solutions to the eCBS equation. These results are simulated through three- and two-dimensional plots.

This paper is devoted to constructing some recollements of additive categories associated to concentric twin cotorsion pairs on an extriangulated category. As an application, this result generalizes the work by W. J. Chen, Z. K. Liu, and X. Y. Yang in a triangulated case [J. Algebra Appl., 2018, 17(5): 1–15]. Moreover, it highlights new phenomena when it applied to an exact category. Finally, we give some applications of our main results. In particular, we obtain Krause's recollement whose proofs are both elementary and very general.

This paper is a subsequent work of [Invent. Math., 2013, 191: 197-253]. The second fundamental theorem in Ahlfors covering surface theory is that, for each set E_q of q (≥3) distinct points in the extended complex plane \overline{ℂ}; there is a minimal positive constant H₀ (E_q) (called Ahlfors constant with respect to E_q), such that the inequality

(q-2)A(\sum )-4\pi ♯(f^{-1}(E_q)\cap U)\le H_0(E_q)L(\partial \sum )

holds for any simply-connected surface{\displaystyle \sum \mathrm{=}（f,\bar{U}）} ; where A(\sum) is the area of\sum; L(\partial \sum) is the perimeter of\sum; and # denotes the cardinality. It is difficult to compute H₀ (E_q) explicitly for general set E_q; and only a few properties of H₀(E_q) are known. The goals of this paper are to prove the continuity and differentiability of H₀ (E_q); to estimate H₀ (E_q); and to discuss the minimum of H₀ (E_q) for fiixed q.

Suppose that the underlying field is of characteristic different from 2 and 3. We first prove that the so-called stem deformations of a free presentation of a finite-dimensional Lie superalgebra L exhaust all the maximal stem extensions of L; up to equivalence of extensions. Then we prove that multipliers and covers always exist for a Lie superalgebra and they are unique up to superalgebra isomorphisms. Finally, we describe the multipliers, covers, and maximal stem extensions of Heisenberg superalgebras and model filiform Lie superalgebras.

We study the well-posedness and decay properties of a onedimensional thermoelastic laminated beam system either with or without structural damping, of which the heat conduction is given by Fourier's law effective in the rotation angle displacements. We show that the system is wellposed by using the Lumer-Philips theorem, and prove that the system is exponentially stable if and only if the wave speeds are equal, by using the perturbed energy method and Gearhart-Herbst-Prüss-Huang theorem. Furthermore, we show that the system with structural damping is polynomially stable provided that the wave speeds are not equal, by using the second-order energy method. When the speeds are not equal, whether the system without structural damping may has polynomial stability is left as an open problem.

We present upper bounds of eigenvalues for finite and infinite dimensional Cauchy-Hankel tensors. It is proved that an m-order infinite dimensional Cauchy-Hankel tensor defines a bounded and positively (m－1)-homogeneous operator from l¹ into l^p (1<p<∞); and two upper bounds of corresponding positively homogeneous operator norms are given. Moreover, for a fourth-order real partially symmetric Cauchy-Hankel tensor, suffcient and necessary conditions of M-positive definiteness are obtained, and an upper bound of M-eigenvalue is also shown.

We focus on the elliptic genera of level N at the cusps of a congruence subgroup for any complete intersection. Writing the first Chern class of a complete intersection as a product of an integral coefficient c₁ and a generator of the 2nd integral cohomology group, we mainly discuss the values of the elliptic genera of level N for the complete intersection in the cases of c₁>, =, or<0, In particular, the values about the Todd genus, \widehat{A}\mathrm{-genus}, and A_k-genus can be derived from the elliptic genera of level N.

Let u=\{u(t,x);(t,x)\in {ℝ}_{+}\times ℝ\}be the solution to a linear stochastic heat equation driven by a Gaussian noise, which is a Brownian motion in time and a fractional Brownian motion in space with Hurst parameterH\in (\mathrm{0},\mathrm{1}): For any givenx\in ℝ(\text{resp}\text{.},t\in {ℝ}_{+}), we show a decomposition of the stochastic processt\to u(t,x)(\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}.,x\to u(t,x))as the sum of a fractional Brownian motion with Hurst parameter H/2 (resp., H) and a stochastic process with C^∞-continuous trajectories. Some applications of those decompositions are discussed.

Using the degeneration formula, we study the change of Gromov-Witten invariants under blow-up for symplectic 4-manifolds and obtain a genus-decreasing relation of Gromov-Witten invariant of symplectic four manifold under blow-up.

We define the Whittaker modules over the simply-connected quantum group U_q（{\mathfrak{s}\mathfrak{l}}_{\mathrm{3}},\wedge ） ; where \wedge is the weight lattice of Lie algebra {\mathfrak{s}\mathfrak{l}}_{\mathrm{3}}: Then we completely classify all those simple ones. Explicitly, a simple Whittaker module over U_q（{\mathfrak{s}\mathfrak{l}}_{\mathrm{3}},\wedge ） is either a highest weight module, or determined by two parametersz\in ℂ\text{} and\gamma \in {ℂ}^{*}\text{} (up to a Hopf automorphism).

The Fourier matrix is fundamental in discrete Fourier transforms and fast Fourier transforms. We generalize the Fourier matrix, extend the concept of Fourier matrix to higher order Fourier tensor, present the spectrum of the Fourier tensors, and use the Fourier tensor to simplify the high order Fourier analysis.

We are concerned with SIR epidemics in a random environment on complete graphs, where edges are assigned with i.i.d. weights. Our main results give large and moderate deviation principles of sample paths of this model. Our results generalize large and moderate deviation principles of the classic SIR models given by E. Pardoux and B. Samegni-Kepgnou [J. Appl. Probab., 2017, 54: 905-920] and X. F. Xue [Stochastic Process. Appl., 2019, 140: 49-80].

This paper is devoted to studying the Marcinkiewicz integral operators associated to polynomial compound curves. Some new bounds for the above operators on the Lebesgue, Triebel-Lizorkin, and Besov spaces are established by assuming that their rough kernels are given by\mathrm{\Omega }\in H^{\mathrm{1}}（{\mathrm{S}}^{n-\mathrm{1}}） andh\in {\mathrm{\Delta }}_{}（{ℝ}_{+}）for some\text{γ>1}: It should be pointed out that the bounds are independent of h,\mathrm{\Omega },\gamma and the coefficients of the polynomials in the definition of the operators.

At each time n\in \mathbb{N},\mathrm{l}\mathrm{e}\mathrm{t}{\overline{Y}}^{(n)}(\xi )=(y_1^{(n)}(\xi ),y_2^{(n)}(\xi ),\cdots ) be a random sequence of non-negative numbers that are ultimately zero in a random environment\xi \text{=}（{\xi }_n{）}_{n\in \mathbb{N}}. The existence and uniqueness of the nonnegative fixed points of the associated smoothing transformation in random environment are considered. These fixed points are solutions to the distributional equation for a.e.\xi ,Z(\xi )\stackrel{\text{d}}{=}{\sum }_{i\in \mathbb{N}+}{y}_i^{(\mathrm{0})}(\xi )Z_i^{(\mathrm{1})}(\xi )，where ｛Z_i^{(\mathrm{1})}:i\in {\mathbb{N}}_{+}｝ are random variables in random environment which satisfy that for any environment\xi; under P_{\xi }; ｛Z_i^{(\mathrm{1})}:i\in {\mathbb{N}}_{+}｝are independent of each other and {\bar{Y}}^{(\mathrm{0})}(\xi ), and have the same conditional distribution P_{\xi }(Z_i^{(1)}(\xi )\in \cdot )=P_{T\xi }(Z(T\xi )\in \cdot ) where T is the shift operator. This extends the classical results of J. D. Biggins [J. Appl. Probab., 1977, 14: 25-37] to the random environment case. As an application, the martingale convergence of the branching random walk in random environment is given as well.