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This paper provides another proof of Fermat's theorem. As in the previous work, a geometric approach is used, namely: instead of integers a, b, c, a triangle with side lengths a, b, c is considered. To preserve the completeness of the proof of the theorem in this work, the proof is repeated for the cases of right and obtuse triangles. In this case, the Fermat equation ap+bp = cp has no solutions for any natural number p > 2 and arbitrary numbers a, b, c. When considering the case when the numbers a, b, c are sides of an acute triangle, it is proven that Fermat's equation has no solutions for any natural number p > 2 and non-zero integer numbers a, b, c. Numbers a = k, b = k+m, c = k+n, where k, m, n are natural numbers that satisfy the inequalities n > m, n < k+m, exhaust all possible variants of natural numbers a, b, c, which are the sides of the triangle. In an acute triangle, the following condition is additionally satisfied: k > n -m + √2n(n - m). To study the Fermat equation, an auxiliary function f(k,p) = kp+(k+m)p-(k+n)p, is introduced, which is a polynomial of natural degree p in the variable k. The equation f(k, p) = 0 has a single positive root for any natural p = 2. A recurrent formula connecting the functions f(k,p+1) and f(k,p) has been proven: f(k,p+1) = kf(k, p)-[n(k+n)p-m(k+m)p]. The proof of the main proposition 2 is based on considering all possible relationships between the assumed integer solution of the equation f(k, p+1) = 0 and the number (ñ -m) corresponding to this solution k. The proof was carried out using the mathematical apparatus of number theory, elements of higher algebra and the foundations of mathematical analysis. These studies are a continuation of the author's works, in which some special cases of Fermat's theorem were proved. [ABSTRACT FROM AUTHOR]

This paper examines two interpretations of a passage in Descartes's text. Johannes Clauberg and Benedict Spinoza comment on the same paragraph in the Principles of Philosophy (1646). Descartes, in the paragraph, argues that the same amount of motion remains in the universe because of God's immutable essence and operation. On the one hand, Clauberg embraces Descartes's physics in general but modifies it to suit the theological tradition of the Reformed church, which held the official confession for where his professional career mattered. Spinoza, on the other hand, gets rid of all traces of the biblical religion from Descartes's physics. While particular theological (or anti-theological) positions of these thinkers dictate their interpretations of Descartes's text, their solutions are surprisingly similar. [ABSTRACT FROM AUTHOR]

Descartes held the following view of declarative memory: to remember is to reconstruct an idea that you intellectually recognize as a reconstruction. Descartes countenanced two overarching varieties of declarative memory. To have an intellectual memory is to intellectually reconstruct a universal idea that you recognize as a reconstruction, and to have a sensory memory is to neurophysiologically reconstruct a particular idea that you recognize as a reconstruction. Sensory remembering is thus a capacity of neither ghosts nor machines, but only of human beings qua mind-body unions. This interpretation unifies Descartes's various remarks (and conspicuous silences) about remembering, from the 1628 Rules for the Direction of the Mind through the suppressed-in-1633 Treatise of Man to the 1649 Passions of the Soul . It also rebuts a prevailing thesis in the current secondary literature—that Cartesian critters can remember—while incorporating the textual evidence for that thesis—Descartes's detailed descriptions of the corporeal mechanisms that construct sensory memories. [ABSTRACT FROM AUTHOR]

I analyze in this article the different ways in which Descartes uses abstraction in the Regulæ, discussing his project of a mathematical physics, the role of the imagination, and the status of numbers. I also try to show that the doctrine of simple natures cannot be well accommodated with the theory of abstraction developed in Rule 14, having instead a greater affinity with Descartes' later theory of abstraction and exclusion (from the period after the Meditationes), in which imagination plays no role and everything happens at a strictly intellectual level. This interpretation is supported by the recent discovery of the Cambridge manuscript, which almost certainly records an early stage of composition and in which the doctrine of simple natures is absent, thus being very probably a later development. [ABSTRACT FROM AUTHOR]

Suppose f(x, y) is a binary form of degree d with coefficients in a field K ⊆ C. The K-rank of f is the smallest number of d-th powers of linear forms over K of which f is a K-linear combination. We prove that for d ≥ 5, there always exists a form of degree d with at least three different ranks over various fields. The K-rank of a form f (such as x3y2) may depend on whether -1 is a sum of two squares in K. [ABSTRACT FROM AUTHOR]

We prove that for any degree d, there exist (families of) finite sequences {λk,d}0≤k≤d of positive numbers such that, for any real polynomial P of degree d, the number of its real roots is less than or equal to the number of the so-called essential tropical roots of the polynomial obtained from P by multiplication of its coefficients by λ0,d,λ1,d,…,λd,d, respectively. In particular, for any real univariate polynomial P(x) of degree d with a non-vanishing constant term, we conjecture that one can take λk,d=e−k2, k=0,…,d. The latter claim can be thought of as a tropical generalization of Descartes’s rule of signs. We settle this conjecture up to degree 4 as well as a weaker statement for arbitrary real polynomials. Additionally, we describe an application of the latter conjecture to the classical Karlin problem on zero-diminishing sequences. [ABSTRACT FROM AUTHOR]

We argue that the idea of embodiment and the strategies for carrying out embodied approaches are some of the most prevalent and interdisciplinary legacies of early modern science. The idea of embodiment is simple: to explain the behavior of bodies, we must understand them as unified wholes in their environments. Embodied approaches eschew explanations in terms of qualitative descriptions of the intrinsic properties of bodies and promote explanation in terms of the interaction between bodies. This idea can be found in Kepler's optics, Descartes' physics, and Newton's physico-mathematics. The Senses Considered as Perceptual Systems (Gibson, 1966) is the culmination of this centuries-long embodiment movement which can be traced back to the 17th century. [ABSTRACT FROM AUTHOR]

Reaction networks taken with mass-action kinetics arise in many settings, such as epidemiology, population biology and systems of chemical reactions. Bistable reaction networks are posited to underlie biochemical switches, which motivates the following question: Which reaction networks have the capacity for multiple steady states? Mathematically, this asks, from among certain parametrized families of polynomial systems, which admit multiple positive roots? No complete answer is known. This work analyzes the smallest networks, i.e., those with only a few chemical species or reactions. For these "smallest" networks, we completely answer the question of multistationarity and in some cases, multistability, thereby extending related work of Boros. Our results highlight the role played by the Newton polytope of a network (the convex hull of the reactant vectors). Also, our work is motivated by recent results that explain how a given network's capacity for multistationarity arises from that of certain related networks which are typically smaller. Hence, we are interested in classifying small multistationary networks, and our work forms a first step in this direction. [ABSTRACT FROM AUTHOR]

We give a Descartes'-like bound on the number of positive solutions to a system of fewnomials that holds when its exponent vectors are not in convex position and a sign condition is satisfied. This was discovered while developing algorithms and software for computing the Gale transform of a fewnomial system, which is our main goal. This software is a component of a package we are developing for Khovanskii-Rolle continuation, which is a numerical algorithm to compute the real solutions to a system of fewnomials. [ABSTRACT FROM AUTHOR]

Given convex polytopes $$P_1,\ldots ,P_r \subset \mathbb {R}^n$$ and finite subsets $$\mathcal {W}_I$$ of the Minkowski sums $$P_I=\sum _{i \in I} P_i$$ , we consider the quantity $$N(\mathbf {W})=\sum _{I \subset \mathbf{[}r \mathbf{]} } {(-1)}^{r-|I|} \big | \mathcal {W}_I \big |$$ . If $$\mathcal {W}_I=\mathbb {Z}^n \cap P_I$$ and $$P_1,\ldots ,P_n$$ are lattice polytopes in $$\mathbb {R}^n$$ , then $$N(\mathbf {W})$$ is the classical mixed volume of $$P_1,\ldots ,P_n$$ giving the number of complex solutions of a general complex polynomial system with Newton polytopes $$P_1,\ldots ,P_n$$ . We develop a technique that we call irrational mixed decomposition which allows us to estimate $$N(\mathbf {W})$$ under some assumptions on the family $$\mathbf {W}=(\mathcal {W}_I)$$ . In particular, we are able to show the nonnegativity of $$N(\mathbf {W})$$ in some important cases. A special attention is paid to the family $$\mathbf {W}=(\mathcal {W}_I)$$ defined by $$\mathcal {W}_I=\sum _{i \in I} \mathcal {W}_i$$ , where $$\mathcal {W}_1,\ldots ,\mathcal {W}_r$$ are finite subsets of $$P_1,\ldots ,P_r$$ . The associated quantity $$N(\mathbf {W})$$ is called discrete mixed volume of $$\mathcal {W}_1,\ldots ,\mathcal {W}_r$$ . Using our irrational mixed decomposition technique, we show that for $$r=n$$ the discrete mixed volume is an upper bound for the number of nondegenerate solutions of a tropical polynomial system with supports $$\mathcal {W}_1,\ldots ,\mathcal {W}_n \subset \mathbb {R}^n$$ . We also prove that the discrete mixed volume associated with $$\mathcal {W}_1,\ldots ,\mathcal {W}_r$$ is bounded from above by the Kouchnirenko number $$\prod _{i=1}^r (|\mathcal {W}_i|-1)$$ . For $$r=n$$ this number was proposed as a bound for the number of nondegenerate positive solutions of any real polynomial system with supports $$\mathcal {W}_1,\ldots ,\mathcal {W}_n \subset \mathbb {R}^n$$ . This conjecture was disproved, but our result show that the Kouchnirenko number is a sharp bound for the number of nondegenerate positive solutions of real polynomial systems constructed by means of the combinatorial patchworking. [ABSTRACT FROM AUTHOR]

The article discusses constant power line sequences and analyzes relative geometric position of the roots as coefficients are allowed to depend on n. Topics include application of Descartes's rule of signs, use of squeeze theorem and continuity of the polynomials in case of zero, and a theorem assuming two sequences of positive real numbers in case of odd numbers.

The article offers information on the use of Descartes' rule of signs and certain properties of polynomial functions to estimate the maximum number of turning points. It states that the number of turning points in a graph is connected to both the oscillatory and complexing behavior of the function. The important focus of the intermediate value theorem, Descartes' rule of signs, and the factor theorem on the roots of polynomial equations is cited.

The folium of Descartes is still studied and understood today. Not only did it provide for the proof of some properties connected to Fermat's Last Theorem, or as Hessian curve associated to an elliptic curve, but it also has a very interesting property over it: a multiplicative group law. While for the elliptic curves, the addition operation is compatible with their geometry (additive group), for the folium of Descartes, the multiplication operation is compatible with its geometry (multiplicative group). The original results in this paper include: points at infinity described by the Legendre symbol, a group law on folium of Descartes, the fundamental isomorphism and adequate algorithms, algorithms for algebraic computation. [ABSTRACT FROM AUTHOR]

The article offers the author's insights on the aspect of teaching Platonic solids. Topics discussed include a brief characteristics of the Platonic solids which is defined as regular polygons with the same number of faces meeting at each vertex, named after Greek philosopher Plato, the aspect of how the student explore the characteristics of the polygons through constructing the polygons, and the exploration of Descartes's theorem of the angular deficiency.