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It is known (see [4, Brändén, Lemma 2.7]) that a necessary condition for T := Q k (x)D k to be hyperbolicity preserving is that Q k (x) and Q k−1 (x) have interlacing zeros. We characterize all quadratic linear operators, as a consequence we find several classes of Pn-multiplier sequence.Our investigation involves such operators that act on polynomials, in particular, we are interested in polynomials with the following property.whose zeros are all real is said to be hyperbolic. Following the convention of G. Pólya and J. Schur [10, p.89], the constant 0 is also deemed to be hyperbolic. Definition 2. A linear operator T : R[x] → R[x] is said to preserve hyperbolicity (or T is a hyperbolicity preserver ) if T [f (x)] is a hyperbolic polynomial, whenever f (x) is a hyperbolic polynomial.Hyperbolicity preserving operators have been studied by virtually every author who has studied hyperbolic polynomials (see [5] and the references contained therein). The focus of our investigation involves the relationship between hyperbolicity preserving operators and hyperbolic polynomials with interlacing zeros.

Problem. Middle Egyptian has played an increasingly important role in Biblical Archaeology and Old Testament studies. Much of biblical chronology is closely connected to Egyptian chronology; many biblical events record Egyptian involvement, and some of these are included on Egyptian monuments. A knowledge of Middle Egyptian has been crucial in understanding the historical background and cultural context of the biblical world because this phase of the Egyptian language influenced the widest range of Egyptian writing, particularly as it pertains to Old Testament history. Unfortunately, students do not have all the necessary tools for searching for unfamiliar words in Middle Egyptian. There are no recent Middle Egyptian dictionaries in English for students. The purpose of this dissertation is to create a concise student dictionary of Middle Egyptian as a teaching tool for biblical archaeology students. Method. A. Gardiner’s Egyptian Grammar was used as the base vocabulary for this dictionary because it is regarded as the standard for teaching Middle Egyptian. Additional definitions for these words were added from R. O. Faulkner’s Concise Dictionary of Middle Egyptian. These definitions were expanded and clarified by adding new definitions, removing unfamiliar British colloquialisms, and replacing them with American Standard English definitions. The glossaries and dictionaries of J. Allen, J. Hoch, M. Collier and B. Manley, B. Ockinga, and G. Englund’s Middle Egyptian grammars were included for additional nuances. There are three parts to this Egyptian dictionary: hieroglyphic, transliteration, and English glossary. In Part I the hieroglyphic entries appear first. They are first organized by their initial phonetic value, second by their first two hieroglyphs, and third, by their phonetic value. In Part II, the transliterated values are organized alphabetically according to thier values in a manner consistent with other standard Egyptian dictionaries. Part III is an alphabetical list of the English definitions of each Egyptian word followed by its hieroglyphic and transliterated values. Each lexicographic entry includes a hieroglyphic and a transliterated form, its grammatical function, and any variations or abbreviations. A phonetic key and hieroglyphic index has also been included to aid students in finding Egyptian words according to their initial hieroglyph. In addition, there are three appendixes. Appendix A is a glossary of Egyptian words that appear in the Old Testament including the Hebrew word and biblical references, the Egyptian word with references, and related Egyptian synonyms. Appendix B is a list of Egyptian ruler names listed by kind. Appendix C is a list of the forty-two nomes of Upper and Lower Egypt and includes the hieroglyphs, transliteration, and common name of each nome as well as their respective capital cities.

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