Volume 30, Issue 3

We explicitly describe the second cohomology of the Lie superalgebra 𝒦 ⁢ ( 1 ) {\mathcal{K}(1)} of contact vector fields on the supercircle S 1 | 1 {S^{1|1}} with coefficients in the spaces of weighted densities. We deduce the second cohomology of 𝒦 ⁢ ( 1 ) {\mathcal{K}(1)} with coefficients in the Poisson algebra of pseudodifferential symbols on S 1 | 1 {S^{1|1}} . We study formal deformations of the standard embedding of 𝒦 ⁢ ( 1 ) {\mathcal{K}(1)} into the Poisson superalgebra of pseudodifferential symbols.

In this paper, we show that, under not too restrictive conditions, results much stronger than those obtained earlier by Hejduk can be established in category bases.

In this paper we show that there exist infinitely many square-free numbers of the form n 2 + n + 1 {n^{2}+n+1} . We achieve this by deriving an asymptotic formula by improving the reminder term from previous results.

This paper presents the robust and stable difference scheme to estimate singularly perturbed Sobolev boundary value problems with Robin type boundary condition. Firstly, the asymptotic behavior of the solution is analyzed. By using interpolating quadrature rules and basis functions, a completely exponentially fitted tree-level difference scheme is constructed on the uniform mesh. Then an error estimation is investigated in a discrete energy norm. Two numerical examples are solved and the computational results are tabulated.

We consider a parametric Neumann problem driven by a nonhomogeneous differential operator, with a reaction consisting of a critical term plus a Carathéodory perturbation which is only locally restricted near zero and is not assumed to be odd. Combining variational methods, with upper-lower solutions and truncation techniques, and flow invariance arguments, we show that the problem admits infinitely many sign-changing solutions.

Consider the equation { - Δ H 3 ⁢ u = f ⁢ ( ξ , u , ∇ H ⁡ u , ∇ H 2 ⁡ u , ∇ H 3 ⁡ u , ∇ H 4 ⁡ u , ∇ H 5 ⁡ u ) in  ⁢ Ω , u > 0 in  ⁢ Ω , u = ∂ ∂ ⁡ ν ⁢ ( ∇ H 2 ⁡ u ) = ∂ ∂ ⁡ ν ⁢ ( ∇ H 3 ⁡ u ) = 0 on  ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle{}{-}\Delta_{H}^{3}u&\displaystyle=f(\xi,u% ,\nabla_{H}u,\nabla_{H}^{2}u,\nabla_{H}^{3}u,\nabla_{H}^{4}u,\nabla_{H}^{5}u)&% &\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle>0&&\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=\dfrac{\partial}{\partial\nu}(\nabla_{H}^{2}u)=% \dfrac{\partial}{\partial\nu}(\nabla_{H}^{3}u)=0&&\displaystyle\phantom{}\text% {on }\partial\Omega,\end{aligned}\right. where Ω is a domain of the finite-dimensional space ℍ n {\mathbb{H}^{n}} and f is a positive and bounded function. We prove the existence of a solution for the above equation. In addition, we prove the uniqueness and the cylindrical symmetry of the solution.

It is known that a translation invariant density differentiation basis B formed of convex sets and its Busemann–Feller extension B BF {B_{\operatorname{BF}}} have similar properties, in particular, the Busemann–Feller extension differentiates the same class of non-negative functions as the basis B . We prove the optimality of the convexity condition in this result.

Let 𝒜 be a prime ∗-algebra. For any A , B ∈ A \mathscr{A},\mathscr{B}\in\mathcal{A} , a product A ⋆ B = A ⁢ B * + B ⁢ A * \mathscr{A}\star\mathscr{B}=\mathscr{A}\mathscr{B}^{*}+\mathscr{B}\mathscr{A}^{*} is called a bi-skew Jordan product. In this paper, it is shown that every multiplicative bi-skew Jordan triple derivation is an additive ∗-derivation on a prime ∗-algebra.

It is shown that, for any uncountable solvable group ( G , ⋅ ) {(G,\cdot\,)} and any nonzero σ-finite left G -quasi-invariant measure μ on G , there exist two G -absolutely negligible subsets A and B of G such that the product A ⋅ B {A\cdot B} is not measurable with respect to μ.

In the classical Fourier analysis, the representation of the double Fourier transform as the integral of f ( x , y ) exp ( − i ⟨ ( x , y ) , ( s 1 , s 2 ) ⟩ f(x,y)\exp(-i\langle(x,y),(s_{1},s_{2})\rangle is usually defined from the Lebesgue integral. Using an improper Kurzweil–Henstock integral, we obtain a similar representation of the Fourier transform of non-Lebesgue integrable functions on R 2 \mathbb{R}^{2} . We prove the Riemann–Lebesgue lemma and the pointwise continuity for the classical Fourier transform on a subspace of non-Lebesgue integrable functions which is characterized by the bounded variation functions in the sense of Hardy–Krause. Moreover, we generalize some properties of the classical Fourier transform defined on L p ⁢ ( R 2 ) L^{p}(\mathbb{R}^{2}) , where 1 < p ≤ 2 1<p\leq 2 , yielding a generalization of the results obtained by E. Hewitt and K. A. Ross. With our integral, we define the space of integrable functions KP ⁢ ( R 2 ) \mathrm{KP}(\mathbb{R}^{2}) which contains a subspace whose completion is isometrically isomorphic to the space of integrable distributions on the plane as defined by E. Talvila [The continuous primitive integral in the plane, Real Anal. Exchange 45 (2020), 2, 283–326]. A question arises about the dual space of the new space KP ⁢ ( R 2 ) \mathrm{KP}(\mathbb{R}^{2}) .

In this paper, we study the question of representing one class of Brownian functionals as a stochastic Itô integral with an explicit form of the integrand. The considered class of functionals also includes functionals that are not smooth in the sense of Malliavin, to which both the well-known Clark–Ocone formula (1984) and its generalization, the Glonti–Purtukhia formula (2017), are inapplicable.

In this paper, we discuss the generalized Bessel matrix functions and their integral representations. Also, we derive some properties using integral transform and fractional calculus of these functions.

In this paper, we study initial and initial-boundary problems for the time-fractional degenerate diffusion equations in bounded and unbounded domains. We obtain the existence and uniqueness of solutions using Fourier methods.

This paper mainly gives some sufficient and necessary conditions for an element in a ring to be EP {\mathrm{EP}} , partial isometry and strongly EP {\mathrm{EP}} by some equalities, using the solutions of certain equations and constructing the invertible elements in a ring.

Let α i \alpha_{i} , β i > 0 \beta_{i}>0 , 1 ≤ i ≤ n 1\leq i\leq n , and for t > 0 t>0 and x = ( x 1 , … , x n ) ∈ R n x=(x_{1},\dots,x_{n})\in\mathbb{R}^{n} , let t ⋅ x = ( t α 1 ⁢ x 1 , … , t α n ⁢ x n ) , t ∘ x = ( t β 1 ⁢ x 1 , … , t β n ⁢ x n ) , t\cdot x=(t^{\alpha_{1}}x_{1},\dots,t^{\alpha_{n}}x_{n}),\quad t\circ x=(t^{\beta_{1}}x_{1},\dots,t^{\beta_{n}}x_{n}), and let α = α 1 + ⋯ + α n , ∥ x ∥ α = ∑ i = 1 n | x i | 1 α i . \alpha=\alpha_{1}+\dots+\alpha_{n},\quad\lVert x\rVert_{\alpha}=\sum_{i=1}^{n}\lvert x_{i}\rvert^{\frac{1}{\alpha_{i}}}. Let φ 1 , … , φ n \varphi_{1},\dots,\varphi_{n} be real functions in C ∞ ⁢ ( R n ∖ { 0 } ) C^{\infty}(\mathbb{R}^{n}\setminus\{0\}) such that φ = ( φ 1 , … , φ n ) \varphi=(\varphi_{1},\dots,\varphi_{n}) is a homogeneous function with respect to these groups of dilations, i.e., φ ⁢ ( t ⋅ x ) = t ∘ φ ⁢ ( x ) \varphi(t\cdot x)=t\circ\varphi(x) . Let γ > 0 \gamma>0 and let 𝜇 be the Borel measure in R 2 ⁢ n \mathbb{R}^{2n} given by μ ⁢ ( E ) = ∫ χ E ⁢ ( x , φ ⁢ ( x ) ) ⁢ ∥ x ∥ α γ − α ⁢ d x . \mu(E)=\int\chi_{E}(x,\varphi(x))\lVert x\rVert_{\alpha}^{\gamma-\alpha}\,dx. Let T μ ⁢ f = μ ∗ f T_{\mu}f=\mu\ast f , f ∈ S ⁢ ( R 2 ⁢ n ) f\in S(\mathbb{R}^{2n}) . In this paper, we study the boundedness of T μ T_{\mu} from L p ⁢ ( ⋅ ) ⁢ ( R 2 ⁢ n ) L^{p(\,{\cdot}\,)}(\mathbb{R}^{2n}) into L q ⁢ ( ⋅ ) ⁢ ( R 2 ⁢ n ) L^{q(\,{\cdot}\,)}(\mathbb{R}^{2n}) for certain variable exponents p ⁢ ( ⋅ ) p(\,{\cdot}\,) and q ⁢ ( ⋅ ) q(\,{\cdot}\,) .

Let A and B be Artin algebras and let M be an ( A , B ) (A,B) -bimodule with M A {}_{A}M and M B M_{B} finitely generated. In this paper, we construct tilting pairs of subcategories and Wakamatsu tilting subcategories over an upper triangular matrix Artin algebra Λ = ( A M 0 B ) {\Lambda=\bigl{(}\begin{smallmatrix}A&M\\ 0&B\\ \end{smallmatrix}\bigr{)}} using tilting pairs and Wakamatsu tiling subcategories over A and B . Let 𝒞 {\mathcal{C}} be a subcategory of A ⁢ -mod {A\mbox{-mod}} and let 𝒟 {\mathcal{D}} be a subcategory of B ⁢ -mod {B\mbox{-mod}} . Consider the subcategory of left Λ-modules 𝔅 𝒟 𝒞 = { ( X Y ) f : f  is a monomorphism,  Y ∈ 𝒟  and  Coker ⁡ f ∈ 𝒞 } {\mathfrak{B}^{\mathcal{C}}_{\mathcal{D}}=\{{\bigl{(}\begin{smallmatrix}{X}\\ {Y}\\ \end{smallmatrix}\bigr{)}_{f}}:\text{$f$ is a monomorphism, $Y\in\mathcal{D}$ % and $\operatorname{Coker}f\in\mathcal{C}$}\}} . We prove the following results: (1) Assume that M ⊗ B 𝒯 ′ ⊆ 𝒯 {M\otimes_{B}\mathcal{T}^{\prime}\subseteq\mathcal{T}} , M ⊗ B 𝒞 ′ ⊆ 𝒞 {M\otimes_{B}\mathcal{C}^{\prime}\subseteq\mathcal{C}} and Tor i B ⁢ ( M , 𝒯 ′ ) = 0 {\mathrm{Tor}^{B}_{i}(M,\mathcal{T}^{\prime})=0} for all i ≥ 1 {i\geq 1} . Then ( 𝒞 , 𝒯 ) {(\mathcal{C},\mathcal{T})} and ( 𝒞 ′ , 𝒯 ′ ) {(\mathcal{C}^{\prime},\mathcal{T}^{\prime})} are n -tilting pairs respectively in A ⁢ - ⁢ mod {A\text{-}\mathrm{mod}} and B ⁢ - ⁢ mod {B\text{-}\mathrm{mod}} if and only if ( 𝔅 𝒞 ′ 𝒞 , 𝔅 𝒯 ′ 𝒯 ) {(\mathfrak{B}^{\mathcal{C}}_{\mathcal{C}^{\prime}},\mathfrak{B}^{\mathcal{T}}% _{\mathcal{T}^{\prime}})} is an n -tilting pair in Λ ⁢ - ⁢ mod {\Lambda\text{-}\mathrm{mod}} . (2) Assume that M ⊗ B 𝒱 ⊆ 𝒲 {M\otimes_{B}\mathcal{V}\subseteq\mathcal{W}} and Tor i B ⁢ ( M , 𝒱 ⊥ ) = 0 {\mathrm{Tor}^{B}_{i}(M,{{}^{\perp}\mathcal{V}})=0} for all i ≥ 1 {i\geq 1} . If 𝒲 {\mathcal{W}} and 𝒱 {\mathcal{V}} are Wakamatsu tilting subcategories respectively in A ⁢ - ⁢ mod {A\text{-}\mathrm{mod}} and B ⁢ - ⁢ mod {B\text{-}\mathrm{mod}} , then 𝔅 𝒱 𝒲 {\mathfrak{B}^{\mathcal{W}}_{\mathcal{V}}} is a Wakamatsu tilting subcategory in Λ ⁢ - ⁢ mod {\Lambda\text{-}\mathrm{mod}} .