Monochromatic Solution Research Articles

On Rado numbers for equations with unit fractions

Let fr(k) be the smallest positive integer n such that every r-coloring of {1,2,…,n} has a monochromatic solution to the nonlinear equation1/x1+⋯+1/xk=1/y, where x1,…,xk are not necessarily distinct. Brown and Rödl [3] proved that f2(k)=O(k6). In this paper, we prove that f2(k)=O(k3). The main ingredient in our proof is a finite set A⊆N such that every 2-coloring of A has a monochromatic solution to the linear equation x1+⋯+xk=y and the least common multiple of A is sufficiently small. This approach can also be used to study fr(k) with r>2. For example, a recent result of Boza et al. [2] implies that f3(k)=O(k43).

On Some Exact Formulas for \(2\)-Color Off-diagonal Rado Numbers

Let \(\varepsilon_{0}\), \(\varepsilon_{1}\) be two linear homogenous equations, each with at least three variables and coefficients not all the same sign. Define the \(2\)-color off-diagonal Rado number \(R_2(\varepsilon_{0},\varepsilon_{1})\) to be the smallest \(N\) such that for any 2-coloring of \([1,N]\), it must admit a monochromatic solution to \(\varepsilon_{0}\) of the first color or a monochromatic solution to \(\varepsilon_{1}\) of the second color. Mayers and Robertson gave the exact \(2\)-color off-diagonal Rado numbers \(R_2(x+qy=z,x+sy=z). \) Xia and Yao established the formulas for \(R_2(3x+3y=z,3x+qy=z) \) and \(R_2(2x+3y=z,2x+2qy=z) \). In this paper, we determine the exact numbers \(R_2(2x+qy=2z,2x+sy=2z)\), where \(q, s\) are odd integers with \(q>s\geq1\).

On Disjunctive Rado Numbers for Some Sets of Equations

Given a set of linear equations $S$ with positive integral parameters $a_1,\ldots,a_k$, $k \ge 2$, the disjunctive Rado number for the set $S$ is the least positive integer $R=R_d(S)$, if it exists, such that every $2$-coloring $\chi$ of the integers in $[1, R]$ admits a monochromatic solution to at least one equation in $S$. We give conditions for the existence of $R_d(S)$, and also give general upper and lower bounds on $R_d(S)$, when $S$ is a set of additive equations $\{y=x+a_1,\ldots,y=x+a_k\}$. We also determine $R_d(S)$ when $\max a_i$ is large enough, or when $a_1,\ldots,a_k$ form an arithmetic or geometric progression. We also give conditions for the existence of $R_d(S)$ when $S$ is a set of multiplicative equations $\{y=a_1 x,\ldots,y=a_k x\}$. Further, we give a general search-based algorithm to determine $R_d(S)$ when $S$ is a system of equations in two variables, given an upper bound on $R_d(S)$ and an algorithm to determine solutions to $S$. This algorithm runs in time $O(ka_k \log a_k)$ for the case of additive equations, which is exponentially better than the brute-force algorithm for the problem.

Nonparaxial accelerating waves as a superposition of nondiffracting Bessel-lattice optical fields.

In the first part of this work, we introduce a monochromatic solution to the scalar wave equation in free space, defined by a superposition of monochromatic nondiffracting half Bessel-lattice optical fields, which is determined by two scalar functions; one is defined on frequency space, and the other is a complete integral to the eikonal equation in free space. We obtain expressions for the geometrical wavefronts, the caustic region, and the Poynting vector. We highlight that this solution is stable under small perturbations because it is characterized by a caustic of the hyperbolic umbilical type. In the second part, we introduce the corresponding solution to the Maxwell equations in free space.

Superposition of nondiffracting beams characterized by a caustic of the hyperbolic umbilical type

The aim of the present work is to introduce two monochromatic solutions to the scalar wave equation in free space, characterized by a caustic with a singularity of the hyperbolic umbilical type. The first solution, is a superposition of half-Mathieu beams, and the second one, is a superposition of parabolic beams. Since these solutions are determined by two particular complete integrals of the eikonal equation in free space, we compute their geometrical wavefronts, the caustic regions, and the corresponding Poynting vectors. Finally, we remark that, under certain conditions, these solutions describe three-dimensional accelerating beams in free space, propagating along semielliptical and parabolic paths, respectively.

Two-color Rado number of x + y + c = kz for large c

For positive integers c and k, we consider the equation L:x+y+c=kz. Two-color Rado number R=R(c,k) of L is the least integer, provided that it exists, such that every two-coloring of 1,2,⋯,R admits a monochromatic solution to L. The Rado number R(c,k) exists if and only if k is odd or c is even.In this paper, we show that if k is odd or c is even with k≥5 and c≥2k3+2k2−k, then the two-color Rado number of L is equal to N=⌈2⌈c+2k⌉+ck⌉, as predicted by Jones and Schaal.

A study of wide unfocused wavefront for convex-array ultrasound imaging

Ultrafast ultrasound imaging modalities have attracted a lot of attention in the ultrasound community. It breaks the compromise between the frame rate and the region of interest by insonifying the whole medium with wide unfocused waves. Coherent compounding can be performed to enhance the image quality at a cost of frame rate. Ultrafast imaging has wide clinical applications, such as vector Doppler imaging and shear elastography. On the other hand, the use of unfocused waves is still marginal with convex-array transducers. For convex array, plane wave imaging is limited by the complicated transmission delay calculation, limited field-of-view, and inefficient coherent compounding. In this article, we study three wide unfocused wavefronts, namely, lateral virtual-source defined diverging wave imaging (latDWI), tilt virtual-source defined diverging wave imaging (tiltDWI), and Archimedean-spiral-based imaging (AMI) for convex-array imaging using the full-aperture transmission. The analytical monochromatic wave solutions to this three imaging are given. The mainlobe width and grating lobe position are given explicitly. Theoretical −6 dB beamwidth and synthetic transmit field response are studied. Simulation studies are carried on with the point targets and hypoechoic cysts. Time-of-flight formulas are given explicitly for beamforming. The conclusions are in good agreement with the theory: latDWI provides the finest lateral resolution but generates the severest axial lobe level for scatterers with large obliquities (i.e., for scatterers located at the image border) which degrades the image contrast. This effect gets worsen as the compound number increases. The tiltDWI and AMI give a very close performance on resolution and image contrast. AMI displays better contrast with a small compound number.

Exact and geometrical optics energy trajectories in Bessel beams via the quantum potential

The aim of the present work is to show that any monochromatic solution to the scalar wave equation in free space defines a conservative Hamiltonian system, describing a particle of mass m = 1 / 2 and energy E = 1 , under the influence of the so-called quantum potential. We remark that the integral curves of its Poynting vector, exact optics energy trajectories, define a particular subset of solutions to the corresponding Hamilton equations. Furthermore, we introduce the zero quantum potential straight lines concept, as the family of tangent lines to the integral curves of the Poynting vector at the zeros of the quantum potential. These general results are applied to a family of plane waves and to Bessel beams. In the case of Bessel beams, we present a detailed study of the trajectories determined by the corresponding Hamiltonian system, and we show that the zero quantum potential straight lines coincide with the geometrical light rays, geometrical optics energy trajectories. Furthermore, we show that the areal velocity, determined by the exact optics energy trajectories, for non-zero order Bessel beams is not a constant of motion. However, its projection along the z ^ direction is a constant of motion because L z is a constant.

The Ether and its Properties. Photons and Light Emitters in Biquaternionic Presentation

Here the ether equation in a biquaternionic representation has been presented which is generalization of Maxwell equations in algebra of biquaternions. We consider the ether as electro-gravimagnetic (EGM) field, which is described by the biquaternion of the intensity of EGM field. Its scalar part is called the ether density. The vector part, by construction, describes the intensity of electric and gravimagnetic fields. The ether equation is the biquaternionic wave (biwave) equation which expresses EGM charges and currents through the bigradient of EGM field intensity. Monochromatic solutions of these equations are constructed on the basis of which biquaternionic representations of photons are presented. Free ether waves are considered. It is shown that they and photon biquaternions contain a gravitational component, which determines the light pressure. The various light emitters (ball, spherical, ring, circular pulsars) are considered and their biquaternions are constructed in analytical form.

Towards characterizing the 2-Ramsey equations of the form ax + by = p(z)

In this paper, we study a Ramsey-type problem for equations of the form ax+by=p(z). We show that if certain technical assumptions hold, then any 2-colouring of the positive integers admits infinitely many monochromatic solutions to the equation ax+by=p(z). This entails the 2-Ramseyness of several notable cases such as the equation ax+y=zn for arbitrary a∈Z+ and n≥2, and also of ax+by=aDzD+…+a1z∈Z[z] such that gcd(a,b)=1, D≥2, a,b,aD>0 and a1≠0.

On improving a Schur-type theorem in shifted primes

We show that if $$N \geq {\rm exp}({\rm exp}({\rm exp} (k^{O(1)})))$$ , then any k-colouring of the primes that are less than N contains a monochromatic solution to $$p_1 - p_2 = p_3 -1$$ .

Machine Learning Non-Reciprocity of a Linear Waveguide With a Local Nonlinear, Asymmetric Gate: Case of Strong Coupling

Abstract We study nonreciprocity in a passive linear waveguide augmented with a local asymmetric, dissipative, and strongly nonlinear gate. Strong coupling between the constituent oscillators of the waveguide is assumed, resulting in broadband capacity for wave transmission. The local nonlinearity and asymmetry at the gate can yield strong global nonreciprocal acoustics, in the sense of drastically different acoustical responses depending on which side of the waveguide a harmonic excitation is applied. Two types of highly nonreciprocal responses are observed: (i) Monochromatic responses without frequency distortion compared to the applied harmonic excitation, and (ii) strongly modulated responses (SMRs) with strong frequency distortion. The complexification averaging (CX-A) method is applied to analytically predict the monochromatic solutions of this strongly nonlinear problem, and a stability analysis is performed to study the governing bifurcations. In addition, we build a machine learning framework where neural net (NN) simulators are trained to predict the performance measures of the gated waveguide in terms of certain transmissibility and nonreciprocity measures. The NN drastically reduces the required simulation time, enabling the determination of parameter ranges for desired performance in a high-dimensional parameter space. In the predicted desirable parameter space for nonreciprocity, the maximum transmissibility reaches 40%, and the transmitted energy varies by up to three orders of magnitude depending on the direction of wave transmission. The machine learning tools along with the analytical methods of this work can inform predictive designs of practical nonreciprocal waveguides and acoustic metamaterials that incorporate local nonlinear gates.

Avoiding monochromatic solutions to 3-term equations

Given any equation and the integers $[n] = \{1,\dots,n\}$, one can ask: does every 2-coloring of $[n]$ contain at least as many monochromatic solutions (asymptotically) as a uniformly random 2-coloring? We show that the answer is no for every 3-term linear equation with integer coefficients, i.e. there always exist 2-colorings with asymptotically fewer monochromatic solutions than uniformly random ones.

The pigeonhole principle and multicolor Ramsey numbers

A standard proof of Schur's Theorem yields that any $r$-coloring of $\{1,2,\dots,R_r-1\}$ yields a monochromatic solution to $x+y=z$, where $R_r$ is the classical $r$-color Ramsey number, the minimum $N$ such that any $r$-coloring of a complete graph on $N$ vertices yields a monochromatic triangle. We explore generalizations and modifications of this result in higher dimensional integer lattices, showing in particular that if $k\geq d+1$, then any $r$-coloring of $\{1,2,\dots,R_r(k)^d-1\}^d$ yields a monochromatic solution to $x_1+\cdots+x_{k-1}=x_k$ with $\{x_1,\dots,x_d\}$ linearly independent, where $R_r(k)$ is the analogous Ramsey number in which triangles are replaced by complete graphs on $k$ vertices. We also obtain computational results and examples in the case $d=2$, $k=3$, and $r\in\{2,3,4\}$.

An asymmetric random Rado theorem: 1-statement

A classical result by Rado characterises the so-called partition-regular matrices A, i.e. those matrices A for which any finite colouring of the positive integers yields a monochromatic solution to the equation Ax=0. We study the asymmetric random Rado problem for the (binomial) random set [n]p in which one seeks to determine the threshold for the property that any r-colouring, r≥2, of the random set has a colour i∈[r] admitting a solution for the matrical equation Aix=0, where A1,…,Ar are predetermined partition-regular matrices pre-assigned to the colours involved.We prove a 1-statement for the asymmetric random Rado property. In the symmetric setting our result retrieves the 1-statement of the symmetric random Rado theorem established in a combination of results by Rödl and Ruciński [34] and by Friedgut, Rödl and Schacht [11]. We conjecture that our 1-statement in fact unveils the threshold for the asymmetric random Rado property, yielding a counterpart to the so-called Kohayakawa-Kreuter conjecture concerning the threshold for the asymmetric random Ramsey problem in graphs.We deduce the aforementioned 1-statement for the asymmetric random Rado property after establishing a broader result generalising the main theorem of Friedgut, Rödl and Schacht from [11]. The latter then serves as a combinatorial framework through which 1-statements for Ramsey-type problems in random sets and (hyper)graphs alike can be established in the asymmetric setting following a relatively short combinatorial examination of certain hypergraphs. To establish this framework we utilise a recent approach put forth by Mousset, Nenadov and Samotij [26] for the Kohayakawa-Kreuter conjecture.

Polynomial Schur’s Theorem

We resolve the Ramsey problem for {x,y,z: x + y = p(z)} for all polynomials p over ℤ. In particular, we characterise all polynomials that are 2-Ramsey, that is, those p(z) such that any 2-colouring of ℕ contains infinitely many monochromatic solutions for x+y=p(z). For polynomials that are not 2-Ramsey, we characterise all 2-colourings of ℕ that are not 2-Ramsey, revealing that certain divisibility barrier is the only obstruction to 2-Ramseyness for x + y = p(z).

Possible quantum effects at the transition from cosmological deceleration to acceleration

The recent transition from decelerated to accelerated expansion can be seen as a reflection (or ``bounce'') in the connection variable, defined by the inverse comoving Hubble length ($b=\stackrel{\ifmmode \dot{}\else \textperiodcentered \fi{}}{a}$, on shell). We study the quantum cosmology of this process. We use a formalism for obtaining relational time variables either through the demotion of the constants of nature to integration constants, or by identifying fluid constants of motion. We extend its previous application to a toy model (radiation and $\mathrm{\ensuremath{\Lambda}}$) to the realistic setting of a transition from dust matter to $\mathrm{\ensuremath{\Lambda}}$ domination. In the dust and $\mathrm{\ensuremath{\Lambda}}$ model two time variables may be defined, conjugate to $\mathrm{\ensuremath{\Lambda}}$ and to the dust constant of motion, and we work out the monochromatic solutions to the Schr\"odinger equation representing the Hamiltonian constraint. As for their radiation and $\mathrm{\ensuremath{\Lambda}}$ counterparts, these solutions exhibit ``ringing,'' whereby the incident and reflected waves interfere, leading to oscillations in the amplitude. In the semiclassical approximation we find that, close to the bounce, the probability distribution becomes double peaked, one peak following a trajectory close to the classical limit but with a Hubble parameter slightly shifted downwards, the other with a value of $b$ stuck at its minimum $b={b}_{\ensuremath{\star}}$. Still closer to the transition, the distribution is better approximated by an exponential distribution, with a single peak at $b={b}_{\ensuremath{\star}}$, and a (more representative) average $b$ biased towards a value higher than the classical trajectory. Thus, we obtain a distinctive prediction for the average Hubble parameter with redshift: slightly lower than its classical value when $z\ensuremath{\approx}0$, but potentially much higher than the classical prediction around $z\ensuremath{\sim}0.64$, where the bounce most likely occurred. The implications for the ``Hubble tension'' have not escaped us.

Two-color Rado number of x + y + c = kz for odd c and k with k ≥ c + 6

For positive integers c and k, 2-color Rado number R=R(c,k) is the least integer, provided it exists, such that every 2-coloring of the positive integers up to R admits a monochromatic solution to x+y+c=kz. It is known that R exists if and only if k is odd or c is even.In this paper, for odd c,k we show that R(c,k)=12(k(k+1)−c+1) when k≥c+6. Moreover, we show that 12(k(k+1)−c+1) is also an upper bound of R(c,k) when k≥c+4.

An asymmetric random Rado theorem for single equations: The 0‐statement

AbstractA famous result of Rado characterizes those integer matrices A which are partition regular, that is, for which any finite coloring of the positive integers gives rise to a monochromatic solution to the equation . Aigner‐Horev and Person recently stated a conjecture on the probability threshold for the binomial random set having the asymmetric random Rado property: given partition regular matrices (for a fixed ), however one r‐colors , there is always a color such that there is an i‐colored solution to . This generalizes the symmetric case, which was resolved by Rödl and Ruciński, and Friedgut, Rödl and Schacht. Aigner‐Horev and Person proved the 1‐statement of their asymmetric conjecture. In this paper, we resolve the 0‐statement in the case where the correspond to single linear equations. Additionally we close a gap in the original proof of the 0‐statement of the (symmetric) random Rado theorem.

Partition regularity of polynomial systems near zero

Recently, S. Kanti Patra and Md. Moid Shaikh proved the existence of monochromatic solutions to systems of polynomial equations near zero for particular dense subsemigroups (S,+) of (mathbb {R}^{+},+). We extend their results to a much larger class of systems whilst weakening the requests on S, using solely basic results about ultrafilters.