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Given a smooth integer q, we use existing upper bounds for character sums to find a lower bound for the size of a multiplicative subgroup of the integers modulo q which contains the image of an interval of consecutive integers I ⊂ Zq under a polynomial f ∊ Z[X]. [ABSTRACT FROM AUTHOR]

We present lower bounds for the orbit length of reduction modulo primes of parametric polynomial dynamical systems defined over the integers, under a suitable hypothesis on its set of preperiodic points over C. Applying recent results of Baker and DeMarco (2011) and of Ghioca, Krieger, Nguyen and Ye (2017), we obtain explicit families of parametric polynomials and initial points such that the reductions modulo primes have long orbits, for all but a finite number of values of the parameters. This generalizes a previous lower bound due to Chang (2015). As a by-product, we also slightly improve a result of Silverman (2008) and recover a result of Akbary and Ghioca (2009) as special extreme cases of our estimates. [ABSTRACT FROM AUTHOR]

In this work, we establish a nontrivial level of distribution for densities on {1,...,N} obtained by a biased coin convolution. As a consequence of sieving theory, one then derives the expected lower bound for the weight of such densities on sets of almost primes. [ABSTRACT FROM AUTHOR]

Let f(x) = ax² + bx + c ∈ Z[x] be a quadratic polynomial with a ≢ 0 mod p. Take z ∈ Fp and let Oz = {fi(z)}i∈Z+ be the orbit of z under f, where fi(z) = f(fi-1(z)) and f0(z) = z. For M < |Oz|, we study the diameter of the partial orbit OM = {z, f(z), f2(z),..., fM-1(z)} and prove that there exists c1 > 0 such that diam OM ≳ min {...}. For a complete orbit C, we prove that diam C ≳ min{p5c1, eT/4}, where T is the period of the orbit. [ABSTRACT FROM AUTHOR]

We experimentally investigate the Lagrangian-Eulerian dynamics of fluid motion and wave-form evolution for a breaking shallow water wave approaching a slope by tracking tracer motions. It is found that, before breaking, the surface element can climb over the crest and exhibits cyclic oscillation with small forward drift. The increasing asymmetric tangential compression (accumulation) and rarefaction (depletion) in the crest front and the crest are the keys for the crest front steepening with the increasing particle cyclic excursion and forward Stoke drift. Eventually, the surface layer cannot climb over the crest with the vertical front. It curls up and forms an overhanging plunging jet leading the crest, while the lower flow still can reach the crest rear. This process leads to wave breaking with the rapid drop of crest height and the transition from slow divergence to rapid divergence of the adjacent fluid trajectories. [ABSTRACT FROM AUTHOR]

Intermittent dust acoustic wave turbulence self-excited by downward ion flow in dissipative dusty plasma is experimentally observed and investigated. The power spectra of the temporal dust density fluctuation show distinct bumps in the low-frequency regime and power-law scaling in the high-frequency regime. The structure-function analysis demonstrates the multifractal dynamics of the wave turbulence. Decreasing dissipation by decreasing neutral pressure leads to a more turbulent state with a less distinct low-frequency bump in the power spectrum, more stretched non-Gaussian tails in the histogram of the wave-height increment at a small time interval τ, and a higher degree of multifractality. The loss of long time memory with increasing τ for a more turbulent state causes a change from the distribution with stretched non-Gaussian tails to Gaussian with increasing τ. [ABSTRACT FROM AUTHOR]

We experimentally investigate the micro-origin of the absence of trough trapping in nonlinear traveling dust acoustic waves self-excited by the downward ion flow in the dissipative dusty plasma. The wave forms of dust density, the drag force from the background neutrals, ions, and dusts, and the effective potential energy for dusts are constructed by tracking dust motion and measuring the velocity and the position-dependent forces. The tilted washboard type potential wave form with a slight phase lead to the dust density wave form is obtained. It provides sufficient kinetic energy to compensate drag dissipation and move dusts from the dust density trough to the crest front. The dusts with sufficient energy overcome the downward pushing by the crest front, climb over the crest, and sustain the oscillatory motion with upward drift. Those dusts with insufficient energy to climb over the potential barrier of the crest are trapped in and move downward with the crest front, until kicked upward by fluctuation. The upward neutral dominated drag force prevents them from sliding down die potential energy hill at the crest front and further oscillating in the trough. It leads to the absence of trough trapping. [ABSTRACT FROM AUTHOR]

The Eulerian/Lagrangian dynamics in the narrow wake field of the dusty plasma bubble is explored by directly tracking dust motion at the microscopic level. The bubble is induced by the focused laser pulse ablation in three-dimensional quiescent dusty plasma liquids operated in the pressure higher than the critical pressure for the self-excitation of dust acoustic wave by the downward ion wind. It is found that, after bubble expansion ceases, the collective excitation maintains its width and travels downward as a solitary wave, led by an ultrasonic rarefaction front contributed by the dust motion below the lower boundary, and trailed by the few compressional crests with descending crest heights and speeds in the narrow wake, under the symmetry breaking by the downward ion flow. The quick damping of the waves propagating along other directions leads to a narrow wake. Increasing the background pressure causes the more isotropic collapsing of the bubble without wake field oscillation. The role played by dust motion on interacting with and sustaining the wake field evolution is identified and discussed. [ABSTRACT FROM AUTHOR]

We study product theorems for matrix spaces. In particular, we prove the following theorems. Theorem 1.For all $\varepsilon0$, there is $\delta0$such that if $A\subset\mathrm{SL}_3(\mathbb{Z})$is a finite set, then either $A$intersects a coset of a nilpotent subgroup in a set of size at least $|A|^{1-\varepsilon}$, or $|A^3||A|^{1+\delta}$. Theorem 2.Let $A$be a finite subset of $\mathrm{SL}_2(\mathbb{C})$. Then either $A$is contained in a virtually abelian subgroup, or $|A^3|c|A|^{1+\delta}$for some absolute constant $\delta0$. Here $A^3=\{a_1a_2a_3:a_i\in A,\ i=1,2,3\}$is the $3$-fold product set of $A$. [ABSTRACT FROM AUTHOR]

We prove the following theorems in incidence geometry. 1. There is δ > 0 such that for any P1, . . ., P4 ∈ C², and Q1, . . .,Qn ∈ C², if there are C² n(1+δ)/2 distinct lines between Pi and Qj for all i, j, then P1, . . ., P4 are collinear. If the number of the distinct lines is < cn1/2, then the cross ratio of the four points is algebraic. 2. Given c > 0, there is δ > 0 such that for any P1, P2, P3 ∈ C² noncollinear, and Q1, . . .,Qn ∈ C², if there are ≤ cn1/2 distinct lines between Pi and Qj for all i, j, then for any P ∈ C² r {P1, P2, P3}, we have δn distinct lines between P and Qj. 3. Given c > 0, there is ∊ > 0 such that for any P1, P2, P3 ∈ C² (respectively, R²) collinear, and Q1, . . .,Qn ∈ C² (respectively, R²), if there are ∊ cn1/2 distinct lines between Pi and Qj for all i, j, then for any P not lying on the line L(P1, P2), we have at least n1-∊ (resp. n/log n) distinct lines between P and Qj. The main ingredients used are the subspace theorem, Balog-Szemerédi-Gowers theorem, and Szemer 'edi-Trotter theorem. We also generalize the theorems to higher dimensions, extend Theorem 1 to F²p, and give the version of Theorem 2 over Q. [ABSTRACT FROM AUTHOR]

Let $A$ be a set of $N$ matrices. Let $g(A)\ccolon=|A+A|+|A\cdot A|$, where $A+A=\{a_1 + a_2 \mid a_i \in A\}$ and $A\cdot A=\{a_1a_2 \mid a_i \in A\}$ are the sum set and product set. We prove that if the determinant of the difference of any two distinct matrices in $A$ is nonzero, then $g(A)$ cannot be bounded below by $cN$ for any constant $c$. We also prove that if $A$ is a set of $d\times d$ symmetric matrices, then there exists $\varepsilon=\varepsilon (d)>0$ such that $g(A)>N^{1+\varepsilon}.$ For the first result, we use the bound on the number of factorizations in a generalized progression. For the symmetric case, we use a technical proposition which provides an affine space $V$ containing a large subset $E$ of $A$, with the property that if an algebraic property holds for a large subset of $E$, then it holds for $V$. Then we show that the system $\{a^2\ccolon a\in V\}$ is commutative, allowing us to decompose ${\mathbb R}^d$ as eigenspaces simultaneously, so we can finish the proof with induction and a variant of the Erdős–Szemerédi argument. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate efficient representations of the residue classes modulo q, by performing sum and product set operations starting from a given subset A of ℤq. We consider the case of very small sets A and composite q for which not much seemed known (non-trivial results were recently obtained when q is prime or when log ∣A∣ ∼ log q). Roughly speaking we show that all residue classes are obtained from a k-fold sum of an r-fold product set of A, where r ⟪ log q and log k ⟪ log q, provided the residue sets πq′,(A) are large for all large divisors q′ of q. Even in the special case of prime modulus q, some results are new, when considering large but bounded sets A. It follows for instance from our estimates that one can obtain r as small as r ∼ log q/log ∣A∣ with similar restriction on k, something not covered by earlier work of Konyagin and Shparlinski. On the technical side, essential use is made of Freiman's structural theorem on sets with small doubling constant. Taking for A = H a possibly very small multiplicative subgroup, bounds on exponential sums and lower bounds on mina∊ℤq* maxx∊H ∥ax/q∥ are obtained. This is an extension to the results obtained by Konyagin, Shparlinski and Robinson on the distribution of solutions of xm = a (mod q) to composite modulus q. [ABSTRACT FROM AUTHOR]

A well-known conjecture of W. Rudin is that the set of squares is a ∧p-set for all p>4. In particular, this implies that for all ε>0, there exists a constant cϵ such that∫Π∑j=1keinj2λ4dx14⩽cϵk12+ϵfor any k distinct integers n1…nk. In this article we give a combinatorial interpretation of the inequality above in the spirit of \|q\|q sum and product sets along graphs as considered by P. Erdo¨s and E. Szemeredi (Studies in Pure Mathematics, pp. 213–218). We also show that the left-hand side of the inequality is bounded by Cϵk34(logk)148−ϵ. [Copyright &y& Elsevier]

A vector bundle has the {\em Bloch-Gieseker property} if all its Chern classes are numerically positive. In this paper we show that the non-ample bundle $\Omega ^{p}_{\mathbb{P}_{n}}(p+1)$ has the Bloch-Gieseker property, except for two cases, in which the top Chern classes are trivial and the other Chern classes are positive. Our method is to reduce the problem to showing, e.g. the positivity of the coefficient of $t^{k}$ in the rational function $\frac{(1+t)^{\binom n p} (1+3t)^{\binom {n}{p-2}} \cdots (1+(p-1)t)^{\binom n2} (1+(p+1)t)}{(1+2t)^{\binom {n}{p-1}} (1+4t)^{\binom {n}{p-3}} \cdots (1+pt)^{\binom {n}{1}}}$ (for $p$ even). [ABSTRACT FROM AUTHOR]

We prove the following analogue of Silverman's results [9] for pairs of maps. Let d≥2 be an integer, K/Q a number field, and N=NK/Q(P) the norm of an ideal POK. Let h(z)K[z]h be non-constant and not of the form h(z)=z, d-1=1. Denote ft(z)=zd+t, gt(z)=zd+h(t), and F(l)F(l) the l-th iteration of F. There are constants c1,c2 depending on d and h such that the following holds. For almost all prime ideals POK, there is a finite subset TFP, |T|≤c1 such that if tFP\T at least one of the sets... (1) consists of distinct elements. [ABSTRACT FROM AUTHOR]

We give a lower bound on the number of distinct dimensions of irreducible components of the Hilbert scheme of codimension 2 subvarieties in $\mathbb{P}^{n}$, for $n \le 5$ (respectively, the moduli space of surfaces or 3-folds) in terms of the Hilbert polynomial (resp. Chern numbers). [ABSTRACT FROM AUTHOR]

In this work, we briefly review our recent experimental studies on the observations and waveform dynamics of spontaneous excitations of low and high amplitude singular objects: low amplitude hole filaments coinciding with the wiggling trajectories of topological defects surrounded by acoustic vortices with helical waveforms, and uncertain rogue wave events, in self-excited weakly disordered dust acoustic waves. The changes of waveform topology, caused by kinking, rupturing and reconnection of sequential wave crests surfaces, and the reversed process, are responsible for the chaotic creation, propagation, and annihilation of acoustic vortex pairs with opposite helicities winding around low amplitude hole filaments. The observed rogue wave events are preceded by a higher probability of surrounding defects. Particle focusing by the transverse electric forces from ruptured and tilted wave crests nearby defects are identified as the major cause for rogue wave generation. [ABSTRACT FROM AUTHOR]

We apply the refined composite multiscale entropy (MSE) method to a one-dimensional directed small-world network composed of nodes whose states are binary and whose dynamics obey the majority rule. We find that the resulting fluctuating signal becomes dynamically complex. This dynamical complexity is caused (i) by the presence of both short-range connections and long-range shortcuts and (ii) by how well the system can adapt to the noisy environment. By tuning the adaptability of the environment and the long-range shortcuts we can increase or decrease the dynamical complexity, thereby modeling trends found in the MSE of a healthy human heart rate in different physiological states. When the shortcut and adaptability values increase, the complexity in the system dynamics becomes uncorrelated. [ABSTRACT FROM AUTHOR]