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A classical question in combinatorial number theory asks whether an equation has a solution inside a particular subset of its domain. The Rado's Theorem gives a set of necessary and sufficient conditions for a systems of linear equations to…
Defective coloring is a variant of traditional vertex-coloring, according to which adjacent vertices are allowed to have the same color, as long as the monochromatic components induced by the corresponding edges have a certain structure.…
Cooperative scattering has been the subject of intense research in the last years. In this article, we discuss the concept of cooperative scattering from a broad perspective. We briefly review the various collective effects that occur when…
Suppose that each number $1,2,...,N$ has one of n colours assigned. We show that if there are no monochromatic solutions to the equation $x_1+x_2+x_3=y_1+y_2$, then $N=O((n!)^{1/2})$, improving upon a result of Cwalina and Schoen. Further,…
We study the number of monochromatic solutions to linear equations in a $2$-coloring of $\{1,\ldots,n\}$. We show that any nontrivial linear equation has a constant fraction of solutions that are monochromatic in any $2$-coloring of…
The set of sums of two squares plays a significant role in elementary number theory. In this article, we establish the existence of several rich monochromatic configurations in the natural numbers by exploiting algebraic structures induced…
The main purpose of this work is to introduce and analyse some generalizations of diverse superposition rules for first-order differential equations to the setting of second-order differential equations. As a result, we find a way to apply…
Phase stability and properties of two-dimensional phosphorus carbide, P$_x$C$_{1-x}$, are investigated using the first-principles method in combination with cluster expansion and Monte Carlo simulation. Monolayer P$_x$C$_{1-x}$ is found to…
We give partial generalizations of the classical Descartes' rule of signs to multivariate polynomials (with real exponents), in the sense that we provide upper bounds on the number of connected components of the complement of a hypersurface…
Depth estimation plays a important role in SLAM, odometry, and autonomous driving. Especially, monocular depth estimation is profitable technology because of its low cost, memory, and computation. However, it is not a sufficiently…
In this paper we prove that for any finite coloring of N there are lambda,rho in N such that infinitely many pairs (x,y),(u,v) in N^2 satisfy the sets {lambda x, lambda y, x y, lambda(x+y)} and {u+rho, v+rho, u v+rho, u+v} being…
Surface roughness of particle monolayer is one of its fundamental characteristics, conventionally described in terms of power spectral density. We derive a general equation for this function and demonstrate its similarity to those applied…
We report measurements of the x-ray scattering intensity as mesoporous glasses are filled with Neon. We outline a simple model to account for the non-linear increase in the intensity of the first peak in a liquid-like diffraction pattern as…
Fix $k \geq 3$, and let $G$ be a $k$-uniform hypergraph with maximum degree $\Delta$. Suppose that for each $l = 2, ..., k-1$, every set of l vertices of G is in at most $\Delta^{(k-l)/(k-1)}/f$ edges. Then the chromatic number of $G$ is…
In this paper, we explicitly construct the Calabi composition of multiple affine hyperspheres possibly including some points viewing as 0-dimensional hypersheres. Then we compute all the basic affine invariants of the composed affine…
In the paper, we search for monochromatic infinite additive structures involving polynomials over $\mathbb{N}$. It is proved that for any $r\in \mathbb{N}$, any two distinct natural numbers $a,b$, and any $2$-coloring of $\mathbb{N}$, there…
Suppose that $\mathbb{F}_p$ is coloured with $r$ colours. Then there is some colour class containing at least $c_r p^2$ quadruples of the form $(x, y , x + y, xy)$.
Let $G=(V,E)$ be a simple graph with $V=\{1,2,\cdots,n\}$ and $\chi(G,x)$ be its chromatic polynomial. For an ordering $\pi=(v_1,v_2,\cdots,v_n)$ of elements of $V$, let $\delta_G(\pi)$ be the number of $i$'s, where $1\le i\le n-1$, with…
Compared to color images captured by conventional RGB cameras, monochrome images usually have better signal-to-noise ratio (SNR) and richer textures due to its higher quantum efficiency. It is thus natural to apply a mono-color dual-camera…
A mixed graph is, informally, an object obtained from a simple undirected graph by choosing an orientation for a subset of its edges. A mixed graph is $(m, n)$-coloured if each edge is assigned one of $m \geq 0$ colours, and each arc is…