Related papers: Another counterexample to Zygmund's conjecture

We extend the notion of (smooth) stable generalized complex structures to allow for an anticanonical section with normal self-crossing singularities. This weakening not only allows for a number of natural examples in higher dimensions but…

We present a creative reimagining of Zolotarev's classical proof of the Law of Quadratic Reciprocity.

Let $d\equiv 2\pmod 4$ be a square-free integer such that $x^2 - dy^2 =- 1$ and $x^2 - dy^2 = 6$ are solvable in integers. We prove the existence of infinitely many quadruples in $\mathbb{Z}[\sqrt{d}]$ with the property $D(n)$ when $n \in…

Efforts to understand the origin of the cosmological constant {\Lambda} and its observed value have led to consider it as a dynamical field rather than as a universal constant. Then the possibility arises that the universe, or regions of…

We construct a counterexample to a conjectured inequality L<2D, relating the diameter D and the least length L of a nontrivial closed geodesic, for a Riemannian metric on the 2-sphere. The construction relies on Guillemin's theorem…

Thanks to recent results on ring homomorphisms of Azumaya algebras and to the following ones about endomorphisms of canonical Poisson algebras and Dirac quantum algebras, and about the reformulation in positive characteristic of these…

We develop a systematic method to derive the Majorana representation of the Dirac equation in (1+3)-dimensions. We compare with similar approach in (2+2)-dimensions . We argue that our formalism can be useful to have a better understanding…

Dumont has conjectured a marvellous identity, which generalizes, in particular, the classical results of Lagrange, Gauss, Jacobi and Kronecker on the sums of two, three and four squares. We give a combinatorial proof of Dumont's conjecture.

We survey some results that provide different versions of classical results through different summability methods. Specifically, in order to adapt such classical results, we analyze which properties should satisfy the summability methods.…

Classical W-algebras in higher dimensions have been recently constructed. In this letter we show that there is a finitely generated subalgebra which is isomorphic to the algebra of local diffeomorphisms in D dimensions. Moreover, there is a…

In this paper we give a counterexample to the conjecture: Let $S\in{\rm SInn}$. Then $z\cdot S$ is onto $U$.

We prove the crepant resolution conjecture for Donaldson-Thomas invariants of hard Lefschetz CY3 orbifolds, formulated by Bryan-Cadman-Young, interpreting the statement as an equality of rational functions. In order to do so, we show that…

We introduce a special class of convex rational polyhedral cones which allows to construct generalized Calabi-Yau varieties of dimension $(d + 2(r-1))$, where $r$ is a positive integer and d is the dimension of critical string vacua with…

We give a constructive counterpart of the theorem of Andrunakievi\v{c} and Rjabuhin, which states that every reduced ring is a subdirect product of domains. As an application, we extract a constructive proof of the fact that every ring $A$…

A $(d-1)$-dimensional simplicial complex is called balanced if its underlying graph admits a proper $d$-coloring. We show that many well-known face enumeration results have natural balanced analogs (or at least conjectural analogs).…

The codimension-three conjecture states that any regular holonomic module extends uniquely beyond an analytic subset with codimension equal to or larger than three. We give a proof of this conjecture.

In this paper, we prove a congruence which confirms a conjecture of Adamchuk. For any prime $p\equiv1\pmod3$ and $a\in\mathbb{Z}^{+}$, we have \begin{align*} \sum_{k=1}^{\frac{2}3(p^a-1)}\binom{2k}k\equiv0\pmod{p^2}. \end{align*}

In this paper we consider Erd\"os-Mordell inequality and its extension in the plane of triangle to the Erd\"os-Mordell curve. Algebraic equation of this curve is derived, and using modern computer tools in mathematics, we verified one…

In this paper, we present a minimal counterexample to a conjecture of Perles that answers a question of Haase and Ziegler. The example is a simple 4-polytope that has an induced 3-connected 3-regular subgraph, whose graph complement is…

We give a new proof of a_4\phi_3 summation due to G.E. Andrews and confirm another_4\phi_3 summation conjectured by him recently. Some variations of these two_4\phi_3 summations are also given.