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Many proofs of the Fundamental Theorem of Algebra, including various proofs based on the theory of analytic functions of a complex variable, are known. To the best of our knowledge, this proof is different from the existing ones.
The S-measure construction from nonstandard analysis is used to prove an extension of a result on the intersection of sets in a finitely-additive measure space. This is then used to give a density-limit version of a representation theorem…
The martingale method is used to establish concentration inequalities for a class of dependent random sequences on a countable state space, with the constants in the inequalities expressed in terms of certain mixing coefficients. Along the…
The Representation Theorem of Lions (RTL) is a version of the Lax--Milgram Theorem where completeness of one of the spaces is not complete. In this paper, RTL is deduced from an operator-theoretical version on normed space. The main point…
We illustrate a process that constructs martingales from raw material that arises naturally from the theory of sampling without replacement.The usefulness of the new martingales is illustrated by the development of maximal inequalities for…
We present a method for constructing countable models of small theories and apply it to prove theorems on the maximal number of countable non-isomorphic models of linearly ordered theories.
Extending the classical result that the roots of a polynomial with coefficients in $\mathbf{C}$ are continuous functions of the coefficients of the polynomial, nonstandard analysis is used to prove that if $\mathcal{F} = \{f_{\lambda}…
In this note, we give a simple proof that the values of the trigonometric functions at any nonzero rational number are transcendental numbers.
We propose the method for obtaining invariants of arbitrary representations of Lie groups that reduces this problem to known problems of linear algebra. The basis of this method is the idea of a special extension of the representation…
Non-probability sampling, for example in the form of online panels, has become a fast and cheap method to collect data. While reliable inference tools are available for classical probability samples, non-probability samples can yield…
We develop a stochastic calculus that makes it easy to capture a variety of predictable transformations of semimartingales such as changes of variables, stochastic integrals, and their compositions. The framework offers a unified treatment…
The first and second representation theorems for sign-indefinite, not necessarily semi-bounded quadratic forms are revisited. New straightforward proofs of these theorems are given. A number of necessary and sufficient conditions ensuring…
In this monograph, nonstandard characteristics for many notions from real analysis are obtained and applied. However, only two simple types of atomic formula are used and almost all of the characteristics are shown to hold for a simple…
There is no general existence theorem for solutions for nonlinear difference equations, so we must prove the existence of solutions in accordance with models one by one. In our work, we found theorems for the existence of analytic solutions…
We consider a possibly degenerate porous media type equation over all of $\R^d$ with $d = 1$, with monotone discontinuous coefficients with linear growth and prove a probabilistic representation of its solution in terms of an associated…
We show that the reduction mod p of an orthogonal linear representation is orthogonal, and we generalize this fact to representations of algebras with involution.The proofs make an essential use of the notion of " middle lattices ".
It has been shown that a functional interpretation of proofs in mathematical analysis can be given by the product of selection functions, a mode of recursion that has an intuitive reading in terms of the computation of optimal strategies in…
We examine the convergence properties of sequences of nonnegative real numbers that satisfy a particular class of recursive inequalities, from the perspective of proof theory and computability theory. We first establish a number of results…
Uncertainty associated with statistical problems arises due to what has not been seen as opposed to what has been seen. Using probability to quantify the uncertainty the task is to construct a probability model for what has not been seen…
The Taylor expansion is a widely used and powerful tool in all branches of Mathematics, both pure and applied. In Probability and Mathematical Statistics, however, a stronger version of Taylor's classical theorem is often needed, but only…