More Like this

1. Asymptotics of singular values for quantum derivatives

   https://doi.org/10.1090/tran/8827  

   Frank, Rupert; Sukochev, Fedor; Zanin, Dmitriy (March 2023, Transactions of the American Mathematical Society)

   We obtain Weyl type asymptotics for the quantised derivative \dj \mkern 1muf of a function f f from the homgeneous Sobolev space W ˙ d 1 ( R d ) \dot {W}^1_d(\mathbb {R}^d) on R d . \mathbb {R}^d. The asymptotic coefficient ‖ ∇ f ‖ L d ( R d ) \|\nabla f\|_{L_d(\mathbb R^d)} is equivalent to the norm of \dj \mkern 1muf in the principal ideal L d , ∞ , \mathcal {L}_{d,\infty }, thus, providing a non-asymptotic, uniform bound on the spectrum of \dj \mkern 1muf. Our methods are based on the C ∗ C^{\ast } -algebraic notion of the principal symbol mapping on R d \mathbb {R}^d , as developed recently by the last two authors and collaborators. 

   more » « less
2. Invariant random subgroups of semidirect products

   https://doi.org/10.1017/etds.2018.46  

   BIRINGER, IAN; BOWEN, LEWIS; TAMUZ, OMER (January 2018, Ergodic Theory and Dynamical Systems)

   We study invariant random subgroups (IRSs) of semidirect products $$G=A\rtimes \unicode[STIX]{x1D6E4}$$ . In particular, we characterize all IRSs of parabolic subgroups of $$\text{SL}_{d}(\mathbb{R})$$ , and show that all ergodic IRSs of $$\mathbb{R}^{d}\rtimes \text{SL}_{d}(\mathbb{R})$$ are either of the form $$\mathbb{R}^{d}\rtimes K$$ for some IRS of $$\text{SL}_{d}(\mathbb{R})$$ , or are induced from IRSs of $$\unicode[STIX]{x1D6EC}\rtimes \text{SL}(\unicode[STIX]{x1D6EC})$$ , where $$\unicode[STIX]{x1D6EC}<\mathbb{R}^{d}$$ is a lattice. 

   more » « less
3. On realizations of the subalgebra 𝒜^{ℝ}(1) of the ℝ-motivic Steenrod algebra

   https://doi.org/10.1090/btran/114  

   Bhattacharya, P.; Guillou, B.; Li, A. (July 2022, Transactions of the American Mathematical Society, Series B)

   In this paper, we show that the finite subalgebra A R ( 1 ) \mathcal {A}^\mathbb {R}(1) , generated by S q 1 \mathrm {Sq}^1 and S q 2 \mathrm {Sq}^2 , of the R \mathbb {R} -motivic Steenrod algebra A R \mathcal {A}^\mathbb {R} can be given 128 different A R \mathcal {A}^\mathbb {R} -module structures. We also show that all of these A \mathcal {A} -modules can be realized as the cohomology of a 2 2 -local finite R \mathbb {R} -motivic spectrum. The realization results are obtained using an R \mathbb {R} -motivic analogue of the Toda realization theorem. We notice that each realization of A R ( 1 ) \mathcal {A}^\mathbb {R}(1) can be expressed as a cofiber of an R \mathbb {R} -motivic v 1 v_1 -self-map. The C 2 {\mathrm {C}_2} -equivariant analogue of the above results then follows because of the Betti realization functor. We identify a relationship between the R O ( C 2 ) \mathrm {RO}({\mathrm {C}_2}) -graded Steenrod operations on a C 2 {\mathrm {C}_2} -equivariant space and the classical Steenrod operations on both its underlying space and its fixed-points. This technique is then used to identify the geometric fixed-point spectra of the C 2 {\mathrm {C}_2} -equivariant realizations of A C 2 ( 1 ) \mathcal {A}^{\mathrm {C}_2}(1) . We find another application of the R \mathbb {R} -motivic Toda realization theorem: we produce an R \mathbb {R} -motivic, and consequently a C 2 {\mathrm {C}_2} -equivariant, analogue of the Bhattacharya-Egger spectrum Z \mathcal {Z} , which could be of independent interest. 

   more » « less
4. Equivalence of codes for countable sets of reals

   https://doi.org/10.4153/S0008439520000661  

   Chan, William (September 2021, Canadian Mathematical Bulletin)

   null (Ed.)

   Abstract A set $$U \subseteq {\mathbb {R}} \times {\mathbb {R}}$$ is universal for countable subsets of $${\mathbb {R}}$$ if and only if for all $$x \in {\mathbb {R}}$$ , the section $$U_x = \{y \in {\mathbb {R}} : U(x,y)\}$$ is countable and for all countable sets $$A \subseteq {\mathbb {R}}$$ , there is an $$x \in {\mathbb {R}}$$ so that $$U_x = A$$ . Define the equivalence relation $$E_U$$ on $${\mathbb {R}}$$ by $$x_0 \ E_U \ x_1$$ if and only if $$U_{x_0} = U_{x_1}$$ , which is the equivalence of codes for countable sets of reals according to U . The Friedman–Stanley jump, $=^+$ , of the equality relation takes the form $$E_{U^*}$$ where $U^*$ is the most natural Borel set that is universal for countable sets. The main result is that $=^+$ and $$E_U$$ for any U that is Borel and universal for countable sets are equivalent up to Borel bireducibility. For all U that are Borel and universal for countable sets, $$E_U$$ is Borel bireducible to $=^+$ . If one assumes a particular instance of $$\mathbf {\Sigma }_3^1$$ -generic absoluteness, then for all $$U \subseteq {\mathbb {R}} \times {\mathbb {R}}$$ that are $$\mathbf {\Sigma }_1^1$$ (continuous images of Borel sets) and universal for countable sets, there is a Borel reduction of $=^+$ into $$E_U$$ . 

   more » « less
5. Degenerate linear parabolic equations in divergence form on the upper half space

   https://doi.org/10.1090/tran/8892  

   Dong, Hongjie; Phan, Tuoc; Tran, Hung (June 2023, Transactions of the American Mathematical Society)

   We study a class of second-order degenerate linear parabolic equations in divergence form in ( − ∞ , T ) × R + d (-\infty , T) \times {\mathbb {R}}^d_+ with homogeneous Dirichlet boundary condition on ( − ∞ , T ) × ∂ R + d (-\infty , T) \times \partial {\mathbb {R}}^d_+ , where R + d = { x ∈ R d : x d > 0 } {\mathbb {R}}^d_+ = \{x \in {\mathbb {R}}^d: x_d>0\} and T ∈ ( − ∞ , ∞ ] T\in {(-\infty , \infty ]} is given. The coefficient matrices of the equations are the product of μ ( x d ) \mu (x_d) and bounded uniformly elliptic matrices, where μ ( x d ) \mu (x_d) behaves like x d α x_d^\alpha for some given α ∈ ( 0 , 2 ) \alpha \in (0,2) , which are degenerate on the boundary { x d = 0 } \{x_d=0\} of the domain. Our main motivation comes from the analysis of degenerate viscous Hamilton-Jacobi equations. Under a partially VMO assumption on the coefficients, we obtain the well-posedness and regularity of solutions in weighted Sobolev spaces. Our results can be readily extended to systems. 

   more » « less