On Some Noteworthy Pairs of Ideals in Mod-R

An additive functor $F \colon {\mathcal A}\to{\mathcal B}$ between preadditive categories $\mathcal A$ and $\mathcal B$ is said to be a local functor if, for every morphism $f\colon A\to A'$ in $\mathcal A$ , F(f) isomorphism in $\mathcal B$ implies f isomorphism in $\mathcal A$ . We show that there exist several pairs $(\mathcal I_1,\mathcal I_2)$ of ideals of $\mathcal A$ for which the canonical functor $\mathcal A\to\mathcal A/\mathcal I_1\times \mathcal A/\mathcal I_2$ is a local functor. In most of our examples, the category $\mathcal A$ is a full subcategory of the category Mod?-R of all right modules over a ring R. These pairs of ideals arise in a surprisingly natural way and enjoy several properties. Ideals are kernels of functors, and most of our examples of ideals are kernels of important and well studied functors. E.g., (1) the kernel Δ of the canonical functor P of Mod?-R into its spectral category Spec(Mod?-R), so that Δ is the ideal of all morphisms with an essential kernel; (2) the kernel Σ of the dual functor F of P, so that Σ is the ideal of all morphisms with a superfluous image; (3) the kernels Δ⁽¹⁾ and Σ₍₁₎ of the first derived functors P ⁽¹⁾ and F ₍₁₎ of P and F, respectively; (4) the kernels of suitable functors Hom and ? and their first derived functors ${\rm Ext}^1_R$ and ${\rm Tor}^R_1$ .     

Semi-Heyting Algebras Term-equivalent to Gödel Algebras

In this paper we investigate those subvarieties of the variety $\mathcal {SH}$ of semi-Heyting algebras which are term-equivalent to the variety $\mathcal L_{\mathcal H}$ of Gödel algebras (linear Heyting algebras). We prove that the only other subvarieties with this property are the variety $\mathcal L^{\rm Com}$ of commutative semi-Heyting algebras and the variety $\mathcal L^{\vee}$ generated by the chains in which a?<?b implies a → b?=?b. We also study the variety $\mathcal C$ generated within $\mathcal{SH}$ by $\mathcal L_{\mathcal H}$ , $\mathcal L_\vee$ and $\mathcal L_{\rm Com}$ . In particular we prove that $\mathcal C$ is locally finite and we obtain a construction of the finitely generated free algebra in $\mathcal C$ .     

A Topologist’s View of Chu Spaces

For a symmetric monoidal-closed category $\mathcal{X}$ and any object K, the category of K-Chu spaces is small-topological over $\mathcal{X}$ and small cotopological over $\mathcal{X}^{{{\text{op}}}}$ . Its full subcategory of $\mathcal{M}$ -extensive K-Chu spaces is topological over $\mathcal{X}$ when $\mathcal{X}$ is $\mathcal{M}$ -complete, for any morphism class $\mathcal{M}$ . Often this subcategory may be presented as a full coreflective subcategory of Diers’ category of affine K-spaces. Hence, in addition to their roots in the theory of pairs of topological vector spaces (Barr) and their connections with linear logic (Seely), the Dialectica categories (Hyland, de Paiva), and with the study of event structures for modeling concurrent processes (Pratt), Chu spaces seem to have a less explored link with algebraic geometry. We use the Zariski closure operator to describe the objects of the *-autonomous category of $\mathcal{M}$ -extensive and $\mathcal{M}$ -coextensive K-Chu spaces in terms of Zariski separation and to identify its important subcategory of complete objects.     

{\boldsymbol(}\boldsymbol{\mathcal{Z}}_{\bf 1}, \boldsymbol{\mathcal{Z}}_{\bf 2}{\boldsymbol)}-Complete Partially Ordered Sets and Their Representations by \boldsymbol{\mathcal{Q}}-Spaces

The present paper proposes a general theory for $\left( \mathcal{Z}_{1}, \mathcal{Z}_{2}\right) $ -complete partially ordered sets (alias $\mathcal{Z} _{1}$ -join complete and $\mathcal{Z}_{2}$ -meet complete partially ordered sets) and their Stone-like representations. It is shown that for suitably chosen subset selections $\mathcal{Z}_{i}$ (i?=?1,...,4) and $\mathcal{Q} =\left( \mathcal{Z}_{1},\mathcal{Z}_{2},\mathcal{Z}_{3},\mathcal{Z} _{4}\right) $ , the category $\mathcal{Q}$ P of $\left( \mathcal{Z}_{1},\mathcal{Z}_{2}\right) $ -complete partially ordered sets and $\left( \mathcal{Z}_{3},\mathcal{Z}_{4}\right) $ -continuous (alias $\mathcal{ Z}_{3}$ -join preserving and $\mathcal{Z}_{4}$ -meet preserving) functions forms a useful categorical framework for various order-theoretical constructs, and has a close connection with the category $\mathcal{Q}$ S of $\mathcal{Q}$ -spaces which are generalizations of topological spaces involving subset selections. In particular, this connection turns into a dual equivalence between the full subcategory $ \mathcal{Q}$ P _s of $\mathcal{Q}$ P of all $\mathcal{Q}$ -spatial objects and the full subcategory $\mathcal{Q}$ S _s of $\mathcal{Q}$ S of all $\mathcal{Q}$ -sober objects. Here $\mathcal{Q}$ -spatiality and $\mathcal{Q}$ -sobriety extend usual notions of spatiality of locales and sobriety of topological spaces to the present approach, and their relations to $\mathcal{Z}$ -compact generation and $\mathcal{Z}$ -sobriety have also been pointed out in this paper.     

Linear recurring sequences and subfield subcodes of cyclic codes

Linear recurring sequences over finite fields play an important role in coding theory and cryptography. It is known that subfield subcodes of linear codes yield some good codes. In this paper, we study linear recurring sequences and subfield subcodes. Let Mqm(f(x)) denote the set of all linear recurring sequences over Fqm with characteristic polynomial f(x) over Fqm . Denote the restriction of Mqm(f(x)) to sequences over Fq and the set after applying trace function to each sequence in Mqm(f(x)) by Mqm(f(x)) | Fq and Tr( Mqm(f(x))), respectively. It is shown that these two sets are both complete sets of linear recurring sequences over Fq with some characteristic polynomials over Fq. In this paper, we firstly determine the characteristic polynomials for these two sets. Then, using these results, we determine the generator polynomials of subfield subcodes and trace codes of cyclic codes over Fqm .     

Centers for the restricted category \mathcal {O} at the critical level over affine Kac–Moody algebras

The restricted category $\mathcal {O}$ at the critical level over an affine Kac–Moody algebra is a certain subcategory of the ordinary BGG-category $\mathcal {O}$ . We study a deformed version introduced by Arakawa and Fiebig and calculate the center of the deformed restricted category $\mathcal {O}$ . This complements a result of Fiebig which describes the center of the non-restricted category $\mathcal {O}$ outside the critical hyperplanes over a symmetrizable Kac–Moody algebra.     

Derived categories of Burniat surfaces and exceptional collections

We construct an exceptional collection $\varUpsilon $ of maximal possible length 6 on any of the Burniat surfaces with $K_X^2=6$ , a 4-dimensional family of surfaces of general type with $p_g=q=0$ . We also calculate the DG algebra of endomorphisms of this collection and show that the subcategory generated by this collection is the same for all Burniat surfaces. The semiorthogonal complement $\mathcal{A }$ of $\varUpsilon $ is an “almost phantom” category: it has trivial Hochschild homology, and $K_0(\mathcal{A })=\mathbb{Z }_2^6$ .     

Hypercentralizing generalized skew derivations on left ideals in prime rings

Let $\mathcal{R }$ be a prime ring of characteristic different from $2, \mathcal{Q }_r$ the right Martindale quotient ring of $\mathcal{R }, \mathcal{C }$ the extended centroid of $\mathcal{R }, \mathcal{I }$ a nonzero left ideal of $\mathcal{R }, F$ a nonzero generalized skew derivation of $\mathcal{R }$ with associated automorphism $\alpha $ , and $n,k \ge 1$ be fixed integers. If $[F(r^n),r^n]_k=0$ for all $r \in \mathcal{I }$ , then there exists $\lambda \in \mathcal{C }$ such that $F(x)=\lambda x$ , for all $x\in \mathcal{I }$ . More precisely one of the following holds: (1) $\alpha $ is an $X$ -inner automorphism of $\mathcal{R }$ and there exist $b,c \in \mathcal{Q }_r$ and $q$ invertible element of $\mathcal{Q }_r$ , such that $F(x)=bx-qxq^{-1}c$ , for all $x\in \mathcal{Q }_r$ . Moreover there exists $\gamma \in \mathcal{C }$ such that $\mathcal{I }(q^{-1}c-\gamma )=(0)$ and $b-\gamma q \in \mathcal{C }$ ; (2) $\alpha $ is an $X$ -outer automorphism of $\mathcal{R }$ and there exist $c \in \mathcal{Q }_r, \lambda \in \mathcal{C }$ , such that $F(x)=\lambda x-\alpha (x)c$ , for all $x\in \mathcal{Q }_r$ , with $\alpha (\mathcal{I })c=0$ .     

A metrical lower bound on the star discrepancy of digital sequences

In this paper we study uniform distribution properties of digital sequences over a finite field of prime order. In 1998 it was shown by Larcher that for almost all $s$ -dimensional digital sequences $\mathcal{S }$ the star discrepancy $D_N^*$ satisfies an upper bound of the form $D_N^*(\mathcal{S })=O((\log N)^s (\log \log N)^{2+\varepsilon })$ for any $\varepsilon >0$ . Generally speaking it is much more difficult to obtain good lower bounds for specific sequences than upper bounds. Here we show that Larchers result is best possible up to some $\log \log N$ term. More detailed, we prove that for almost all $s$ -dimensional digital sequences $\mathcal{S }$ the star discrepancy satisfies $D_N^*(\mathcal{S }) \ge c(q,s) (\log N)^s \log \log N$ for infinitely many $N \in \mathbb{N }$ , where $c(q,s)>0$ only depends on $q$ and $s$ but not on $N$ .     

Characterization of Lie higher derivations on triangular algebras

Let $\mathcal{A}$ and $\mathcal{B}$ be unital rings, and $\mathcal{M}$ be an $\left( {\mathcal{A},\mathcal{B}} \right)$ -bimodule, which is faithful as a left $\mathcal{A}$ -module and also as a right $\mathcal{B}$ -module. Let $\mathcal{U} = Tri\left( {\mathcal{A},\mathcal{M},\mathcal{B}} \right)$ be the triangular algebra. In this paper, we give some different characterizations of Lie higher derivations on $\mathcal{U}$ .     

Ranks of backward shift invariant subspaces of the Hardy space over the bidisk

Let $\{\varphi _n(z)\}_{n\ge 0}$ be a sequence of inner functions satisfying that $\zeta _n(z):=\varphi _n(z)/\varphi _{n+1}(z)\in H^\infty (z)$ for every $n\ge 0$ and $\{\varphi _n(z)\}_{n\ge 0}$ has no nonconstant common inner divisors. Associated with it, we have a Rudin type invariant subspace $\mathcal{M }$ of $H^2(\mathbb{D }^2)$ . The ranks of $\mathcal{M }\ominus w\mathcal{M }$ for $\mathcal{F }_z$ and $\mathcal{F }^*_z$ respectively are determined, where $\mathcal{F }_z$ is the fringe operator on $\mathcal{M }\ominus w\mathcal{M }$ . Let $\mathcal{N }= H^2(\mathbb{D }^2)\ominus \mathcal{M }$ . It is also proved that the rank of $\mathcal{M }\ominus w\mathcal{M }$ for $\mathcal{F }^*_z$ equals to the rank of $\mathcal{N }$ for $T^*_z$ and $T^*_w$ .     

On the first Gaussian map for Prym-canonical line bundles

We prove by degeneration to Prym-canonical binary curves that the first Gaussian map $\mu _A$ of the Prym canonical line bundle $\omega _C \otimes A$ is surjective for the general point $[C,A] \in \mathcal{R }_g$ if $g \ge 12$ , while it is injective if $g \le 11$ .     

Some results on metric n-Lie algebras

We study the structure of a metric n-Lie algebra G over the complex field C. Let G = SR be the Levi decomposition, where R is the radical of G and S is a strong semisimple subalgebra of G. Denote by m(G) the number of all minimal ideals of an indecomposable metric n-Lie algebra and R ⊥ the orthogonal complement of R. We obtain the following results. As S-modules, R ⊥ is isomorphic to the dual module of G/R. The dimension of the vector space spanned by all nondegenerate invariant symmetric bilinear forms on G is equal to that of the vector space of certain linear transformations on G; this dimension is greater than or equal to m(G) + 1. The centralizer of R in G is equal to the sum of all minimal ideals; it is the direct sum of R ⊥ and the center of G. Finally, G has no strong semisimple ideals if and only if R⊥■R.     

AW*-algebras Which are Enveloping C*-algebras of JC-algebras

In the given article, enveloping C*-algebras of AJW-algebras are considered. Conditions are given, when the enveloping C*-algebra of an AJW-algebra is an AW*-algebra, and corresponding theorems are proved. In particular, we proved that if $\mathcal{A}$ is a real AW*-algebra, $\mathcal{A}_{sa}$ is the JC-algebra of all self-adjoint elements of $\mathcal{A}$ , $\mathcal{A}+i\mathcal{A}$ is an AW*-algebra and $\mathcal{A}\cap i\mathcal{A} = \{0\}$ then the enveloping C*-algebra $C^*(\mathcal{A}_{sa})$ of the JC-algebra $\mathcal{A}_{sa}$ is an AW*-algebra. Moreover, if $\mathcal{A}+i\mathcal{A}$ does not have nonzero direct summands of type I₂, then $C^*(\mathcal{A}_{sa})$ coincides with the algebra $\mathcal{A}+i\mathcal{A}$ , i.e. $C^*(\mathcal{A}_{sa})= \mathcal{A}+i\mathcal{A}$ .     

Geodesics,distance, and the CAT(0) property for the manifold of Riemannian metrics

Let $\mathcal F ^a_\lambda $ be the PBW degeneration of the flag varieties of type $A_{n-1}$ . These varieties are singular and are acted upon with the degenerate Lie group $SL_n^a$ . We prove that $\mathcal F ^a_\lambda $ have rational singularities, are normal and locally complete intersections, and construct a desingularization $R_\lambda $ of $\mathcal F ^a_\lambda $ . The varieties $R_\lambda $ can be viewed as towers of successive $\mathbb{P }^1$ -fibrations, thus providing an analogue of the classical Bott–Samelson–Demazure–Hansen desingularization. We prove that the varieties $R_\lambda $ are Frobenius split. This gives us Frobenius splitting for the degenerate flag varieties and allows to prove the Borel–Weil type theorem for $\mathcal F ^a_\lambda $ . Using the Atiyah–Bott–Lefschetz formula for $R_\lambda $ , we compute the $q$ -characters of the highest weight $\mathfrak sl _n$ -modules.     

Geometric rigidity for sequences of W^{2,2} conformal immersions

We analyse sequences of discs conformally immersed in $ \mathbb{R }^ n$ with energy $ \int _{ D} |A_k |^ 2 \le \gamma _n$ , where $ \gamma _n = 8\pi $ if $ n=3$ and $ \gamma _n = 4 \pi $ when $n\ge 4$ . We show that if such sequences do not weakly converge to a conformal immersion, then by a sequence of dilations we obtain a complete minimal surface with bounded total curvature, either Enneper’s minimal surface if $ n=3$ or Chen’s minimal graph if $ n \ge 4$ . In the papers, (Kuwert and Li, Comm Anal Geom 20(2), 313–340, 2012; Rivière, Adv Calculus Variations 6(1), 1–31, 2013) it was shown that if a sequence of immersed tori diverges in moduli space then $\liminf _ {k\rightarrow \infty } \mathcal W ( f_k )\ge 8\pi $ . We apply the above analysis to show that in $ \mathbb{R }^3$ if the sequence diverges so that $ \lim _{ k \rightarrow \infty } \mathcal W (f_k) =8\pi $ then there exists a sequence of Möbius transforms $ \sigma _{k}$ such that $ \sigma _k\circ f _k$ converges weakly to a catenoid.     

A unified approach for minimizing composite norms

We propose a first-order augmented Lagrangian algorithm (FALC) to solve the composite norm minimization problem $$\begin{aligned} \begin{array}{ll} \min \limits _{X\in \mathbb{R }^{m\times n}}&\mu _1\Vert \sigma (\mathcal{F }(X)-G)\Vert _\alpha +\mu _2\Vert \mathcal{C }(X)-d\Vert _\beta ,\\ \text{ subject} \text{ to}&\mathcal{A }(X)-b\in \mathcal{Q }, \end{array} \end{aligned}$$ where $\sigma (X)$ denotes the vector of singular values of $X \in \mathbb{R }^{m\times n}$ , the matrix norm $\Vert \sigma (X)\Vert _{\alpha }$ denotes either the Frobenius, the nuclear, or the $\ell _2$ -operator norm of $X$ , the vector norm $\Vert .\Vert _{\beta }$ denotes either the $\ell _1$ -norm, $\ell _2$ -norm or the $\ell _{\infty }$ -norm; $\mathcal{Q }$ is a closed convex set and $\mathcal{A }(.)$ , $\mathcal{C }(.)$ , $\mathcal{F }(.)$ are linear operators from $\mathbb{R }^{m\times n}$ to vector spaces of appropriate dimensions. Basis pursuit, matrix completion, robust principal component pursuit (PCP), and stable PCP problems are all special cases of the composite norm minimization problem. Thus, FALC is able to solve all these problems in a unified manner. We show that any limit point of FALC iterate sequence is an optimal solution of the composite norm minimization problem. We also show that for all $\epsilon >0$ , the FALC iterates are $\epsilon $ -feasible and $\epsilon $ -optimal after $\mathcal{O }(\log (\epsilon ^{-1}))$ iterations, which require $\mathcal{O }(\epsilon ^{-1})$ constrained shrinkage operations and Euclidean projection onto the set $\mathcal{Q }$ . Surprisingly, on the problem sets we tested, FALC required only $\mathcal{O }(\log (\epsilon ^{-1}))$ constrained shrinkage, instead of the $\mathcal{O }(\epsilon ^{-1})$ worst case bound, to compute an $\epsilon $ -feasible and $\epsilon $ -optimal solution. To best of our knowledge, FALC is the first algorithm with a known complexity bound that solves the stable PCP problem.     

The Calabi homomorphism, Lagrangian paths and special Lagrangians

Let $\mathcal{O }$ be an orbit of the group of Hamiltonian symplectomorphisms acting on the space of Lagrangian submanifolds of a symplectic manifold $(X,\omega ).$ We define a functional $\mathcal{C }:\mathcal{O } \rightarrow \mathbb{R }$ for each differential form $\beta $ of middle degree satisfying $\beta \wedge \omega = 0$ and an exactness condition. If the exactness condition does not hold, $\mathcal{C }$ is defined on the universal cover of $\mathcal{O }.$ A particular instance of $\mathcal{C }$ recovers the Calabi homomorphism. If $\beta $ is the imaginary part of a holomorphic volume form, the critical points of $\mathcal{C }$ are special Lagrangian submanifolds. We present evidence that $\mathcal{C }$ is related by mirror symmetry to a functional introduced by Donaldson to study Einstein–Hermitian metrics on holomorphic vector bundles. In particular, we show that $\mathcal{C }$ is convex on an open subspace $\mathcal{O }^+ \subset \mathcal{O }.$ As a prerequisite, we define a Riemannian metric on $\mathcal{O }^+$ and analyze its geodesics. Finally, we discuss a generalization of the flux homomorphism to the space of Lagrangian submanifolds, and a Lagrangian analog of the flux conjecture.     

Rigid objects, triangulated subfactors and abelian localizations

We show that the abelian category $\mathsf{mod}\text{-}\mathcal{X }$ of coherent functors over a contravariantly finite rigid subcategory $\mathcal{X }$ in a triangulated category $\mathcal{T }$ is equivalent to the Gabriel–Zisman localization at the class of regular maps of a certain factor category of $\mathcal{T }$ , and moreover it can be calculated by left and right fractions. Thus we generalize recent results of Buan and Marsh. We also extend recent results of Iyama–Yoshino concerning subfactor triangulated categories arising from mutation pairs in $\mathcal{T }$ . In fact we give a classification of thick triangulated subcategories of a natural pretriangulated factor category of $\mathcal{T }$ and a classification of functorially finite rigid subcategories of $\mathcal{T }$ if the latter has Serre duality. In addition we characterize $2$ -cluster tilting subcategories along these lines. Finally we extend basic results of Keller–Reiten concerning the Gorenstein and the Calabi–Yau property for categories arising from certain rigid, not necessarily cluster tilting, subcategories, as well as several results of the literature concerning the connections between $2$ -cluster tilting subcategories of triangulated categories and tilting subcategories of the associated abelian category of coherent functors.     

On the unique representation of very strong algebraic geometry codes

This paper addresses the question of retrieving the triple ${(\mathcal X,\mathcal P, E)}$ from the algebraic geometry code ${\mathcal C = \mathcal C_L(\mathcal X, \mathcal P, E)}$ , where ${\mathcal X}$ is an algebraic curve over the finite field ${\mathbb F_q, \,\mathcal P}$ is an n-tuple of ${\mathbb F_q}$ -rational points on ${\mathcal X}$ and E is a divisor on ${\mathcal X}$ . If ${\deg(E)\geq 2g+1}$ where g is the genus of ${\mathcal X}$ , then there is an embedding of ${\mathcal X}$ onto ${\mathcal Y}$ in the projective space of the linear series of the divisor E. Moreover, if ${\deg(E)\geq 2g+2}$ , then ${I(\mathcal Y)}$ , the vanishing ideal of ${\mathcal Y}$ , is generated by ${I_2(\mathcal Y)}$ , the homogeneous elements of degree two in ${I(\mathcal Y)}$ . If ${n >2 \deg(E)}$ , then ${I_2(\mathcal Y)=I_2(\mathcal Q)}$ , where ${\mathcal Q}$ is the image of ${\mathcal P}$ under the map from ${\mathcal X}$ to ${\mathcal Y}$ . These three results imply that, if ${2g+2\leq m < \frac{1}{2}n}$ , an AG representation ${(\mathcal Y, \mathcal Q, F)}$ of the code ${\mathcal C}$ can be obtained just using a generator matrix of ${\mathcal C}$ where ${\mathcal Y}$ is a normal curve in ${\mathbb{P}^{m-g}}$ which is the intersection of quadrics. This fact gives us some clues for breaking McEliece cryptosystem based on AG codes provided that we have an efficient procedure for computing and decoding the representation obtained.