Related papers: A theorem with constructive and non-constructive p…
We provide examples of contractible complexes which fail to have non-positive immersions and weak non-positive immersions, answering a conjecture of Wise in the negative.
Existence theorem is proven for the generating equations of the split involution constraint algebra. The structure of the general solution is established, and the characteristic arbitrariness in generating functions is described.
This paper introduces a space of variable lotteries and proves a constructive version of the expected utility theorem. The word ``constructive'' is used here in two senses. First, as in constructive mathematics, the logic underlying proofs…
The aim of this short note is to present an elementary, self-contained, and direct proof for the classical Lebesgue decomposition theorem.
We prove that any finite set of real numbers can be split into two parts, one part being highly non-additive and the other highly non-multiplicative.
We show that when certain statements are provable in subsystems of constructive analysis using intuitionistic predicate calculus, related sequential statements are provable in weak classical subsystems. In particular, if a $\Pi^1_2$…
Theoretical and computational properties of a vector equation $Ax-\|x\|_1x=b$ are investigated, where $A$ is an invertible $M$-matrix and $b$ is a nonnegative vector. Existence and uniqueness of a nonnegative solution is proved. Fixed-point…
We construct a simple C*-algebra with nuclear dimension zero that is not isomorphic to its tensor product with the Jiang-Su algebra Z, and a hyperfinite II_1 factor not isomorphic to its tensor product with the separable hyperfinite II_1…
We give a procedure for counting the number of different proofs of a formula in various sorts of propositional logic. This number is either an integer (that may be 0 if the formula is not provable) or infinite.
We prove that any smooth Riemannian manifold of non-negative scalar curvature and with a strictly mean convex and compact boundary component can be (C^2) extended beyond the component to have non-negative scalar curvature and to enjoy…
Based on various strategies, we obtain several simple proofs of the celebrated Sharkovsky cycle coexistence theorem.
In this paper, we give a counter-example, in the general case, Kronecker theorem will derive contradiction. Kronecker theorem be correct after removing some conditions.
It is established that there exists an absolute constant $c>0$ such that for any finite set $A$ of positive real numbers $$|AA+A| \gg |A|^{\frac{3}{2}+c}.$$ On the other hand, we give an explicit construction of a finite set $A \subset…
In this note, simple proofs of certain well-known results involving the positive square root of positive matrices are given.
We give a new proof of the existence of designs, which is much shorter and gives better bounds.
In measure theory, Steinhaus theorem is a result that deals with a property of the difference between two sets of positive measure. We give a simple elementary proof of the result.
A problem of completing a linear map on C*-algebras to a completely positive map is analyzed. It is shown that whenever such a completion is feasible there exists a unique minimal completion. This theorem is used to show that under some…
In this paper, we prove several theorems on systems of polynomials with at least one positive real zero based on the theory of conceive polynomials. These theorems provide sufficient conditions for systems of multivariate polynomials…
In this article, we give a proof for positive mass theorem of asymptotically flat manifolds with arbitrary ends when the dimension is no greater than seven. As an application, we also show a positive mass theorem for asymptotically locally…
We present an incomplete proof synthesis method for the Calculus of Constructions which is always terminating and a complete Vernacular for the Calculus of Constructions based on this method.