Viorel Nitica
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Viorel Nitica has a total of 32 co-authors. Here are the top co-authors by number of shared publications:
- Andrei Török
Department of Mathematics, University of Houston, Houston, TX 77204, USA
11 shared publications - Anatole Katok
Pennsylvania State University
6 shared publications - Ivan Singer
Department of Mathematics Institute of Mathematics P. O. Box 1-764 Bucharest Romania
5 shared publications - Sergei Sergeev
School of Mathematics University of Birmingham Edgbaston B15 2TT
5 shared publications I. Singer
4 shared publications- Ian Melbourne
Department of Mathematics, University of Surrey, Guildford, Surrey GU2 7XH, UK
4 shared publications
Latest Publications
- Open Access
On a Semigroup Problem II
We consider the following semigroup problem: is the closure of a semigroup S in a topological vector space X a group when S does not lie on “one side” of any closed hyperplane of X? Whereas for finite dimensional spaces, the answer is positive, we give a new example of infinite dimensional spaces where the answer is negative.
Open AccessOn a Semigroup Problem II
We consider the following semigroup problem: is the closure of a semigroup S in a topological vector space X a group when S does not lie on “one side” of any closed hyperplane of X? Whereas for finite dimensional spaces, the answer is positive, we give a new example of infinite dimensional spaces where the answer is negative.
- Open Access
About the Orbit Structure of Sequences of Maps of Integers
Motivated by connections to the study of sequences of integers, we study, from a dynamical systems point of view, the orbit structure for certain sequences of maps of integers. We find sequences of maps for which all individual orbits are bounded and periodic and for which the number of periodic orbits of fixed period is finite. This allows the introduction of a formal -function for the maps in these sequences, which are actually polynomials. We also find sequences of maps for which the orbit structure is more complicated, as they have both bounded and unbounded orbits, both individual and global. Most of our results are valid in a general numeration base.Open AccessAbout the Orbit Structure of Sequences of Maps of Integers
Motivated by connections to the study of sequences of integers, we study, from a dynamical systems point of view, the orbit structure for certain sequences of maps of integers. We find sequences of maps for which all individual orbits are bounded and periodic and for which the number of periodic orbits of fixed period is finite. This allows the introduction of a formal -function for the maps in these sequences, which are actually polynomials. We also find sequences of maps for which the orbit structure is more complicated, as they have both bounded and unbounded orbits, both individual and global. Most of our results are valid in a general numeration base. - Open Access
- 1 January 2019
- Published by American Institute of Mathematical Sciences (AIMS) in Discrete and Continuous Dynamical Systems - S
- Vol. 12 (8) , 2365-2377
On a semigroup problem
If$ S $ is a semigroup in$ \mathbb{R}^n $ that is not separated by a linear functional, then it is known that the closure of$ S $ is a group. We investigate a similar statement in an infinite dimensional topological vector space$ X $ . We show that if$ X $ is an infinite dimensional Banach space, then there exists a semigroup$ S\subset X $ , not separated by the continuous functionals supported by the closed linear span of$ S $ , for which the closure of the semigroup is not a group. If$ X $ is an infinite dimensional Fréchet space, then the closure of a semigroup that is not separated is always a group if and only if$ X $ is$ \mathbb{R}^{\omega} $ , the countably infinite direct product of lines. Other infinite dimensional topological vector spaces, such as$ \mathbb{R}^{\infty} $ , the countably infinite direct sum of lines, are discussed. The Semigroup Problem has applications to the study of certain dynamical systems, in particular for the construction of topologically transitive extensions of hyperbolic systems. Some examples are shown in the paper.- 1 January 2019
- Published by American Institute of Mathematical Sciences (AIMS) in Discrete and Continuous Dynamical Systems - S
- Vol. 12 (8) , 2365-2377
Open AccessOn a semigroup problem
If$ S $ is a semigroup in$ \mathbb{R}^n $ that is not separated by a linear functional, then it is known that the closure of$ S $ is a group. We investigate a similar statement in an infinite dimensional topological vector space$ X $ . We show that if$ X $ is an infinite dimensional Banach space, then there exists a semigroup$ S\subset X $ , not separated by the continuous functionals supported by the closed linear span of$ S $ , for which the closure of the semigroup is not a group. If$ X $ is an infinite dimensional Fréchet space, then the closure of a semigroup that is not separated is always a group if and only if$ X $ is$ \mathbb{R}^{\omega} $ , the countably infinite direct product of lines. Other infinite dimensional topological vector spaces, such as$ \mathbb{R}^{\infty} $ , the countably infinite direct sum of lines, are discussed. The Semigroup Problem has applications to the study of certain dynamical systems, in particular for the construction of topologically transitive extensions of hyperbolic systems. Some examples are shown in the paper. - Open Access
Infinite sets of $b$-additive and $b$-multiplicative Ramanujan-Hardy numbers
Let $b$ be a numeration base. A $b$-additive Ramanujan-Hardy number $N$ is an integer for which there exists at least an integer $M$, called additive multiplier, such that the product of $M$ and the sum of base $b$ digits of $N$, added to the reversal of the product, gives $N$. We show that for any $b$ there exists an infinity of $b$-additive Ramanujan-Hardy numbers and an infinity of additive multipliers. A $b$-multiplicative Ramanujan-Hardy number $N$ is an integer for which there exists at least an integer $M$, called multiplicative multiplier, such that the product of $M$ and the sum of base $b$ digits of $N$, multiplied by the reversal of the product, gives $N$. We show that for an even $b$, $b\equiv 1 \pmod {3}$, and for $b=2$, there exists an infinity of $b$-multiplicative Ramanujan-Hardy numbers and an infinity of multiplicative multipliers. These results completely answer two questions and partially answer two other questions among those asked in V. Ni\c{t}ic\u{a}, \emph{About some relatives of the taxicab number}, arXiv:1805.10739v4.Open AccessInfinite sets of $b$-additive and $b$-multiplicative Ramanujan-Hardy numbers
Let $b$ be a numeration base. A $b$-additive Ramanujan-Hardy number $N$ is an integer for which there exists at least an integer $M$, called additive multiplier, such that the product of $M$ and the sum of base $b$ digits of $N$, added to the reversal of the product, gives $N$. We show that for any $b$ there exists an infinity of $b$-additive Ramanujan-Hardy numbers and an infinity of additive multipliers. A $b$-multiplicative Ramanujan-Hardy number $N$ is an integer for which there exists at least an integer $M$, called multiplicative multiplier, such that the product of $M$ and the sum of base $b$ digits of $N$, multiplied by the reversal of the product, gives $N$. We show that for an even $b$, $b\equiv 1 \pmod {3}$, and for $b=2$, there exists an infinity of $b$-multiplicative Ramanujan-Hardy numbers and an infinity of multiplicative multipliers. These results completely answer two questions and partially answer two other questions among those asked in V. Ni\c{t}ic\u{a}, \emph{About some relatives of the taxicab number}, arXiv:1805.10739v4. - Open Access
About Some Relatives of the Taxicab Number
The taxicab number, $1729$, has the following property. If we add its digits we obtain $19$. The number obtained from $19$ by reversing the order of its digits is $91$. If we multiply $19$ by $91$ we obtain again $1729$. In the paper we study various generalizations of this property.Open AccessAbout Some Relatives of the Taxicab Number
The taxicab number, $1729$, has the following property. If we add its digits we obtain $19$. The number obtained from $19$ by reversing the order of its digits is $91$. If we multiply $19$ by $91$ we obtain again $1729$. In the paper we study various generalizations of this property. - 23 January 2018
- Published by Institute of Electrical and Electronics Engineers (IEEE) in IEEE Transactions on Information Theory
- Vol. 64 (4) , 3162-3169
Revisiting a Tiling Hierarchy
We revisit Golomb's tiling hierarchy for finite sets of polyominoes. We consider the case when only translations are allowed for the tiles in a tile set. In this classification, for several levels in Golomb's hierarchy, more types need to be considered. First, we show that there is no general relationship among the types corresponding to a fixed level for the existence of tilings. Second, we review the relationships among the levels from Golomb's hierarchy that are still valid in this setup, and third, we give counterexamples for several relationships in Golomb's hierarchy that fail. As a consequence of the results, we show alternative tiling hierarchies. We also show some unexpected additions to all of the above hierarchies and discuss various implications of tiling capabilities if deficient regions are added to the pool.
- 23 January 2018
- Published by Institute of Electrical and Electronics Engineers (IEEE) in IEEE Transactions on Information Theory
- Vol. 64 (4) , 3162-3169
Revisiting a Tiling Hierarchy
We revisit Golomb's tiling hierarchy for finite sets of polyominoes. We consider the case when only translations are allowed for the tiles in a tile set. In this classification, for several levels in Golomb's hierarchy, more types need to be considered. First, we show that there is no general relationship among the types corresponding to a fixed level for the existence of tilings. Second, we review the relationships among the levels from Golomb's hierarchy that are still valid in this setup, and third, we give counterexamples for several relationships in Golomb's hierarchy that fail. As a consequence of the results, we show alternative tiling hierarchies. We also show some unexpected additions to all of the above hierarchies and discuss various implications of tiling capabilities if deficient regions are added to the pool.
- Open Access
Tropical convexity over max-min semiring
This is a survey on an analogue of tropical convexity developed over the max-min semiring, starting with the descriptions of max-min segments, semispaces, hyperplanes and an account of separation and non-separation results based on semispaces. There are some new results. In particular, we give new "colorful" extensions of the max-min Caratheodory theorem. In the end of the paper, we list some consequences of the topological Radon and Tverberg theorems (like Helly and Centerpoint theorems), valid over a more general class of max-T semirings, where multiplication is a triangular norm.
Open AccessTropical convexity over max-min semiring
This is a survey on an analogue of tropical convexity developed over the max-min semiring, starting with the descriptions of max-min segments, semispaces, hyperplanes and an account of separation and non-separation results based on semispaces. There are some new results. In particular, we give new "colorful" extensions of the max-min Caratheodory theorem. In the end of the paper, we list some consequences of the topological Radon and Tverberg theorems (like Helly and Centerpoint theorems), valid over a more general class of max-T semirings, where multiplication is a triangular norm.
- Open Access
Signed tilings by ribbon L n-ominoes, n odd, via Groebner bases
We show that a rectangle can be signed tiled by ribbon L n-ominoes, n odd, if and only if it has a side divisible by n. A consequence of our technique, based on the exhibition of an explicit Groebner basis, is that any k-inflated copy of the skewed L n-omino has a signed tiling by skewed L n-ominoes. We also discuss regular tilings by ribbon L n-ominoes, n odd, for rectangles and more general regions. We show that in this case obstructions appear that are not detected by signed tilings.
Open AccessSigned tilings by ribbon L n-ominoes, n odd, via Groebner bases
We show that a rectangle can be signed tiled by ribbon L n-ominoes, n odd, if and only if it has a side divisible by n. A consequence of our technique, based on the exhibition of an explicit Groebner basis, is that any k-inflated copy of the skewed L n-omino has a signed tiling by skewed L n-ominoes. We also discuss regular tilings by ribbon L n-ominoes, n odd, for rectangles and more general regions. We show that in this case obstructions appear that are not detected by signed tilings.
- Open Access
Signed tilings by ribbon L n-ominoes, n even, via Groebner bases
Let $\mathcal{T}_n$ be the set of ribbon $L$-shaped $n$-ominoes for some $n\ge 4$ even, and let $\mathcal{T}_n^+$ be $\mathcal{T}_n$ with an extra $2\times 2$ square. We investigate signed tilings of rectangles by $\mathcal{T}_n$ and $\mathcal{T}_n^+$. We show that a rectangle has a signed tiling by $\mathcal{T}_n$ if and only if both sides of the rectangle are even and one of them is divisible by $n$, or if one of the sides is odd and the other side is divisible by $n\left (\frac{n}{2}-2\right ).$ We also show that a rectangle has a signed tiling by $\mathcal{T}_n^+, n\ge 6$ even, if and only if both sides of the rectangle are even, or if one of the sides is odd and the other side is divisible by $n\left (\frac{n}{2}-2\right ).$ Our proofs are based on the exhibition of explicit Gr\"obner bases for the ideals generated by polynomials associated to the tiling sets. In particular, we show that some of the regular tiling results in \emph{ V.~Nitica, Every tiling of the first quadrant by ribbon $L$ $n$-ominoes follows the rectangular pattern. Open Journal of Discrete Mathematics, {\em 5}, (2015) 11--25,} cannot be obtained from coloring invariants.Open AccessSigned tilings by ribbon L n-ominoes, n even, via Groebner bases
Let $\mathcal{T}_n$ be the set of ribbon $L$-shaped $n$-ominoes for some $n\ge 4$ even, and let $\mathcal{T}_n^+$ be $\mathcal{T}_n$ with an extra $2\times 2$ square. We investigate signed tilings of rectangles by $\mathcal{T}_n$ and $\mathcal{T}_n^+$. We show that a rectangle has a signed tiling by $\mathcal{T}_n$ if and only if both sides of the rectangle are even and one of them is divisible by $n$, or if one of the sides is odd and the other side is divisible by $n\left (\frac{n}{2}-2\right ).$ We also show that a rectangle has a signed tiling by $\mathcal{T}_n^+, n\ge 6$ even, if and only if both sides of the rectangle are even, or if one of the sides is odd and the other side is divisible by $n\left (\frac{n}{2}-2\right ).$ Our proofs are based on the exhibition of explicit Gr\"obner bases for the ideals generated by polynomials associated to the tiling sets. In particular, we show that some of the regular tiling results in \emph{ V.~Nitica, Every tiling of the first quadrant by ribbon $L$ $n$-ominoes follows the rectangular pattern. Open Journal of Discrete Mathematics, {\em 5}, (2015) 11--25,} cannot be obtained from coloring invariants. - 1 January 2016
- Published by Elsevier in Linear Algebra and its Applications
- Vol. 489, 144-166
The geometric structure of max-plus hemispaces
- 1 January 2016
- Published by Elsevier in Linear Algebra and its Applications
- Vol. 489, 144-166
The geometric structure of max-plus hemispaces