arXiv subjects

Babak Jabbar Nezhad

Publications and source records attributed to Babak Jabbar Nezhad.

5 recordsLinked to original sources

arXiv2025

A Paradox on the Law of Excluded Middle in the framework of category of set

In this paper, we present a paradox arising from the acceptance of the Law of Excluded Middle (LEM) within classical mathematics. Specifically, we construct a nonzero analytic function on a connected open subset of the complex plane whose zeros are not isolated. This contradicts a fundamental theorem in complex analysis, thereby revealing an inconsistency tied to LEM. Unlike traditional critiques that reject LEM in favor of intuitionistic or constructive mathematics, we argue that LEM is instrumental in discovering \textbf{relations} between objects and facts rather than the objects themselves. Since we are not always in direct attachment with objects, this relational perspective introduces \textbf{inherent uncertainty} in mathematical reasoning. Consequently, we propose that the logical framework of the world is undecidable, making contradictions possible in more complex contexts. Our findings suggest that LEM, while powerful, may not be universally reliable in all mathematical frameworks. This work has implications for foundational mathematics, particularly in relation to the limits of classical logic and the necessity of alternative logical paradigms.

math.GM↗

arXiv2025

Paradox on the Countable Axiom of Choice

Bishop's constructive mathematics school rejects the Law of Excluded Middle, but instead vastly makes use of weaker versions of the Choice. In this paper we pioneer an example, which shows that this road is not consistent, as our example provides a paradox. Therefore, rejecting the Law of Excluded Middle, and as an alternative using the Countable Axiom of Choice and the Axiom of Dependent Choice, still does not create a consistent structure. Actually, constructively; the Countable Axiom of Choice is an implication of the Axiom of Dependent Choice.

math.LO↗

arXiv2023

Equations of the multi-Rees algebra of fattened coordinate subspaces

In this paper we describe the equations defining the multi-Rees algebra $k[x_1,\dots,x_n][I_1^{a_1}t_1,\dots,I_r^{a_r}t_r]$, where the ideals are generated by subsets of $x_1,\dots,x_n$. We also show that a family of binomials whose leading terms are squrefree, form a Gröbner basis for the defining equations with lexicographic order. We show that if we remove binomials that include $x$'s, then remaining binomials form a Gröbner basis for the toric ideal associated to the multi-fiber ring. However binomials, including $x$'s, in Gröbner basis of defining equations of the multi-Rees algebra are not necessarily defining equations of corresponding symmetric algebra. Despite this fact, we show that this family of ideals is of multi-fiber type.

math.AC↗

arXiv2022

Multi-Rees algebras on principal ideal rings

When $R$ is a Noetherian ring and we have a family of ideals in which every ideal contains at least one nonzero divisor, then it is already known that the defining ideal of the multi-Rees algebra of these ideals is equal to a saturated ideal. In such a case to get the defining ideal of the multi-Rees algebra we only need to saturate the first syzygies of direct sum of this family of ideals. However, this fact is not true, when at least one of these ideals does not contain any nonzero divisor. In this paper we show that the defining ideal of the multi-Rees algebra of a family of ideals of a polynomial ring over a principal ideal ring, is equal to another kind of saturated ideal, where we saturate more explicit polynomials other than just first syzygies. Please notice that in general some of these ideals may not contain nonzero divisors. Given this explicit formula, we can compute Gröbner basis of the defining ideal using an elimination order, which we talk about in the present paper.

math.AC↗

arXiv2020

Koszul multi-Rees algebras of principal $L$-Borel Ideals

Given a monomial $m$ in a polynomial ring and a subset $L$ of the variables of the polynomial ring, the principal $L$-Borel ideal generated by $m$ is the ideal generated by all monomials which can be obtained from $m$ by successively replacing variables of $m$ by those which are in $L$ and have smaller index. Given a collection $\mathcal{I}=\{I_1,\ldots,I_r\}$ where $I_i$ is $L_i$-Borel for $i=1,\ldots,r$ (where the subsets $L_1,\ldots,L_r$ may be different for each ideal), we prove in essence that if the bipartite incidence graph among the subsets $L_1,\ldots,L_r$ is chordal bipartite, then the defining equations of the multi-Rees algebra of $\mathcal{I}$ has a Gröbner basis of quadrics with squarefree lead terms under lexicographic order. Thus the multi-Rees algebra of such a collection of ideals is Koszul, Cohen-Macaulay, and normal. This significantly generalizes a theorem of Ohsugi and Hibi on Koszul bipartite graphs. As a corollary we obtain that the multi-Rees algebra of a collection of principal Borel ideals is Koszul. To prove our main result we use a fiber-wise Gröbner basis criterion for the kernel of a toric map and we introduce a modification of Sturmfels' sorting algorithm.

math.AC↗