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In this work, we construct charged AdS black holes with a global monopole charge in diverse dimensions and study the thermodynamics. We find a critical monopole charge below which the solution exhibits Van-der Waals like behaviors. In the context of holography, this could be intepreted using the boundary degrees of freedoms. As an example, we study the phase diagram in the four dimensions analytically. We further analyze the microstructures of the solution using Ruppeiner geometry. We find that repulsive interactions dominates for black holes in a wide range of temperatures. However, around the critical point, attractive interactions is dominant and the Ruppeiner scalar curvature shows universal behaviors: it has a critical exponent $2$ and coefficient $-1/8$ in diverse dimensions. Universality of the results is interpreted from the scaling behavior of free energy near the critical point for Van-der Waals like fluids., Comment: minor revisions;21pages, 9figures

Employing extended phase space formalism, we study critical phenomenon of A-charge and C-charge for holographic theories dual to Gauss-Bonnet black holes. We find a universal critical Gauss-Bonnet coupling, giving rise to a universal ratio between the two central charges at the critical point. This leads to a new intepretation for critical behavior of Gauss-Bonnet black holes in terms of the boundary degrees of freedoms, although the solutions are electrically neutral. Another novel feature is for either of the central charges, the transition temperature is beyond the critical point but is upper bounded by causality of the boundary theories., Comment: 12pages, 3figures; minor revisions, reference added

Bessenrodt and Ono, Chen, Wang and Jia, DeSalvo and Pak were the first to discover the log-subadditivity, log-concavity, and the third-order Tur\'{a}n inequality of partition function, respectively. Many other important partition statistics are proved to enjoy similar properties. This paper focuses on the partition function $\overline{p}_k(n)$, which counts the number of overpartitions of $n$ with no parts divisible by $k$. We provide a combinatorial proof to establish that for any $k\geq2$, the partition function $\overline{p}_k(n)$ exhibits strict log-subadditivity. Specifically, we show that $\overline{p}_k(a)\overline{p}_k(b)>\overline{p}_k(a+b)$ for integers $a\geq b\geq1$ and $a+b\geq k$. Furthermore, we investigate the log-concavity and the satisfaction of the third-order Tur\'{a}n inequality for $\overline{p}_k(n)$, where $2\leq k\leq9$., Comment: 28 pages

In this paper, we studied a set of generalised Krylov complexity for operator growth. We demonstrate their universal features at both initial times and long times using half-analytical technique as well as numerical results. In particular, by using the logarithmic relation to the Krylov entropy, we establish an inequality (\ref{master}) between the variance of the K-complexity and the generalised notions which holds in the long time limit. Extending the result to finite (but long) times, we show that for fast scramblers, the K-complexity constrains the growth of generalised complexity more stringently than the dispersion bound. However, for slow scramblers, the growth rate of K-complexity is tighter bounded by the generalised complexity in the other way around. Our results enlarge the zoo of Krylov quantities and may shed new light on the future research in this field., Comment: 16pages,5 figures; minor corrections

In this paper, we obtain asymptotic formulas for $k$-crank of $k$-colored partitions. Let $M_k(a, c; n)$ denote the number of $k$-colored partitions of $n$ with a $k$-crank congruent to $a$ mod $c$. For the cases $k=2,3,4$, Fu and Tang derived several inequality relations for $M_k(a, c; n)$ using generating functions. We employ the Hardy-Ramanujan Circle Method to extend the results of Fu and Tang. Furthermore, additional inequality relations for $M_k(a, c; n)$ have been established, such as logarithmic concavity and logarithmic subadditivity., Comment: 40 pages. arXiv admin note: text overlap with arXiv:1311.4344 by other authors

Recently the deep learning has shown its advantage in representation learning and clustering for time series data. Despite the considerable progress, the existing deep time series clustering approaches mostly seek to train the deep neural network by some instance reconstruction based or cluster distribution based objective, which, however, lack the ability to exploit the sample-wise (or augmentation-wise) contrastive information or even the higher-level (e.g., cluster-level) contrastiveness for learning discriminative and clustering-friendly representations. In light of this, this paper presents a deep temporal contrastive clustering (DTCC) approach, which for the first time, to our knowledge, incorporates the contrastive learning paradigm into the deep time series clustering research. Specifically, with two parallel views generated from the original time series and their augmentations, we utilize two identical auto-encoders to learn the corresponding representations, and in the meantime perform the cluster distribution learning by incorporating a k-means objective. Further, two levels of contrastive learning are simultaneously enforced to capture the instance-level and cluster-level contrastive information, respectively. With the reconstruction loss of the auto-encoder, the cluster distribution loss, and the two levels of contrastive losses jointly optimized, the network architecture is trained in a self-supervised manner and the clustering result can thereby be obtained. Experiments on a variety of time series datasets demonstrate the superiority of our DTCC approach over the state-of-the-art.

In this work, we develop a universal picture from topology in the thermodynamic parameters space to describe first order phase transitions of black holes. By employing an off-shell internal energy, we find two types of topological defects. The first is normal, describing black holes which move with a nonzero velocity and acceleration, in a central force field. The second type of defects are exotic: they are static in the space, not describing black holes but encoding information about first order transitions. For each defect, we assign a winding number and an inertial mass. By studying neutral and charged black holes in asymptotically anti-de Sitter space, we show that first order transitions can be viewed as once or twice interchange of winding numbers between black holes and the exotic defects, through wired action at a distance. This corresponds to the usual notion: a smaller black hole grows into a larger black hole or vice versa. However, our topological analysis illustrates that the transition can also be locally interpreted as virtual collisions between black holes and the exotic defects. In this interpretation, a smaller black hole first grows into a new exotic defect whereas an original exotic defect grows into a larger black hole. All the defects simply change their positions and momentums, rather than interchanging the winding numbers. Critical point of the transition can be extracted when all the defects meet in the parameters space. Certain quantities, such as the Jacobians and the velocities of normal defects show universal behaviors near the critical point., Comment: 5pages, 4figures

Bessenrodt and Ono initially found the strict log-subadditivity of partition function $p(n)$, that is, $p(a+b)< p(a)p(b)$ for $a,b>1$ and $a+b>9$. Many other important statistics of partitions are proved to enjoy similar properties. Lovejoy introduced the overpartition rank as an analog of Dyson's rank for partitions from the $q$-series perspective. Let $\overline{N}(a,c,n)$ denote the number of overpartitions with rank congruent to $a$ modulo $c$. Ciolan computed the asymptotic formula of $\overline{N}(a,c,n)$ and showed that $\overline{N}(a, c, n) > \overline{N}(b, c, n)$ for $c\geq7$ and $n$ large enough. In this paper, we derive an upper bound and a lower bound of $\overline{N}(a,c,n)$ for each $c\geq3$ by using the asymptotics of Ciolan. Consequently, we establish the strict log-subadditivity of $\overline{N}(a,c,n)$ analogous to the partition function $p(n)$., Comment: 19 pages

Let $\overline{N}_2(a,c,n)$ be the number of overpartitions of $n$ whose the $M_2$-rank is congruent to $a$ modulo $c$. In this paper, we obtain the asymptotic formula of $\overline{N}_2(a,c,n)$ utilizing the Ingham Tauberian Theorem. As applications, we derive inequalities concerning with $\overline{N}_2(a,c,n)$ including its strict concavity and log-concavity.

We study upper bounds on the growth of operator entropy $S_K$ in operator growth. Using uncertainty relation, we first prove a dispersion bound on the growth rate $|\partial_t S_K|\leq 2b_1 \Delta S_K$, where $b_1$ is the first Lanczos coefficient and $\Delta S_K$ is the variance of $S_K$. However, for irreversible process, this bound generally turns out to be too loose at long times. We further find a tighter bound in the long time limit using a universal logarithmic relation between Krylov complexity and operator entropy. The new bound describes the long time behavior of operator entropy very well for physically interesting cases, such as chaotic systems and integrable models., Comment: minor corrections,19 pages, 4 figures

Let $\bar{p}(n)$ denote the overpartition function. In this paper, we study the asymptotic higher order $\log$-concavity property of the overpatition function in a similar framework done by Hou and Zhang for the partition function. This will enable us to move on further in order to prove $\log$-concavity of overpartitions, explicitly by studying the asymptotic expansion of the quotient $\bar{p}(n-1)\bar{p}(n+1)/\bar{p}(n)^2$ upto a certain order so that one can finally ends up with the phenomena of $2$-$\log$-concavity and higher order Tur\'{a}n property of $\bar{p}(n)$ by following a sort of unified approach.

We study Krylov complexity $C_K$ and operator entropy $S_K$ in operator growth. We find that for a variety of systems, including chaotic ones and integrable theories, the two quantities always enjoy a logarithmic relation $S_K\sim \log{C_K}$ at long times, where dissipative behavior emerges in unitary evolution. Otherwise, the relation does not hold any longer. Universality of the relation is deeply connected to irreversibility of operator growth., Comment: Ppublication version.Minor revisions, conclusions unchanged. 22pages,4figures

Unsupervised Deep Learning (DL) techniques have been widely used in various security-related anomaly detection applications, owing to the great promise of being able to detect unforeseen threats and superior performance provided by Deep Neural Networks (DNN). However, the lack of interpretability creates key barriers to the adoption of DL models in practice. Unfortunately, existing interpretation approaches are proposed for supervised learning models and/or non-security domains, which are unadaptable for unsupervised DL models and fail to satisfy special requirements in security domains. In this paper, we propose DeepAID, a general framework aiming to (1) interpret DL-based anomaly detection systems in security domains, and (2) improve the practicality of these systems based on the interpretations. We first propose a novel interpretation method for unsupervised DNNs by formulating and solving well-designed optimization problems with special constraints for security domains. Then, we provide several applications based on our Interpreter as well as a model-based extension Distiller to improve security systems by solving domain-specific problems. We apply DeepAID over three types of security-related anomaly detection systems and extensively evaluate our Interpreter with representative prior works. Experimental results show that DeepAID can provide high-quality interpretations for unsupervised DL models while meeting the special requirements of security domains. We also provide several use cases to show that DeepAID can help security operators to understand model decisions, diagnose system mistakes, give feedback to models, and reduce false positives., Comment: Accepted by 2021 ACM SIGSAC Conference on Computer and Communications Security (CCS '21)