Examples and Counterexamples最新文献

The $A$-energy of a graph $G$, denoted by $E_A\left(G\right)$, is defined as sum of the absolute values of eigenvalues of adjacency matrix of $G$. Nikiforov in Nikiforov (2016) proved that $E_A\left(\overline{G}\right)-E_A\left(G\right)\le 2{\overline{\mu }}_1$ and $E_A\left(G\right)-E_A\left(\overline{G}\right)\le 2{\mu }_1$ for any graph $G$ and posed a problem to find best possible upper bound for $E_A\left(G\right)-E_A\left(\overline{G}\right)$, where ${\mu }_1$ and $\overline{{\mu }_1}$ are the largest adjacency eigenvalues of $G$ and its complement $\overline{G}$ respectively. We attempt to provide an answer by giving an improved upper bound on a class of graphs where regular graphs become particular case. As a consequence, it is proved that there is no strongly regular graph with negative eigenvalues greater than $-1$. The obtained results also improves some of the other existing results.

Minimal codes are being intensively studied in last years. ${[n,k]}_q$-minimal linear codes are in bijection with strong blocking sets of size $n$ in $PG(k-1,q)$ and a lower bound for the size of strong blocking sets is given by $(k-1)(q+1)\le n$. In this note we show that all strong blocking sets of length 9 in $PG(3,2)$ are the hyperbolic quadrics $Q^{+}(3,2)$.

Let $ℱ$ be a finite nonempty family of finite nonempty sets. We prove the following: (1) $ℱ$ satisfies the condition of the title if and only if for every pair of distinct subfamilies $\{A_1,\dots ,A_r\}$, $\{B_1,\dots ,B_s\}$ of $ℱ$, $\bigcup _{i=1}^r{A}_i\ne \bigcup _{i=1}^s{B}_i$. (2) If $ℱ$ satisfies the condition of the title, then the number of subsets of $\bigcup _{A\in ℱ}A$ containing at least one set of $ℱ$ is odd. We give two applications of these results, one to number theory and one to commutative algebra.

An incompressible MHD nanofluid boundary layer flow over a vertical stretching permeable surface employing Buongiorno’s design investigated by considering the convective states. The Brownian motion and thermophoresis effects are used to implement the nanofluid model. Operating the similarity transmutations, to transform the governing partial differential equations into ordinary differential equations consisting of the momentum, energy, and concentration fields and later worked by using a program written together with the stiffness shifting in Wolfram Language. The consequences of various physical parameters on the velocity, temperature, and concentration fields are analyzed, such as magnetic parameter $M$, Brownian motion parameter $Nb$, thermophoresis parameter $Nt$, Lewis number $Le$, temperature Biot number $B{i}_{\theta }$, concentration Biot number $B{i}_{\varphi }$, and suction parameter $f_w$. Furthermore, the Skin friction coefficient, local Nusselt, and local Sherwood numbers concerning magnetic parameter for various values of physical parameters (i.e. $f_w$, $Nb$) are obtained graphically, then the outcome is validated with other recent works. Finally, introduced a new environment to employ machine learning by performing the sensitivity analysis based on the iterative method for predicting the Skin friction coefficient, reduced Nusselt number, and Sherwood number with respect to magnetic parameter for suction parameter and Brownian motion parameter. Machine learning algorithms provide a strong and quick data processing structure to enhance the actual research procedures and industrial application of fluid mechanics. These techniques have been upgraded and organized for fluid flow characteristics. The present optimization process has the potential for a new perspective on the metallurgical process, heat exchangers in electronics, and some medicinal applications.

Some series of Group divisible designs using generalized Bhaskar Rao designs over Dihedral, Symmetric and Alternating groups are obtained.

Rinne et al. (2018) conduct a detailed analysis of the impact of wind turbine technology and land-use on wind power potentials, which allows important insights into each factor’s contribution to overall potentials. The paper presents a detailed and very valuable model of site-specific wind turbine investment cost (i.e. road- and grid access costs), complemented by a model used to estimate site-independent costs.

However, the site-independent cost model is flawed in our opinion. This flaw most likely does not impact the results on cost supply-curves of wind power presented in the paper. However, we expect a considerable generalization error. Thus the application of the wind turbine cost model in other contexts may lead to unreasonable results. More generally, the derivation of the wind turbine cost model serves as an example of how applications of automated regression analysis can go wrong.

A high-resolution compact discretization scheme for the numerical approximation of two-point nonlinear fractal boundary value problems is presented to study the stationary anomalous diffusion process. Hausdorff derivative is applied to derive the models in fractal media. The proposed scheme solves the nonlinear fractal model and achieves an accuracy of order four by employing only three mesh points in a stencil and consumes short computing time. Numerical simulations with heat conduction in polar bear, convection–diffusion, boundary layer, Bessel’s and Burgers equation in a fractal medium are carried out to illustrate the utility of the scheme and their numerical rate of convergence.

Finite element approximations of poroelastic materials are nowadays used within multiple applications. Due to wide variation of possible material parameters, robustness of the considered discretization is important. Within this contribution robust of discretization schemes, initially developed for Biot’s theory, will be applied within the Theory of Porous Media. Selected numerical test-cases, special attention will be paid to incompressible and impermeable regimes, are conducted.

In this work, we solve partial differential equations using the Aboodh transform and the homotopy perturbation method (HPM). The Swift–Hohenberg equation accurately describes pattern development and evolution. The Swift–Hohenberg (S–H) model is linked to fluid dynamics, temperature, and thermal convection, and it can be used to describe how liquid surfaces with a horizontally well-conducting boundary form.