Search

We investigate the notion of quasi-invariant states introduced in [2, 3] from an analytic viewpoint.We give the structures of quasi-invariant states with uniformly bounded cocycles. As a consequence, we can apply a Theorem of Kovacs and Szucs to get a conditional expectation on fixed points and another of Stormer to get an invariant semifinite trace under extra assumptions.

We consider a class of external potentials on the complex plane $\mathbb{C}$ for which the coincidence set to the obstacle problem contains a Jordan curve in the exterior of the droplet. We refer to this curve as a spectral outpost. We study the corresponding Coulomb gas at $\beta=2$. Generalizing recent work in the radially symmetric case, we prove that the number of particles which fall near the spectral outpost has an asymptotic Heine distribution, as the number of particles $n\to\infty$.We also consider a class of potentials with disconnected droplets whose connected components are separated by a ring-shaped spectral gap. We prove that the fluctuations of the number of particles that fall near a given component has an asymptotic discrete normal distribution, which depends on $n$. For the case of disconnected droplets we also consider fluctuations of general smooth linear statistics and show that they tend to distribute as the sum of a Gaussian field and an independent, oscillatory, discrete Gaussian field. Our techniques involve a new asymptotic formula on the norm of monic orthogonal polynomials in the bifurcation regime and a variant of the method of limit Ward identities of Ameur, Hedenmalm, and Makarov., Comment: 37 pages

We investigate a cops and robber game on directed graphs, where the robber moves along the arcs of the graph, while the cops can select any position at each time step. Our main focus is on the cop number: the minimum number of cops required to guarantee the capture of the robber. We prove that deciding whether the cop number of a digraph is equal to 1 is NP-hard, whereas this is decidable in polynomial time for tournaments. Furthermore, we show that computing the cop number for general digraphs is fixed parameter tractable when parameterized by a generalization of vertex cover. However, for tournaments, tractability is achieved with respect to the minimum size of a feedback vertex set. Among our findings, we prove that the cop number of a digraph is equal to that of its reverse digraph, and we draw connections to the matrix mortality problem.

We study model near-critical polymer gelling systems made of gluten proteins dispersions stabilized at different distances from the gel point. We impose different shear rates and follow the time evolution of the stress. For sufficiently large shear rates, an intermediate stress overshoot is measured before reaching the steady state. We evidence self-similarity of the stress overshoot as a function of the applied shear rate for samples with various distances from the gel point, which is related to the elastic energy stored by the samples, as for dense systems close to the jamming transition. In concordance with the findings for glassy and jammed systems, we also measure that the stress after flow cessation decreases as a power law with time with a characteristic relaxation time that depends on the shear rate previously imposed. These features revealed in non-linear rheology could be the signature of a mesoscopic dynamics, which would depend on the extent of gelation., Comment: 10.1021/acsmacrolett.4c00121

The consumption of a video requires a considerable amount of energy during the various stages of its life-cycle. With a billion hours of video consumed daily, this contributes significantly to the greenhouse gas emission. Therefore, reducing the end-to-end carbon footprint of the video chain, while preserving the quality of experience at the user side, is of high importance. To contribute in an impactful manner, we propose 3R-INN, a single light invertible network that does three tasks at once: given a high-resolution grainy image, it Rescales it to a lower resolution, Removes film grain and Reduces its power consumption when displayed. Providing such a minimum viable quality content contributes to reducing the energy consumption during encoding, transmission, decoding and display. 3R-INN also offers the possibility to restore either the high-resolution grainy original image or a grain-free version, thanks to its invertibility and the disentanglement of the high frequency, and without transmitting auxiliary data. Experiments show that, while enabling significant energy savings for encoding (78%), decoding (77%) and rendering (5% to 20%), 3R-INN outperforms state-of-the-art film grain synthesis and energy-aware methods and achieves state-of-the-art performance on the rescaling task on different test-sets.

In this paper, we study products of asymmetric Toeplitz matrices, we give necessary and sufficient conditions for the product of two asymmetric Toeplitz matrices compatible sizes is asymmetric Toeplitz matrix. We also give some results related to the isometry.

Bladder cancer ranks within the top 10 most diagnosed cancers worldwide and is among the most expensive cancers to treat due to the high recurrence rates which require lifetime follow-ups. The primary tool for diagnosis is cystoscopy, which heavily relies on doctors' expertise and interpretation. Therefore, annually, numerous cases are either undiagnosed or misdiagnosed and treated as urinary infections. To address this, we suggest a deep learning approach for bladder cancer detection and segmentation which combines CNNs with a lightweight positional-encoding-free transformer and dual attention gates that fuse self and spatial attention for feature enhancement. The architecture suggested in this paper is efficient making it suitable for medical scenarios that require real time inference. Experiments have proven that this model addresses the critical need for a balance between computational efficiency and diagnostic accuracy in cystoscopic imaging as despite its small size it rivals large models in performance., Comment: 7 pages, 5 figures

We discuss the martingales in relevance with $G$-strongly quasi-invariant states on a $C^*$-algebra $\mathcal A$, where $G$ is a separable locally compact group of $*$-automorphisms of $\mathcal A$. In the von Neumann algebra $\mathfrak A$ of the GNS representation, we define a unitary representation of the group and define a group $\hat G$ of $*$-automorphisms of $\mathfrak A$, which is homomorphic to $G$. For the case of compact $G$, under some mild condition, we find a $\hat G$-invariant state on $\mathfrak A$ and define a conditional expectation with range the $\hat G$-fixed subalgebra. Moving to the separable locally compact group $G=\cup_NG_N$, which is the union of increasing compact groups, we construct a sequence of conditional expectations and thereby construct (decreasing) martingales, which have limits by the martingale convergence theorem. We provide with an example for the group of finite permutations on the set of nonnegative integers acting on a $C^*$-algebra of infinite tensor product., Comment: 10 pages, 1 figure

In this note we prove equicontinuity for the family of one-point densities with respect to a two-dimensional Coulomb gas at an inverse temperature $\beta\ge 1/2$ confined by an external potential of Hele-Shaw (or quasi-harmonic) type. As a consequence, subsequential limiting Lipschitz continuous densities are defined on the microscopic scale. There are several additional results, for example comparing the one-point density with the thermal equilibrium density., Comment: This version contains some very minor corrections and cosmetic changes

The classification of maximal left algebras of quaternion Toeplitz matrices is a harder problem that has received little attention up to now. In this paper, we introduce certain families of maximal left algebras of Toeplitz matrices with entries from an algebra of quaternions that cover various classes of the left algebras of quaternion Toeplitz matrices.

The verification of liveness conditions is an important aspect of state-based rigorous methods. This article investigates this problem in a fragment $\square$LTL of the logic LTL(EB), the integration of the UNTIL-fragment of Pnueli's linear time temporal logic (LTL) and the logic of Event-B, in which the most commonly used liveness conditions can be expressed. For this fragment a sound set of derivation rules is developed, which is also complete under mild restrictions for Event-B machines., Comment: 22 pages

We study large $n$ expansions for the partition function of a Coulomb gas $$Z_n=\frac 1 {\pi^n}\int_{\mathbb{C}^n}\prod_{1\le i

For a $*$-automorphism group $G$ on a $C^*$- or von Neumann algebra, we study the $G$-quasi invariant states and their properties. The $G$-quasi invariance or $G$-strongly quasi invariance are weaker than the $G$-invariance and have wide applications. We develop several properties for $G$-strongly quasi invariant states. Many of them are the extensions of the already developed theories for $G$-invariant states. Among others, we consider the relationship between the group $G$ and modular automorphism group, invariant subalgebras, ergodicity, modular theory, and abelian subalgebras. We provide with some examples to support the results., Comment: 25 pages