Tensor rank is not multiplicative under the tensor product

The tensor rank of a tensor t is the smallest number r such that t can be decomposed as a sum of r simple tensors. Let s be a k-tensor and let t be an l-tensor. The tensor product of s and t is a (k + l)-tensor. Tensor rank is sub-multiplicative under the tensor product. We revisit the connection between restrictions and degenerations. A result of our study is that tensor rank is not in general multiplicative under the tensor product. This answers a question of Draisma and Saptharishi. Specifically, if a tensor t has border rank strictly smaller than its rank, then the tensor rank of t is not multiplicative under taking a sufficiently hight tensor product power. The "tensor Kronecker product" from algebraic complexity theory is related to our tensor product but different, namely it multiplies two k-tensors to get a k-tensor. Nonmultiplicativity of the tensor Kronecker product has been known since the work of Strassen. It remains an open question whether border rank and asymptotic rank are multiplicative under the tensor product. Interestingly, lower bounds on border rank obtained from generalised flattenings (including Young flattenings) multiply under the tensor product

A note on the gap between rank and border rank

We study the tensor rank of the tensor corresponding to the algebra of n-variate complex polynomials modulo the dth power of each variable. As a result we find a sequence of tensors with a large gap between rank and border rank, and thus a counterexample to a conjecture of Rhodes. At the same time we obtain a new lower bound on the tensor rank of tensor powers of the generalised W-state tensor. In addition, we exactly determine the tensor rank of the tensor cube of the three-party W-state tensor, thus answering a question of Chen et al.Comment: To appear in Linear Algebra and its Application

The Asymptotic Rank Conjecture and the Set Cover Conjecture are not Both True

Strassen's asymptotic rank conjecture [Progr. Math. 120 (1994)] claims a strong submultiplicative upper bound on the rank of a three-tensor obtained as an iterated Kronecker product of a constant-size base tensor. The conjecture, if true, most notably would put square matrix multiplication in quadratic time. We note here that some more-or-less unexpected algorithmic results in the area of exponential-time algorithms would also follow. Specifically, we study the so-called set cover conjecture, which states that for any $\epsilon>0$ there exists a positive integer constant $k$ such that no algorithm solves the $k$-Set Cover problem in worst-case time $\mathcal{O}((2-\epsilon)^n|\mathcal F|\operatorname{poly}(n))$. The $k$-Set Cover problem asks, given as input an $n$-element universe $U$, a family $\mathcal F$ of size-at-most-$k$ subsets of $U$, and a positive integer $t$, whether there is a subfamily of at most $t$ sets in $\mathcal F$ whose union is $U$. The conjecture was formulated by Cygan et al. in the monograph Parameterized Algorithms [Springer, 2015] but was implicit as a hypothesis already in Cygan et al. [CCC 2012, ACM Trans. Algorithms 2016], there conjectured to follow from the Strong Exponential Time Hypothesis. We prove that if the asymptotic rank conjecture is true, then the set cover conjecture is false. Using a reduction by Krauthgamer and Trabelsi [STACS 2019], in this scenario we would also get a $\mathcal{O}((2-\delta)^n)$-time randomized algorithm for some constant $\delta>0$ for another well-studied problem for which no such algorithm is known, namely that of deciding whether a given $n$-vertex directed graph has a Hamiltonian cycle

Self-organisation processes in (magneto)hydrodynamic turbulence

Self-organising processes occurring in isotropic turbulence and homogeneous magnetohydrodynamic (MHD) turbulence are investigated in relation to the stability of helical flow structures. A stability analysis of helical triad interactions shows that compared to hydrodynamics, equilibria of the triadic evolution equations have more instabilities with respect to perturbations on scales larger than the characteristic scale of the system. Some of these instabilities can be mapped to Stretch-Twist-Fold dynamo action and others to the inverse cascade of magnetic helicity. High levels of cross-helicity are found to constrain small-scale instabilities more than large scale instabilities and are thus expected to have an asymmetric damping effect on forward and inverse energy transfer. Results from a numerical investigation into the influence of helicity on energy transfer and dissipation are consistent with this observation. The numerical work also confirms the predictions of an approximate method describing the Reynolds number dependence of the dimensionless dissipation coefficient for MHD turbulence. These predictions are complemented by the derivation of mathematically rigorous upper bounds on the dissipation rates of total energy and cross-helicity in terms of applied external forces. Large-scale helical flows are also found to emerge in relaminarisation events in direct numerical simulations of isotropic hydrodynamic turbulence at low Reynolds number, where the turbulent fluctuations suddenly collapse in favour of a large-scale helical flow, which was identified as a phase-shifted ABC-flow. A statistical investigation shows similarities to relaminarisation of localised turbulence in wall-bounded parallel shear flows. The turbulent states have an exponential survival probability indicating a memoryless process with a characteristic lifetime, which is found to depend super-exponentially on Reynolds number akin to well-established results for pipe and plane Couette flow. These and further similarites suggest that the phase space dynamics of isotropic turbulence and wall-bounded shear flows are qualitatively similar and that the relaminarisation of isotropic turbulence can also be explained by the escape from a chaotic saddle

Study of scattering from aperiodic set-ups with the use of local parity

We study the breaking of global discrete symmetries -specifically inversion and translation- in one-dimensional scattering set-ups. We focus on the case where the broken global symmetry is retained locally, in arbitrary domains of finite spatial extent and we find a class of space invariant, non-local currents, which are remnants of the broken global symmetry. The proposed method addresses successfully any combination of translation and inversion symmetry and can be applied to the study of wave propagation in aperiodic and quasi-periodic media

Study of scattering from aperiodic set-ups with the use of local parity

Μελετάται η διαδικασία θραύσης διακριτών συμμετριών -συγκεκριμένα των συμμετριών μετάθεσης και αντιστροφής- κατά τη διαδικασία σκέδασης από μονοδιάστατα συστήματα. Το ενδιαφέρον επικεντρώνεται στην περίπτωση όπου η σπασμένη καθολική συμμετρία εξακολουθεί να ισχύει τοπικά, σε χωρία πεπερασμένης έκτασης, εντός των οποίων διαπιστώνεται η ύπαρξη μιας κλάσης μη-τοπικών, χωρικά αναλλοίωτων ρευμάτων. Τα ρεύματα αυτά επιτρέπουν την απεικόνιση της κυματοσυνάρτησης από μία αυθαίρετη υποπεριοχή-πηγή σε μία άλλη υποπεριοχή- στόχο, όταν αυτές συνδέονται με μία σχέση συμμετρίας μετάθεσης ή αντιστροφής. Η προκύπτουσα σχέση απεικόνισης της κυματοσυνάρτησης από τη μία υποπεριοχή στην άλλη αποτελεί μία γενίκευση των θεωρημάτων Bloch και ομοτιμίας, για πεπερασμένα, εν γένει απεριοδικά και κατοπτρικώς μη-συμμετρικά συστήματα τα οποία διατηρούν τις εν λόγω συμμετρίες σε τοπικό επίπεδο. Τα παραπάνω τοπικά διατηρούμενα ρεύματα μπορούν να θεωρηθούν ως κατάλοιπα της αντίστοιχης καθολικής συμμετρίας, προσφέροντας ένα συστηματικό τρόπο μελέτης της διαδικασίας θραύσης διακριτών, καθολικών συμμετριών. Η προτεινόμενη μεθοδολογία μπορεί να εφαρμοστεί σε οποιοδήποτε συνδυασμό συμμετριών μετάθεσης και αντιστροφής και ενδείκνυται για τη μελέτη απεριοδικών και οιονεί-περιοδικών διατάξεων. Επιπλέον, το γεγονός ότι στηρίζεται σε πολύ γενικές αρχές συμμετρίας σε συνδυασμό με τον ισομορφισμό των εξισώσεων Helmholtz και Schrodinger, οδηγεί σε ένα γενικό πλαίσιο για την αντιμετώπιση περιπτώσεων κβαντικής και κλασικής σκέδασης. Στην συνέχεια, επικεντρωνόμαστε στη μελέτη της συμμετρίας αντιστροφής όταν αυτή ισχύει σε τοπικά, σε πεπερασμένα χωρία του συνολικού συστήματος. Εισάγοντας την έννοια της τοπικής ομοτιμίας και του αντίστοιχου τελεστή, θεωρούμε απεριοδικές και οιονεί-περιοδικές μονοδιάστατες διατάξεις οι οποίες μπορούν να αναλυθούν πλήρως σε υποπεριοχές που ισχύει η τοπική ομοτιμία. Αποδεικνύεται ότι η επίδραση της συγκεκριμένης τοπικής συμμετρίας στις διαδικασίες σκέδασης είναι ιδιαίτερα σημαντική, κυρίως όσον αφορά τους συντονισμούς πλήρους διέλευσης (ΣΠΔ). Με έμφαση στην εμφάνιση ΣΠΔ και χρησιμοποιώντας τα προαναφερθέντα μη-τοπικά ρεύματα, προτείνεται μία κατηγοριοποίηση των ΣΠΔ, βασισμένη στη γεωμετρική ερμηνεία τους στο μιγαδικό επίπεδο. Η προτεινόμενη κατηγοριοποίηση αίρει τις επικαλύψεις που προέκυπταν από αντίστοιχες προσεγγίσεις στην υπάρχουσα βιβλιογραφία, προσφέροντας έναν ξεκάθαρο διαχωρισμό ανάμεσα στους ΣΠΔ, βασισμένο σε βασικές αρχές (τοπικών) συμμετριών. Τέλος, αναπτύσσεται μία “κατασκευαστική αρχή” η οποία αξιοποιεί τις πολλαπλές κλίμακες συμμετρίας που μπορεί να υπάρχουν στην ίδια διάταξη, για το σχεδιασμό διατάξεων κυματικής σκέδασης με συγκεκριμένες ιδιότητες ΣΠΔ. Η εφαρμογή της κατασκευαστικής αρχής σε κβαντικά, φωτονικά και ακουστικά συστήματα υποδεικνύει την ευρεία εφαρμοσιμότητά της.We study the breaking of global discrete symmetries -specifically inversion and translation- in one-dimensional scattering set-ups. We focus on the case where the broken global symmetry is retained locally, in arbitrary domains of finite spatial extent and we find a class of space invariant, non-local currents, which are remnants of the broken global symmetry. These currents provide a mapping of the wave function from an arbitrary spatial domain, considered as source, to a target domain. The two domains are linked through the corresponding symmetry transform. The derived mapping constitutes a generalization of the Bloch and parity theorems for arbitrary finite systems which can be aperiodic or non-inversion symmetric. The obtained invariant currents are identified as remnants of the respective global symmetry, providing a systematic approach to the breaking of discrete global symmetries. The proposed method addresses successfully any combination of translation and inversion symmetry and can be applied to the study of wave propagation in aperiodic and quasi-periodic media. The fact that it lies on very general symmetry arguments, combined with the Helmholtz-Schrodinger isomorphism, provide a unified framework which can be applied to the quantum optical and acoustic systems, notwithstanding their essential differences. Focusing on the parity transformation when applied to finite mirror symmetric domains, we introduce the concept of local parity (LP) and the corresponding operator. We consider non-symmetric, aperiodic scattering set-ups which can be completely decomposed in space domains where the LP symmetry is satisfied and reveal its impact on their transport properties and particularly on perfect transmission resonances (PTRs). With emphasis on the PTR manifestation and by employing the aforementioned invariant currents, we propose a classification of PTRs, based on their geometric representation on the complex plane. This classification lifts certain overlaps of alternative approaches in the existing literature and provides an unambiguous distinction between resonances based on fundamental, local symmetry principles. Finally we develop a construction principle which utilizes the simultaneous existence of different LP symmetry scales in the same set-up for the design of aperiodic wave mechanical devices with prescribed PTR properties. The implementation of this construction principle on quantum, photonic and acoustic systems, reveals its applicability in diverse systems

Model Order Reduction

An increasing complexity of models used to predict real-world systems leads to the need for algorithms to replace complex models with far simpler ones, while preserving the accuracy of the predictions. This three-volume handbook covers methods as well as applications. This third volume focuses on applications in engineering, biomedical engineering, computational physics and computer science