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Broken links in websites external resources pose a serious threat to cybersecurity and the credibility of websites. They can be hijacked to eavesdrop user traffic or to inject malicious software. In this paper, we present the first result of an ongoing research. We focus on the prevalence of broken links in external resources on home pages of the most visited websites in the world. The analysis was conducted on the top 88 000 homepages extracted from the Majestic Million rankings. 35,2% of them have at least one broken link. We also identify the common causes of these broken links and highlight improper implementation of testing phases to prevent such errors. We provide a formal model for the distribution of external links. At the next research step, we are exploring the potential impact on privacy of broken links by analyzing inherited traffic of purchasable expired domains.

We develop the fundamentals of a new theory of convex geometry -- which we call "broken line convex geometry". This is a theory of convexity where the ambient space is the rational tropicalization of a cluster variety, as opposed to an ambient vector space. In this theory, line segments are replaced by broken line segments, and we adopt the notion of convexity in [CMN21]. We state and prove broken line convex geometry versions of many standard results from usual convex geometry.

Given a matroid $M=(E,{\cal I})$, and a total ordering over the elements $E$, a broken circuit is a circuit where the smallest element is removed and an NBC independent set is an independent set in ${\cal I}$ with no broken circuit. The set of NBC independent sets of any matroid $M$ define a simplicial complex called the broken circuit complex which has been the subject of intense study in combinatorics. Recently, Adiprasito, Huh and Katz showed that the face of numbers of any broken circuit complex form a log-concave sequence, proving a long-standing conjecture of Rota. We study counting and optimization problems on NBC bases of a generic matroid. We find several fundamental differences with the independent set complex: for example, we show that it is NP-hard to find the max-weight NBC base of a matroid or that the convex hull of NBC bases of a matroid has edges of arbitrary large length. We also give evidence that the natural down-up walk on the space of NBC bases of a matroid may not mix rapidly by showing that for some family of matroids it is NP-hard to count the number of NBC bases after certain conditionings.

Reassembling 3D broken objects is a challenging task. A robust solution that generalizes well must deal with diverse patterns associated with different types of broken objects. We propose a method that tackles the pairwise assembly of 3D point clouds, that is agnostic on the type of object, and that relies solely on their geometrical information, without any prior information on the shape of the reconstructed object. The method receives two point clouds as input and segments them into regions using detected closed boundary contours, known as breaking curves. Possible alignment combinations of the regions of each broken object are evaluated and the best one is selected as the final alignment. Experiments were carried out both on available 3D scanned objects and on a recent benchmark for synthetic broken objects. Results show that our solution performs well in reassembling different kinds of broken objects.

When a (super) conformal field theory is placed on a non-trivial manifold, the (super) conformal symmetry is broken. However, it is still possible to derive broken Ward identities for these broken symmetries, which provide additional constraints on the theory. We derive and apply the broken Ward identities associated with the (super) conformal group on the thermal manifold $\mathcal{M}_β= S_β^1 \times \mathbb{R}^{d-1}$ and $\mathcal{M} = T^2 \times \mathbb{R}^{d-2}$. The novel constraints not only systematically reproduce known results, including an implicit formulation of the generalized Cardy formula, but also elegantly relate the thermal energy spectrum with the conformal spectrum.

Stack Overflow hosts valuable programming-related knowledge with 11,926,354 links that reference to the third-party websites. The links that reference to the resources hosted outside the Stack Overflow websites extend the Stack Overflow knowledge base substantially. However, with the rapid development of programming-related knowledge, many resources hosted on the Internet are not available anymore. Based on our analysis of the Stack Overflow data that was released on Jun. 2, 2019, 14.2% of the links on Stack Overflow are broken links. The broken links on Stack Overflow can obstruct viewers from obtaining desired programming-related knowledge, and potentially damage the reputation of the Stack Overflow as viewers might regard the posts with broken links as obsolete. In this paper, we characterize the broken links on Stack Overflow. 65% of the broken links in our sampled questions are used to show examples, e.g., code examples. 70% of the broken links in our sampled answers are used to provide supporting information, e.g., explaining a certain concept and describing a step to solve a problem. Only 1.67% of the posts with broken links are highlighted as such by viewers in the posts' c

We survey recent developments in the theory and applications of the broken ray transforms. Furthermore, we discuss some open problems.

This PhD thesis studies the broken ray transform, a generalization of the geodesic X-ray transform where geodesics are replaced with broken rays that reflect on a part of the boundary. The fundamental question is whether this transform is injective. We employ four different methods to approach this question, and each of them gives interesting results. Direct calculation can be used in a ball, where the geometry is particularly simple. If the reflecting part of the boundary is (piecewise) flat, a reflection argument can be used to reduce the problem to the usual X-ray transform. In some geometries one can use broken rays near the boundary to determine the values of the unknown function at the reflector, and even construct its Taylor series. One can also use energy estimates -- which in this context are known as Pestov identities -- to show injectivity in the presence of one convex reflecting obstacle. Many of these methods work also on Riemannian manifolds. We also discuss the periodic broken ray transform, where the integrals are taken over periodic broken rays. The broken ray transform and its periodic version have applications in other inverse problems, including Calderón's probl

In condensed matter physics broken symmetries and emergence of quasi-particles are intimately linked to each other. Whenever a symmetry is broken, it leaves its fingerprints, and that may be observed indirectly via its influence on the other quasi-particles. Here, we report the strong signature of broken rotational symmetry induced due to long range-ordering of spins in Mn - sublattice of LiMn1.5Ni0.5O4 below Tc ~ 113 K reflected with the marked changes in the lattice vibrations using Raman scattering. In particular, the majority of the observed first-order phonon modes show a sharp shift in frequency in the vicinity of long range magnetic-ordering temperature. Phonons exist in a crystalline system because of broken translational symmetry, therefore any renormalization in the phonon-spectrum could be a good gauge for broken translational symmetry. Anomalous evolution of the few modes associated with stretching of Mn/NiO6 octahedra in the intermediate temperature range (~ 60-260 K) marked the broken translational symmetry attributed to the charge ordering. Interestingly same modes also show strong coupling with magnetic degrees of freedom, suggesting that charge-ordering and magneti

Inspired by the work of Amdeberhan, Can, and Moll on broken necklaces, we define a broken bracelet as a linear arrangement of marked and unmarked vertices and introduce a generalization called $n$-stars, which is a collection of $n$ broken bracelets whose final (unmarked) vertices are identified. Through these combinatorial objects, we provide a new framework for the study of Kostant's partition function, which counts the number of ways to express a vector as a nonnegative integer linear combination of the positive roots of a Lie algebra. Our main result establishes that (up to reflection) the number of broken bracelets with a fixed number of unmarked vertices with nonconsecutive marked vertices gives an upper bound for the value of Kostant's partition function for multiples of the highest root of a Lie algebra of type $A$. We connect this work to multiplex juggling sequences, as studied by Benedetti, Hanusa, Harris, Morales, and Simpson, by providing a correspondence to an equivalence relation on $n$-stars.

Topological edge states arise in parity-time ($\mathcal{PT}$)-symmetric non-unitary quantum dynamics but have so far only been discussed in the $\mathcal{PT}$-symmetry-unbroken regime. Here we report the experimental detection of robust topological edge states in one-dimensional photonic quantum walks with spontaneously broken $\mathcal{PT}$ symmetry, thus establishing the existence of topological phenomena therein. We theoretically prove and experimentally confirm that the global Berry phase in non-unitary quantum-walk dynamics unambiguously defines topological invariants of the system in both the $\mathcal{PT}$-symmetry-unbroken and broken regimes. As topological edge states exist in both $\mathcal{PT}$ unbroken and broken regimes, we reveal that topological phenomena are not driven by $\mathcal{PT}$ symmetry.

Let $(X,T)$ be a topological dynamical system, $n\geq 2$ and $\mathcal{F}$ be a Furstenberg family of subsets of $\mathbb{Z}_+$. $(X,T)$ is called broken $\mathcal{F}$-$n$-sensitive if there exist $δ>0$ and $F\in\mathcal{F}$ such that for every opene (non-empty open) subset $U$ of $X$ and every $l\in\mathbb{N}$, there exist $x_1^l,x_2^l,\dotsc,x_n^l\in U$ and $m_l\in \mathbb{Z}_+$ satisfying $d(T^k x_i^l, T^k x_j^l)> δ,\ \forall 1\leq i<j\leq n, k\in m_l+ F\cap[1,l]$. We investigate broken $\mathcal{F}$-$n$-sensitivity for the family of all piecewise syndetic subsets ($\mathcal{F}_{ps}$), the family of all positive upper Banach density subsets ($\mathcal{F}_{pubd}$) and the family of all infinite subsets ($\mathcal{F}_{inf}$). We show that a transitive system $(X,T)$ is broken $\mathcal{F}$-$n$-sensitive for $\mathcal{F}=\mathcal{F}_{ps}\ \text{or}\ \mathcal{F}_{pubd}$ if and only if there exists an essential $n$-sensitive tuple which is an $\mathcal{F}$-recurrent point of $(X^n, T^{(n)})$; is broken $\mathcal{F}_{inf}$-$n$-sensitive if and only if there exists an essential $n$-sensitive tuple $(x_1,x_2,\dotsc,x_n)$ such that $\limsup_{k\to\infty}\min_{1\leq i<j\leq n}d

SHORTENED ABSTRACT: A system exhibiting multiple simultaneously broken symmetries offers the opportunity to influence physical phenomena such as tunneling currents by means of external control parameters. In this paper, we consider the broken SU(2) (internal spin) symmetry of ferromagnetic systems coexisting with \textit{i)} the broken U(1) symmetry of superconductors and \textit{ii)} the broken spatial inversion symmetry induced by a Rashba term in a spin-orbit coupling Hamiltonian. In order to study the effect of these broken symmetries, we consider tunneling currents that arise in two different systems; tunneling junctions consisting of non-unitary spin-triplet ferromagnetic superconductors and junctions consisting of ferromagnets with spin-orbit coupling.

The symmetries that govern the laws of nature can be spontaneously broken, enabling the occurrence of ordered states. Crystals arise from the breaking of translation symmetry, magnets from broken spin rotation symmetry and massive particles break a phase rotation symmetry. Time translation symmetry can be spontaneously broken in exactly the same way. The order associated with this form of spontaneous symmetry breaking is characterised by the emergence of quantum state reduction: systems which spontaneously break time translation symmetry act as ideal measurement machines. In this review the breaking of time translation symmetry is first compared to that of other symmetries such as spatial translations and rotations. It is then discussed how broken time translation symmetry gives rise to the process of quantum state reduction and how it generates a pointer basis, Born's rule, etc. After a comparison between this model and alternative approaches to the problem of quantum state reduction, the experimental implications and possible tests of broken time translation symmetry in realistic experimental settings are discussed.

We reduce the broken ray transform on some Riemannian manifolds (with corners) to the geodesic ray transform on another manifold, which is obtained from the original one by reflection. We give examples of this idea and present injectivity results for the broken ray transform using corresponding earlier results for the geodesic ray transform. Examples of manifolds where the broken ray transform is injective include Euclidean cones and parts of the spheres $S^n$. In addition, we introduce the periodic broken ray transform and use the reflection argument to produce examples of manifolds where it is injective. We also give counterexamples to both periodic and nonperiodic cases. The broken ray transform arises in Calderón's problem with partial data, and we give implications of our results for this application.

Isometric fluctuation relations are deduced for the fluctuations of the order parameter in equilibrium systems of condensed-matter physics with broken discrete or continuous symmetries. These relations are similar to their analogues obtained for non-equilibrium systems where the broken symmetry is time reversal. At equilibrium, these relations show that the ratio of the probabilities of opposite fluctuations goes exponentially with the symmetry-breaking external field and the magnitude of the fluctuations. These relations are applied to the Curie-Weiss, Heisenberg, and $XY$~models of magnetism where the continuous rotational symmetry is broken, as well as to the $q$-state Potts model and the $p$-state clock model where discrete symmetries are broken. Broken symmetries are also considered in the anisotropic Curie-Weiss model. For infinite systems, the results are calculated using large-deviation theory. The relations are also applied to mean-field models of nematic liquid crystals where the order parameter is tensorial. Moreover, their extension to quantum systems is also deduced.

We generalize the well-known broken stick problem in several ways, including a discrete "brick" analogue and a sequential "pick-up sticks/bricks" version. The limit behavior of the broken brick problem gives a combinatorial proof of the broken stick problem. The pick-up version gives a variation on those scenarios, and we conclude by showing a greater context---namely, that the broken stick/brick problem and the pick-up sticks/bricks problem are two extremes in a family of interesting, and largely open, questions.

We consider the broken ray transform on Riemann surfaces in the presence of an obstacle, following earlier work of Mukhometov. If the surface has nonpositive curvature and the obstacle is strictly convex, we show that a function is determined by its integrals over broken geodesic rays that reflect on the boundary of the obstacle. Our proof is based on a Pestov identity with boundary terms, and it involves Jacobi fields on broken rays. We also discuss applications of the broken ray transform.

We study the full phase diagram of a non-Hermitian PT-symmetric generalization of the paradigmatic two-dimensional massive Thirring model. Employing the non-perturbative functional renormalization-group, we find that the model hosts a regime where PT symmetry is spontaneously broken. This new phase is characterized by a relevant imaginary mass, corresponding to monstronic excitations displaying exponentially growing amplitudes for time-like intervals and tachyonic (Lieb-Robison-bound breaking, oscillatory) excitations for space-like intervals. Furthermore, since the phase manifests itself as an unconventional attractive spinodal fixed point, which is typically unreachable in finite real-life systems, we find that the effective renormalized mass reached can be tuned through the microscopic parameters of the model. Our results further predict that the new phase is robust to external gauge fields, contrary to the celebrated BKT phase in the PT unbroken sector. The gauge field then provides an effective and easy means to tune the renormalized imaginary mass through a wide range of values, and therefore the amplitude growth/oscillation rate of the corresponding excitations.