We introduce a concept called refinement and develop two different ways of refining metrics. By a... more We introduce a concept called refinement and develop two different ways of refining metrics. By applying these methods we produce several refinements of the shortest-path distance on the collaboration graph and hence a couple new versions of the Erdős number.
We generalize and unify the proofs of several results on algebraic in- dependence of arithmetic f... more We generalize and unify the proofs of several results on algebraic in- dependence of arithmetic functions and Dirichlet series by a theorem of Ax on differential Schanuel conjecture. Along the way, we find counter-examples to some results in the literature.
We study the set of monomial ideals in a polynomial ring as an ordered set, with the ordering giv... more We study the set of monomial ideals in a polynomial ring as an ordered set, with the ordering given by reverse inclusion. We give a short proof of the fact that every antichain of monomial ideals is finite. Then we investigate ordinal invariants for the complexity of this ordered set. In particular, we give an interpretation of the height function in terms of the Hilbert-Samuel polynomial, and we compute upper and lower bounds on the maximal order type.
For any Q-linearly independent complex numbers α1,...,αn, there are at least n numbers among α1,.... more For any Q-linearly independent complex numbers α1,...,αn, there are at least n numbers among α1,...,αn, e α1,..., eαn that are algebraically independent over Q.
Two numbers are spectral equivalent if they have the same length spectrum. We show how to compute... more Two numbers are spectral equivalent if they have the same length spectrum. We show how to compute the equivalence classes of this relation. Moreover, we show that these classes can only have either 1,2 or infinitely many elements.
While looking for exercises for a number theory class, I recently came across the following quest... more While looking for exercises for a number theory class, I recently came across the following question in a book by André Weil [4, Question III.4]: Which natural numbers can be written as the sum of two or more consecutive integers? The origin of this question is unknown to me, but one can easily believe that it is part of the mathematical folklore. Solutions to this problem and some generalizations maybe found in [1] and [2], for example. We will give a somewhat different proof here, one that we hope readers will find intuitively appealing. To get a feeling for the problem, let us consider some small values. The list below gives one attempt at expressing the numbers up to 16 as sums of two or more consecutive integers. We leave the right-hand side of the equation blank if there is no such expression for the number on the left-hand side.
We study the set of monomial ideals in a polynomial ring as an ordered set, with the ordering giv... more We study the set of monomial ideals in a polynomial ring as an ordered set, with the ordering given by reverse inclusion. We give a short proof of the fact that every antichain of monomial ideals is finite. Then we investigate ordinal invariants for the complexity of this ordered set. In particular, we give an interpretation of the height function in terms of the Hilbert-Samuel polynomial, and we compute upper and lower bounds on the maximal order type.
We introduce a concept called refinement and provide two different ways of refining metrics. By a... more We introduce a concept called refinement and provide two different ways of refining metrics. By applying these methods we produce several refinements of the shortest-path distance on the collaboration graph and hence a couple of new versions of the Erdős number.
We give a new proof of the fact that the vanishing of generalized Wronskians implies linear depen... more We give a new proof of the fact that the vanishing of generalized Wronskians implies linear dependence of formal power series in serveral variables. Our results are also valid for quotients of germs of analytic functions.
We give a new proof of the fact that the vanishing of generalized Wronskians implies linear depen... more We give a new proof of the fact that the vanishing of generalized Wronskians implies linear dependence of formal power series in several variables. Our results are also valid for quotients of germs of analytic functions.
We introduce a concept called refinement and develop two different ways of refining metrics. By a... more We introduce a concept called refinement and develop two different ways of refining metrics. By applying these methods we produce several refinements of the shortest-path distance on the collaboration graph and hence a couple new versions of the Erdős number.
We introduce a concept called refinement and develop two different ways of refining metrics. By a... more We introduce a concept called refinement and develop two different ways of refining metrics. By applying these methods we produce several refinements of the shortest-path distance on the collaboration graph and hence a couple new versions of the Erdős number.
We generalize and unify the proofs of several results on algebraic in- dependence of arithmetic f... more We generalize and unify the proofs of several results on algebraic in- dependence of arithmetic functions and Dirichlet series by a theorem of Ax on differential Schanuel conjecture. Along the way, we find counter-examples to some results in the literature.
We study the set of monomial ideals in a polynomial ring as an ordered set, with the ordering giv... more We study the set of monomial ideals in a polynomial ring as an ordered set, with the ordering given by reverse inclusion. We give a short proof of the fact that every antichain of monomial ideals is finite. Then we investigate ordinal invariants for the complexity of this ordered set. In particular, we give an interpretation of the height function in terms of the Hilbert-Samuel polynomial, and we compute upper and lower bounds on the maximal order type.
For any Q-linearly independent complex numbers α1,...,αn, there are at least n numbers among α1,.... more For any Q-linearly independent complex numbers α1,...,αn, there are at least n numbers among α1,...,αn, e α1,..., eαn that are algebraically independent over Q.
Two numbers are spectral equivalent if they have the same length spectrum. We show how to compute... more Two numbers are spectral equivalent if they have the same length spectrum. We show how to compute the equivalence classes of this relation. Moreover, we show that these classes can only have either 1,2 or infinitely many elements.
While looking for exercises for a number theory class, I recently came across the following quest... more While looking for exercises for a number theory class, I recently came across the following question in a book by André Weil [4, Question III.4]: Which natural numbers can be written as the sum of two or more consecutive integers? The origin of this question is unknown to me, but one can easily believe that it is part of the mathematical folklore. Solutions to this problem and some generalizations maybe found in [1] and [2], for example. We will give a somewhat different proof here, one that we hope readers will find intuitively appealing. To get a feeling for the problem, let us consider some small values. The list below gives one attempt at expressing the numbers up to 16 as sums of two or more consecutive integers. We leave the right-hand side of the equation blank if there is no such expression for the number on the left-hand side.
We study the set of monomial ideals in a polynomial ring as an ordered set, with the ordering giv... more We study the set of monomial ideals in a polynomial ring as an ordered set, with the ordering given by reverse inclusion. We give a short proof of the fact that every antichain of monomial ideals is finite. Then we investigate ordinal invariants for the complexity of this ordered set. In particular, we give an interpretation of the height function in terms of the Hilbert-Samuel polynomial, and we compute upper and lower bounds on the maximal order type.
We introduce a concept called refinement and provide two different ways of refining metrics. By a... more We introduce a concept called refinement and provide two different ways of refining metrics. By applying these methods we produce several refinements of the shortest-path distance on the collaboration graph and hence a couple of new versions of the Erdős number.
We give a new proof of the fact that the vanishing of generalized Wronskians implies linear depen... more We give a new proof of the fact that the vanishing of generalized Wronskians implies linear dependence of formal power series in serveral variables. Our results are also valid for quotients of germs of analytic functions.
We give a new proof of the fact that the vanishing of generalized Wronskians implies linear depen... more We give a new proof of the fact that the vanishing of generalized Wronskians implies linear dependence of formal power series in several variables. Our results are also valid for quotients of germs of analytic functions.
We introduce a concept called refinement and develop two different ways of refining metrics. By a... more We introduce a concept called refinement and develop two different ways of refining metrics. By applying these methods we produce several refinements of the shortest-path distance on the collaboration graph and hence a couple new versions of the Erdős number.
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