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We prove a curious identity for the Bernoulli numbers.
We investigate the possibilities of global versions of Chang's Conjecture that involve singular cardinals. We show some $\mathrm{ZFC}$ limitations on such principles, and prove relative to large cardinals that Chang's Conjecture can…
We study an extension to the uniqueness conjecture for Markov numbers. For any three positive integers $m\geq a$ and $m\geq b$ satisfying $a^2+b^2+m^2=3abm$, this conjecture states that the triple $(a,m,b)$ is uniquely determined by the…
The theory of random attractors has different notions of attraction, amongst them pullback attraction and weak attraction. We investigate necessary and sufficient conditions for the existence of pullback attractors as well as of weak…
We establish the validity of a strong unique continuation property for weakly coupled elliptic systems, including competitive ones. Our proof exploits the system structure and uses Carleman estimates. We apply this result to obtain some…
We consider Sinai's walk in i.i.d. random scenery and focus our attention on a conjecture of R\'ev\'esz \cite{r05} concerning the upper limits of Sinai's walk in random scenery when the scenery is bounded from above. A close study of the…
We study a one-dimensional random walk among random conductances, with unbounded jumps. Assuming the ergodicity of the collection of conductances and a few other technical conditions (uniform ellipticity and polynomial bounds on the tails…
We collect here various conjectures on congruences made by the author in a series of papers, some of which involve binary quadratic forms and other advanced theories. Part A consists of 100 unsolved conjectures of the author while…
We prove the Baum-Connes conjecture for hyperbolic groups and their subgroups.
First, we prove a \emph{local almost sure central limit theorem} for lattice random walks in the plane. The corresponding version for random walks in the line was considered by the author in \cite{5}. This gives us a quantitative version of…
It has been long congectured that the crossing number of $C_m\times C_n$ is $(m-2)n$ for $2<m<=n$. In this paper we proved that conjecture is true for all but finitely many $n$ for each $m$. More specifically we proved conjecture for…
Exploration and trapping properties of random walkers that may evanesce at any time as they walk have seen very little treatment in the literature, and yet a finite lifetime is a frequent occurrence, and its effects on a number of random…
We give certain generalization of Niederreiter's result concerning famous Zaremba's conjecture on existence of rational numbers with bounded partial quotients.
There are no known failures of Bounded Negativity in characteristic 0. In the light of recent work showing the Bounded Negativity Conjecture fails in positive characteristics for rational surfaces, we propose new characteristic free…
We provide a proof of a variant of the Landau-Siegel Zeros conjecture.
We derive an interesting congruence relation motivated by an Indian Olympiad problem. We give three different proofs of the theorem and mention a few interesting related results.
New cases of the multiplicity conjecture are considered.
Using Easton collapses, we give a simplified construction of a model in which Chang's Conjecture for triples holds.
A conjecture is given that, if true, could lead to an algorithm for computing definite sums of rational functions.
The Langberg-M\'{e}dard multiple unicast conjecture claims that for a strongly reachable $k$-pair network, there exists a multi-flow with rate $(1,1,\dots,1)$. In this paper, we show that the conjecture holds true for {\em stable} $3$-pair…