Odd Order Research Articles

Blocks whose defect groups are Suzuki 2-groups

We classify up to Morita equivalence all blocks whose defect groups are Suzuki 2-groups. The classification holds for blocks over a suitable discrete valuation ring as well as for those over an algebraically closed field, and in fact holds up to basic Morita equivalence. As a consequence Donovan's conjecture holds for Suzuki 2-groups. A corollary of the proof is that Suzuki Sylow 2-subgroups of finite groups with no nontrivial odd order normal subgroup are trivial intersection.

Comparison of the Impacts of Background Distortion with Even and Odd Orders in the Supraharmonic Range on Grid-Connected Converters

In this paper, a comparison of the emissions of grid-connected converters under grid distortion in the supraharmonic range is performed. The investigated supraharmonic grid distortion is divided into two types, namely odd-integer and even-integer multiples of the grid frequency. It is found that the first distortion type does not affect the emissions of the grid-connected converters, while the second distortion is responsible for the 2N-harmonic emissions that appear in the frequency spectra of the grid-connected converters, which, in turn, interact with the supraharmonic primary emissions of these converters, resulting in additional supraharmonic emissions. To explain this behavior, a mathematical analysis is provided for two types of grid-connected converters with an inverter interface and with a diode bridge rectifier interface. Simulations and experimental studies are performed to verify the main findings of the mathematical analysis.

Infinite locally finite groups groups with the given properties of the norm of Abelian non-cyclic subgroups

In the paper, the properties of infinite locally finite groups with non-Dedekind locally nil\-potent norms of Abelian non-cyclic subgroups are studied. It is proved that such groups are finite extensions of a quasicyclic subgroup and contain Abelian non-cyclic $p$-subgroups for a unique prime $p$. In particular, in the paper is prove the following assertions: 1) Let $G$ be an infinite locally finite group and contain the locally nilpotent norm $N_{G}^{A}$ with the non-Hamiltonian Sylow $p$-subgroup $(N_{G}^{A})_{p}$. Then $G$ is a finite extension of a quasicyclic $p$-subgroup, all Sylow $p'$-subgroups are finite and do not contain Abelian non-cyclic subgroups. In particular, Sylow $q$-subgroups ($q$ is an odd prime, $q\in \pi(G)$, $q\neq p$) are cyclic, Sylow $2$-subgroups ($p\neq 2$) are either cyclic or finite quaternion $2$-groups (Theorem 1). 2) Let $G$ be a locally finite non-locally nilpotent group with the infinite locally nilpotent non-Dedekind norm $N_{G}^{A}$ of Abelian non-cyclic subgroups. Then $G=G_{p} \leftthreetimes H,$ where $G_{p}$ is an infinite $\overline{HA}_{p}$-group of one of the types (1)--(4) of Proposition~2 in present paper, which coincides with the Sylow $p$-subgroup of the norm $N_{G}^{A}$, $H$ is a finite group, all Abelian subgroups of which are cyclic, and $(|H|,p)=1$. Any element $h\in H$ of odd order that centralizes some Abelian non-cyclic subgroup $M\subset N_{G}^{A}$ is contained in the centralizer of the norm $N_{G}^{A}$. (Theorem 2).3) Let $G$ be an infinite locally finite non-locally nilpotent group with the finite nilpotent non-Dedekind norm $N_{G}^{A}$ of Abelian non-cyclic subgroups. Then$G=H\leftthreetimes K,$ where $H$ is a finite group, all Abelian subgroups of which are cyclic,$\left(\left|H\right|,2\right)=1$, $K$ is an infinite 2-group of one of the types (5)--(6) of Proposition~2 (in present paper). Moreover, the norm $N_{K}^{A}$ of Abelian non-cyclic subgroups of the group $K$ is finite, $K\cap N_{G}^{A}=N_{K}^{A}$ and coincides with the Sylow 2-subgroup $(N_{G}^{A})_2$ of the norm $N_{G}^{A}$ of a group $G$.Moreover, any element $h\in H$ of the centralizer of some Abelian non-cyclic subgroup $M \subset N_{G}^{A}$ is contained in the centralizer of the norm $N_{G}^{A}$. (Theorem 4).

On Nielsen realization and manifold models for classifying spaces

We consider the problem of whether, for a given virtually torsionfree discrete group Γ \Gamma , there exists a cocompact proper topological Γ \Gamma -manifold, which is equivariantly homotopy equivalent to the classifying space for proper actions. This problem is related to Nielsen Realization. We will make the assumption that the expected manifold model has a zero-dimensional singular set. Then we solve the problem in the case, for instance, that Γ \Gamma contains a normal torsionfree subgroup π \pi such that π \pi is hyperbolic and π \pi is the fundamental group of an aspherical closed manifold of dimension greater or equal to five and Γ / π \Gamma /\pi is a finite cyclic group of odd order.

Off-diagonally symmetric domino tilings of the Aztec diamond of odd order

We study the enumeration of off-diagonally symmetric domino tilings of odd-order Aztec diamonds in two directions: (1) with one boundary defect, and (2) with maximally-many zeroes on the diagonal. In the first direction, we prove a symmetry property which states that the numbers of off-diagonally symmetric domino tilings of the Aztec diamond of order 2n−1 are equal when the boundary defect is at the kth position and the (2n−k)th position on the boundary, respectively. This symmetry property proves a special case of a recent conjecture by Behrend, Fischer, and Koutschan.In the second direction, a Pfaffian formula is obtained for the number of “nearly” off-diagonally symmetric domino tilings of odd-order Aztec diamonds, where the entries of the Pfaffian satisfy a simple recurrence relation. The numbers of domino tilings mentioned in the above two directions do not seem to have a simple product formula, but we show that these numbers satisfy simple matrix equations in which the entries of the matrix are given by Delannoy numbers. The proof of these results involves the method of non-intersecting lattice paths and a modification of Stembridge's Pfaffian formula for families of non-intersecting lattice paths. Finally, we propose conjectures concerning the log-concavity and asymptotic behavior of the number of off-diagonally symmetric domino tilings of odd-order Aztec diamonds.

Exploring redundant trees in bipartite graphs

Luo et al. conjectured that for a tree T with bipartition X and Y, if a k-connected bipartite graph G with minimum degree at least k+max⁡{|X|,|Y|}, then G has a subtree TG isomorphic to T such that G−V(TG) is k-connected. Although this conjecture has been validated for spiders and caterpillars in cases where k≤3, and also for paths with odd order, its general applicability has remained an open question. In this paper, we establish the validity of this conjecture for k≤3 with the girth under of G at least the diameter of G minus one.

Representation-reduced stated skein modules and algebras

For any marked three manifold (M,N) and any quantum parameter q12 (a nonzero complex number), we use Sq1/2(M,N) to denote the stated skein module of (M,N). When q12 is a root of unity of odd order, the commutative algebra S1(M,N) acts on Sq1/2(M,N). For any maximal ideal ρ of S1(M,N), define Sq1/2(M,N)ρ=Sq1/2(M,N)⊗S1(M,N)(S1(M,N)/ρ).We prove the splitting map for Sq1/2(M,N) respects the S1(M,N)-module structure, so it reduces to the splitting map for Sq1/2(M,N)ρ. We prove the splitting map for Sq1/2(M,N)ρ is injective if there exists at least one component of N such that this component and the boundary of the splitting disk belong to the same component of ∂M. We also prove the representation-reduced stated skein module of the marked handlebody is an irreducible Azumaya representation of the stated skein algebra of its boundary.Let M be an oriented connected closed three manifold. For any positive integer k, we use Mk to denote the marked three manifold obtained from M by removing k open three dimensional balls and adding one marking to each newly created sphere boundary component. We prove dimCSq1/2(Mk)ρ=1 for any maximal ideal ρ of S1(Mk).

Investigation of multifunctional grid-tied PV inverter with even and odd order harmonic mitigation using active and harmonic power weight control strategy

Investigation of multifunctional grid-tied PV inverter with even and odd order harmonic mitigation using active and harmonic power weight control strategy

On the relation of character codegrees and the minimal faithful quasi-permutation representation degree of p-groups

For a finite group G, we denote by c ( G ) , the minimal degree of a faithful representation of G by quasi-permutation matrices over C . For an irreducible character χ of G, the codegree of χ is defined as cod ( χ ) = | G / ker ( χ ) | / χ ( 1 ) . In this article, we establish equality between c ( G ) and a Q ≥ 0 -sum of codegrees of some irreducible characters of a non-abelian p-group G of odd order. We also study the relation between c ( G ) and irreducible character codegrees for various classes of non-abelian p-groups, such as, p-groups with cyclic center, maximal class p-groups, GVZ p-groups, and others.

On parallelisms of PG(3,4) with automorphisms of order 2

Let PG(n,q) be the n-dimensional projective space over the finite field Fq. A spread in PG(n,q) is a set of mutually skew lines which partition the point set. A parallelism is a partition of the set of lines by spreads. The classification of parallelisms in small finite projective spaces is of interest for problems from projective geometry, design theory, network coding, error-correcting codes, and cryptography. All parallelisms of PG(3,2) and PG(3,3) are known. Parallelisms of PG(3,4) which are invariant under automorphisms of odd prime orders have also been classified. The present paper contributes to the classification of parallelisms in PG(3,4) with automorphisms of even order. We focus on cyclic groups of order four and the group of order two generated by a Baer involution. We examine invariants of the parallelisms such as the full automorphism group, the type of spreads and questions of duality. The results given in this paper show that the number of parallelisms of PG(3,4) is at least 8675365. Some future directions of research are outlined.

Time-domain constraints for passive materials: The Brendel-Bormann model revisited

This paper presents a systematic approach to derive physical bounds for passive systems, or equivalently for positive real (PR) functions, directly in the time-domain (TD). As a generic, canonical example we explore the TD dielectric response of a passive material. We will furthermore revisit the theoretical foundation regarding the Brendel-Bormann (BB) oscillator model which is reportedly very suitable for the modeling of thin metallic films in high-speed optoelectronic devices. To this end, an important result here is to re-establish the physical realizability of the BB model by showing that it represents a passive and causal system. The theory is based on Cauer's representation of an arbitrary PR function together with associated sum rules (moments of the measure) and exploits the unilateral Laplace transform to derive rigorous bounds on the TD response of a passive system. Similar bounds have recently been reported for more general casual systems with other assumptions. To this end, it is important to note here that the existence of useful sum rules and related physical bounds rely heavily on an assumption about the PR functions having a low- or high-frequency asymptotic expansion at least of odd order 1. As a particular numerical example, we consider here the electric susceptibility of gold (Au) which is commonly modeled by well established Drude or BB models. Explicit physical bounds are given as well as an efficient fast-Fourier transform -based numerical procedure to compute the TD impulse response associated with the nonrational BB model. Published by the American Physical Society 2024

Bernoulli numbers in embedding constants for Sobolev spaces with diverse boundary conditions

The Sobolev spaces W 2 n [ 0 ; 1 ] W^n_2[0;1] are treated with five different boundary conditions (periodic, antiperiodic, of even and odd orders, and of even-odd order). Sharp estimates for the derivatives of order k = 0 , 1 , … , n − 1 k=0,1,\dots , n-1 are obtained, sharp constants for the embedding of W 2 n [ 0 ; 1 ] W^n_2[0;1] into W ∞ k [ 0 ; 1 ] W^k_\infty [0;1] are found, it is shown that they are rationally expressed in terms of Bernoulli numbers and, therefore, are rational. The exact embedding constants can also be expressed in terms of the Riemann ζ \zeta -function. The reproducing kernels of these spaces are calculated.

Symmetric 2‐(36,15,6) $(36,15,6)$ designs with an automorphism of order two

AbstractBouyukliev, Fack and Winne classified all 2‐ designs that admit an automorphism of odd prime order, and gave a partial classification of such designs that admit an automorphism of order 2. In this paper, we give the classification of all symmetric 2‐ designs that admit an automorphism of order two. It is shown that there are exactly nonisomorphic such designs, of which are self‐dual designs. The ternary linear codes spanned by the incidence matrices of these designs are computed. Among these codes, there are near‐extremal self‐dual codes with previously unknown weight distributions.

Effects of edge addition or removal on the nullity of a graph

Lights Out is a game which can be played on any graph G. Initially we have a configuration which assigns one of the two states on or off to each vertex. The aim of the game is to turn all vertices to off state for an initial configuration by activating some vertices where each activation switches the state of the vertex and all of its neighbors. If the aim of the game can be accomplished for all initial configurations then G is called always solvable. We call the dimension of the kernel of the closed neighborhood matrix of the graph over the field Z2, nullity of G. It turns out that G is always solvable if and only if its nullity is zero. Moreover, the number of solutions of a given configuration is also determined by the nullity. We investigate the problem of how nullity changes when an edge is added to or removed from a graph. As a result we show that for every graph with positive nullity there exists an edge whose removal decreases the nullity. Conversely, we show that for every always solvable graph which is not an even graph with odd order, there exists an edge whose addition increases the nullity. We also show that if an always solvable graph is not even, then there is an edge whose removal increases the nullity.

Unrestricted Quantum Moduli Algebras, II: Noetherianity and Simple Fraction Rings at Roots of 1

We prove that the quantum graph algebra and the quantum moduli algebra associated to a punctured sphere and complex semisimple Lie algebra $\mathfrak{g}$ are Noetherian rings and finitely generated rings over $\mathbb{C}(q)$. Moreover, we show that these two properties still hold on $\mathbb{C}\big[q,q^{-1}\big]$ for the integral version of the quantum graph algebra. We also study the specializations $\mathcal{L}_{0,n}^\epsilon$ of the quantum graph algebra at a root of unity $\epsilon$ of odd order, and show that $\mathcal{L}_{0,n}^\epsilon$ and its invariant algebra under the quantum group $U_\epsilon(\mathfrak{g})$ have classical fraction algebras which are central simple algebras of PI degrees that we compute.

The associative algebra of derivations of a group algebra

In this paper, necessary and sufficient conditions on a group algebra of a finitely generated group [Formula: see text] over a finite field [Formula: see text] are determined such that the set of derivations of the group algebra form an associative [Formula: see text]-algebra. The derivations of [Formula: see text] form a nontrivial associative [Formula: see text]-algebra if and only if [Formula: see text] has characteristic 2 and [Formula: see text] is the direct product of a finite abelian group of odd order with either a cyclic 2-group or an infinite cyclic group. In this special case, the Jacobson radical of the resulting [Formula: see text]-algebra is determined.

Next-to-eikonal corrected double graviton dressing and gravitational wave observables at OG2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O}\\left({G}^2\\right) $$\\end{document}

Following a recent proposal to describe inelastic eikonal scattering processes in terms of gravitationally dressed elastic eikonal amplitudes, we motivate a collinear double graviton dressing and investigate its properties. This is derived from a generalized Wilson line operator in the worldline formalism by integrating over fluctuations of the eikonal trajectories of external particles in gravitationally interacting theories. The dressing can be expressed as a product of exponential terms — a coherent piece with contributions to all odd orders in the gravitational coupling constant and a term quadratic in graviton modes, with the former providing classical gravitational wave observables. In particular, the coherent dressing involves Oκ3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O}\\left({\\kappa}^3\\right) $$\\end{document} subleading double graviton corrections to the Weinberg soft factor. We use this dressing to derive expressions for the waveform, radiative momentum spectrum and angular momentum. In a limiting case of the waveform, we derive the nonlinear memory effect resulting from the emission of nearly soft gravitons from a scattering process.

Fundamental role of nonlocal orders in 1D extended Bose-Hubbard model.

Nonlocal order parameters capture the presence of correlated fluctuations between specific degrees of freedom, in otherwise disordered quantum matter. Here, we provide a further example of their fundamental role, deriving the ground state phase diagram of the filling one extended Bose-Hubbard model, exclusively in terms of their ordering. By means of a density matrix renormalization group numerical analysis, we show that in addition to the (even) parity order characteristic of the Mott insulating phase and the string order nonvanishing in the Haldane insulator, the recently proposed odd parity order completes the picture, becoming nonvanishing at the transition from the normal superfluid to the paired superfluid phase. The above three nonlocal parameters capture all the distinct phases, including the density wave phase, in which the local order is seen as the simultaneous presence of correlated fluctuations in different channels. They provide a unique tool for the experimental observation of the full phase diagram of strongly correlated quantum matter, by means of local density measurements.

The Symmetric Dual Inductor Hybrid Converter for Direct 48V-to-PoL Conversion

This work introduces the symmetric dual inductor hybrid (SDIH) dc-dc converter topology, which is suitable for large conversion ratios where regulation is required, such as direct 48 V to point-of-load (PoL) applications. A dickson-type switched capacitor network is used to effectively produce two interleaved PWM outputs with a greatly reduced voltage amplitude relative to the input voltage, allowing the subsequent magnetic volume to be reduced while retaining modest switching frequencies. Distinct from related variations, part count is significantly reduced while both even and odd order switched capacitor networks can be used with straightforward split-phase control; allowing either network type to achieve complete soft-charging of all flying capacitors. Additionally, charge flow is uniformly distributed through all elements, with equal capacitor and inductor values being preferred. Subsequently this topology is expected to simplify component selection, improve electrical and thermal performance, and reduce cost. Furthermore, analysis is presented that calculates precise phase durations without making small ripple assumptions, revealing up to a 75% timing error in cases where either inductor or capacitor ripple is ignored. Finally, a discrete prototype validates this analysis and demonstrates very high measured power densities of 1,029 W/in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$^{3}$</tex-math></inline-formula> , 754 W/in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$^{3}$</tex-math></inline-formula> , and 663 W/in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$^{3}$</tex-math></inline-formula> for 48 V input and regulated output voltages of 3 V, 2 V, and 1 V, respectively, while switching at a frequency of 750 kHz.

On the Bloch eigenvalues, band functions and bands of the differential operator of odd order with the periodic matrix coefficients

On the Bloch eigenvalues, band functions and bands of the differential operator of odd order with the periodic matrix coefficients