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A classical t-tensor product expander is a natural way of formalising correlated walks of t particles on a regular expander graph. A quantum t-tensor product expander is a completely positive trace preserving map that is a straightforward analogue of a classical t-tensor product expander. Interest in these maps arises from the fact that iterating a quantum t-tensor product expander gives us a unitary t-design, which has many applications to quantum computation and information. We show that the zigzag product of a high dimensional quantum expander (i.e. t = 1) of moderate degree with a moderate dimensional quantum t-tensor product expander of low degree gives us a high dimensional quantum t-tensor product expander of low degree. Previously such a result was known only for quantum expanders i.e. t = 1. Using the zigzag product we give efficient constructions of quantum t-tensor product expanders in dimension D where t = polylog(D). We then show how replacing the zigzag product by the generalised zigzag product leads to almost-Ramanujan quantum tensor product expanders i.e. having near-optimal almost quadratic tradeoff between the degree and the second largest singular value. Both the

Product states, unentangled tensor products of single qubits, are a ubiquitous ansatz in quantum computation, including for state-of-the-art Hamiltonian approximation algorithms. A natural question is whether we should expect to efficiently solve product state problems on any interesting families of Hamiltonians. We completely classify the complexity of finding minimum-energy product states for Hamiltonians defined by any fixed set of allowed 2-qubit interactions. Our results follow a line of work classifying the complexity of solving Hamiltonian problems and classical constraint satisfaction problems based on the allowed constraints. We prove that estimating the minimum energy of a product state is in P if and only if all allowed interactions are 1-local, and NP-complete otherwise. Equivalently, any family of non-trivial two-body interactions generates Hamiltonians with NP-complete product-state problems. Our hardness constructions only require coupling strengths of constant magnitude. A crucial component of our proofs is a collection of hardness results for a new variant of the Vector Max-Cut problem, which should be of independent interest. Our definition involves sums of distan

In recent years, with rising consumer demand, fresh products have gained increasing attention, leading to rapid growth in the fresh food market. However, due to their perishable nature and sensitivity to storage conditions, fresh products are vulnerable to damage during transportation. Improper handling, excessive transit times, and physical impacts can result in significant losses. As a result, enhancing the efficiency of fresh product distribution while maintaining quality has become critical to the further development of the fresh food industry. Using Y chain supermarket as a case study, this paper investigates the logistics of fresh product distribution, identifying current challenges and inefficiencies. Through literature review, expert interviews, and comparative analysis, the study offers strategic recommendations for optimizing fresh product delivery routes to improve distribution efficiency and product quality.

The Odd Hadwiger number of a graph $G$ is the largest integer $r$ such that $G$ has a clique of size $r$ as an odd minor. In this paper, we investigate how large is the Odd Hadwiger number of the product of two graphs, when considering any of the four standard graph products: Cartesian, direct, lexicographic, strong. We provide an optimal lower bound in the cases of the strong and lexicographic products.

We define a two-variable $p$-adic Asai $L$-function for a finite-slope family of Hilbert modular forms over a real quadratic field (with one component of the weight, and the cyclotomic twist variable, varying independently); and a two-variable ``twisted triple product'' $L$-function, interpolating the central $L$-value of the tensor product of such a family with a family of elliptic modular forms. The former construction generalizes a construction due to Grossi, Zerbes and the second author for ordinary families; the latter is a counterpart of the twisted triple product $L$-function of arXiv:2401.13230, but differs in that it interpolates classical $L$-values in a different range of weights, in which the dominant weight comes from the Hilbert modular form. Our construction relies on a ``nearly-overconvergent'' version of higher Coleman theory for Hilbert modular surfaces.

Developing equivariant neural networks for the E(3) group plays an important role in modeling 3D data across real-world applications. Enforcing this equivariance primarily involves the tensor products of irreducible representations (irreps). However, the computational complexity of such operations increases significantly as higher-order tensors are used. In this work, we propose a systematic approach to substantially accelerate the computation of the tensor products of irreps. We mathematically connect the commonly used Clebsch-Gordan coefficients to the Gaunt coefficients, which are integrals of products of three spherical harmonics. Through Gaunt coefficients, the tensor product of irreps becomes equivalent to the multiplication between spherical functions represented by spherical harmonics. This perspective further allows us to change the basis for the equivariant operations from spherical harmonics to a 2D Fourier basis. Consequently, the multiplication between spherical functions represented by a 2D Fourier basis can be efficiently computed via the convolution theorem and Fast Fourier Transforms. This transformation reduces the complexity of full tensor products of irreps from

This report details our methodology and results developed for the Multilingual E-commerce Search Competition. The problem aims to recognize relevance between user queries versus product items in a multilingual context and improve recommendation performance on e-commerce platforms. Utilizing Large Language Models (LLMs) and their capabilities in other tasks, our data-centric method achieved the highest score compared to other solutions during the competition. Final leaderboard is publised at https://alibaba-international-cikm2025.github.io. The source code for our project is published at https://github.com/nhtlongcs/e-commerce-product-search.

Modern neural models capture rich priors and have complementary knowledge over shared data domains, e.g., images and videos. Integrating diverse knowledge from multiple sources -- including visual generative models, visual language models, and sources with human-crafted knowledge such as graphics engines and physics simulators -- remains under-explored. We propose a Product of Experts (PoE) framework that performs inference-time knowledge composition from heterogeneous models. This training-free approach samples from the product distribution across experts via Annealed Importance Sampling (AIS). Our framework shows practical benefits in image and video synthesis tasks, yielding better controllability than monolithic methods and additionally providing flexible user interfaces for specifying visual generation goals.

The "back-stabilization number" for products of Schubert polynomials is the distance the corresponding permutations must be shifted before the structure constants stabilize. We give an explicit formula for this number and thereby prove a conjecture of N. Li in a strengthened form. This leads to an additional result: a formula for the smallest $n$ such that a given Schubert product expands completely over $S_n$. Our method is to explore back-stable fundamental slide polynomials and their products combinatorially, in the context of their associated words. We use three main tools: (i) an algebra consisting of "colored words", with a modified shuffle product, and which contains the rings of back (quasi)symmetric functions as subquotients; (ii) the combinatorics of increasing suffixes of reduced words; and (iii) the lift of differential operators to the space of colored words.

We define a tensor product of linear sites, and a resulting tensor product of Grothendieck categories based upon their representations as categories of linear sheaves. We show that our tensor product is a special case of the tensor product of locally presentable linear categories, and that the tensor product of locally coherent Grothendieck categories is locally coherent if and only if the Deligne tensor product of their abelian categories of finitely presented objects exists. We describe the tensor product of non-commutative projective schemes in terms of Z-algebras, and show that for projective schemes our tensor product corresponds to the usual product scheme.

In this paper, we prove a dihedral extremality and rigidity theorem for a large class of codimension zero submanifolds with polyhedral boundary in warped product manifolds. We remark that the spaces considered in this paper are not necessarily warped product manifolds themselves. In particular, the results of this paper are applicable to submanifolds (of warped product manifolds) with faces that are neither orthogonal nor parallel to the radial direction of the warped product metric. Generally speaking, the dihedral rigidity results require the leaf of the underlying warped space to have positive Ricci curvature and the warping function to be strictly log-concave. Nevertheless, we prove a dihedral rigidity theorem for a large class of hyperbolic polyhedra, where the leaf of the underlying warped product space is flat and the warping function is not strictly log-concave.

The aim of this note is to introduce a notion of dynamical entropy, which we call infinite-product entropy, for probability measures on (countable) infinite cartesian product of any measurable space with itself. The idea behind the definition is that any infinite product space may be considered as a type of dynamical object. We have considered in a previous note a similar idea in topological dynamics to define a notion of dynamical entropy for arbitrary subsets of infinite products of compact topological spaces. We consider some basic properties of infinite-product entropy, e.g. shift invariance, convexity, subadditivity with respect to product of probability measures, the behavior with respect to dilation and restriction. We show that for a translation invariant probability measure the infinite-product entropy coincides with the usual entropy of a shift transformation. We consider some basic examples and computations. We also consider a variational inequality related to infinite-product entropy and topological entropy of subsets of infinite product spaces.

Private Information Retrieval (PIR) was first proposed by B. Chor, O. Goldreich, E. Kushilevitz and M. Sudan in their 1995 FOCS paper. For MDS coded distributed storage system private information retrieval was proposed and the capacity of PIR schemes for MDS coded distributed storage was studied. Star product PIR schemes from general coded distributed storage system with colluding servers were constructed over general finite fields. These star product schemes has no restriction on the sizes of fields and can be constructed for coded distributed storage across large number of servers. In this paper we first propose and prove the Singleton type upper bound on the storage rate, ratio of colluding servers and the retrieval rate of the star product PIR schemes. Secondly star product PIR schemes for coded distributed storage from algebraic geometry (AG) codes are analysed. We prove that when the number of the servers goes to the infinity, star product PIR schemes with colluding servers for AG-coded distributed storage have parameters closing to the Singleton type upper bound if the field is large. Comparing with the star product PIR schemes for Reed-Solomon coded and Reed-Muller coded di

We study operators acting on a tensor product Hilbert space and investigate their product numerical range, product numerical radius and separable numerical range. Concrete bounds for the product numerical range for Hermitian operators are derived. Product numerical range of a non-Hermitian operator forms a subset of the standard numerical range containing the barycenter of the spectrum. While the latter set is convex, the product range needs not to be convex nor simply connected. The product numerical range of a tensor product is equal to the Minkowski product of numerical ranges of individual factors.

We show that for a C*-algebra A and a discrete group G with an action of G on A, the reduced crossed product C*-algebra possesses a natural generalization of the convolution product, which we suggest should be named the Hadamard product. We show that this product has a natural Stinespring representation and we lift some known results on block Schur products to this setting, but we also show that the block Schur product is a special case of the Hadamard product in a crossed product algebra.

In an arbitrary complete differential graded Lie algebra, we construct a group operation $\bullet$ on $L_1$ such that the differential of the product of two elements is the Baker-Campbell-Hausdorff product of their differentials, i.e., $d(x\bullet y)=dx\ast dy$. We study some properties of this new structure and some applications, especially in homotopy theory, where this operation can be used to construct a Lie model for the 4-simplex. In particular, this solves, in dimension 4, a problem proposed by Lawrence and Sullivan.

Product Question Answering (PQA) systems are key in e-commerce applications to provide responses to customers' questions as they shop for products. While existing work on PQA focuses mainly on English, in practice there is need to support multiple customer languages while leveraging product information available in English. To study this practical industrial task, we present xPQA, a large-scale annotated cross-lingual PQA dataset in 12 languages across 9 branches, and report results in (1) candidate ranking, to select the best English candidate containing the information to answer a non-English question; and (2) answer generation, to generate a natural-sounding non-English answer based on the selected English candidate. We evaluate various approaches involving machine translation at runtime or offline, leveraging multilingual pre-trained LMs, and including or excluding xPQA training data. We find that (1) In-domain data is essential as cross-lingual rankers trained on other domains perform poorly on the PQA task; (2) Candidate ranking often prefers runtime-translation approaches while answer generation prefers multilingual approaches; (3) Translating offline to augment multilingual

In this paper, research on AI based modeling technique to optimize development of new alloys with necessitated improvements in properties and chemical mixture over existing alloys as per functional requirements of product is done. The current research work novels AI in lieu of predictions to establish association between material and product customary. Advanced computational simulation techniques like CFD, FEA interrogations are made viable to authenticate product dynamics in context to experimental investigations. Accordingly, the current research is focused towards binding relationships between material design and product design domains. The input to feed forward back propagation prediction network model constitutes of material design features. Parameters relevant to product design strategies are furnished as target outputs. The outcomes of ANN shows good sign of correlation between material and product design domains. The study enriches a new path to illustrate material factors at the time of new product development.

We give an example showing that the product and linearization formulas for the Wick product versions of the $q$-Charlier polynomials in (Anshelevich 2004) are incorrect. Next, we observe that the relation between monomials and several families of Wick polynomials is governed by "incomplete" versions of familiar posets. We compute Möbius functions for these posets, and prove a general poset product formula. These provide new proofs and new inversion and product formulas for Wick product versions of Hermite, Chebyshev, Charlier, free Charlier, and Laguerre polynomials. By different methods, we prove inversion formulas for the Wick product versions of the free Meixner polynomials.