Related papers: The operator formula for monotone triangles - simp…
Let $A(n,r;3)$ be the total weight of the alternating sign matrices of order $n$ whose sole `1' of the first row is at the $r^{th}$ column and the weight of an individual matrix is $3^k$ if it has $k$ entries equal to -1. Define the…
We lower bound the rank of a tensor by a linear combination of the ranks of three of its unfoldings, using Sylvester's rank inequality. In a similar way, we lower bound the symmetric rank by a linear combination of the symmetric ranks of…
In many classification tasks there is a requirement of monotonicity. Concretely, if all else remains constant, increasing (resp. decreasing) the value of one or more features must not decrease (resp. increase) the value of the prediction.…
Operator monotone functions, introduced by Lowner in 1934, are an important class of real-valued functions. They arise naturally in matrix and operator theory and have various applications in other branches of mathematics and related…
Consider two horizontal lines in the plane. A pair of a point on the top line and an interval on the bottom line defines a triangle between two lines. The intersection graph of such triangles is called a simple-triangle graph. This paper…
In this paper, we prove that the set of triangulations of a polygon can be equipped with an order to become a lattice. First, we define this order. In [HN99], authors defined the flip operator and then prove some properties of the graph of…
We consider the problem of counting straight-edge triangulations of a given set $P$ of $n$ points in the plane. Until very recently it was not known whether the exact number of triangulations of $P$ can be computed asymptotically faster…
This paper concerns three classes of real-valued functions on intervals, operator monotone functions, operator convex functions, and strongly operator convex functions. Strongly operator convex functions were previously treated in [3] and…
We derive exact and sharp lower bounds for the number of monochromatic generalized Schur triples $(x,y,x+ay)$ whose entries are from the set $\{1,\dots,n\}$, subject to a coloring with two different colors. Previously, only asymptotic…
We introduce and study a mathematical framework for a broad class of regularization functionals for ill-posed inverse problems: Regularization Graphs. Regularization graphs allow to construct functionals using as building blocks linear…
The number of $n \times n$ matrices whose entries are either -1, 0, or 1, whose row- and column- sums are all 1, and such that in every row and every column the non-zero entries alternate in sign, is proved to be $[1!4! >...…
The paper is devoted to the problem of exact calculation of the norms in ideal spaces for monotone operators on the cones of functions with monotonicity properties. We implement a general approach to this problem that covers many concrete…
Let $P$ be a set of $n$ points in the plane. A crossing-free structure on $P$ is a plane graph with vertex set $P$. Examples of crossing-free structures include triangulations of $P$, spanning cycles of $P$, also known as polygonalizations…
We shall present a ``linear algebraic'' proof (involving some calculations in the algebra of linear operators on a vector space of polynomials and some manipulations of determinants) of the formula for the enumeration of symmetric…
We consider the problem of minimal correction of the training set to make it consistent with monotonic constraints. This problem arises during analysis of data sets via techniques that require monotone data. We show that this problem is…
We identify subsets of the joint numerical range of an operator tuple in terms of its joint spectrum. This result helps us to transfer weak convergence of operator orbits into certain approximation and interpolation properties for powers in…
In this article we study decreasing and increasing factorisations of the cycle, which are decompositions of the cycle $(1~2\dots n)$ into a product of $n-1$ transpositions satisfying monotonicity conditions. We explicit a bijection between…
We enumerate and show tables of minimal diagrams for all prime knots up to the triple-crossing number equal to five. We derive a minimal generating set of oriented moves connecting triple-crossing diagrams of the same oriented knot. We also…
The relationships between five classes of monotonicity, namely 3^*-, 3-cyclic, strictly, para-, and maximal monotonicity, are explored for linear operators and linear relations in Hilbert space. Where classes overlap, examples are given;…
A simple-triangle graph is the intersection graph of triangles that are defined by a point on a horizontal line and an interval on another horizontal line. The time complexity of the recognition problem for simple-triangle graphs was a…