Related papers: Monotone unitary families
We prove generalizations of L\"owner's results on matrix monotone functions to several variables. We give a characterization of when a function of $d$ variables is locally monotone on $d$-tuples of commuting self-adjoint $n$-by-$n$…
We study the number of queries needed to identify a monotone Boolean function $f:\{0,1\}^n \rightarrow \{0,1\}$. A query consists of a 0-1-sequence, and the answer is the value of $f$ on that sequence. It is well-known that the number of…
In this short paper, the commutator of monomials of operators obeying constant commutation relations is expressed in terms of anticommutators. The formula involves Bernoulli numbers or Euler polynomials evaluated in zero. The role of…
We study uniformity conditions for parameterized Boolean circuit families. Uniformity conditions require that the infinitely many circuits in a circuit family are in some sense easy to construct from one shared description. For shallow…
A monotone drawing of a graph G is a straight-line drawing of G such that every pair of vertices is connected by a path that is monotone with respect to some direction. Trees, as a special class of graphs, have been the focus of several…
Let $\mathcal{H}$ be a complex, separable Hilbert space (of finite or infinite dimension), and let $\mathcal{U}(\mathcal{H})$ denote the group of unitary operators on $\mathcal{H}$. A symmetry is, by definition, a unitary operator $J$ with…
Unitary graphs are arc-transitive graphs with vertices the flags of Hermitian unitals and edges defined by certain elements of the underlying finite fields. They played a significant role in a recent classification of a class of…
We demonstrate that when a graph exhibits a specific type of symmetry, it satisfies the Union Closed Conjecture(UCC). Additionally, we show that certain graph classes, such as Cylindrical Grid Graphs and Torus Grid Graphs also satisfy the…
We develop an abstract look at linear optical networks from the viewpoint of combinatorics and permanents. In particular we show that calculation of matrix elements of unitarily transformed photonic multi-mode states is intimately linked to…
The uniform structure on a differential space defined by a family of generators is considered.
We show that certain determinantal functions of multiple matrices, when summed over the symmetries of the cube, decompose into functions of the original matrices. These are shown to be true in complete generality; that is, no properties of…
We introduce unitary network, an oriented architecture for tensor network unitaries. Compared to existing architectures, in a unitary network each local tensor is required to be a unitary matrix upon suitable reshaping. Global unitarity is…
We consider the unitary group $\U$ of complex, separable, infinite-dimensional Hilbert space as a discrete group. It is proved that, whenever $\U$ acts by isometries on a metric space, every orbit is bounded. Equivalently, $\U$ is not the…
We demonstrate new abstract characterizations for unital and non-unital operator spaces. We characterize unital operator spaces in terms of the cone of accretive operators (operators whose real part is positive). Defining the gauge of an…
Let $U$ be a unitary operator defined on some infinite-dimensional complex Hilbert space ${\cal H}$. Under some suitable regularity assumptions, it is known that a local positive commutation relation between $U$ and an auxiliary…
The Union Closed Sets Conjecture is one of the most renowned problems in combinatorics. Its appeal lies in the simplicity of its statement contrasted with the potential complexity of its resolution. The conjecture posits that, in any union…
A family of sets is called union-closed if whenever $A$ and $B$ are sets of the family, so is $A\cup B$. The long-standing union-closed conjecture states that if a family of subsets of $[n]$ is union-closed, some element appears in at least…
We give a description of unital operads in a symmetric monoidal category as monoids in a monoidal category of unital $\Lambda$-sequences. This is a new variant of Kelly's old description of operads as monoids in the monoidal category of…
We consider unimodular matrices $M$ such that neither $M$ nor $M^{-1}$ contain zero entries. Matrices typically exhibit a trade-off: small $M$ imply large $M^{-1}$. We investigate rare cases where both remain small, classify these matrices…
We take the trace of Von-Neumann's ergodic theorem and get a trace formula of a unitary matrix family. It is an extension of Poisson summation formula in higher dimension. We also construct a family of crystalline measure with complex…