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Tanya Khovanova

Publications and source records attributed to Tanya Khovanova.

At least 109 records · Page 6Linked to original sources
arXiv2011

Saturated Domino Coverings

A domino covering of a board is saturated if no domino is redundant. We introduce the concept of a fragment tiling and show that a minimal fragment tiling always corresponds to a maximal saturated domino covering. The size of a minimal fragment tiling is the domination number of the board. We define a class of regular boards and show that for these boards the domination number gives the size of a minimal X-pentomino covering. Natural sequences that count maximal saturated domino coverings of square and rectangular boards are obtained. These include the new sequences A193764, A193765, A193766, A193767, and A193768 of OEIS.

math.CO↗
arXiv2011

Jewish Problems

This is a special collection of problems that were given to select applicants during oral entrance exams to the math department of Moscow State University. These problems were designed to prevent Jews and other undesirables from getting a passing grade. Among problems that were used by the department to blackball unwanted candidate students, these problems are distinguished by having a simple solution that is difficult to find. Using problems with a simple solution protected the administration from extra complaints and appeals. This collection therefore has mathematical as well as historical value.

math.HO↗
arXiv2011

Volume bounds for shadow covering

For n >= 2 a construction is given for a large family of compact convex sets K and L in n-dimensional Euclidean space such that the orthogonal projection L_u onto the subspace u^\perp contains a translate of the corresponding projection K_u for every direction u, while the volumes of K and L satisfy V_n(K) > V_n(L). It is subsequently shown that, if the orthogonal projection L_u onto the subspace u^\perp contains a translate of K_u for every direction u, then the set (n/(n-1))L contains a translate of K. If follows that V_n(K) <= (n/(n-1))^n V_n(L). In particular, we derive a universal constant bound V_n(K) <= 2.942 V_n(L), independent of the dimension n of the ambient space. Related results are obtained for projections onto subspaces of some fixed intermediate co-dimension. Open questions and conjectures are also posed.

math.MG↗
arXiv2011

Sequences of Integers with Missing Quotients and Dense Points Without Neighbors

Let A be a pre-defined set of rational numbers. We say a set of natural numbers S is an A-quotient-free set if no ratio of two elements in S belongs to A. We find the maximal asymptotic density and the maximal upper asymptotic density of A-quotient-free sets when A belongs to a particular class. It is known that in the case A = {p, q}, where p, q are coprime integers greater than one, the latest problem is reduced to evaluation of the largest number of lattice non-adjacent points in a triangle whose legs lie on coordinate axis. We prove that this number is achieved by choosing points of the same color in the checkerboard coloring.

math.CO↗
arXiv2011

Martin Gardner's Mistake

When Martin Gardner first presented the Two-Children problem, he made a mistake in its solution. Later he corrected the mistake in another publication, but unfortunately his incorrect solution is more widely known than his correction. In fact, a Tuesday-Child variation of this problem went viral in 2010, and the same flaw keeps reappearing in solutions for that problem as well. In this article, I would like to popularize Martin Gardner's correction and conduct a detailed discussion of the new problem.

math.PR↗
arXiv2010

Baron Munchhausen Redeems Himself: Bounds for a Coin-Weighing Puzzle

We investigate a coin-weighing puzzle that appeared in the Moscow Math Olympiad in 1991. We generalize the puzzle by varying the number of participating coins, and deduce an upper bound on the number of weighings needed to solve the puzzle that is noticeably better than the trivial upper bound. In particular, we show that logarithmically-many weighings on a balance suffice.

math.CO↗
arXiv2010

Odd One Out

This article covers my second talk at the Gathering for Gardner in March, 2010. It is about an Odd One Out puzzle I invented, after having been inspired by Martin Gardner. I do not like Odd One Out questions; that is why I invented one.

math.HO↗
arXiv2010

Baron Munchhausen's Sequence

We investigate a coin-weighing puzzle that appeared in the all-Russian math Olympiad in 2000. We liked the puzzle because the methods of analysis differ from classical coin-weighing puzzles. We generalize the puzzle by varying the number of participating coins, and deduce a complete solution, perhaps surprisingly, the objective can be achieved in no more than two weighings regardless of the number of coins involved.

math.CO↗
arXiv2008

Clifford Algebras and Graphs

I show how to associate a Clifford algebra to a graph. I describe the structure of these Clifford graph algebras and provide many examples and pictures. I describe which graphs correspond to isomorphic Clifford algebras and also discuss other related sets of graphs. This construction can be used to build models of representations of simply-laced compact Lie groups.

math.CO↗
arXiv2008

Jamming as Information: a Geometric Approach

In this paper I discuss the kinds of information that can be extracted by our enemy if our jamming is too precise. I show geometric solutions for reconstructing linear routes given certain information about them, such as the shortest distance to a point or the times of entering and exiting a circle.

math.HO↗
arXiv2008

Number Gossip

This article covers my talk at the Gathering for Gardner 2008, with some additions.

math.CO↗
arXiv2008

Autobiographical Numbers

I introduce autobiographical numbers as defined in A046043 (see Online Encyclopedia of Integer Sequences). I continue by defining and analyzing biographies, curricula vitae and complete life stories of numbers. I end with the definition of mutually-praising number pairs.

math.CO↗
arXiv2007

How to Create a New Integer Sequence

There are several standard procedures used to create new sequences from a given sequence or from a given pair of sequences. In this paper I discuss the most popular of these procedures. For each procedure, I give a definition and provide examples based on three famous sequences: the natural numbers, the prime numbers and the Fibonacci numbers. I also add my thoughts on what makes a sequence interesting. My goal is to help my readers invent new sequences, differentiate interesting sequences from boring ones, and better understand sequences they encounter.

math.CO↗
arXiv2007

Unique Tournaments and Radar Tracking

The sequence counting the number of unique tournaments with n people is the same as the sequence counting non-tracking binary strings corresponding to n-2 radar observations with the tracking rule "3 out of 5 with loss 2." This fact allows us to build a bijection between unique tournaments and non-tracking binary strings.

math.CO↗
arXiv1997

Quantum groups and representations with highest weight

We consider a special category of Hopf algebras, depending on parameters $Σ$ which possess properties similar to the category of representations of simple Lie group with highest weight $λ$. We connect quantum groups to minimal objects in this categories---they correspond to irreducible representations in the category of representations with highest weight $λ$. Moreover, we want to correspond quantum groups only to finite dimensional irreducible representations. This gives us a condition for $λ$: $λ$--- is dominant means the minimal object in the category of representations with highest weight $λ$ is finite dimensional. We put similar condition for $Σ$. We call $Σ$ dominant if the minimal object in corresponding category has polynomial growth. Now we propose to define quantum groups starting from dominant parameters $Σ$.

q-alg↗
arXiv1997

On quantum group GL_{p,q}(2)

In the Hopf algebra GL_{p,q}(2) the determinant is central iff p=q. In this case we put determinant to be equal to 1 to get SL_q(2). In this paper I consider the case when p/q is a root of unity; and, consequently, a power of the determinant is central.

q-alg↗
arXiv1994

On quantum group SL_q(2)

We start with the observation that the quantum group SL_q(2), described in terms of its algebra of functions has a quantum subgroup, which is just a usual Cartan group. Based on this observation we develop a general method of constructing quantum groups with similar property. We also describe this method in the language of quantized universal enveloping algebras, which is another common method of studying quantum groups. We carry our method in detail for root systems of type SL(2); as a byproduct we find a new series of quantum groups - metaplectic groups of SL(2)-type. Representations of these groups can provide interesting examples of bimodule categories over monoidal category of representations of SL_q(2).

hep-th↗