The Milnor-Hirzebruch class of a locally complete intersection X in an algebraic manifold M measures the difference between the (Poincaré dual of the) Hirzebruch class of the virtual tangent bundle of X and, respectively, the Brasselet-Schürmann-Yokura (homology) Hirzebruch class of X. In this note, we calculate the Milnor-Hirzebruch class of a globally defined algebraic hypersurface X in terms...

We introduce partial Esakia morphisms, well partial Esakia morphisms, and strong partial Esakia morphisms between Esakia spaces and show that they provide the dual description of (∧,→) homomorphisms, (∧,→, 0) homomorphisms, and (∧,→,∨) homomorphisms between Heyting algebras, thus establishing a generalization of Esakia duality. This yields an algebraic characterization of Zakharyaschev’s subred...

we introduce the notions of t-dual rickart and strongly t-dual rickart modules. we provide several characterizations and investigate properties of each of these concepts. it is shown that every free (resp. finitely generated free) $r$-module is t-dual rickart if and only if $overline{z}^2(r)$ is a  direct summand of $r$ and end$(overline{z}^2(r))$ is a semisimple (resp. regular) ring. it is sho...

We formulate and analyze a goal-oriented adaptive finite element method for semilinear elliptic PDE linear goal functional. The discretization is based on elements of arbitrary (but fixed) polynomial degree involves linearized dual problem. linearization part the proposed algorithm, which employs marking strategy different to that standard methods. Moreover, unlike well-known dual-weighted resi...

In this paper, we study the codes over matrix ring $\mathbb{Z}_4$, which is perhaps first time structure $M_2(\mathbb{Z}_4)$ considered as a code alphabet. This isomorphic to $\mathbb{Z}_4[w]+U\mathbb{Z}_4[w]$, where $w$ root of irreducible polynomial $x^2+x+1 \in \mathbb{Z}_2[x]$ and $U\equiv$ ${11}\choose{11}$. We discuss then focus on algebraic cyclic self-dual $M_2(\mathbb{Z}_4)$. obtain ge...

Noetherian operators are differential that encode primary components of a polynomial ideal. We develop framework, as well algorithms, for computing with local dual spaces, both symbolically and numerically. For ideal, such provide an alternative representation to one given by set generators. This description fits numerical algebraic geometry, taking step toward the goal decomposition.

The aim of this work is to study the dual and algebraic an evaluation code using standard monomials indicator functions. We show that dual. develop algorithm for computing a basis Let $C_1$ $C_2$ be linear codes spanned by monomials. give combinatorial condition monomial equivalence $C_2^\perp$. Moreover, we explicit description generator matrix $C_2^\perp$ in terms coefficients For Reed--Mulle...

Using the parallel between the preframe and the suplattice approach to locale theory it is shown that the patch construction, as an action on topologies, is the same thing as the process of recovering a discrete poset from its algebraic dcpo (ideal completion).

In this article, the minimum distance of the dual C⊥ of a functional code C on an arbitrary–dimensional variety X over a finite field Fq is studied. The approach is based on problems à la Cayley–Bacharach and consists in describing the minimal configurations of points on X which fail to impose independent conditions on forms of some fixed degree m. If X is a curve, the result improves in some s...

Three primal-dual interior-point algorithms for homogeneous cone programming are presented. They are a short-step algorithm, a large-update algorithm, and a predictor-corrector algorithm. These algorithms are described and analyzed based on a characterization of homogeneous cone via T -algebra. The analysis show that the algorithms have polynomial iteration complexity.