### Well-Posedness and Asymptotic Stability of a Non-Linear Porous System with a Delay Term

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**Authors:**H. MAKHELOUFI, N. MEZOUAR AND M. BAHLIL

**DOI:**10.46793/KgJMat2105.751E

**Abstract:**

We prove that there is no circulant Hadamard matrix H with ﬁrst row [h

_{1},…,h

_{n}] of order n > 4, under some linear conditions on the h

_{i}’s. All these conditions hold in the known case n = 4, so that our results can be thought as characterizations of properties that only hold when n = 4. Our ﬁrst conditions imply that some eigenvalue λ of H is a sum of terms h

_{ j}ω

^{j}, where ω is a primitive n-th root of 1. The same conclusion holds also if some complex arithmetic means associated to λ are algebraic integers (second conditions). Moreover, our third conditions, related to the recent notion of robust Hadamard matrices, implies also the nonexistence of these circulant Hadamard matrices. If some of the conditions fail, it appears (to us) very diﬃcult to be able to prove the result.

**Keywords:**

Circulant matrices, Hadamard matrices, eigenvalues, unit circle, cyclotomic ﬁelds.

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