Irreducibility for cellular algebras can be investigated by solving problems in linear algebra. In this work, we discuss a large family of cell modules over Fuss-Catalan algebras and describe dimensions and basis diagrams of these cell modules. Also, conditions for these modules to be irreducible are determined by their Gram matrices and determinants.

Physiologically based pharmacokinetic (PBPK) models can be used to develop frameworks for risk assessment and predictive toxicology testing routines. However, the predictive power of these models is only as good as the confidence in the parameters within the model itself. Sensitivity analysis, or the study of the effect of propagated error on the predictive power of the model, can be used to determine which model parameters are most likely to affect change in the model. This is important when considering optimization routines, as optimizing non-sensitive parameters may lead to biologically incorrect parameter estimates. This study explores the sensitivity of physiological, metabolic, and chemical-specific parameters for a published dermal exposure PBPK model of bromochloromethane.

We present complete antiderivative formulas not reduction formulas) for $ \int x^{n} \sqrt{a^2 - x^2} \ dx $, $ \int x^{n}/\sqrt{a^2 - x^2} \ dx $, $ \int \sqrt{a^2 - x^2}/x^{n} \ dx $, and $ \int 1/(x^{n}\sqrt{a^2 - x^2}) \ dx $, where $ n $ is any nonnegative integer. We shall provide a complete proof of one formula, the proofs of the others being quite similar.

We determine a formula for all coefficients of the Taylor series for the function $ \frac{1}{1+x^2} $ centered at $ x=a $ for any real number $ a $ along with the radius of convergence of the series. We utilize complex numbers to extend a standard calculus technique involving geometric series, but we write the coefficients using real numbers only.

Suppose $L/F$ is a finite extension of fields of characteristic 0. We study when $L$ can be defined by an irreducible power compositional polynomial; that is, a polynomial of the form $f(x^k)\in F[x]$. If $L$ contains the $k$th roots of unity, then Kummer theory provides an answer. In this case, $f(x^k)$ can be constructed explicitly using automorphisms of $L/F$. If $L$ does not contain the $k$th roots of unity, we use a result of Kang to give a complete answer when $k=3$ and $[L:F]=6$, and we construct a polynomial $x^6+ax^3+b$ defining $L/F$. As an application, we give a simple method for determining the Galois group of an irreducible polynomial $x^6+ax^3+b\in F[x]$.

The elliptical range theorem asserts that the field of values (or numerical range) of a two-by-two matrix with complex entries is an elliptical disk, the foci of which are the eigenvalues of the given matrix. Many proofs of this result are available in the literature, but most, with one exception, are computational and quite involved. In this note, it is shown that the elliptical range theorem follows from the properties of plane algebraic curves and a straightforward application of a well-known result due to Kippenhahn.

In this paper, we define $\mathfrak{I}$-Menger, $\mathfrak{I}$-star Menger, and $\mathfrak{I}$-strongly star Menger spaces using ideals, and give their relations to related spaces. We also investigate some properties of them. Finally, we show that the concepts of $\mathfrak{I}$-Menger, $\mathfrak{I}$-star Menger, and $\mathfrak{I}$-strongly star Menger are equivalent in the class of paracompact spaces.

In this paper, we give some useful characterizations of right weakly regular semigroups of generalized bipolar fuzzy subsemigroups. Based on these characterizations, we also characterize right weakly regular semigroups in terms of generalized bipolar fuzzy semigroups.

We show that the matching book thickness of the Cartesian product of two odd-length cycle-graphs is five, if at least one of the cycles has length 3 or 5.

The middle coefficient of a polynomial of an even degree $d$ is the coefficient of $x^{\frac{d}{2}}$. In this note, we compute the middle coefficient of the cyclotomic polynomial $\Phi_{3p_2p_3}(x)$ when $p_3\equiv \pm 7,\pm 8, \pm10 \mod3p_2$. Also, we compute the middle coefficient of the cyclotomic polynomial $\Phi_{3p_2p_3}(x)$ for some general cases.

A commutative ring $A$ is called présimplifiable (resp., domain-like) if all its zero divsors are contained in its Jacobson radical (resp., $0$ is a primary ideal of $A$). Let $A$ and $B$ be two commutative rings with identity, $J$ be an ideal of $B$, and $\rho: A\rightarrow B$ be a ring homomorphism. In this paper, we give a characterization for the amalgamation of $A$ with $B$ along $J$ with respect to $\rho$ (denoted by $R\Join ^\rho J$) to be présimplifiable (resp., domain-like).

In this paper, we introduce the concept of cubic quasi-ideal semigroups and we study the basic properties of cubic quasi-ideals. Also, we find necessary and sufficient conditions for a cubic bi-ideal to be a cubic quasi-ideal of a semigroup. We present characterizations of regular and intra-regular semigroups in terms of cubic quasi-ideals.