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We propose the representation principle to study physical systems with a given symmetry. In the context of symmetry enriched topological orders, we give the appropriate representation category, the category of SET orders, which include SPT orders and symmetry breaking orders as special cases. For fusion n-category symmetries, we show that the category of SET orders encodes almost all information about the interplay between symmetry and topological orders, in a natural and canonical way. These information include defects and boundaries of SET orders, symmetry charges, explicit and spontaneous symmetry breaking, stacking of SET orders, gauging of generalized symmetry, as well as quantum currents (SymTFT or symmetry TO). We also provide a detailed categorical algorithm to compute the generalized gauging. In particular, we proved that gauging is always reversible, as a special type of Morita-equivalence. The explicit data for ungauging, the inverse to gauging, is given.

There is a well-known factorization of the number $2^{2m}+1$, with $m$ odd, related to the orders of tori of simple Suzuki groups: $2^{2m}+1$ is a product of $a=2^m+2^{(m+1)/2}+1$ and $b=2^m-2^{(m+1)/2}+1$. By the Bang-Zsigmondy theorem, there is a primitive prime divisor of $2^{4m}-1$, that is, a prime $r$ that divides $2^{4m}-1$ and does not divide $2^i-1$ for any $i<4m$. It is easy to see that $r$ divides $2^{2m}+1$, and so it divides one of the numbers $a$ and $b$. The main objective of this paper is to show that for every $m>5$, each of $a$ and $b$ is divisible by some primitive prime divisor of $2^{4m}-1$. Also we prove similar results for primitive prime divisors related to the simple Ree groups. As an application, we find the independence and 2-independence numbers of the prime graphs of almost simple Suzuki-Ree groups.

The two main approaches to the study of irreducible representations of orders (via traces and Poisson orders) have so far been applied in a completely independent fashion. We define and study a natural compatibility relation between the two approaches leading to the notion of Poisson trace orders. It is proved that all regular and reduced traces are always compatible with any Poisson order structure. The modified discriminant ideals of all Poisson trace orders are proved to be Poisson ideals and the zero loci of discriminant ideals are shown to be unions of symplectic cores, under natural assumptions (maximal orders and Cayley--Hamilton algebras). A base change theorem for Poisson trace orders is proved. A broad range of Poisson trace orders are constructed based on the proved theorems: quantized universal enveloping algebras, quantum Schubert cell algebras and quantum function algebras at roots of unity, symplectic reflection algebras, 3 and 4-dimensional Sklyanin algebras, Drinfeld doubles of pre-Nichols algebras of diagonal type, and root of unity quantum cluster algebras.

We study the analytical properties of a one-side order book model in which the flows of limit and market orders are Poisson processes and the distribution of lifetimes of cancelled orders is exponential. Although simplistic, the model provides an analytical tractability that should not be overlooked. Using basic results for birth-and-death processes, we build an analytical formula for the shape (depth) of a continuous order book model which is both founded by market mechanisms and very close to empirically tested formulas. We relate this shape to the probability of execution of a limit order, highlighting a law of conservation of the flows of orders in an order book. We then extend our model by allowing random sizes of limit orders, hereby allowing to study the relationship between the size of the incoming limit orders and the shape of the order book. Our theoretical model shows that, for a given total volume of incoming limit orders, the less limit orders are submitted (i.e. the larger the average size of these limit orders), the deeper is the order book around the spread. This theoretical relationship is finally empirically tested on several stocks traded on the Paris stock excha

We prove the conjecture that the higher Tamari orders of Dimakis and Müller-Hoissen coincide with the first higher Stasheff--Tamari orders. To this end, we show that the higher Tamari orders may be conceived as the image of an order-preserving map from the higher Bruhat orders to the first higher Stasheff--Tamari orders. This map is defined by taking the first cross-section of a cubillage of a cyclic zonotope. We provide a new proof that this map is surjective and show further that the map is full, which entails the aforementioned conjecture. We explain how order-preserving maps which are surjective and full correspond to quotients of posets. Our results connect the first higher Stasheff--Tamari orders with the literature on the role of the higher Tamari orders in integrable systems.

We provide a method of constructing better-quasi-orders by generalising a technique for constructing operator algebras that was developed by Pouzet. We then generalise the notion of $σ$-scattered to partial orders, and use our method to prove that the class of $σ$-scattered partial orders is better-quasi-ordered under embeddability. This generalises theorems of Laver, Corominas and Thomassé regarding $σ$-scattered linear orders and trees, countable forests and N-free partial orders respectively. In particular, a class of countable partial orders is better-quasi-ordered whenever the class of indecomposable subsets of its members satisfies a natural strengthening of better-quasi-order.

Starting with on-shell amplitudes compatible with the scattering of Kerr black holes, we produce the gravitational waveform and memory effect including spin at their leading post-Minkowskian orders to all orders in the spins of both scattering objects. For the memory effect, we present results at next-to-leading order as well, finding a closed form for all spin orders when the spins are anti-aligned and equal in magnitude. Considering instead generically oriented spins, we produce the next-to-leading-order memory to sixth order in spin. Compton-amplitude contact terms up to sixth order in spin are included throughout our analysis.

Bounded-input bounded-output stability conditions for fractional-order linear time-invariant (LTI) system with multiple noncommensurate orders have been established in this paper. The orders become noncommensurate orders when they do not have a common divisor. Sufficient and necessary conditions of stability for this kind of fractional-order LTI system with multiple noncommensurate orders. Based on the numerical inverse Laplace transform technique, time-domain responses for a fractional-order system with double noncommensurate orders are presented to illustrate the obtained stability results.

Learning high-dimensional distributions is often done with explicit likelihood modeling or implicit modeling via minimizing integral probability metrics (IPMs). In this paper, we expand this learning paradigm to stochastic orders, namely, the convex or Choquet order between probability measures. Towards this end, exploiting the relation between convex orders and optimal transport, we introduce the Choquet-Toland distance between probability measures, that can be used as a drop-in replacement for IPMs. We also introduce the Variational Dominance Criterion (VDC) to learn probability measures with dominance constraints, that encode the desired stochastic order between the learned measure and a known baseline. We analyze both quantities and show that they suffer from the curse of dimensionality and propose surrogates via input convex maxout networks (ICMNs), that enjoy parametric rates. We provide a min-max framework for learning with stochastic orders and validate it experimentally on synthetic and high-dimensional image generation, with promising results. Finally, our ICMNs class of convex functions and its derived Rademacher Complexity are of independent interest beyond their applic

We introduce a new optimal control problem where the controlled dynamical system depends on multi-order (incommensurate) fractional differential equations. The cost functional to be maximized is of Bolza type and depends on incommensurate Caputo fractional-orders derivatives. We establish continuity and differentiability of the state solutions with respect to perturbed trajectories. Then, we state and prove a Pontryagin maximum principle for incommensurate Caputo fractional optimal control problems. Finally, we give an example, illustrating the applicability of our Pontryagin maximum principle.

Much real-world data come with explicitly defined domain orders; e.g., lexicographic order for strings, numeric for integers, and chronological for time. Our goal is to discover implicit domain orders that we do not already know; for instance, that the order of months in the Chinese Lunar calendar is Corner < Apricot < Peach. To do so, we enhance data profiling methods by discovering implicit domain orders in data through order dependencies. We enumerate tractable special cases and proceed towards the most general case, which we prove is NP-complete. We show that the general case nevertheless can be effectively handled by a SAT solver. We also devise an interestingness measure to rank the discovered implicit domain orders, which we validate with a user study. Based on an extensive suite of experiments with real-world data, we establish the efficacy of our algorithms, and the utility of the domain orders discovered by demonstrating significant added value in three applications (data profiling, query optimization, and data mining).

A bosonic topological order on $d$-dimensional closed space $Σ^d$ may have degenerate ground states. The space $Σ^d$ with different shapes (different metrics) form a moduli space ${\cal M}_{Σ^d}$. Thus the degenerate ground states on every point in the moduli space ${\cal M}_{Σ^d}$ form a complex vector bundle over ${\cal M}_{Σ^d}$. It was suggested that the collection of such vector bundles for $d$-dimensional closed spaces of all topologies completely characterizes the topological order. Using such a point of view, we propose a direct relation between two seemingly unrelated properties of 2+1-dimensional topological orders: (1) the chiral central charge $c$ that describes the many-body density of states for edge excitations (or more precisely the thermal Hall conductance of the edge), (2) the ground state degeneracy $D_g$ on closed genus $g$ surface. We show that $c D_g/2 \in \mathbb{Z},\ g\geq 3$ for bosonic topological orders. We explicitly checked the validity of this relation for over 140 simple topological orders. For fermionic topological orders, let $D_{g,σ}^{e}$ ($D_{g,σ}^{o}$) be the degeneracy with even (odd) number of fermions for genus-$g$ surface with spin structure

Succinctness is a natural measure for comparing the strength of different logics. Intuitively, a logic L_1 is more succinct than another logic L_2 if all properties that can be expressed in L_2 can be expressed in L_1 by formulas of (approximately) the same size, but some properties can be expressed in L_1 by (significantly) smaller formulas. We study the succinctness of logics on linear orders. Our first theorem is concerned with the finite variable fragments of first-order logic. We prove that: (i) Up to a polynomial factor, the 2- and the 3-variable fragments of first-order logic on linear orders have the same succinctness. (ii) The 4-variable fragment is exponentially more succinct than the 3-variable fragment. Our second main result compares the succinctness of first-order logic on linear orders with that of monadic second-order logic. We prove that the fragment of monadic second-order logic that has the same expressiveness as first-order logic on linear orders is non-elementarily more succinct than first-order logic.

In this paper, we study the relation between topological orders and their gapped boundaries. We propose that the bulk for a given gapped boundary theory is unique. It is actually a consequence of a microscopic definition of a local topological order, which is a (potentially anomalous) topological order defined on an open disk. Using this uniqueness, we show that the notion of "bulk" is equivalent to the notion of center in mathematics. We achieve this by first introducing the notion of a morphism between two local topological orders of the same dimension, then proving that the bulk satisfying the same universal property as that of the center in mathematics. We propose a classification (formulated as a macroscopic definition) of $n+$1D local topological orders by unitary multi-fusion $n$-categories, and explain that the notion of a morphism between two local topological orders is compatible with that of a unitary monoidal $n$-functor in a few low dimensional cases. We also explain in some low dimensional cases that this classification is compatible with the result of "bulk = center". In the end, we explain that above boundary-bulk relation is only the first layer of a hierarchical s

In this note we construct a poset map from a Boolean algebra to the Bruhat order which unveils an interesting connection between subword complexes, sorting orders, and certain totally nonnegative spaces. This relationship gives a new proof of Björner and Wachs' result \cite{BW} that the proper part of Bruhat order is homotopy equivalent to the proper part of a Boolean algebra --- that is, to a sphere. We also obtain a geometric interpretation for sorting orders. We conclude with two new results: that the intersection of all sorting orders is the weak order, and the union of sorting orders is the Bruhat order.

We define toric partial orders, corresponding to regions of graphic toric hyperplane arrangements, just as ordinary partial orders correspond to regions of graphic hyperplane arrangements. Combinatorially, toric posets correspond to finite posets under the equivalence relation generated by converting minimal elements into maximal elements, or sources into sinks. We derive toric analogues for several features of ordinary partial orders, such as chains, antichains, transitivity, Hasse diagrams, linear extensions, and total orders.

We describe families of nonassociative finite unital rings that occur as quotients of natural nonassociative orders in generalized nonassociative cyclic division algebras over number fields. These natural orders have already been used to systematically construct fully diverse fast-decodable space-time block codes. We show how the quotients of natural orders can be employed for coset coding. Previous results by Oggier and Sethuraman involving quotients of orders in associative cyclic division algebras are obtained as special cases.