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In this article, following an insight of Kontsevich, we extend the famous Weil conjecture (as well as the strong form of the Tate conjecture) from the realm of algebraic geometry to the broad noncommutative setting of dg categories. As a…

We present an astonishingly simple and elegant proof of the celebrated Basel problem.

We give an elementary proof of Kelley's theorem based on a minimax argument. Some applications to related problems are also developed.

We state conjectures on the relationships between automorphic representations and Galois representations, and give evidence for them.

We prove a partition identity conjectured by Lassalle (Adv. in Appl. Math. 21 (1998), 457-472).

We prove existence of conjugate Galois representations, and we use it to derive a simple method of weight reduction. As a consequence, an alternative proof of the level 1 case of Serre's conjecture follows.

We give a one-sentence elementary proof of the combinatorial Fa\`a di Bruno's formula.

Watson proved Kirkman's hypothesis (partially solved by Cayley). Using Lagrange Inversion, we drastically shorten Watson's computations and generalize his results at the same time.

Based on the results people have obtained, we try to prove the Jacobian conjecture, but there is a gap in the proof.

Several results about the union-closed sets conjecture are presented.

We present the proofs of the conjectures mentioned in the paper published in the proceedings of the 2024 AAAI conference [1], and discovered by the decomposition methods presented in the same paper.

The main aim of the present paper is to represent an exact and simple proof for FLT by using properties of the algebra identities and linear algebra.

In this paper a new conjecture equivalent to Collatz conjecture is presented. In particural, showing that (all) the solution(s) of newly introduced iterative functional equation(s) have a given property is equivalent to prove Collatz…

We prove Burkholder inequality using Bregman divergence.

We prove an analytic Bertini theorem, generalizing a previous result of Fujino and Matsumura.

We establish a relative Bertini type theorem for multiplier ideal sheaves. Then we prove a relative version of the Koll\'ar--Nadel type vanishing theorem as an application.

In this paper we discuss analogs of F\"uredi-Hajnal and Stanley-Wilf conjectures for $t$-dimensional matrices with $t>2$.

We prove a conjecture of Ohba which says that every graph $G$ on at most $2\chi(G)+1$ vertices satisfies $\chi_\ell(G)=\chi(G)$.

We consider the conjecture of Brutman and Pasow on a totality divided differences and prove the conjecture for continuous functions.

In the present article, we study Bell based Euler polynomial of order {\alpha} and investigate some useful correlation formula, summation formula and derivative formula. Also, we introduce some relation of string number of the second kind.…