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We prove several rigidity results on multiplier and length spectrum. For example, we show that for every non-exceptional rational map $f:\mathbb{P}^1(\mathbb{C})\to\mathbb{P}^1(\mathbb{C})$ of degree $d\geq2$, the $\mathbb{Q}$-vector space generated by the characteristic exponents (that are not $-\infty$) of periodic points of $f$ has infinite dimension. This answers a stronger version of a question of Levy and Tucker. Our result can also be seen as a generalization of recent results of Ji-Xie and of Huguin which proved Milnor's conjecture about rational maps having integer multipliers. We also get a characterization of postcritically finite maps by using its length spectrum. Finally as an application of our result, we get a new proof of the Zariski dense orbit conjecture for endomorphisms on $(\mathbb{P}^1)^N, N\geq 1$., Comment: 27 pages

A sequence $a=(a_n)_{n=1}^\infty$ of non-negative integers is called realizable if there is a self-map $T:X\to X$ on a set $X$ such that $a_n$ is equal to the number of periodic points of $T$ in $X$ of (not necessarily exact) period $n$, for all $n\geq1$. The sequence $a$ is called almost realizable if there exists a positive integer $m$ such that $(ma_n)_{n=1}^\infty$ is realizable. In this article, we show that certain wide classes of integer sequences are realizable, which contain many famous combinatorial sequences, such as the sequences of Ap\'ery numbers of both kinds, central Delannoy numbers, Franel numbers, Domb numbers, Zagier numbers, and central trinomial coefficients. We also show that the sequences of Catalan numbers, Motzkin numbers, and large and small Schr\"oder numbers are not almost realizable., Comment: 25 pages

Deep learning recommendation models (DLRMs) have been widely applied in Internet companies. The embedding tables of DLRMs are too large to fit on GPU memory entirely. We propose a GPU-based software cache approaches to dynamically manage the embedding table in the CPU and GPU memory space by leveraging the id's frequency statistics of the target dataset. Our proposed software cache is efficient in training entire DLRMs on GPU in a synchronized update manner. It is also scaled to multiple GPUs in combination with the widely used hybrid parallel training approaches. Evaluating our prototype system shows that we can keep only 1.5% of the embedding parameters in the GPU to obtain a decent end-to-end training speed.

In 2016 J. Koenigsmann refined a celebrated theorem of J. Robinson by proving that $\mathbb Q\setminus\mathbb Z$ is diophantine over $\mathbb Q$, i.e., there is a polynomial $P(t,x_1,\ldots,x_{n})\in\mathbb Z[t,x_1,\ldots,x_{n}]$ such that for any rational number $t$ we have $$t\not\in\mathbb Z\iff \exists x_1\cdots\exists x_{n}[P(t,x_1,\ldots,x_{n})=0]$$ where variables range over $\mathbb Q$, equivalently $$t\in\mathbb Z\iff \forall x_1\cdots\forall x_{n}[P(t,x_1,\ldots,x_{n})\not=0].$$ In this paper we prove that we may take $n=32$. Combining this with a result of Z.-W. Sun, we show that there is no algorithm to decide for any $f(x_1,\ldots,x_{41})\in\mathbb Z[x_1,\ldots,x_{41}]$ whether $$\forall x_1\cdots\forall x_9\exists y_1\cdots\exists y_{32}[f(x_1,\ldots,x_9,y_1,\ldots,y_{32})=0],$$ where variables range over $\mathbb Q$., Comment: 13 pages

In search, exploration, and reconnaissance tasks performed with autonomous ground vehicles, an image classification capability is needed for specifically identifying targeted objects (relevant classes) and at the same time recognize when a candidate image does not belong to anyone of the relevant classes (irrelevant images). In this paper, we present an open-set low-shot classifier that uses, during its training, a modest number (less than 40) of labeled images for each relevant class, and unlabeled irrelevant images that are randomly selected at each epoch of the training process. The new classifier is capable of identifying images from the relevant classes, determining when a candidate image is irrelevant, and it can further recognize categories of irrelevant images that were not included in the training (unseen). The proposed low-shot classifier can be attached as a top layer to any pre-trained feature extractor when constructing a Convolutional Neural Network.

The clock transitions (CTs) of central spins have long coherence times because their frequency fluctuations vanish in the linear order of external field noise (such as Overhauser fields from nuclear spin baths). Therefore, CTs are useful for quantum technologies. Also, the quadratic dependence of frequencies on noises makes the CT decoherence an interesting physics problem. Thus we are motivated to study the decoherence of CTs. We consider noise from spin baths, which is one of the most relevant mechanisms of qubit decoherence. Various quantum many-body methods have been developed to study the decoherence of a central spin in spin baths. In particular, the cluster-correlation expansion (CCE) systematically accounts for the many-body correlations that cause the central spin decoherence. However, the CCE can not be straightforwardly applied to CTs in spin baths, for the expansion may fail to converge due to the effective long-range interactions resulting from the quadratic term of the noise (e.g., the second-order interaction mediated by hyperfine interactions for a nuclear spin bath). In this work, we develop a modified CCE method to tackle this class of decoherence problems. By diagonalizing the central spin Hamiltonian for each bath eigenstate of the hyperfine interaction, we find that the effects of long-range interactions are absorbed as fluctuations of central spin eigenenergies in the form of single-spin correlations. We apply the method to two specific systems, namely, nitrogen vacancy center electron spins in near zero magnetic field and singlet-triplet transition of two electrons in a double quantum dot. The numerical simulation shows that the modified CCE converges rapidly for the CTs.

Exceptional points (EPs), the degeneracy point of non-Hermitian systems, have recently attracted great attention after its ability to greatly enhance the sensitivity of micro-cavities is demonstrated experimentally. Unlike the usual degeneracies in Hermitian systems, at EPs, both the eigenenergies and eigenvectors coalesce. Although some of the exotic properties and potential applications of EPs are explored, the range of EPs studied is largely limited by the experimental capability. Some of the systems, e.g. with higher-order EPs, are hard to achieve with conventional simulations. Here we propose an extendable method to simulate non-Hermitian systems on the quantum circuits, where a wide range of EPs can be studied. The system is inherently parity-time (PT) broken due to the non-symmetric controlling effects and post-selection. A sample circuit is implemented in a quantum programming framework, and the phase transition at EP is demonstrated. Considering the scalable and flexible nature of quantum circuits, our model is capable of simulating large scale systems with higher-order EPs. We believe this work may lead to broader applications of quantum computers and provide a tool to the studies for non-Hermitian systems and the associated EPs.