Related papers: Definable Hamel bases and $AC_\omega(R)$
We show that for a given pure contraction $T_7$ acting on a Hilbert space $\mathcal{H}$, if $(\tilde{F}_1, \dots, \tilde{F}_6) \in \mathcal{B}(\mathcal{D}_{T^*_7})$ with $[\tilde{F}_i, \tilde{F}_j] = 0, [\tilde{F}^*_i, \tilde{F}_{7-j}] =…
The standard treatment of sets and definable classes in first-order Zermelo-Fraenkel set theory accords in many respects with the Fregean foundational framework, such as the distinction between objects and concepts. Nevertheless, in set…
We show there exists a complete theory in a language of size continuum possessing a unique atomic model which is not constructible. We also show it is consistent with $ZFC + \aleph_1 < 2^{\aleph_0}$ that there is a complete theory in a…
We prove that some holomorphic continuations of functions in the classes $\mathbf{an}^*$ and $\mathcal{G}$ are definable in the o-minimal structures $\mathbb{R}_{\mathrm{an}^*}$ and $\mathbb{R}_{\mathcal{G}}$ respectively. More…
Motivated by showing that in ZFC we cannot construct a special Aronszajn tree on some cardinal greater than $\aleph_1$, we produce a model in which the approachability property fails (hence there are no special Aronszajn trees) at all…
We establish the undecidability of conditional affine information inequalities, the undecidability of the conditional independence implication problem with a constraint that one random variable is binary, and the undecidability of the…
In this paper we construct representations of certain graded double affine Hecke algebras (DAHA) with possibly unequal parameters from geometry. More precisely, starting with a simple Lie algebra $\mathfrak{g}$ together with a…
We calculate some infinite sums containing the digamma function in closed-form. These sums are related either to the incomplete beta function or to the Bessel functions. The calculations yield interesting new results as by-products, such as…
Let $[\gamma]$ be the conformal boundary of a warped product $C^{3,\alpha}$ AHE metric $g=g_M+u^2h$ on $N=M \times F$, where $(F,h)$ is compact with unit volume and nonpositive curvature. We show that if $[\gamma]$ has positive Yamabe…
We develop the notion of coherent ultrafilters (extenders without normality or well-foundedness). We then use definable coherent ultraproducts to characterize any extension of a model $M$ in any fragment of $\mathbb{L}_{\infty, \omega}$…
In this paper, deformations of $L_\infty$-algebras are defined in such a way that the bases of deformations are $L_\infty$-algebras, as well. A universal and a semiuniversal deformation is constructed for $L_\infty$-algebras, whose…
Two solvable Hamiltonians for describing the dynamic gamma deformation, are proposed. The limiting case of each of them is the X(5) Hamiltonian. Analytical solutions for both energies and wave functions, which are periodic in $\gamma$, are…
We give a full description of the structure under inclusion of all finite level Borel classes of functions, and provide an elementary proof of the well-known fact that not every Borel function can be written as a countable union of…
By restricting the variables running over various (possibly different) subfields, we introduce the notion of a partial zeta function. We prove that the partial zeta function is rational in an interesting case, generalizing Dwork's well…
We investigate the family of surfaces defined by the affine equation $$Y^2 + Z^2 = (aT^2 + b)(cT^2 +d)$$ where $\vert ad-bc \vert=1$ and develop an asymptotic formula for the frequency of Hasse principle failures. We show that a positive…
Several relations are obtained among the Riemann zeta and Hurwitz zeta functions, as well as their products. A particular case of these relations give rise to a simple re-derivation if the important results of [11]. Also, a relation derived…
We force over the constructible universe to obtain a model of the $\Pi^1_3$-reduction property, thus lowering the best known large cardinal strength from the existence of $M_1^{\#}$ to just ZFC. In this model the $\Pi^1_3$-uniformization…
Given a finite abelian $p$-group $F$, we prove an efficient recursive formula for $\sigma_a(F)=\sum_{\substack{H\leq F}}|H|^a$ where $H$ ranges over the subgroups of $F$. We infer from this formula that the $p$-component of the…
For a ring R, denote by Spec^R_kappa(Gamma) the kappa-spectrum of the Gamma-invariant of strongly uniform right R-modules. Recent realization techniques of Goodearl and Wehrung show that Spec^R_{aleph_1}(Gamma) is full for suitable von…
We combine several folklore observations to provide a working framework for iterating constructions which contradict the axiom of choice. We use this to define a model in which any kind of structural failure must fail with a proper class of…