Related papers: Excision and a Theorem of Popa

We prove Polya's conjecture of 1943: For a real entire function of order greater than 2, with finitely many non-real zeros, the number of non-real zeros of the n-th derivative tends to infinity with n. We use the saddle point method and…

We construct an example of a simple nuclear separable unital stably finite Z-stable C*-algebra along with an action of the circle such that the crossed product is simple but not Z-stable.

We put forward a proof of Solomon's rule, in terms of matrices, for multiplication in the descent algebra of the symmetric group. Our proof exploits the graphs that we can obtain from all the subsets of the set of transpositions,…

We present a new `elementary' proof of the irrationality of $\zeta(3)$ based on some recent `hypergeometric' ideas of Yu.Nesterenko, T.Rivoal, and K.Ball, and on Zeilberger's algorithm of creative telescoping.

This paper is an extension program of the notion of circle of partition developed in our first paper \cite{CoP}. As an application we prove the Erd\H{o}s-Tur\'{a}n additive base conjecture.

We prove a $p$-converse to the theorem of Gross-Zagier and Kolyvagin for elliptic curves $E/\mathbf{Q}$ at primes $p>3$ of multiplicative reduction. Two key ingredients in the argument are an extension to this setting of a $p$-adic formula…

In this paper, we develop a quantitative K-theory for filtered C*-algebras. Particularly interesting examples of filtered C*-algebras include group C*-algebras, crossed product C*-algebras and Roe algebras. We prove a quantitative version…

We study the Cuntz semigroup for non-simple $\text{C}^*$-algebras in this paper. In particular, we use the extended Elliott invariant to characterize the Cuntz comparison for $\text{C}^*$-algebras with the projection property which have…

For an inclusion of the form $\Bbb C\subseteq M_n(\Bbb C)$, where $M_n(\Bbb C)$ is endowed with a state with diagonal weights $\lambda=(\lambda_1, ..., \lambda_n)$, we use Popa's construction, for non-tracial states, to obtain an…

We study positive supersolutions to an elliptic equation $(*)$: $-\Delta u=c|x|^{-s}u^p$, $p,s\in\bf R$ in cone-like domains in $\bf R^N$ ($N\ge 2$). We prove that in the sublinear case $p<1$ there exists a critical exponent $p_*<1$ such…

This paper proves that the equational theory of the class $RA_{\alpha}^{csp}$ of representable polyadic algebras is finitely axiomatizable over its substitution-free reduct $RA_{\alpha}^{cp}$, for finite $\alpha$. That is, substitutions of…

We give a short proof of a result of G. Paouris on the tail behaviour of the Euclidean norm $|X|$ of an isotropic log-concave random vector $X\in\R^n$, stating that for every $t\geq 1$, $P(|X|\geq ct\sqrt n)\leq \exp(-t\sqrt n)$. More…

We generalize a result of Matom\"aki, Radziwi{\l}{\l}, and Tao, by proving an averaged version of a conjecture of Chowla and a conjecture of Elliott regarding correlations of the Liouville function, or more general bounded multiplicative…

This paper demonstrates that P \not= NP. The way was to generalize the traditional definitions of the classes P and NP, to construct an artificial problem (a generalization to SAT: The XG-SAT, much more difficult than the former) and then…

We prove an explicit finite-sample version of the Borel--Cantelli lemma under $m$-dependence. Given any $m$-dependent sequence of events $(A_k)_{1\leq k\leq N}$, we show that \[ \mathbb{P}\Bigl(\bigcup_{k=1}^N A_k\Bigr) \ge 1 -…

We present a self-contained account of Woodin's extender algebra and its use in proving absoluteness results, including a proof of the $\Sigma^2_1$-absoluteness theorem. We also include a proof that the existence of an inner model with…

A new theory of programming is proposed. The theory consists of OE (Operation Expression), SP (Semantic Predicate) and A (Axiom), abbreviated as OESPA. OE is for programming: its syntax is given by BNF formulas and its semantics is defined…

We prove a formula which generalizes both Onn's colorful determinantal formula, related to Rota's basis conjecture, and Svrtan's $n!$ formula, related to the Atiyah-Sutcliffe problem. In some cases, our formula allows us to prove some…

We provide a new proof of Vivinai's Theorem using what George Polya calls a 'leading particular case.' Our proof highlights the role of generalization in mathematics.

In 1967, Kadison asked "if $N$ is a subfactor of the factor $M$ for which $N' \cap M$ consists of scalars, will some maximal abelian *-subalgebra of $N$ be a maximal abelian subalgebra of $M$?". Generalizing a theorem of Popa in the type…