Related papers: On an example of LaBuz
This article re-examines Lawvere's abstract, category-theoretic proof of the fixed-point theorem whose contrapositive is a `universal' diagonal argument. The main result is that the necessary axioms for both the fixed-point theorem and the…
There are several extensions of the classical Banach Fixed Point Theorem in technical literature. A branch of generalizations replaces usual contractivity by weaker but still effective assumptions. Our note follows this stream, presenting…
This paper explores multiple closely related themes: bounding the complexity of Diophantine equations over the integers and developing mathematical proofs in parallel with formal theorem provers. Hilbert's Tenth Problem (H10) asks about the…
A collection of orthonormal bases for a complex dXd Hilbert space is called mutually unbiased (MUB) if for any two vectors v and w from different bases the square of the inner product equals 1/d: |<v,w>| ^{2}=1/d. The MUB problem is to…
The Steinitz exchange lemma is a basic theorem in linear algebra used, for example, to show that any two bases for a finite-dimensional vector space have the same number of elements. The result is named after the German mathematician Ernst…
These lecture notes survey the emerging area of Universal Proof Theory, which investigates general questions about the existence, equivalence, and characterization of good proof systems for broad classes of logics. In particular, the notes…
In this paper we review a general proof for the irrationality property of numbers which take a certain form of infinite sums.
For a class of equations generalizing the model case \[ \Delta _p u-a(r)u^{p-1}+b(r)u^q=0 \; \; \mbox{in $B$}, \; \; u=0 \; \; \mbox{on $\partial B$}, \] where $B$ is the unit ball in $R^n$, $n \geq 1$, $r=|x|$, $p,q>1$, and $\Delta _p$…
In this paper some proof theory for propositional Lax Logic is developed. A cut free terminating sequent calculus is introduced for the logic, and based on that calculus it is shown that the logic has uniform interpolation. Furthermore, a…
This paper explains why internal and external validity cannot be simultaneously maximised. It introduces "evidential states" to represent the information available for causal inference and shows that routine study operations (restriction,…
Maximal sets of mutually unbiased bases are useful throughout quantum physics, both in a foundational context and for applications. To date, it remains unknown if complete sets of mutually unbiased bases exist in Hilbert spaces of…
We present an alternative procedure to eliminate irregular contributions in the perturbation expansion of c=0-matrix models representing the sum over triangulations of random surfaces, thereby reproducing the results of Tutte [1] and Brezin…
This article deals with universal deformations of dihedral representations with a particular focus on the question when the universal deformation is dihedral. Results are obtained in three settings: (1) representation theory, (2) algebraic…
We describe the non-minimal Standard Model, consisting of minimalistic extensions of the Standard Model, which for all we know is the theory of the universe, able to describe all of the universe from the beginning of time. Extensions…
For an integer $x$, an integer of the form $P_5(x)=\frac{3x^2-x}2$ is called a generalized pentagonal number. For positive integers $\alpha_1,\dots,\alpha_k$, a sum…
We show that in a vector space over Z_3, the union of any four linear bases is an additive basis, thus proving the Additive Basis Conjecture for p=3, and providing an alternative proof of the weak 3-flow conjecture.
Natural deduction systems, as proposed by Gentzen and further studied by Prawitz, is one of the most well known proof-theoretical frameworks. Part of its success is based on the fact that natural deduction rules present a simple…
Mutually unbiased bases encapsulate the concept of complementarity - the impossibility of simultaneous knowledge of certain observables - in the formalism of quantum theory. Although this concept is at the heart of quantum mechanics, the…
We prove that all Euclidean lattices of dimension $n\le 9$ which are generated by their minimal vectors, also possess a basis of minimal vectors. By providing a new counterexample, we show that this is not the case for all dimensions $n\ge…
Generics express generalizations about the world (e.g., birds can fly) that are not universally true (e.g., newborn birds and penguins cannot fly). Commonsense knowledge bases, used extensively in NLP, encode some generic knowledge but…