Testing Equality in Communication Graphs

Let $G=(V,E)$ be a connected undirected graph with $k$ vertices. Suppose that on each vertex of the graph there is a player having an $n$-bit string. Each player is allowed to communicate with its neighbors according to an agreed communication protocol, and the players must decide, deterministically, if their inputs are all equal. What is the minimum possible total number of bits transmitted in a protocol solving this problem ? We determine this minimum up to a lower order additive term in many cases (but not for all graphs). In particular, we show that it is $kn/2+o(n)$ for any Hamiltonian $k$-vertex graph, and that for any $2$-edge connected graph with $m$ edges containing no two adjacent vertices of degree exceeding $2$ it is $mn/2+o(n)$. The proofs combine graph theoretic ideas with tools from additive number theory

Comment on "Fitting the annual modulation in DAMA with neutrons from muons and neutrinos''

We estimate rates of solar neutrino-induced neutrons in a DAMA/LIBRA-like detector setup, and find that the needed contribution to explain the annual modulation would require neutrino-induced neutron cross sections several orders of magnitude larger than current calculations indicate. Although these cross sections have never been measured, it is likely that the solar-neutrino effect on DAMA/LIBRA is negligible.Comment: Comment submitted to PR

Pulsed plasma deposition of Fe-C-Cr-W coating on high-Cr-cast iron: Effect of layered morphology and heat treatment on the microstructure and hardness

Pulsed plasma treatment was applied for surface modification and laminated coating deposition on 14.5 wt%-Cr cast iron. The scopes of the research were: (a) to obtain a microstructure gradient, (b) to study the relationship between cathode material and coating layer microstructure/hardness, and (c) to improve coating quality by applying post-deposition heat treatment. An electrothermal axial plasma accelerator with a gas-dynamic working regime was used as plasma source (4.0 kV, 10 kA). The layered structure was obtained by alternation of the cathode material (T1 - 18 wt% W high speed steel and 28 wt% Cr-cast iron). It was found that pulsed plasma treatment led to substrate sub-surface modification by the formation of an 11–18 μm thick remelted layer with very fine carbide particles that provided a smooth transition from the substrate into the coating (80–120 μm thick). The as-deposited coating of 500–655 HV0.05 hardness consisted of “martensite/austenite” layers which alternated with heat-affected layers (layers the microstructure of which was affected by the subsequent plasma pulses). Post-deposition heat treatment (isothermal holding at 950 °C for 2 h followed by oil quenching) resulted in precipitation of carbides M7C3, M3C2, M3C (in Cr-rich layers) and M6C, M2C (in W-rich layers). These carbides were found to be Cr/W depleted in favor of Fe. The carbide precipitation led to a substantial increase in the coating hardness to 1240–1445 HV0.05. The volume fraction of carbides in the coating notably increased relatively to the electrode materials

Barriers for Rank Methods in Arithmetic Complexity

Arithmetic complexity, the study of the cost of computing polynomials via additions and multiplications, is considered (for many good reasons) simpler to understand than Boolean complexity, namely computing Boolean functions via logical gates. And indeed, we seem to have significantly more lower bound techniques and results in arithmetic complexity than in Boolean complexity. Despite many successes and rapid progress, however, foundational challenges, like proving super-polynomial lower bounds on circuit or formula size for explicit polynomials, or super-linear lower bounds on explicit 3-dimensional tensors, remain elusive. At the same time (and possibly for similar reasons), we have plenty more excuses, in the form of "barrier results" for failing to prove basic lower bounds in Boolean complexity than in arithmetic complexity. Efforts to find barriers to arithmetic lower bound techniques seem harder, and despite some attempts we have no excuses of similar quality for these failures in arithmetic complexity. This paper aims to add to this study. In this paper we address rank methods, which were long recognized as encompassing and abstracting almost all known arithmetic lower bounds to-date, including the most recent impressive successes. Rank methods (under the name of flattenings) are also in wide use in algebraic geometry for proving tensor rank and symmetric tensor rank lower bounds. Our main results are barriers to these methods. In particular, 1. Rank methods cannot prove better than (2^d)*n^(d/2) lower bound on the tensor rank of any d-dimensional tensor of side n. (In particular, they cannot prove super-linear, indeed even >8n tensor rank lower bounds for any 3-dimensional tensors.) 2. Rank methods cannot prove (d+1)n^(d/2) on the Waring rank of any n-variate polynomial of degree d. (In particular, they cannot prove such lower bounds on stronger models, including depth-3 circuits.) The proofs of these bounds use simple linear-algebraic arguments, leveraging connections between the symbolic rank of matrix polynomials and the usual rank of their evaluations. These techniques can perhaps be extended to barriers for other arithmetic models on which progress has halted. To see how these barrier results directly inform the state-of-art in arithmetic complexity we note the following. First, the bounds above nearly match the best explicit bounds we know for these models, hence offer an explanations why the rank methods got stuck there. Second, the bounds above are a far cry (quadratically away) from the true complexity (e.g. of random polynomials) in these models, which if achieved (by any methods), are known to imply super-polynomial formula lower bounds. We also explain the relation of our barrier results to other attempts, and in particular how they significantly differ from the recent attempts to find analogues of "natural proofs" for arithmetic complexity. Finally, we discuss the few arithmetic lower bound approaches which fall outside rank methods, and some natural directions our barriers suggest

Parallel Search with no Coordination

We consider a parallel version of a classical Bayesian search problem. $k$ agents are looking for a treasure that is placed in one of the boxes indexed by $\mathbb{N}^+$ according to a known distribution $p$. The aim is to minimize the expected time until the first agent finds it. Searchers run in parallel where at each time step each searcher can "peek" into a box. A basic family of algorithms which are inherently robust is \emph{non-coordinating} algorithms. Such algorithms act independently at each searcher, differing only by their probabilistic choices. We are interested in the price incurred by employing such algorithms when compared with the case of full coordination. We first show that there exists a non-coordination algorithm, that knowing only the relative likelihood of boxes according to $p$, has expected running time of at most $10+4(1+\frac{1}{k})^2 T$, where $T$ is the expected running time of the best fully coordinated algorithm. This result is obtained by applying a refined version of the main algorithm suggested by Fraigniaud, Korman and Rodeh in STOC'16, which was designed for the context of linear parallel search.We then describe an optimal non-coordinating algorithm for the case where the distribution $p$ is known. The running time of this algorithm is difficult to analyse in general, but we calculate it for several examples. In the case where $p$ is uniform over a finite set of boxes, then the algorithm just checks boxes uniformly at random among all non-checked boxes and is essentially $2$ times worse than the coordinating algorithm.We also show simple algorithms for Pareto distributions over $M$ boxes. That is, in the case where $p(x) \sim 1/x^b$ for $0< b < 1$, we suggest the following algorithm: at step $t$ choose uniformly from the boxes unchecked in ${1, . . . ,min(M, \lfloor t/\sigma\rfloor)}$, where $\sigma = b/(b + k - 1)$. It turns out this algorithm is asymptotically optimal, and runs about $2+b$ times worse than the case of full coordination

Development of the international monetary system conditions of globalization

This paper outlines the modern trends in the development of the international monetary system in conditions of financial globalization. The paper establishes that the international monetary system has evolved through different stages necessitated by various economic crises at different periodic intervals. It is noted in the paper that despite the inefficiencies in the current dollar denominated international monetary system developed countries are reluctant to do any reforms. However, International monetary system reform is obvious and most likely will happen under the growing crisis in the international financial systems and credit relations. It is proposed that countries look at new innovations that will make international transactions cheaper and less risky.peer-reviewe