Related papers: $(P,Q)$ complex hypercontractivity

Let $H(q,p)$ be a Hamiltonian on $T^*T^n$. We show that the sequence $H_{k}(q,p)=H(kq,p)$ converges for the $\gamma$ topology defined by the author, to $\bar{H}(p)$. This is extended to the case where only some of the variables are…

D. Khavinson and G. Swiatek proved that harmonic polynomials p(z)+q(z), where p is holomorphic, q is antiholomorphic, and deg p = n > 1 = deg q, can have at most 3n-2 complex zeros. We show that this bound is sharp for all n by proving a…

Let $HD_d(p,q)$ denote the minimal size of a transversal that can always be guaranteed for a family of compact convex sets in $\mathbb{R}^d$ which satisfy the $(p,q)$-property ($p \geq q \geq d+1$). In a celebrated proof of the…

Let $\mathbb{H}^{n}$ be the Heisenberg group. For $0 \leq \alpha < Q=2n+2$ and $N \in \mathbb{N}$ we consider exponent functions $p(\cdot) : \mathbb{H}^{n} \to (0, +\infty)$, which satisfies H\"older conditions, such that $\frac{Q}{Q+N} <…

It is known that the notion of a transitive subgroup of a permutation group $G$ extends naturally to subsets of $G$. We consider subsets of the general linear group $\operatorname{GL}(n,q)$ acting transitively on flag-like structures, which…

Let $X^N$ be a family of $N\times N$ independent GUE random matrices, $Z^N$ a family of deterministic matrices, $P$ a self-adjoint non-commutative polynomial, that is for any $N$, $P(X^N)$ is self-adjoint, $f$ a smooth function. We prove…

It is shown that two vectors with coordinates in the finite $q$-element field of characteristic $p$ belong to the same orbit under the natural action of the symmetric group if each of the elementary symmetric polynomials of degree…

Each infinitesimally faithful representation of a reductive complex connected algebraic group $G$ induces a dominant morphism $\Phi$ from the group to its Lie algebra $\g$ by orthogonal projection in the endomorphism ring of the…

For $q\in\mathbb{R}$, the $Q$-matrix $Q=Q_q$ of a connected simple graph $G=(V,E)$ is $Q_q=(q^{\partial(x,y)})_{x,y\in V}$, where $\partial$ denotes the path-length distance. Describing the set $\pi(G)$ consisting of those $q\in \mathbb{R}$…

The functional equation f(p(z))=g(q(z)) is studied, where p,q are polynomials and f,g are trancendental meromorphic functions in C. We find all the pairs p,q for which there exist nonconstant f,g satisfying our equation and there exist no…

We develop a theory of average sizes of kernels of generic matrices with support constraints defined in terms of graphs and hypergraphs. We apply this theory to study unipotent groups associated with graphs. In particular, we establish…

We show that $k$-ary functions giving the measure of the intersection of multi-parametric families of sets in probability spaces, e.g. $(x,y,z) \in X \times Y \times Z \mapsto \mu(P_{x,y} \cap Q_{x,z} \cap R_{y,z})$, satisfy a particularly…

For two $n \times n$ complex matrices $A$ and $B$, we define the $q$-deformed commutator as $[ A, B ]_q := A B - q BA$ for a real parameter $q$. In this paper, we investigate a generalization of the B\"{o}ttcher-Wenzel inequality which…

We prove that fractional Helly and $(p,q)$-theorems imply $(\aleph_0,q)$-theorems in an entirely abstract setting. We give a plethora of applications, including reproving almost all earlier $(\aleph_0,q)$-theorems about geometric…

For a finite loop $Q$, let $P (Q)$ be the set of elements that can be represented as a product containing each element of $Q$ precisely once. Motivated by the recent proof of the Hall-Paige conjecture, we prove several universal…

Let $p \in (1,\infty)$ and $\Omega \subset \mathbb{R}^N$ be a domain. Let $ A: =(a_{ij}) \in L^{\infty}_{\text{loc}}(\Omega; \mathbb{R}^{N\times N})$ be a symmetric and locally uniformly positive definite matrix. Set $|\xi|_A^2:=…

The hypercontractivity is proved for the Markov semigroup associated to a class of finite/infinite dimensional stochastic Hamiltonian systems. Consequently, the Markov semigroup is exponentially convergent to the invariant probability…

We introduce a generalized definition of the isothermal compressibility ($\kappa_{T,\sigma_Q^2}$) calculable by keeping net conserved charge fluctuations rather than total number densities constant. We present lattice QCD results for this…

\begin{abstract} In this paper, we partly solve the generalized Khavinson conjecture in the setting of hyperbolic harmonic mappings in Hardy space. Assume that $u=\mathcal{P}_{\Omega}[\phi]$ and $\phi\in L^{p}(\partial\Omega, \mathbb{R})$,…

We introduce the notion of P-critical connections for hermitian holomorphic vector bundles over compact balanced manifolds: integrable hermitian connections whose curvature solves a polynomial equation. Such connections include HYM and dHYM…