Volume 74 , issue 4 ( 2022 ) | back |

NORMALIZED LAPLACIAN ENERGY AND NORMALIZED LAPLACIAN-ENERGY-LIKE INVARIANT OF SOME DERIVED GRAPHS | 229$-$241 |

**Abstract**

For a connected graph $G$, the smallest normalized Laplacian eigenvalue is 0 while all others are positive and the largest cannot exceed the value 2. The sum of absolute deviations of the eigenvalues from 1 is called the normalized Laplacian energy, denoted by $\mathbb{LE}(G)$. In analogy with Laplacian-energy-like invariant of $G$, we define here the normalized Laplacian-energy-like as the sum of square roots of normalized Laplacian eigenvalues of $G$, denoted by $\mathbb{LEL}(G)$.

**Keywords:** Normalized Laplacian energy; normalized Laplacian-energy-like invariant; double graph; extended double cover; Mycielskian.

**MSC:** 05C50

ON SOLVABILITY OF QUADRATIC HAMMERSTEIN INTEGRAL EQUATIONS IN HÖLDER SPACES | 242$-$248 |

**Abstract**

Using Schauder's fixed point theorem we consider the solvability of a quadratic Hammerstein integral equation in the space of functions satisfying a Hölder condition. An example is included to illustrate our results.

**Keywords:** Quadratic Hammerstein integral equation; Hölder condition; Schauder fixed point theorem.

**MSC:** 45G10, 45M99, 47H09

CANTOR SETS AND FIELDS OF REALS | 249$-$259 |

**Abstract**

Our main result is a construction of four families ${\cal C}_1,{\cal C}_2,{\cal B}_1,{\cal B}_2$ which are equipollent with the power set of ${\Bbb R}$ and satisfy the following properties. (i) The members of the families are proper subfields $K$ of ${\Bbb R}$ where ${\Bbb R}$ is algebraic over $K$. (ii) Each field in ${\cal C}_1\cup{\cal C}_2$ contains a {\it Cantor set}. (iii) Each field in ${\cal B}_1\cup{\cal B}_2$ is a {\it Bernstein set}. (iv) All fields in ${\cal C}_1\cup{\cal B}_1$ are isomorphic. (v) If $K,L$ are fields in ${\cal C}_2\cup{\cal B}_2$ then $K$ is isomorphic to some subfield of $L$ only in the trivial case $K=L$.

**Keywords:** Transcendental extensions; descriptive set theory.

**MSC:** 12F20, 54H05

NEW FIXED POINT RESULTS FOR ASYMPTOTIC CONTRACTIONS AND ITS APPLICATION TO CANTILEVER BEAM PROBLEMS | 260$-$271 |

**Abstract**

In this article, we deal with some interesting variants of asymptotic contractions, namely Reich type and Chatterjea type weak asymptotic contractions defined on the usual metric spaces. We derive a couple of fixed point results concerning such contractions. Moreover, we look over the existence of solutions to a fourth-order two-point boundary value problem which is a particular type of cantilever beam problems. Furthermore, we construct numerical examples to justify our obtained results.

**Keywords:** Asymptotic contractions; orbital continuity; cantilever beam problems; boundary value problems.

**MSC:** 47H10, 54H25

A GENERALIZATION OF NONSINGULAR REGULAR MAGIC SQUARES | 272$-$279 |

**Abstract**

A generalization of regular magic squares with magic sum $\mu$ is an sq-corner (or square corner) magic square. It is a magic square satisfying the condition that the sum of 4 entries, square symmetrically placed with respect to the center, equals $\frac{4\mu}{n}$. Using the sq-corner magic squares of order $n$, a construction of sq-corner magic squares of order {$n+2$} is derived. Moreover, this construction provides some nonsingular classical sq-corner magic squares of all orders. In particular, a nonsingular regular magic square of any odd order can be constructed under this new method, as well.

**Keywords:** Nonsingular matrices; determinants; magic squares.

**MSC:** 15A15, 15F10

UNIQUENESS OF THE SOLUTION OF A NONLINEAR ALGEBRAIC SYSTEM | 280$-$288 |

**Abstract**

In this article we give a sufficient condition for a nonlinear algebraic system of some classes of hypersurfaces to intersect in a unique point and we express the corresponding unique solution in exact form, as well as for the corresponding nonlinear functional system of equations. We conclude extending our results for the functional case in a Banach space of Bochner measurable functions.

**Keywords:** Nonlinear systems; intersection of hypersurfaces; Bochner integrable functions; equality in the triangle inequality.

**MSC:** 14P99, 46E30, 26D15, 39B05, 97G70

NAPOLEON'S THEOREM FROM THE VIEW POINT OF GRÖBNER BASES | 289$-$300 |

**Abstract**

In this article, we present a new proof of the Napoleon's theorem using algorithmic commutative algebra and algebraic geometry. We also show that, by using the same technique, several related theorems, with the same basic set of objects can be proved. Thus, from the new proof of Napoleon's theorem, we prove the Relative of Napoleon's theorem (result given by B. Grünbaum). Then, we present a new theorem related to Napoleon's theorem. In this theorem the existence of two more quadruplets of equilateral triangles associated with a given triangle was established.

**Keywords:** Napoleon's theorem; automatic theorem proving; elementary geometry; Gröbner bases.

**MSC:** 13P10, 68T15, 51M04