Volume 10, Issue 4

An odd-quadratic Leibniz superalgebra is a (left or right) Leibniz superalgebra with an odd, supersymmetric, non-degenerate and invariant bilinear form. In this paper, we prove that a left (resp. right) Leibniz superalgebra that carries this structure is symmetric (meaning that it is simultaneously a left and a right Leibniz superalgebra). Moreover, we show that any non-abelian (left or right) Leibniz superalgebra does not possess simultaneously a quadratic and an odd-quadratic structure. Further, we obtain an inductive description of odd-quadratic Leibniz superalgebras using the procedure of generalized odd double extension and we reduce the study of this class of Leibniz superalgebras to that of odd-quadratic Lie superalgebras. Finally, several non-trivial examples of odd-quadratic Leibniz superalgebras are included.

In this note, we consider the semilinear heat system ∂t⁡u=Δ⁢u+f⁢(v),∂t⁡v=μ⁢Δ⁢v+g⁢(u),μ>0,\partial_{t}u=\Delta u+f(v),\quad\partial_{t}v=\mu\Delta v+g(u),\quad\mu>0, where the nonlinearity has no gradient structure taking of the particular form f⁢(v)=v⁢|v|p-1 and g⁢(u)=u⁢|u|q-1 with ⁢p,q>1,f(v)=v\lvert v\rvert^{p-1}\quad\text{and}\quad g(u)=u\lvert u\rvert^{q-1}\quad% \text{with }p,q>1, or f⁢(v)=ep⁢v and g⁢(u)=eq⁢u with ⁢p,q>0.f(v)=e^{pv}\quad\text{and}\quad g(u)=e^{qu}\quad\text{with }p,q>0. We exhibit type I blowup solutions for this system and give a precise description of its blowup profiles. The method relies on a two-step procedure: the reduction of the problem to a finite-dimensional one via a spectral analysis, and then solving the finite-dimensional problem by a classical topological argument based on index theory. As a consequence of our technique, the constructed solutions are stable under a small perturbation of initial data. The results and the main arguments presented in this note can be found in our papers [T.-E. Ghoul, V. T. Nguyen and H. Zaag, Construction and stability of blowup solutions for a non-variational semilinear parabolic system, Ann. Inst. H. Poincaré Anal. Non Linéaire 35 2018, 6, 1577–1630] and [M. A. Herrero and J. J. L. Velázquez, Generic behaviour of one-dimensional blow up patterns, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 19 1992, 3, 381–450].

In this work, an operator version of Popoviciu’s inequality for positive operators on Hilbert spaces under positive linear maps for superquadratic functions is proved. Analogously, using the same technique, an operator version of Popoviciu’s inequality for convex functions is obtained. Some other related inequalities are also presented.

We propose a finite element approximation for a fourth-order Steklov eigenvalue problem by means of the virtual elements. In this setting we derive error estimates for the eigenvalues and eigenfunctions under standard assumptions on the domain.

In this paper, we introduce an iterative scheme involving an inertial term and a step size independent of the operator norm for approximating a solution to a split variational inequality problem in a real Hilbert space. Furthermore, we prove a convergence theorem for the sequence generated by the proposed operator norm independent inertial accelerated iterative scheme. We applied our result to solve split convex minimization problems, split zero problem and further give a numerical example to demonstrate the efficiency of the proposed algorithm.

In this paper, we investigate the deterministic and stochastic prey-predator system with refuge. The basic local stability results for the deterministic model have been performed. It is found that all the equilibrium points except the positive coexisting equilibrium point of the deterministic model are independent of the prey refuge. The trivial equilibrium point, predator free equilibrium point and prey free equilibrium point are always unstable (saddle point). The existence and local stability of the coexisting equilibrium point is related to the prey refuge. The permanence and extinction conditions of the proposed biological model have been studied rigourously. It is observed that the stochastic effect may be seen in the form of decaying of the species. The numerical simulations for different values of the refuge values have also been included for understanding the behavior of the model graphically.

Let ℍ{\mathbb{H}} be a separable Hilbert space. In this paper, we establish a generalization of Walnut’s representation and Janssen’s representation of the ℍ{\mathbb{H}}-valued Gabor frame operator on ℍ{\mathbb{H}}-valued weighted amalgam spaces Wℍ⁢(Lp,Lvq){W_{\mathbb{H}}(L^{p},L^{q}_{v})}, 1≤p,q≤∞{1\leq p,q\leq\infty}. Also, we show that the frame operator is invertible on Wℍ⁢(Lp,Lvq){W_{\mathbb{H}}(L^{p},L^{q}_{v})}, 1≤p,q≤∞{1\leq p,q\leq\infty}, if the window function is in the Wiener amalgam space Wℍ⁢(L∞,Lw1){W_{\mathbb{H}}(L^{\infty},L^{1}_{w})}. Further, we obtain the Walnut representation and invertibility of the frame operator corresponding to Gabor superframes and multi-window Gabor frames on Wℍ⁢(Lp,Lvq){W_{\mathbb{H}}(L^{p},L^{q}_{v})}, 1≤p,q≤∞{1\leq p,q\leq\infty}, as a special case by choosing the appropriate Hilbert space ℍ{\mathbb{H}}.

We give the system of Laguerre–Freud equations for the recurrence coefficients βn{\beta_{n}}, γn+1{\gamma_{n+1}}, n≥0{n\geq 0}, of orthogonal polynomials with respect to a D-Laguerre–Hahn form (linear functional) of class one. The system is solved in the case when β0=-θ0{\beta_{0}=-\theta_{0}}, βn+1=θn-θn+1{\beta_{n+1}=\theta_{n}-\theta_{n+1}} and γn+1=-θn2{\gamma_{n+1}=-\theta_{n}^{2}} with θn≠0{\theta_{n}\neq 0}, n≥0{n\geq 0}. There are essentially three canonical cases.

We establish a weighted inequality for fractional maximal and convolution type operators, between weak Lebesgue spaces and Wiener amalgam type spaces on ℝ{\mathbb{R}} endowed with a measure which needs not to be doubling.

We set a definition of a (0,2){(0,2)}-type tensor on the generalized Sasakian-space-forms. The necessary and sufficient conditions for W -semisymmetric generalized Sasakian-space forms are studied. Certain results of the Ricci solitons, the Killing vector fields and the closed 1-form on the generalized Sasakian-space-forms are derived. We also verify our results by taking non-trivial examples of the generalized Sasakian-space-forms.

We prove some important properties of the p -resolvent mapping recently introduced by B. J. Choi and U. C. Ji, The proximal point algorithm in uniformly convex metric spaces, Commun. Korean Math. Soc. 31 2016, 4, 845–855, in p -uniformly convex metric space. Furthermore, we introduce a modified Mann-type PPA involving nonexpansive mapping and prove that the sequence generated by the algorithm converges to a common solution of a finite family of minimization problems, which is also a fixed point of a nonexpansive mapping in the framework of a complete p -uniformly convex metric space.

In this paper, we are interested in the study of the existence of positive solutions for the following nonlinear boundary value problem on the half-line: {-u′′⁢(x)=q⁢(x)⁢f⁢(x,u,u′),x∈(0,+∞),u′⁢(0)=u′⁢(+∞)=0,\left\{\begin{aligned} \displaystyle-u^{\prime\prime}(x)&\displaystyle=q(x)f(x% ,u,u^{\prime}),&&\displaystyle x\in(0,+\infty),\\ \displaystyle u^{\prime}(0)&\displaystyle=u^{\prime}(+\infty)=0,\end{aligned}\right. where q:ℝ+→ℝ+{q:\mathbb{R^{+}}\rightarrow\mathbb{R^{+}}} is a positive measurable function such that ∫0+∞q⁢(x)⁢𝑑x=1{\int_{0}^{+\infty}q(x)\,dx=1} and f:ℝ+×ℝ2→ℝ{f:\mathbb{R}^{+}\times\mathbb{R}^{2}\rightarrow\mathbb{R}} is q -Carathéodory.

A non-trivial multi-norm structure based on the enveloping C*{C^{*}}-algebras will be discussed in this paper. The main example is group C*{C^{*}}-algebras on compact groups. Also a (generalizing) non-trivial multi-norm structure based on the Fourier and Fourier–Stieltjes algebras on compact groups will be given.

The singular values of products of standard complex Gaussian random matrices, or sub-blocks of Haar distributed unitary matrices, have the property that their probability distribution has an explicit, structured form referred to as a polynomial ensemble. It is furthermore the case that the corresponding bi-orthogonal system can be determined in terms of Meijer G-functions, and the correlation kernel given as an explicit double contour integral. It has recently been shown that the Hermitised product XM⁢⋯⁢X2⁢X1⁢A⁢X1T⁢X2T⁢⋯⁢XMT{X_{M}\cdots X_{2}X_{1}AX_{1}^{T}X_{2}^{T}\cdots X_{M}^{T}}, where each Xi{X_{i}} is a standard real Gaussian matrix and A is real anti-symmetric, exhibits analogous properties. Here we use the theory of spherical functions and transforms to present a theory which, for even dimensions, includes these properties of the latter product as a special case. As an example we show that the theory also allows for a treatment of this class of Hermitised product when the Xi{X_{i}} are chosen as sub-blocks of Haar distributed real orthogonal matrices.