Normal approximation and fourth moment theorems for monochromatic triangles

Given a graph sequence denote by T₃(G_n) the number of monochromatic triangles in a uniformly random coloring of the vertices of G_n with colors. In this paper we prove a central limit theorem (CLT) for T₃(G_n) with explicit error rates, using a quantitative version of the martingale CLT. We then relate this error term to the well-known fourth-moment phenomenon, which, interestingly, holds only when the number of colors satisfies . We also show that the convergence of the fourth moment is necessary to obtain a Gaussian limit for any , which, together with the above result, implies that the fourth-moment condition characterizes the limiting normal distribution of T₃(G_n), whenever . Finally, to illustrate the promise of our approach, we include an alternative proof of the CLT for the number of monochromatic edges, which provides quantitative rates for the results obtained in [7].     

Calculations of the Structural,Elastic, and Magnetic Properties of the Novel Full Heusler Alloys Ru2XY with X = Nb,Mn and Y = Te,Sb

JETP Letters - The electronic, elastic, and magnetic properties of the new full Heusler alloys Ru2XY with X = Nb, Mn and Y = Te, Sb are investigated using the Full Potential Nonorthogonal...     

Experimental and Numerical Simulation Research on Boundary Layer Transition Front over a Swept Wing at M = 6

Fluid Dynamics - To predict the boundary layer transition on swept wings, experimental and numerical researches were conducted to obtained the accurate boundary layer transition front and to...     

Profile Monitor for Electron Beam in the High-Voltage Electron Cooling System of the COSY Synchrotron

Technical Physics - A high-voltage system for electron cooling of the COSY (Germany) synchrotron is featured by the presence of a profile monitor that is used for measurement of current density...     

Polarization-Modulation Methods for Determining the Roll and Bearing Angles of an Aircraft using a Rotating Quarter-Wave Phase Plate

Russian Physics Journal - A polarization-modulation method for simultaneous determination of the navigation bearing and roll parameters of an aircraft from orthogonally linearly polarized beacon...     

Self-Similarity in the Problem of Laminar Film Boiling on a Vertical Surface Submerged in a Large Volume of a Liquid

Technical Physics - Using the methods of self-similarity for a laminar free-convective film boiling on a vertical finite-length surface submerged into a large volume of an arbitrarily subcooled...     

Optimization of heat transfer in the thermal Marangoni convective flow of a hybrid nanomaterial with sensitivity analysis

Applied Mathematics and Mechanics - The heat transfer rate of the thermal Marangoni convective flow of a hybrid nanomaterial is optimized by using the response surface methodology (RSM). The...     

Operator Interpolation and Systems of Linear Equations and Inequalities in Euclidean Spaces

Ukrainian Mathematical Journal - We establish new criteria of compatibility for a linear system of equations (equivalent to the Kronecker–Capelli theorem) or inequalities (equivalent to the...     

Raman Scattering Study of Natural and Synthetic Compounds Containing Polyene Chains

Doklady Physics - The dependence of Raman spectra of the molecules containing polyene chains on the conjugation length, structure of side and end groups, and isomer type is analyzed theoretically.     

Multiplication Factor for Subcritical Multiplying System and the Efficiency of Source Neutrons

Physics of Atomic Nuclei - In this paper, the analysis of the relation between the multiplication factor of neutrons of the multiplying system measured in critical experiments and the system...