Examples and Counterexamples最新文献

A recent classification of flag-transitive 2-designs with parameters $(v,k,\lambda )$ whose replication number $r$ is coprime to $\lambda$ gives rise to eight possible infinite families of 2-designs, some of which are with new parameters. In this note, we give explicit constructions for two of these families of 2-designs, and show that for a given positive integer $q=3^{2n+1}⩾27$, there exist 2-designs with parameters $(q^3+1,q^i,q^i-1)$, for $i=1,2$, admitting the Ree group ${}^2{G}_2\left(q\right)$ as their automorphism groups.

In this note we show that for each positive integer $a⩾2$ there exist infinitely many trees whose spectral radius is equal to $\sqrt{2a}$. Such trees are obtained by replacing the central edge of the double star $S(a,2a-2)$ with suitable bidegreed caterpillars.

In this corrigendum, we correct some errors in the proof of Theorem 2.1 in “On the conjecture of Sombor energy of a graph” [Examples and Counterexamples 3 (2023) 100115].

In this note it is shown that there is a bounded linear operator $T$ on the Hardy Hilbert space $H^2$ and a vector $f$ in $H^2$ such that the closure of the set $\{\alpha T^{n}f:\alpha \in ℂ,n\ge 0\}$ is not $H^2$, but for every subsequence ${\left(n_k\right)}_{k=1}^{\infty }$ the closed linear span of $\{T^{n_k}f:k\ge 1\}$ is the whole space $H^2$. Furthermore, the closure of $\{T^{n}g:n\ge 0\}$ is $H^2$ for some $g\in H^2$.

We termed the Pál type interpolation problem as PTIP. Here we studied the regularity of $(0,1)$-PTIP and $(0,2)$-PTIP, where we omitted a real or complex node from a set of zeros of polynomials with complex coefficients.

In this paper, we give a main example indicating the ineffectiveness of the local fractional derivatives on the Riemann curvature tensor that is a common tool in calculating curvature of a Riemannian manifold. For this, first we introduce a general local fractional derivative operator that involves the mostly used ones in the literature as conformable, alternative, truncated $M-$ and $V-$fractional derivatives. Then, according to this general operator, a particular Riemannian metric on the real affine space $R^n$ that is different from the Euclidean one is defined. In conclusion, our main example states that the Riemann curvature tensor of $R^n$ endowed with this particular metric is identically 0, that is, one is locally isometric to Euclidean space.

We consider a convex combination of two classes of Lotka–Volterra operators defined on 2-dimensional simplex. Earlier, the dynamics of a particular case of the considered operators has been investigated. However, its bijective property was not studied. In this paper, we are able to establish that such maps are homeomorphism of the simplex.

In a recent paper (Araya and Harada, 2023), Araya and Harada gave examples of self-dual near-extremal ternary codes of length 48 for 145 distinct values of the number $A_{12}$ of codewords of minimum weight 12, and raised the question about the existence of codes for other values of $A_{12}$. In this note, we use symmetric 2-$(47,23,11)$ designs with an automorphism group of order 6 to construct self-dual near-extremal ternary codes of length 48 for 150 new values of $A_{12}$.

In this paper, we determine some degree-based topological indices, such as the Sombor index, the first and second Zagreb indices, the forgotten topological index, the Narumi–Katayama index, the first and second multiplicative Zagreb indices, the atom-bond connectivity index, and eccentricity-based topological indices such as total eccentricity, the first and second Zagreb eccentricity indices, and the eccentricity connectivity index of the zero divisor graph with vertex set non-zero zero divisors of the reduced ring of the direct product of three finite fields.

With multiple time delays, we investigated a Caputo fractional order dynamical system involving susceptible, exposed, infected, and recovered individuals. Positivity and boundedness are also theoretically demonstrated using Laplace transform and Mittag-Leffler function. The stability of the disease-free and epidemic equilibrium points has been studied for both delayed and non-delayed model. For generating numerical solutions to the model system, we used the Adam-Bashforth-Moulton predictor-corrector technique. With the help of MATLAB (2018a), we were able to conduct graphical demonstrations and numerical simulations. The system displays Hopf bifurcation and the solutions are no longer periodic beyond a certain threshold value of the time delay parameters.