Vadim Ponomarenko - Academia.edu (original) (raw)

Papers by Vadim Ponomarenko

Research paper thumbnail of On length densities

arXiv (Cornell University), Aug 15, 2020

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Research paper thumbnail of Pollock’s Generalized Tetrahedral Numbers Conjecture

The American Mathematical Monthly, 2021

The nth tetrahedral number Ten = ( n+2 3 ) represents the sum of the first n triangular numbers. ... more The nth tetrahedral number Ten = ( n+2 3 ) represents the sum of the first n triangular numbers. In the song “The Twelve Days of Christmas,” Ten counts the total number of gifts received after day n. A longstanding conjecture of Pollock (from [4]) is that every positive integer may be expressed as the sum of at most five tetrahedral numbers. To date, only 241 positive integers have been found requiring five tetrahedral numbers (see [3]). Recently, progress has been made (in [1]) on a related conjecture of Pollock from the same 19th century paper. Here we instead consider generalized tetrahedral numbers Ten = (n+2)(n+1)n 6 , defined for all integers n. These are the generalized binomial coefficients ( n+2 3 ) , as popularized in [2]. With these we can prove the following.

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Research paper thumbnail of Help with the Jumble

The American Mathematical Monthly, 2019

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Research paper thumbnail of Characterizing Pseudodendrimers. I. Graph Representations of Pseudodendrimers Formed by Enhancements to Propagation of Linear Units

Graph theory was used to analyze a series of small pseudodendrimeric structures. Descriptive indi... more Graph theory was used to analyze a series of small pseudodendrimeric structures. Descriptive indices were developed to characterize the pseudodendrimer graphs. The relative proportion of these dendrimers in typical samples was estimated based on three growth models. Weighted average values for the descriptive indices over typical aggregate samples were found to differ only slightly from values for perfect dendrimers.

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Research paper thumbnail of The multidimensional Frobenius problem

Involve, Dec 31, 2011

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Research paper thumbnail of The Golden Supercircle

The Mathematical Intelligencer

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Research paper thumbnail of Count to Six in Six Languages

The American Mathematical Monthly, 2019

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Research paper thumbnail of Elasticity in Apery sets

arXiv (Cornell University), Aug 18, 2019

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Research paper thumbnail of Membership and elasticity in certain affine monoids

arXiv (Cornell University), Dec 6, 2018

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Research paper thumbnail of Diversity in monoids

Czechoslovak Mathematical Journal, 2012

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Research paper thumbnail of Third International Meeting on Integer-Valued Polynomials

Third International Meeting on Integer-Valued Polynomials and Problems in Commutative Algebra Com... more Third International Meeting on Integer-Valued Polynomials and Problems in Commutative Algebra Combinatorial, arithmetical, algebraic, topological and dynamical aspects CIRM International center of mathematics meetings, 163 avenue de luminy 13288 MARSEILLE Organizing committee Sabine EVRARD AMIENS Youssef FARES AMIENS Amandine LERICHE AMIENS Jean-Luc CHABERT AMIENS Paul-Jean CAHEN MARSEILLE Scientific committee Paul-Jean CAHEN FRANCE Jean-Luc CHABERT FRANCE Stefania GABELLI ITALY Byung KANG SOUTH COREA Roger WIEGAND USA MONDAY THUESDAY WEDNESDAY THURSDAY FRIDAY 9H 9H/9H30 9H/9H30 9H40/10H 9H40/10H 9H50/10H30 9H50/10H30 Alice FABBRI Valentina BARUCCI 10H 10H10/10H30 10H10/10H30 Said El BAGHDADI Marco FONTANA 11H 11H/11H40 11H/11H30 11H/11H30 11H/11H30 11H/11H30 11H40/12H 11H40/12H10 11H40/12H 11H40/12H 11H50/12H30 Vadim PONOMARENKO Gabriele FUSACCHIA Gabriel PICAVET 12H 12H10/12H30 12H10/12H30 12H10/12H30 Faten KHOUJA Driss KARIM Amor HAOUAOUI 14H/14H20 Mohamed KHALIFA 14H30 14H30/15H...

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Research paper thumbnail of The Power Mean Test

We devise a new test for convergence or divergence of an infinite series — the Power Mean Test. W... more We devise a new test for convergence or divergence of an infinite series — the Power Mean Test. We explore the strength of this test relative to that of the Ratio and Root Tests and provide a family of series where the Power Mean Test is the most useful of the three tests.

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Research paper thumbnail of Diversity in inside factorial monoids

Czechoslovak Mathematical Journal, 2012

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Research paper thumbnail of INVOLVE 3:3(2010) Distinct solution to a linear congruence

inv lve a journal of mathematics mathematical sciences publishers Distinct solution to a linear c... more inv lve a journal of mathematics mathematical sciences publishers Distinct solution to a linear congruence

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Research paper thumbnail of Complementary Numerical Sets

arXiv: Combinatorics, 2020

A numerical set SSS is a cofinite subset of mathbbN\\mathbb{N}mathbbN which contains 000. We use the natural ... more A numerical set SSS is a cofinite subset of mathbbN\\mathbb{N}mathbbN which contains 000. We use the natural bijection between numerical sets and Young diagrams to define a numerical set widetildeS\\widetilde{S}widetildeS, such that their Young diagrams are complements. We determine various properties of widetildeS\\widetilde{S}widetildeS, particularly with an eye to closure under addition (for both SSS and widetildeS\\widetilde{S}widetildeS), which promotes a numerical set to become a numerical semigroup.

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Research paper thumbnail of On the set of catenary degrees of finitely generated cancellative commutative monoids

The catenary degree of an element n of a cancellative commutative monoid S is a nonnegative integ... more The catenary degree of an element n of a cancellative commutative monoid S is a nonnegative integer measuring the distance between the irreducible factorizations of n. The catenary degree of the monoid S, defined as the supremum over all catenary degrees occurring in S, has been heavily studied as an invariant of nonunique factorization. In this paper, we investigate the set C(S) of catenary degrees achieved by elements of S as a factorization invariant, focusing on the case where S in finitely generated (where C(S) is known to be finite). Answering an open question posed by García-Sánchez, we provide a method to compute the smallest nonzero element of C(S) that parallels a well-known method of computing the maximum value. We also give several examples demonstrating certain extremal behavior for C(S), and present some open questions for further study.

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Research paper thumbnail of The Frobenius Number of Geometric Sequences

The Frobenius problem is about finding the largest integer that is not contained in the numerical... more The Frobenius problem is about finding the largest integer that is not contained in the numerical semigroup generated by a given set of positive integers. In this paper, we derive a solution to the Frobenius problem for sets of the form {mk,mk−1n,mk−2n2,..., nk}, where m,n are relatively prime positive integers. 1.

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Research paper thumbnail of The Probability that Two Semigroup Elements Commute Can Be Anything

In a recent article [2] in this Journal, Givens defined a finite semigroup’s commuting probabilit... more In a recent article [2] in this Journal, Givens defined a finite semigroup’s commuting probability as the probability that x? y = y? x when x and y are chosen at random (independently and uniformly) from the semigroup elements. She asked which commuting probabilities can be achieved, and partially answered this question by showing that the achievable commuting probabilites are dense in (0, 1]. We extend this result to prove that every rational number in (0, 1] can be achieved. We begin by recalling Lagrange’s celebrated four-square theorem (found in, e.g., [1]). It states that every natural number can be expressed as the sum of four integer squares; furthermore, three squares suffice unless the number is of the form 4k(8m+ 7). The proof proceeds with four constructions; it is unknown if a single semigroup family can answer this question. Claim 1 Every rational in (0, 1/3] is an achievable commuting probability. Proof. For positive integers a, b, c and nonnegative integer k, we consi...

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Research paper thumbnail of Multivariate analysis of functional metagenomes

Frontiers in Genetics, 2013

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Research paper thumbnail of On length densities

For a commutative cancellative monoid M, we introduce the notion of the length density of both a ... more For a commutative cancellative monoid M, we introduce the notion of the length density of both a nonunit x∈ M, denoted LD(x), and the entire monoid M, denoted LD(M). This invariant is related to three widely studied invariants in the theory of non-unit factorizations, L(x), ℓ(x), and ρ(x). We consider some general properties of LD(x) and LD(M) and give a wide variety of examples using numerical semigroups, Puiseux monoids, and Krull monoids. While we give an example of a monoid M with irrational length density, we show that if M is finitely generated, then LD(M) is rational and there is a nonunit element x∈ M with LD(M)=LD(x) (such a monoid is said to have accepted length density). While it is well-known that the much studied asymptotic versions of L(x), ℓ (x) and ρ (x) (denoted L(x), ℓ(x), and ρ (x)) always exist, we show the somewhat surprising result that LD(x) = lim_n→∞LD(x^n) may not exist. We also give some finiteness conditions on M that force the existence of LD(x).

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Research paper thumbnail of On length densities

arXiv (Cornell University), Aug 15, 2020

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Research paper thumbnail of Pollock’s Generalized Tetrahedral Numbers Conjecture

The American Mathematical Monthly, 2021

The nth tetrahedral number Ten = ( n+2 3 ) represents the sum of the first n triangular numbers. ... more The nth tetrahedral number Ten = ( n+2 3 ) represents the sum of the first n triangular numbers. In the song “The Twelve Days of Christmas,” Ten counts the total number of gifts received after day n. A longstanding conjecture of Pollock (from [4]) is that every positive integer may be expressed as the sum of at most five tetrahedral numbers. To date, only 241 positive integers have been found requiring five tetrahedral numbers (see [3]). Recently, progress has been made (in [1]) on a related conjecture of Pollock from the same 19th century paper. Here we instead consider generalized tetrahedral numbers Ten = (n+2)(n+1)n 6 , defined for all integers n. These are the generalized binomial coefficients ( n+2 3 ) , as popularized in [2]. With these we can prove the following.

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Research paper thumbnail of Help with the Jumble

The American Mathematical Monthly, 2019

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Research paper thumbnail of Characterizing Pseudodendrimers. I. Graph Representations of Pseudodendrimers Formed by Enhancements to Propagation of Linear Units

Graph theory was used to analyze a series of small pseudodendrimeric structures. Descriptive indi... more Graph theory was used to analyze a series of small pseudodendrimeric structures. Descriptive indices were developed to characterize the pseudodendrimer graphs. The relative proportion of these dendrimers in typical samples was estimated based on three growth models. Weighted average values for the descriptive indices over typical aggregate samples were found to differ only slightly from values for perfect dendrimers.

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Research paper thumbnail of The multidimensional Frobenius problem

Involve, Dec 31, 2011

Bookmarks Related papers MentionsView impact

Research paper thumbnail of The Golden Supercircle

The Mathematical Intelligencer

Bookmarks Related papers MentionsView impact

Research paper thumbnail of Count to Six in Six Languages

The American Mathematical Monthly, 2019

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Research paper thumbnail of Elasticity in Apery sets

arXiv (Cornell University), Aug 18, 2019

Bookmarks Related papers MentionsView impact

Research paper thumbnail of Membership and elasticity in certain affine monoids

arXiv (Cornell University), Dec 6, 2018

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Research paper thumbnail of Diversity in monoids

Czechoslovak Mathematical Journal, 2012

Bookmarks Related papers MentionsView impact

Research paper thumbnail of Third International Meeting on Integer-Valued Polynomials

Third International Meeting on Integer-Valued Polynomials and Problems in Commutative Algebra Com... more Third International Meeting on Integer-Valued Polynomials and Problems in Commutative Algebra Combinatorial, arithmetical, algebraic, topological and dynamical aspects CIRM International center of mathematics meetings, 163 avenue de luminy 13288 MARSEILLE Organizing committee Sabine EVRARD AMIENS Youssef FARES AMIENS Amandine LERICHE AMIENS Jean-Luc CHABERT AMIENS Paul-Jean CAHEN MARSEILLE Scientific committee Paul-Jean CAHEN FRANCE Jean-Luc CHABERT FRANCE Stefania GABELLI ITALY Byung KANG SOUTH COREA Roger WIEGAND USA MONDAY THUESDAY WEDNESDAY THURSDAY FRIDAY 9H 9H/9H30 9H/9H30 9H40/10H 9H40/10H 9H50/10H30 9H50/10H30 Alice FABBRI Valentina BARUCCI 10H 10H10/10H30 10H10/10H30 Said El BAGHDADI Marco FONTANA 11H 11H/11H40 11H/11H30 11H/11H30 11H/11H30 11H/11H30 11H40/12H 11H40/12H10 11H40/12H 11H40/12H 11H50/12H30 Vadim PONOMARENKO Gabriele FUSACCHIA Gabriel PICAVET 12H 12H10/12H30 12H10/12H30 12H10/12H30 Faten KHOUJA Driss KARIM Amor HAOUAOUI 14H/14H20 Mohamed KHALIFA 14H30 14H30/15H...

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Research paper thumbnail of The Power Mean Test

We devise a new test for convergence or divergence of an infinite series — the Power Mean Test. W... more We devise a new test for convergence or divergence of an infinite series — the Power Mean Test. We explore the strength of this test relative to that of the Ratio and Root Tests and provide a family of series where the Power Mean Test is the most useful of the three tests.

Bookmarks Related papers MentionsView impact

Research paper thumbnail of Diversity in inside factorial monoids

Czechoslovak Mathematical Journal, 2012

Bookmarks Related papers MentionsView impact

Research paper thumbnail of INVOLVE 3:3(2010) Distinct solution to a linear congruence

inv lve a journal of mathematics mathematical sciences publishers Distinct solution to a linear c... more inv lve a journal of mathematics mathematical sciences publishers Distinct solution to a linear congruence

Bookmarks Related papers MentionsView impact

Research paper thumbnail of Complementary Numerical Sets

arXiv: Combinatorics, 2020

A numerical set SSS is a cofinite subset of mathbbN\\mathbb{N}mathbbN which contains 000. We use the natural ... more A numerical set SSS is a cofinite subset of mathbbN\\mathbb{N}mathbbN which contains 000. We use the natural bijection between numerical sets and Young diagrams to define a numerical set widetildeS\\widetilde{S}widetildeS, such that their Young diagrams are complements. We determine various properties of widetildeS\\widetilde{S}widetildeS, particularly with an eye to closure under addition (for both SSS and widetildeS\\widetilde{S}widetildeS), which promotes a numerical set to become a numerical semigroup.

Bookmarks Related papers MentionsView impact

Research paper thumbnail of On the set of catenary degrees of finitely generated cancellative commutative monoids

The catenary degree of an element n of a cancellative commutative monoid S is a nonnegative integ... more The catenary degree of an element n of a cancellative commutative monoid S is a nonnegative integer measuring the distance between the irreducible factorizations of n. The catenary degree of the monoid S, defined as the supremum over all catenary degrees occurring in S, has been heavily studied as an invariant of nonunique factorization. In this paper, we investigate the set C(S) of catenary degrees achieved by elements of S as a factorization invariant, focusing on the case where S in finitely generated (where C(S) is known to be finite). Answering an open question posed by García-Sánchez, we provide a method to compute the smallest nonzero element of C(S) that parallels a well-known method of computing the maximum value. We also give several examples demonstrating certain extremal behavior for C(S), and present some open questions for further study.

Bookmarks Related papers MentionsView impact

Research paper thumbnail of The Frobenius Number of Geometric Sequences

The Frobenius problem is about finding the largest integer that is not contained in the numerical... more The Frobenius problem is about finding the largest integer that is not contained in the numerical semigroup generated by a given set of positive integers. In this paper, we derive a solution to the Frobenius problem for sets of the form {mk,mk−1n,mk−2n2,..., nk}, where m,n are relatively prime positive integers. 1.

Bookmarks Related papers MentionsView impact

Research paper thumbnail of The Probability that Two Semigroup Elements Commute Can Be Anything

In a recent article [2] in this Journal, Givens defined a finite semigroup’s commuting probabilit... more In a recent article [2] in this Journal, Givens defined a finite semigroup’s commuting probability as the probability that x? y = y? x when x and y are chosen at random (independently and uniformly) from the semigroup elements. She asked which commuting probabilities can be achieved, and partially answered this question by showing that the achievable commuting probabilites are dense in (0, 1]. We extend this result to prove that every rational number in (0, 1] can be achieved. We begin by recalling Lagrange’s celebrated four-square theorem (found in, e.g., [1]). It states that every natural number can be expressed as the sum of four integer squares; furthermore, three squares suffice unless the number is of the form 4k(8m+ 7). The proof proceeds with four constructions; it is unknown if a single semigroup family can answer this question. Claim 1 Every rational in (0, 1/3] is an achievable commuting probability. Proof. For positive integers a, b, c and nonnegative integer k, we consi...

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Research paper thumbnail of Multivariate analysis of functional metagenomes

Frontiers in Genetics, 2013

Bookmarks Related papers MentionsView impact

Research paper thumbnail of On length densities

For a commutative cancellative monoid M, we introduce the notion of the length density of both a ... more For a commutative cancellative monoid M, we introduce the notion of the length density of both a nonunit x∈ M, denoted LD(x), and the entire monoid M, denoted LD(M). This invariant is related to three widely studied invariants in the theory of non-unit factorizations, L(x), ℓ(x), and ρ(x). We consider some general properties of LD(x) and LD(M) and give a wide variety of examples using numerical semigroups, Puiseux monoids, and Krull monoids. While we give an example of a monoid M with irrational length density, we show that if M is finitely generated, then LD(M) is rational and there is a nonunit element x∈ M with LD(M)=LD(x) (such a monoid is said to have accepted length density). While it is well-known that the much studied asymptotic versions of L(x), ℓ (x) and ρ (x) (denoted L(x), ℓ(x), and ρ (x)) always exist, we show the somewhat surprising result that LD(x) = lim_n→∞LD(x^n) may not exist. We also give some finiteness conditions on M that force the existence of LD(x).

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