True Theorem Research Articles

A new concept of QM-models and truth in quantum mechanics (QM), the new hybrid-epistemic model of QM and its observer independence, the proof of the invalidity of no-go theorems

We define a concept of a QM-model. The true theorem in QM is a statement which is true in all QM-models. In some QM-models the Bell’s theorem can be proved (e.g. in the standard model of QM) while in other QM-models the Bell’s theorem cannot be proved (e.g. in the hybrid-epistemic model of QM). The same situation is true when other no-go theorems are considered. Thus no-go theorems are not true in QM and the proof of the non-locality of QM is invalid.Then the axiomatic definition of the hybrid-epistemic (HE) model of QM is presented in all details. At the end the recent proof of the inconsistency of the standard QM-model is discussed.Our main program is to start the axiomatic study of QM (in the sense of the Hilbert’s sixth problem), to prove the invalidity of no-go theorems and to identify the right (acceptable) QM-model.

Quotient polynomials with positive coefficients

In a recent paper [1] Michael Hirschhorn and the present author had to prove that the polynomialis positive for all integers n ≥ 4. We did so by writing down the identitySince all the coefficients of the quotient polynomial are positive as well as the remainder, this shows by inspection that f(n) > 0 for n ≥ 5 and computing f(4) = 12025 completes the proof. This example illustrates a useful and efficient method for proving that polynomials have strictly positive values for all real numbers exceeding a given one. We leamed this method from a paper by Chen [2], (see [3] and the references cited there), and we have subsequently used it in our own researches [1,4].We also conjectured in [1] that the theoretical basis of the method is a true theorem and the present paper is dedicated to proving this conjecture.Theorem 1: Let f(x) be a polynomial with real coefficients whose leading coefficient is positive and with at least one positive root x = a. Then there exists an x = b ≥ a such thatwhere f(b), and all the coefficients of the quotient polynomial g(x), are non-negative.

THE THEORY OF INTERSTELLAR TRADE

This article extends interplanetary trade theory to an interstellar setting. It is chiefly concerned with the following question: how should interest charges on goods in transit be computed when the goods travel at close to the speed of light? This is a problem because the time taken in transit will appear less to an observer traveling with the goods than to a stationary observer. A solution is derived from economic theory, and two useless but true theorems are proved. (JEL F10, F30)

What was Fisher’s fundamental theorem of natural selection and what was it for?

Fisher’s ‘fundamental theorem of natural selection’ is notoriously abstract, and, no less notoriously, many take it to be false. In this paper, I explicate the theorem, examine the role that it played in Fisher’s general project for biology, and analyze why it was so very fundamental for Fisher. I defend Ewens (1989) and Lessard (1997) in the view that the theorem is in fact a true theorem if, as Fisher claimed, ‘the terms employed’ are ‘used strictly as defined’ (1930, p. 38). Finally, I explain the role that projects such as Fisher’s play in the progress of scientific inquiry.

Constructing a Minimal Counterexample in Group Theory

Desmond MacHale's intriguing article [2] neatly illustrates one mode of interplay between proof and example in group theory by describing the commonly used inductive technique of minimal counterexamples. In this Note, we use some ideas from group theory and linear algebra to lead the reader through a geometric and algebraic construction of one of the minimal counterexamples sought by MacHale. Recall that a group is called nilpotent if each of its Sylow subgroups is normal. One of MacHale's list of known-to-be-false conjectures is this: if G is a finite group having a fixed-pointfree automorphism 4, then G is nilpotent. The true theorem on which the conjecture is based assumes that 4 is of prime order; it was established by Thompson in his dissertation in 1959 [4]. His proof so excited the mathematical community that even the New York Times reported his result and the reaction to it [3]. MacHale indicates that [1, p. 336] contains details of a counterexample to the conjecture with IGI = 147 = 72 3 and 1j1 = 4. In our minimal counterexample to be constructed, I4 = 6 and IGI = 48 = 24 43. In what follows, we use some standard conventions of notation in group theory. In particular, we write operators on the right and denote both the action of a group homomorphism and conjugation using superscripts. Recall that if H and K are subgroups of a group G such that G = HK, H n K = 1, and H is normal in G, then G is called the semidirect product of H by K. In this situation, since H is normal in G, each element of K acts via conjugation as an automorphism of H. Of particular interest here is the situation in which each nontrivial element of K acts as a nontrivial (i.e., nonidentity) automorphism on H. In this case K is isomorphic to a subgroup of Aut H, the group of all automorphisms of H. A well-known geometric example leads directly to an important link between group theory and geometry. If A is the group of rotations of a regular tetrahedron, then A contains eight elements of order 3, three of order 2, and an identity element. Each element of order 3 is a 120?rotation of the tetrahedron about an axis passing through one of the 4 vertices and perpendicular to the opposite face. Each element of order 2 is a rotation of 1800 about one of three axes joining the midpoints of two nonadjacent edges of the tetrahedron. (See FIGURE 1.) If the vertices are labeled, and each rotation is identified with the permutation of the labels it produces, this identification provides a natural isomorphism between A and A4, the group of even permutations on 4 letters. Thus it is possible to use the algebra of permutations or geometry to analyze A.

Some bounds for the Ramsey-Paris-Harrington numbers

It has recently been discovered that a certain variant of Ramsey's theorem cannot be proved in first-order Peano arithmetic although it is in fact a true theorem. In this paper we give some bounds for the “Ramsey-Paris-Harrington numbers” associated with this variant of Ramsey's theorem, involving coloring of pairs. In the course of the investigation we also study certain weaker and stronger partition relations.

The truth about Venn diagrams

Venn diagrams are usually considered to be a method for visualising set-theoretic identities and suggesting true theorems. It is frowned upon to prove a set-theoretic identity using them, such proofs being considered non-rigorous. One of the better kept secrets of the trade is that they can be made rigorous by proving the right kind of ‘metatheorem’. A proof is quite instructive, and might be of interest to set-theoretically minded sixth formers.

True Theorems From False Conjectures

In 1933 R. K. Morley noted that there are just four proper fractions with denominators less than 100 where illegitimate cancellation of a digit produces the correct answer.1

A remark on reversible matrices

It is further stated in [1, p. 50] that the Ck are bounded. This is questioned in [4], where it is pointed out that if the Ck were generally bounded they would have to be almost all zero, but this remark does not dispose of the matter, for it might conceivably be a true theorem that for each reversible matrix the Ck are almost all zero. (For rowfinite matrices, all Ck vanish; [3].) The example given in [4, p. 47] seems inconclusive. The purpose of this note is to show by a very simple example that in fact the ck need not be bounded. Consider the transformation