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In this paper we present a new extension of the theory of well-bounded operators to cover operators with complex spectrum. In previous work a new concept of the class of absolutely continuous functions on a nonempty compact subset $\sigma$…

In this paper, we introduce and investigate a new subclass of the function class $\Sigma$ of bi-univalent functions defined in the open unit disk, which are associated with the S\u{a}l\u{a}gean type $q-$ difference operator and satisfy some…

Potential functional approximations are an intriguing alternative to density functional approximations. The potential functional that is dual to the Lieb density functional is defined and properties given. The relationship between…

In this study, we first define the local potential associated to a weakly positive closed supercurrent in analogy to the one investigated by Ben Messaoud and El Mir in the complex setting. Next, we study the definition and the continuity of…

The theory of $q$-analogs frequently occurs in a number of areas, including the fractals and dynamical systems. The $q$-derivatives and $q$-integrals play a prominent role in the study of $q$-deformed quantum mechanical simple harmonic…

We consider a Schr\"odinger operator $H=-\Delta+V(\vec x)$ in dimension two with a quasi-periodic potential $V(\vec x)$. We prove that the absolutely continuous spectrum of $H$ contains a semiaxis and there is a family of generalized…

We construct Lipschitz $Q$-valued functions which approximate carefully integral currents when their cylindrical excess is small and they are almost minimizing in a suitable sense. This result is used in two subsequent works to prove the…

This is an essay on potential theory for geometric plurisubharmonic functions. It begins with a given closed subset G of the Grassmann bundle $G(p,TX)$ of tangent $p$-planes to a riemannian manifold $X$. This determines a nonlinear partial…

In this paper we propose an alternative construction of a certain class of Deformed Double Current Algebras. We construct them as spherical subalgebras of symplectic reflection algebras in the Deligne category. They can also be thought of…

Let S be a Pfaff system of dimension 1, on a compact complex manifold M. We prove that there is a positive ddbar-closed current T of mass 1 directed by the Pfaff system S. There is no integrability assumption. We also show that local…

We give a local characterization of the class of functions having positive distributional derivative with respect to $\bar{z}$ that are almost everywhere equal to one of finitely many analytic functions and satisfy some mild non-degeneracy…

In [9] a question is raised: if a power bounded operator is quasisimilar to a singular unitary operator, is it similar to this unitary operator? For polynomially bounded operators, a positive answer to this question is known [1], [13]. In…

Let $X$ be a complex manifold of dimension $k,$ and $(V,\omega)$ be a K\"ahler submanifold of dimension $l$ in $X,$ and $B\Subset V$ be a domain with $\mathcal{C}^2$-smooth boundary. Let $T$ be a positive plurisubharmonic current on $X$…

Let $D_j\subset\Bbb C^{n_j}$ be a pseudoconvex domain and let $A_j\subset D_j$ be a locally pluriregular set, $j=1,...,N$. Put $$ X:=\bigcup_{j=1}^N A_1\times...\times A_{j-1}\times D_j\times A_{j+1}\times ...\times A_N\subset\Bbb…

Given a smooth complex toric variety we will compare real Lagerberg forms and currents on its tropicalization with invariant complex forms and currents on the toric variety. Our main result is a correspondence theorem which identifies the…

Let $T$ be a positive closed current of bidimension (1,1) and unit mass on the complex projective space ${\Bbb P}^n$. We prove that the set $V_\alpha(T)$ of points where $T$ has Lelong number larger than $\alpha$ is contained in a complex…

Suppose that $X$ is an analytic subvariety of a Stein manifold $M$ and that $\varphi$ is a plurisubharmonic (psh) function on $X$ which is dominated by a continuous psh exhaustion function $u$ of $M$. Given any number $c>1$, we show that…

In a graph $\Gamma=(V,E)$, we consider the common closed neighbourhood of a subset of vertices and use this notion to introduce a Moore closure operator in $V.$ We also consider the closed twin equivalence relation in which two vertices are…

It is proved that, if $(P_n)$ is a sequence of polynomials with complex coefficients having unbounded valences and tending to infinity at sufficiently many points, then there is an infinite dimensional closed subspace of entire functions,…

"Quasi-harmonic" (QH) theory should not be considered a low-order theory of anharmonic effects in crystals, but should be recognized as an important effect separate from "true" anharmonicity. The original and widely used meaning of QH…