We consider cyclic codes $\mathcal{C}_\mathcal{L}$ associated to quadratic trace forms in $m$ variables $Q_R(x) = \operatorname{Tr}_{q^m/q}(xR(x))$ determined by a family $\mathcal{L}$ of $q$-linearized polynomials $R$ over $\mathbb{F}_{q^m}$, and three related $\mathcal{C}_{\mathcal{L},0}$, $\mathcal{C}_{\mathcal{L},1}$ $\mathcal{C}_{\mathcal{L},2}$. describe the spectra for all these when is ...

We classify all essential extensions of the form $$ 0 \rightarrow \mathcal{B} \mathcal{D} C(X) where $\mathcal{B}$ is a nonunital, simple, separable, finite, real rank zero, $\mathcal{Z}$-stable $C^\*$-algebra with continuous scale, and $X$ finite CW complex. In fact, we prove that there group isomorphism \mathbf{Ext}(C(X), \mathcal{B}) KK(C(X), \mathcal{M}(\mathcal{B})/\mathcal{B}).

A sequence ${T_n}_{n=1}^{infty}$ of bounded linear  operators on a separable infinite dimensional Hilbert space $mathcal{H}$ is called subspace-diskcyclic with respect to the closed subspace $Msubseteq mathcal{H},$ if there exists a vector $xin mathcal{H}$ such that the disk-scaled orbit ${alpha T_n x: nin mathbb{N}, alpha inmathbb{C}, | alpha | leq 1}cap M$ is dense in $M$. The goal of t...

Abstract The moduli stack $\mathcal{A}_2$ of principally polarized abelian surfaces comes equipped with the universal surface $\pi: \mathcal{X}_2 \to \mathcal{A}_2$. fiber π over a point corresponding to an A in is itself. We determine $\ell$-adic cohomology $\mathcal{X}_2$ as Galois representation. Similarly, we consider bundles $\mathcal{X}_2^n \mathcal{A}_2$ and $\mathcal{X}_2^{\textrm{Sym}(...

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 ...

We relate commutative algebras in braided tensor categories to braid-reversed equivalences, motivated by vertex algebra representation theory. First, for $\mathcal{C}$ a category, we give detailed construction of the canonical $\mathcal{C}\boxtimes\mathcal{C}^\text{rev}$: if is semisimple but not necessarily finite or rigid, then $\bigoplus_{X\in\text{Irr}(\mathcal{C})}X'\boxtimes X$ algebra, w...

Distributed optimization algorithms are essential for training machine learning models on very large-scale datasets. However, they often suffer from communication bottlenecks. Confronting this issue, a communication-efficient primal-dual coordinate ascent framework (CoCoA) and its improved variant CoCoA+ have been proposed, achieving a convergence rate of $\mathcal{O}(1/t)$ for solving empirica...

Hierarchical classification is supervised multi-class classification problem over the set of class labels organized according to a hierarchy. In this report, we study the work by Ramaswamy et. al. on hierarchical classification over symmetric tree distance loss. We extend the consistency of hierarchical classification algorithm over asymmetric tree distance loss. We design a $\mathcal{O}(nk\log...

Let $A$ be a right coherent ring and $\mathcal{X}$ contravariantly finite subcategory of ${\rm{mod}}\mbox{-}A$ containing projectives. In this paper, we construct recollement abelian categories $({\rm{mod}}_{0}\mbox{-}\mathcal{X}, {\rm{mod}}\mbox{-}\mathcal{X}, {\rm{mod}}\mbox{-}A)$, where ${\rm{mod}}_{0}\mbox{-}\mathcal{X}$ is full ${\rm{mod}}\mbox{-}\mathcal{X}$ consisting all functors vanish...