A striking feature of the traditional armchair method of philosophy is the use of imaginary examples: for instance, of Gettier cases as counterexamples to the justified true belief analysis of knowledge. The use of such examples is often thought to involve some sort of a priori rational intuition, which crude rationalists regard as a virtue and crude empiricists as a vice. It is argued here tha...

Let E be an elliptic curve defined over Q and P ∈ E(Q) a rational point of infinite order. Suppose that E has complex multiplication by an order in the quadratic imaginary field k. Denote by ME,P the set of rational primes ` such that ` splits in k, E has good reduction at `, and P is a primitive point modulo `. Under the generalized Riemann hypothesis, we can determine the positivity of the de...

A representation of complex rational numbers in quantum mechanics is described that is not based on logical or physical qubits. It stems from noting that the 0s in a product qubit state do not contribute to the number. They serve only as place holders. The representation is based on the distribution of four types of systems on an integer lattice. The four types, labelled as positive real, negat...

In 1961 Youla published his paper 'On the factorization of rational matrices'. He proved that any proper rational parahermitian matrix, positive definite on the imaginary axis can be factorized as the product of a proper rational matrix, stable with respect to the closed right half plane, and its adjoint. In this paper I prove that for any positive definite, nonstrictly proper matrix this facto...

The problem of J-factorization of rational matrices, which have zeros and poles on the imaginary axis, is reduced to construction of the solutions of two algebraic Riccati equations. For construction of these solutions, it is offered to use appropriate algorithms. These algorithms permit to find the solutions in cases when the Hamiltonian matrices, which are corresponding to these equations, ha...

The paper by R. H. Korogui, A. R. Fioravanti, and J. C. Geromel focuses on the H∞ control design for time-delay linear systems. The results are based on the use of a comparison system obtained via Rekasius’ substitution. As it is known from the delay systems’ literature, this transformation allows to approximate the non-rational transfer functions by rational ones. Moreover the two transfer fun...

whereX0(N) is the complete curve overQ which classifies pairs of elliptic curves related by a cyclic N -isogeny. The curve E is equipped with the collection of Heegner points defined over ring class fields of suitable imaginary quadratic fields. More precisely, let K be an imaginary quadratic field in which all rational primes dividing N are split and let O be the order of K of conductor c prim...

Rational approximations of p 1 + x are useful for one-way wave propagation modeling, where a square root operator must be approximated. The standard Pad e approximants incorrectly propagate the evanescent modes corresponding to x < ?1. A modiication of the Pad e approximation is introduced in this paper. Our new approximants have positive real imaginary parts for x < ?1, which give the evanesce...