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Register a new userDensity of sequences of the form $x_n=f(n)^n$ in [0,1]
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Authors
J.C. Saunders
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In 2013, Strauch asked how various sequences of real numbers defined from trigonometric functions such as $x_n=(cos n)^n$ distributed themselves$pmod 1$. Strauchs inquiry is motivated by several such distribution results. For instance, Luca proved that the sequence $x_n=(cos alpha n)^npmod 1$ is dense in $[0,1]$ for any fixed real number $alpha$ such that $alpha/pi$ is irrational. Here we generalise Lucas results to other sequences of the form $x_n=f(n)^npmod 1$. We also examine the size of the set $|{nleq N:r<|cos(npialpha)|^n}|$ where $0<r<1$ and $alpha$ are fixed such that $alpha/pi$ is irrational.
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Let $p$ be a prime, $k$ a positive integer and let $mathbb{F}_q$ be the finite field of $q=p^k$ elements. Let $f(x)$ be a polynomial over $mathbb F_q$ and $ainmathbb F_q$. We denote by $N_{s}(f,a)$ the number of zeros of $f(x_1)+cdots+f(x_s)=a$. In this paper, we show that $$sum_{s=1}^{infty}N_{s}(f,0)x^s=frac{x}{1-qx} -frac{x { M_f^{prime}}(x)}{qM_f(x)},$$ where $$M_f(x):=prod_{minmathbb F_q^{ast}atop{S_{f, m} e 0}}Big(x-frac{1}{S_{f,m}}Big)$$ with $S_{f, m}:=sum_{xin mathbb F_q}zeta_p^{{rm Tr}(mf(x))}$, $zeta_p$ being the $p$-th primitive unit root and ${rm Tr}$ being the trace map from $mathbb F_q$ to $mathbb F_p$. This extends Richmans theorem which treats the case of $f(x)$ being a monomial. Moreover, we show that the generating series $sum_{s=1}^{infty}N_{s}(f,a)x^s$ is a rational function in $x$ and also present its explicit expression in terms of the first $2d+1$ initial values $N_{1}(f,a), ..., N_{2d+1}(f,a)$, where $d$ is a positive integer no more than $q-1$. From this result, the theorems of Chowla-Cowles-Cowles and of Myerson can be derived.
We show that for some $kle 3570$ and all $k$ with $442720643463713815200|k$, the equation $phi(n)=phi(n+k)$ has infinitely many solutions $n$, where $phi$ is Eulers totient function. We also show that for a positive proportion of all $k$, the equation $sigma(n)=sigma(n+k)$ has infinitely many solutions $n$. The proofs rely on recent progress on the prime $k$-tuples conjecture by Zhang, Maynard, Tao and PolyMath.
The two-dimensional minimal supersymmetric sigma models with homogeneous target spaces $G/H$ and chiral fermions of the same chirality are revisited. We demonstrate that the Moore-Nelson consistency condition revealing a global anomaly in CP(N-1) (with N>2 and ${mathcal N}=(0,2)$ supersymmetry) due to a nontrivial first Pontryagin class is in one-to-one correspondence with the local anomalies of isometries in these models. These latter anomalies are generated by fermion loop diagrams which we explicitly calculate. In the case of O}(N) sigma models the first Pontryagin class vanishes, so there is no global obstruction for the minimal ${mathcal N}=(0,1)$ supersymmetrization of these models. We show that at the local level isometries in these models are anomaly free. Thus, there are no obstructions to quantizing the minimal ${mathcal N}=(0,1)$ models with the $S^{N-1}= SO(N)/SO(N-1)$ target space. This also includes CP(1) (equivalent to $S^{2}$) which is an exceptional case from the CP(N-1) series. We also discuss a relation between the geometric and gauged formulations of the CP}(N-1) models.
Suppose that $n$ is a positive integer. In this paper, we show that the exponential Diophantine equation $$(n-1)^{x}+(n+2)^{y}=n^{z}, ngeq 2, xyz eq 0$$ has only the positive integer solutions $(n,x,y,z)=(3,2,1,2), (3,1,2,3)$. The main tools on the proofs are Bakers theory and Bilu-Hanrot-Voutiers result on primitive divisors of Lucas numbers.
The influence of an asymmetric in-plane magnetic anisotropy on the thermally activated spin current is studied theoretically for two different systems; (i) the system consisting of a ferromagnetic insulator in a direct contact with a nonmagnetic metal, and the sandwich structure consisting of a ferromagnetic insulating part sandwiched between two nonmagnetic metals. It is shown that when the difference between the temperatures of the two nonmagnetic metals in a structure is not large, the spin pumping currents from the magnetic part to the nonmagnetic ones are equal in amplitude and have opposite directions, so only the spin torque current contributes to the total spin current. The spin current flows then from the nonmagnetic metal with the higher temperature to the nonmagnetic metal having a lower temperature. Its amplitude varies linearly with the difference in temperatures. In addition, we have found that if the magnetic anisotropy is in the layer plane, then the spin current increases with the magnon temperature, while in the case of an out-of-plane magnetic anisotropy the spin current decreases when the magnon temperature enhances. Enlarging the difference between the temperatures of the nonmagnetic metals, the linear response becomes important, as confirmed by analytical expressions inferred from the Fokker-Planck approach and by the results obtained upon a full numerical integration of the stochastic Landau-Lifshitz-Gilbert equation.
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