Descriptive set theory was originally developed on Polish spaces. It was later extended to $\omega$-continuous domains [Selivanov 2004] and recently to quasi-Polish spaces [de Brecht 2013]. All these spaces are countably-based. Extending descriptive set theory and its effective counterpart to general represented spaces, including non-countably-based spaces has been started in [Pauly, de Brecht ...

We present a general framework for balancing expressions (terms) in form of so called tree straight-line programs. The latter can be seen as circuits over the free term algebra extended by contexts (terms with a hole) and the operations which insert terms/contexts into contexts. It is shown that for every term one can compute in DLOGTIME-uniform TC$^0$ a tree straight-line program of logarithmi...

A unital $C^*$ -- algebra $mathcal A,$ endowed withthe Lie product $[x,y]=xy- yx$ on $mathcal A,$ is called a Lie$C^*$ -- algebra. Let $mathcal A$ be a Lie $C^*$ -- algebra and$g,h:mathcal A to mathcal A$ be $Bbb C$ -- linear mappings. A$Bbb C$ -- linear mapping $f:mathcal A to mathcal A$ is calleda Lie $(g,h)$ -- double derivation if$f([a,b])=[f(a),b]+[a,f(b)]+[g(a),h(b)]+[h(a),g(b)]$ for all ...

A coarse structure $ \mathcal{E}$ on a set $X$ is called finitary if, for each entourage $E\in \mathcal{E}$, there exists natural number $n$ such that E[x]< n $x\in X$. By approximation of \mathcal{E}^\prime$, we mean any \mathcal{E}\subseteq \mathcal{E}^\prime$.If $\mathcal{E}^\prime$ has countable base and $E[x]$ finite X$ then \mathcal{E}^\prime$has cellular the relations linkness subsets...

Let $X$ be a set and $\mathcal{T}_X$ the full transformation semigroup on $X$. For partition $\mathcal{P}$ of $X$, we consider semigroups $T(X, \mathcal{P}) = \{f\in \mathcal{T}_X| (\forall X_i\in (\exists X_j \in \mathcal{P})\;X_i f \subseteq X_j\}$, $\Sigma(X, T(X, \mathcal{P})|(\forall X_i \mathcal{P})\; Xf \cap \neq \emptyset\}$, $\Gamma(X, \mathcal{T}_X|(\forall \mathcal{P})(\exists X_j\in...

Let $R$ be a commutative Noetherian ring, $fa$ anideal of $R$ and $mathcal{D}(R)$ denote the derived category of$R$-modules. For any homologically bounded complex $X$, we conjecture that$sup {bf L}Lambda^{fa}(X)leq$ mag$_RX$. We prove thisin several cases. This generalize the main result of Hatamkhani and Divaani-Aazar cite{HD} for complexes.

Let $\mathcal{X}$ be a separable Hilbert space with norm $\|\cdot\|$ and let $T>0$. $Q$ linear, self-adjoint, positive, trace class operator on $\mathcal{X}$, $F:\mathcal{X}\rightarrow \mathcal{X}$ (smooth enough) function $W(t)$ $\mathcal{X}$-valued cylindrical Wiener process. For $\alpha\in [0,1/2]$ we consider the $A:=-(1/2)Q^{2\alpha-1}:Q^{1-2\alpha}(\mathcal{X})\subseteq \mathcal{X}\righta...