Online Graph Exploration on Trees, Unicyclic Graphs and Cactus Graphs

We study the problem of exploring all vertices of an undirected weighted graph that is initially unknown to the searcher. An edge of the graph is only revealed when the searcher visits one of its endpoints. Beginning at some start node, the searcher's goal is to visit every vertex of the graph before returning to the start node on a tour as short as possible. We prove that the Nearest Neighbor algorithm's competitive ratio on trees with $n$ vertices is $\Theta(\log n)$, i.e. no better than on general graphs. Furthermore, we examine the algorithm Blocking for a range of parameters not considered previously and prove it is 3-competitive on unicyclic graphs as well as $5/2+\sqrt{2}\approx 3.91$-competitive on cactus graphs. The best known lower bound for these two graph classes is 2.Comment: 10 pages, 5 figure

Concentrated Liquidity in Automated Market Makers

We examine how the introduction of concentrated liquidity has changed the liquidity provision market in automated market makers such as Uniswap. To this end, we compare average liquidity provider returns from trading fees before and after its introduction. Furthermore, we quantify the performance of a number of fundamental concentrated liquidity strategies using historical trade data. We estimate their possible returns and evaluate which perform best for certain trading pairs and market conditions

Arbitrageurs' profits, LVR, and sandwich attacks: batch trading as an AMM design response

We consider an automated market maker (AMM) in which all trades are batched and executed at a price equal to the marginal price (i.e., the price of an arbitrarily small trade) after the batch trades. We show that such an AMM is a function maximizing AMM (or FM-AMM): for given prices, it trades to reach the highest possible value of a given function. Competition between arbitrageurs guarantees that an FM-AMM always trades at a fair, equilibrium price, and arbitrage profits (also known as LVR) are eliminated. Sandwich attacks are also eliminated because all trades occur at the exogenously-determined equilibrium price. We use Binance price data to simulate the lower bound to the return of providing liquidity to an FM-AMM. We show that this bound is very close to the empirical returns of providing liquidity on Uniswap v3 (at least for the token pairs and the period we consider).Comment: Arbitrage profits, Loss-vs-Rebalancing (LVR), MEV, Sandwich attacks, AMM, Mechanism design, Batch tradin

A Note on Optimal Fees for Constant Function Market Makers

We suggest a framework to determine optimal trading fees for constant function market makers (CFMMs) in order to maximize liquidity provider returns. In a setting of multiple competing liquidity pools, we show that no race to the bottom occurs, but instead pure Nash equilibria of optimal fees exist. We theoretically prove the existence of these equilibria for pools using the constant product trade function used in popular CFMMs like Uniswap. We also numerically compute the equilibria for a number of examples and discuss the effects the equilibrium fees have on capital allocation among pools. Finally, we use our framework to compute optimal fees for real world pools using past trade data.Comment: 11 pages, 5 figure

Post-translational modifications of histones H3 and H4 associated with the histone methyltransferases Suv39h1 and G9a

Mass spectrometry analysis of the post-transcriptional modifications of histones H3 and H4 that were co-purified with histone methyltransferases Suv39h1 and G9a shows that, in HeLa cells, histone methyltransferases can be physically associated with acetylated histones, which normally mark transcriptionally active chromatin

The Core Binding Factor CBF Negatively Regulates Skeletal Muscle Terminal Differentiation

BACKGROUND: Core Binding Factor or CBF is a transcription factor composed of two subunits, Runx1/AML-1 and CBF beta or CBFbeta. CBF was originally described as a regulator of hematopoiesis. METHODOLOGY/PRINCIPAL FINDINGS: Here we show that CBF is involved in the control of skeletal muscle terminal differentiation. Indeed, downregulation of either Runx1 or CBFbeta protein level accelerates cell cycle exit and muscle terminal differentiation. Conversely, overexpression of CBFbeta in myoblasts slows terminal differentiation. CBF interacts directly with the master myogenic transcription factor MyoD, preferentially in proliferating myoblasts, via Runx1 subunit. In addition, we show a preferential recruitment of Runx1 protein to MyoD target genes in proliferating myoblasts. The MyoD/CBF complex contains several chromatin modifying enzymes that inhibits MyoD activity, such as HDACs, Suv39h1 and HP1beta. When overexpressed, CBFbeta induced an inhibition of activating histone modification marks concomitant with an increase in repressive modifications at MyoD target promoters. CONCLUSIONS/SIGNIFICANCE: Taken together, our data show a new role for Runx1/CBFbeta in the control of the proliferation/differentiation in skeletal myoblasts