A variation of the prime k-tuples conjecture with applications to quantum limits

Let $$\mathcal {H}^{*}=\{h_1,h_2,\ldots \}$$ H ∗ = { h 1 , h 2 , … } be an ordered set of integers. We give sufficient conditions for the existence of increasing sequences of natural numbers $$a_j$$ a j and $$n_k$$ n k such that $$n_k+h_{a_j}$$ n k + h a j is a sum of two squares for every $$k\ge 1$$ k ≥ 1 and $$1\le j\le k.$$ 1 ≤ j ≤ k . Our method uses a novel modification of the Maynard–Tao sieve together with a second moment estimate. As a special case of our result, we deduce a conjecture due to D. Jakobson which has several implications for quantum limits on flat tori.

Applications of analytic newvectors for $$\mathrm {GL}(n)$$

We provide a few natural applications of the analytic newvectors, initiated in Jana and Nelson (Analytic newvectors for $$\text {GL}_n(\mathbb {R})$$ GL n ( R ) , arXiv:1911.01880 [math.NT], 2019), to some analytic questions in automorphic forms for $$\mathrm {PGL}_n(\mathbb {Z})$$ PGL n ( Z ) with $$n\ge 2$$ n ≥ 2 , in the archimedean analytic conductor aspect. We prove an orthogonality result of the Fourier coefficients, a density estimate of the non-tempered forms, an equidistribution result of the Satake parameters with respect to the Sato–Tate measure, and a second moment estimate of the central L-values as strong as Lindelöf on average. We also prove the random matrix prediction about the distribution of the low-lying zeros of automorphic L-function in the analytic conductor aspect. The new ideas of the proofs include the use of analytic newvectors to construct an approximate projector on the automorphic spectrum with bounded conductors and a soft local (both at finite and infinite places) analysis of the geometric side of the Kuznetsov trace formula.

A Dynamic Adam Based Deep Neural Network for Fault Diagnosis of Oil-Immersed Power Transformers

This paper presents a Dynamic Adam and dropout based deep neural network (DADDNN) for fault diagnosis of oil-immersed power transformers. To solve the problem of incomplete extraction of hidden information with data driven, the gradient first-order moment estimate and second-order moment estimate are used to calculate the different learning rates for all parameters with stable gradient scaling. Meanwhile, the learning rate is dynamically attenuated according to the optimal interval. To prevent over-fitted, we exploit dropout technique to randomly reset some neurons and strengthen the information exchange between indirectly-linked neurons. Our proposed approach was utilized on four datasets to learn the faults diagnosis of oil-immersed power transformers. Besides, four benchmark cases in other fields were also utilized to illustrate its scalability. The simulation results show that the average diagnosis accuracies on the four datasets of our proposed method were 37.9%, 25.5%, 14.6%, 18.9%, and 11.2%, higher than international electro technical commission (IEC), Duval Triangle, stacked autoencoders (SAE), deep belief networks (DBN), and grid search support vector machines (GSSVM), respectively.

Considerations for Oversampling in Azimuth on the Phased Array Weather Radar

When spectral moments in the azimuth are spaced by less than a beamwidth, it is called oversampling. Superresolution is a type of oversampling that refers to sampling at half a beamwidth on the national network of Doppler weather radars [Weather Surveillance Radar-1988 Doppler (WSR-88D)]. Such close spacing is desirable because it extends the range at which small severe weather features, such as tornadoes or microbursts, can be resolved. This study examines oversampling for phased array radars. The goal of the study is to preserve the same effective beamwidth as on the WSR-88D while obtaining smaller spectral moment estimate errors at the same or faster volume update times. To that effect, a weighted average of autocorrelations of radar signals from three consecutive radials is proposed. Errors in three spectral moments obtained from these autocorrelations are evaluated theoretically. Methodologies on how to choose weights that preserve the desirable effective beamwidth are presented. The results are demonstrated on the fields of spectral moments obtained with the National Weather Radar Testbed (NWRT), a phased array weather radar at NOAA’s National Severe Storms Laboratory (NSSL).