We look back to concept and dependency graphs of Euclid’s Elements, Book 1, that show the deductive relationships among its propositions. We claim that Book 1 does not have one overall coherent structure but rather is organized around two propositions—1.45 and 1.47. In other words, Book 1 has a dual core. For the latter proposition, the Pythagorean Theorem, we constructed a rooted in-tree graph...

We prove the Weierstrass-type approximation theorem that states every C1 curve in the 2-dimensional or 3-dimensional Euclidean space or in the 3-dimensional Minkowski space can be uniformly approximated by Pythagorean hodograph curves in the corresponding space. This abundance of PH curves is another theoretical confirmation of the usefulness and the versatility of the PH curves. We also addres...

The logicians . . . are too much caught up in psychology . . . Logic is in no way a part of psychology. The Pythagorean theorem expresses the same thought for all men, while each person has its own representations, feelings and resolutions that are different from those of every other person. Thoughts are not psychic structures, and thinking is not an inner producing and forming, but an apprehen...

It is shown that the most important effects of special and general theory of relativity can be understood in a simple and straightforward way. The system of units in which the speed of light c is the unit of velocity allows to cast all formulas in a very simple form.The Pythagorean theorem graphically relates energy, momentum and mass. The paper is addressed to those who teach and popularize th...

We give a constructive proof of Carpenter’s Theorem due to Kadison [14, 15]. Unlike the original proof our approach also yields the real case of this theorem. 1. Kadison’s theorem In [14] and [15] Kadison gave a complete characterization of the diagonals of orthogonal projections on a Hilbert space H. Theorem 1.1 (Kadison). Let {di}i∈I be a sequence in [0, 1]. Define a = ∑ di<1/2 di and b = ∑ d...

Suppose a right triangle has edges A,B,C with corresponding lengths a, b, c. Suppose C is the hypotenuse. Let θ be the angle of the triangle formed by the edges A and C. From the definitions of trigonometric functions that we used in high school, cos θ = a/c and sin θ = b/c. Also, from an identity that we learned in high school, (cos θ) + (sin θ) = 1, so (a/c) + (b/c) = 1, i.e. a + b = c. Howev...

Pythagorean angles, that is angles with rational sines and cosines, provide an interesting environment for studying the question of characterizing trisectable angles. The main result of this paper shows that a Pythagorean angle is trisectable if and only if it is three times some other Pythagorean angle. Using the Euclidean parametrization of Pythagorean angles, this result allows an effective ...

In this paper, we present two Pythagorean fuzzy averaging aggregation operators such as, Pythagorean fuzzy weighted averaging (PFWA) operator, Pythagorean fuzzy ordered weighted averaging (PFOWA) operator and also introduce some of their basic properties.

Abstract. A function is exponentially concave if its exponential is concave. We consider exponentially concave functions on the unit simplex. It is known that gradient maps of exponentially concave functions are solutions of a MongeKantorovich optimal transport problem and allow for a better gradient approximation than those of ordinary concave functions. The approximation error, called L-diver...