Arbitrary Rank Research Articles

A higher Gross–Zagier formula and the structure of Selmer groups

We describe a Kolyvagin system-theoretic refinement of Gross–Zagier formula by comparing Heegner point Kolyvagin systems with Kurihara numbers when the root number of a rational elliptic curve E E over an imaginary quadratic field K K is − 1 -1 . When the root number of E E over K K is 1, we first establish the structure theorem of the p ∞ p^\infty -Selmer group of E E over K K . The description is given by the values of certain families of quaternionic automorphic forms, which is a part of bipartite Euler systems. By comparing bipartite Euler systems with Kurihara numbers, we also obtain an analogous refinement of Waldspurger formula. No low analytic rank assumption is imposed in both refinements. We also prove the equivalence between the non-triviality of various “Kolyvagin systems” and the corresponding main conjecture localized at the augmentation ideal. As consequences, we obtain new applications of (weaker versions of) the Heegner point main conjecture and the anticyclotomic main conjecture to the structure of p ∞ p^\infty -Selmer groups of elliptic curves of arbitrary rank. In particular, the Heegner point main conjecture localized at the augmentation ideal implies the strong rank one p p -converse to the theorem of Gross–Zagier and Kolyvagin.

Schur indices for N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 super-Yang-Mills with more general gauge groups

We study the unflavored Schur indices in the N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 super-Yang-Mills theory for the Bn, Cn, Dn, G2 gauge groups. We explore two methods, namely the character expansion method and the Fermi gas method, to efficiently compute the q-series expansion of the Schur indices to some high orders. Using the available data and the modular properties, we are able to fix the exact formulas for the general gauge groups up to some high ranks and discover some interesting new features. We also identify some empirical modular anomaly equations, but unlike the case of An groups, they are quite complicated and not sufficiently useful to fix exact formulas for gauge groups of arbitrary rank.

Demushkin groups of uncountable rank

AbstractWe extend the theory of countably generated Demushkin groups to Demushkin groups of arbitrary rank. We investigate their algebraic properties and invariants, count their isomorphism classes, and study their realization as absolute Galois group. At the end, we compute their profinite completion and conclude with some results on profinite completion of absolute Galois groups.

The pro-supersolvable topology on a free group: Deciding denseness

Let F be a free group of arbitrary rank and let H be a finitely generated subgroup of F. Given a pseudovariety V of finite groups, i.e. a class of finite groups closed under taking subgroups, quotients and finitary direct products, we endow F with its pro-V topology. Our main result states that it is decidable whether H is pro-Su dense, where Su⊂S denote respectively the pseudovarieties of all finite supersolvable groups and all finite solvable groups. Our motivation stems from the following open problem: is it decidable whether H is pro-S dense?

Leveraging tensor kernels to reduce objective function mismatch in deep clustering

Objective Function Mismatch (OFM) occurs when the optimization of one objective has a negative impact on the optimization of another objective. In this work we study OFM in deep clustering, and find that the popular autoencoder-based approach to deep clustering can lead to both reduced clustering performance, and a significant amount of OFM between the reconstruction and clustering objectives. To reduce the mismatch, while maintaining the structure-preserving property of an auxiliary objective, we propose a set of new auxiliary objectives for deep clustering, referred to as the Unsupervised Companion Objectives (UCOs). The UCOs rely on a kernel function to formulate a clustering objective on intermediate representations in the network. Generally, intermediate representations can include other dimensions, for instance spatial or temporal, in addition to the feature dimension. We therefore argue that the naïve approach of vectorizing and applying a vector kernel is suboptimal for such representations, as it ignores the information contained in the other dimensions. To address this drawback, we equip the UCOs with structure-exploiting tensor kernels, designed for tensors of arbitrary rank. The UCOs can thus be adapted to a broad class of network architectures. We also propose a novel, regression-based measure of OFM, allowing us to accurately quantify the amount of OFM observed during training. Our experiments show that the OFM between the UCOs and the main clustering objective is lower, compared to a similar autoencoder-based model. Further, we illustrate that the UCOs improve the clustering performance of the model, in contrast to the autoencoder-based approach. The code for our experiments is available at https://github.com/danieltrosten/tk-uco.

Hodge duality transformations in tensor-hierarchy formulations

We show in arbitrary space-time dimensions that duality-transformations are possible in general tensor-hierarchy with field-strengths of arbitrary ranks. We first consider the non-Abelian tensor field-strength of the potential , and its compensator field-strength of the potential , where is the adjoint index. We next perform duality transformations into their respective Hodge-dual field-strength of the potential and the field-strength of the potential . As a typical application, we show this to take place in dimensions with an N = 1 supersymmetric set consisting of a Yang–Mills vector-multiplet , an extra vector-multiplet , and a compensating chiral multiplet . The λ I and χ I are Majorana spinors, and is an extra vector with the Proca–Stückelberg scalar CI , while EI is a pseudo-scalar. We perform duality-transformations on and to their Hodge-dual fields and , respectively, ending up with the new set of multiplets and . Another example is the set of multiplets and in . Even though has the maximal 4th rank field-strength, its supersymmetric dual system exists with and after a duality-transformation. Our duality-transformations provide previously-unknown-links among supersymmetric tensor-hierarchy formulations.

Representations of Smith algebras which are free over the Cartan subalgebra

In this paper, we study the category of modules over the Smith algebra which are free of finite rank over the unital polynomial subalgebra generated by the Cartan element h and obtain families of such simple modules of arbitrary rank. In the case of rank one we obtain a full description of the isomorphism classes, a simplicity criterion, and an algorithm to produce all composition series. We show that all such modules have finite length and describe the composition factors and their multiplicity.

Channel Customization for Joint Tx-RISs-Rx Design in Hybrid mmWave Systems

In strong line-of-sight millimeter-wave (mmWave) wireless systems, the rank-deficient channel severely hampers spatial multiplexing. To address this inherent deficiency, distributed reconfigurable intelligent surfaces (RISs) are introduced in this study to customize the wireless channel. Capitalizing on the ability of the RIS to reshape electromagnetic waves, we theoretically show that a favorable channel with an arbitrary tunable rank and a minimized truncated condition number can be established by elaborately designing the placement and reflection matrix of RISs. Different from existing works on distributed RISs, the number of elements needed for each RIS to combat the path loss and the limited phase control is also considered in this research. On the basis of the proposed channel customization, a joint transmitter-RISs-receiver (Tx-RISs-Rx) design under a hybrid mmWave system is investigated to maximize the downlink spectral efficiency. Using the proposed scheme, the optimal singular value decomposition-based hybrid beamforming at the Tx and Rx can be easily obtained without matrix decomposition for the digital and analog beamforming. The bottoms of the sub-channel mode in the water-filling power allocation algorithm, which are conventionally uncontrollable when the noise power is fixed, are proven to be independently adjustable by RISs. Moreover, the transmit power required for realizing multi-stream transmission is derived. Numerical results are presented to verify our theoretical analysis and exhibit substantial gains over systems without RISs.

Column cyclic rank metric codes and linear complementary dual rank metric codes

We introduce and study column cyclic rank metric (CCRM) codes over finite fields. We present natural questions for the existence and characterization of CCRM codes. We completely solve these questions only in the first nontrivial case: The case of the minimum rank distance [Formula: see text] and the number of rows [Formula: see text]. We also consider linear complementary dual rank metric (LCDRM) codes both in the setting of CCRM codes and in the general setting of rank metric codes. We also ask the natural questions for the existence and characterization of CCRM codes in the subclass of LCDRM codes. We also completely solve these questions for this subclass of LCDRM codes only in the first nontrivial case that [Formula: see text] and [Formula: see text]. Moreover we extend a characterization of linear complementary dual (LCD) codes of Massey to arbitrary rank metric codes in the general setting.

Sources of torsion in Poincarè gauge gravity

We study sources for torsion in Poincarè gauge theory in any dimension, signature, and spin. We find that symmetric kinetic terms for non-Yang–Mills bosonic fields of arbitrary rank drive torsion. Our detailed discussion of spin-3/2 Rarita–Schwinger fields shows that they source all independent parts of the torsion. We develop systematic notation for spin-(2k+1)/2 fields and find the spin tensor for arbitrary k in n ge 2k+1 dimensions. For k>0 there is a novel direct coupling between torsion and spinor fields. We also cast the well-known gauge relation between the canonical and Belinfante–Rosenfield energy tensors in terms of different choices of independent variables.

Module structures on U(S−) for the Schrödinger algebra

In this paper, we classify modules which are free of rank 1 over the negative part for the Schrödinger algebra S=sl2⋉H. Sufficient and necessary conditions for such modules to be simple are given. From these modules, we construct examples of simple S-modules which are free of arbitrary rank over U(Ch).

Linear-Scaling Quantum Circuits for Computational Chemistry.

We have recently constructed compact, CNOT-efficient, quantum circuits for Fermionic and qubit excitations of arbitrary many-body rank [Magoulas, I.; Evangelista, F. A. J. Chem. Theory Comput. 2023, 19, 822]. Here, we present approximations of these circuits that substantially reduce the CNOT counts even further. Our preliminary numerical data, using the selected projective quantum eigensolver approach, show up to a 4-fold reduction in CNOTs. At the same time, there is practically no loss of accuracy in the energies compared to the parent implementation, while the ensuing symmetry breaking is essentially negligible.

Cohomogeneity one actions on symmetric spaces of noncompact type and higher rank

We develop a new structural result for cohomogeneity one actions on (not necessarily irreducible) symmetric spaces of noncompact type and arbitrary rank. We apply this result to classify cohomogeneity one actions on the spaces SLn(R)/SOn, n≥2, up to orbit equivalence. We also reduce the classification problem on a reducible space to the classification on each one of its irreducible factors, which in particular allows to classify cohomogeneity one actions on any finite product of hyperbolic spaces.

$$2$$-Adic 1-Lipschitz Maps-Based Nonlinear Pseudorandom Generators of Arbitrary Rank Having the Longest Period

$$2$$-Adic 1-Lipschitz Maps-Based Nonlinear Pseudorandom Generators of Arbitrary Rank Having the Longest Period

Boundedness of Fourier integral operators on classical function spaces

We investigate the global boundedness of Fourier integral operators with amplitudes in the general Hörmander classes Sρ,δm(Rn), ρ,δ∈[0,1] and non-degenerate phase functions of arbitrary rank κ∈{0,1,…,n−1} on Besov-Lipschitz Bp,qs(Rn) and Triebel-Lizorkin Fp,qs(Rn) of order s and 0<p≤∞, 0<q≤∞. The results that are obtained are all up to the end-point and sharp and are also applied to the regularity of Klein-Gordon-type oscillatory integrals in the aforementioned function spaces.

Some examples of separable convex‐cocompact subgroups

AbstractReid asked whether all convex‐cocompact subgroups of mapping class groups are separable. Using a construction of Manning–Mj–Sageev, we give examples of separable convex‐cocompact subgroups that are free of arbitrary finite rank, while prior examples seem to all be virtually cyclic.

Minimum Cut and Minimum k -Cut in Hypergraphs via Branching Contractions

On hypergraphs with m hyperedges and n vertices, where p denotes the total size of the hyperedges, we provide the following results: We give an algorithm that runs in \(\widetilde{O}(mn^{2k-2})\) time for finding a minimum k -cut in hypergraphs of arbitrary rank. This algorithm betters the previous best running time for the minimum k -cut problem, for k &gt; 2. We give an algorithm that runs in \(\widetilde{O}(n^{\max \lbrace r,2k-2\rbrace })\) time for finding a minimum k -cut in hypergraphs of constant rank r . This algorithm betters the previous best running times for both the minimum cut and minimum k -cut problems for dense hypergraphs. Both of our algorithms are Monte Carlo, i.e., they return a minimum k -cut (or minimum cut) with high probability. These algorithms are obtained as instantiations of a generic branching randomized contraction technique on hypergraphs, which extends the celebrated work of Karger and Stein on recursive contractions in graphs. Our techniques and results also extend to the problems of minimum hedge-cut and minimum hedge- k -cut on hedgegraphs, which generalize hypergraphs.

Ulrich bundles of arbitrary rank on Segre-Veronese varieties

We generalize the results by Eisenbud and Schreyer about Ulrich bundles over Veronese varieties to Segre-Veronese varieties. We discuss the range where we have natural cohomology and we construct multigraded resolutions and monads for Ulrich bundles of any rank. Moreover we give cohomological characterizations for significant families of bundles.

Generation function for one-loop tensor reduction

For loop integrals, reduction is the standard method. Having an efficient way to find reduction coefficients is an important topic in scattering amplitudes. In this paper, we present the generation functions of reduction coefficients for general one-loop integrals with an arbitrary tensor rank in their numerator.

Bidiagonal decompositions of Vandermonde-type matrices of arbitrary rank

We present a method to derive new explicit expressions for bidiagonal decompositions of Vandermonde and related matrices such as the (q-, h-) Bernstein-Vandermonde ones, among others. These results generalize the existing expressions for nonsingular matrices to matrices of arbitrary rank. For totally nonnegative matrices of the above classes, the new decompositions can be computed efficiently and to high relative accuracy componentwise in floating point arithmetic. In turn, matrix computations (e.g., eigenvalue computation) can also be performed efficiently and to high relative accuracy.