scholarly journals Catacondensed Chemical Hexagonal Complexes: A Natural Generalisation of Benzenoids

2020 ◽  
Vol 93 (4) ◽  
Author(s):  
Cate S. Anstöter ◽  
Nino Bašić ◽  
Patrick W. Fowler ◽  
Tomaž Pisanski
Keyword(s):  
Natural Generalisation

Multi-particle Spaces

2020 ◽  
pp. 187-200
Author(s):  
Daniel Canarutto
Keyword(s):  
Operator Algebra ◽  
Particle Physics ◽  
Arbitrary Number ◽  
Exterior Algebra ◽  
Linear Combinations ◽  
Absorption And Emission ◽  
Dirac Type ◽  
Natural Generalisation ◽  
Finite Linear ◽  
Number Of Particles

The fundamental algebraic notions needed in many-particle physics are exposed. Spaces of free states containing an arbitrary number of particles of many types are introduced. The operator algebra generated by absorption and emission operators is studied as a natural generalisation of standard exterior algebra. The link between the discrete and the distributional formalisms is provided by the spaces of finite linear combinations of semi-densities of Dirac type.

ON A GENERALISATION OF A RESTRICTED SUM FORMULA FOR MULTIPLE ZETA VALUES AND FINITE MULTIPLE ZETA VALUES

2019 ◽  
Vol 101 (1) ◽  
pp. 23-34
Author(s):  
HIDEKI MURAHARA ◽  
TAKUYA MURAKAMI
Keyword(s):  
Linear Relation ◽  
Analogous Result ◽  
Multiple Zeta Values ◽  
Multiple Zeta ◽  
Zeta Values ◽  
Natural Generalisation ◽  
Sum Formula

We prove a new linear relation for multiple zeta values. This is a natural generalisation of the restricted sum formula proved by Eie, Liaw and Ong. We also present an analogous result for finite multiple zeta values.

scholarly journals On Higher Special Elements of p-adic Representations

2020 ◽  
Author(s):  
David Burns ◽  
Takamichi Sano ◽  
Kwok-Wing Tsoi
Keyword(s):  
Galois Cohomology ◽  
Euler System ◽  
Cohomology Groups ◽  
Exterior Power ◽  
Concrete Application ◽  
Natural Generalisation ◽  
Higher Rank ◽  
Derivatives Of

Abstract As a natural generalisation of the notion of “higher rank Euler system”, we develop a theory of “higher special elements” in the exterior power biduals of the Galois cohomology of $p$-adic representations. We show, in particular, that such elements encode detailed information about the structure of Galois cohomology groups and are related by families of congruences involving natural height pairings on cohomology. As a first concrete application of the approach, we use it to refine, and extend, a variety of existing results and conjectures concerning the values of derivatives of Dirichlet $L$-series.

scholarly journals A note on principal sequences

1964 ◽  
Vol 6 (3) ◽  
pp. 133-135 ◽  
Author(s):  
B. Rotman
Keyword(s):  
Fundamental Problem ◽  
Real Numbers ◽  
The Real ◽  
Natural Generalisation ◽  
Number Class

A fundamental problem in the theory of ordinals is the assignation of principal sequences to limit numbers of the second number class.It is our main object here to show that a certain class of methods, which are a natural generalisation of those used in the solution of the corresponding problem for the real numbers (the description of which we omit), must fail to solve the problem. The methods are those which rest on the following assumption: the principal sequence assigned to any limit number of the second number class is determined once the first i terms of that sequence are known.

scholarly journals Parameter Dependent Optimal Thresholds, Indifference Levels and Inverse Optimal Stopping Problems

2014 ◽  
Vol 51 (02) ◽  
pp. 492-511
Author(s):  
Martin Klimmek
Keyword(s):  
Optimal Stopping ◽  
Infinite Horizon ◽  
Gittins Index ◽  
Dynamic Allocation ◽  
One Dimensional ◽  
Optimal Thresholds ◽  
The Family ◽  
Natural Generalisation ◽  
Optimal Stopping Problems ◽  
Parameter Dependent

Consider the classic infinite-horizon problem of stopping a one-dimensional diffusion to optimise between running and terminal rewards, and suppose that we are given a parametrised family of such problems. We provide a general theory of parameter dependence in infinite-horizon stopping problems for which threshold strategies are optimal. The crux of the approach is a supermodularity condition which guarantees that the family of problems is indexable by a set-valued map which we call the indifference map. This map is a natural generalisation of the allocation (Gittins) index, a classical quantity in the theory of dynamic allocation. Importantly, the notion of indexability leads to a framework for inverse optimal stopping problems.

scholarly journals A new exponential separation between quantum and classical one-way communication complexity

2011 ◽  
Vol 11 (7&8) ◽  
pp. 574-591
Author(s):  
Ashley Montanaro
Keyword(s):  
Fourier Analysis ◽  
Boolean Function ◽  
Communication Complexity ◽  
Quantum Communication ◽  
Membership Problem ◽  
Classical Communication ◽  
Exponential Separation ◽  
Quantum Communication Complexity ◽  
Natural Generalisation ◽  
Subgroup Membership Problem

We present a new example of a partial boolean function whose one-way quantum communication complexity is exponentially lower than its one-way classical communication complexity. The problem is a natural generalisation of the previously studied Subgroup Membership problem: Alice receives a bit string $x$, Bob receives a permutation matrix $M$, and their task is to determine whether $Mx=x$ or $Mx$ is far from $x$. The proof uses Fourier analysis and an inequality of Kahn, Kalai and Linial.

scholarly journals Groups isomorphic to their non-nilpotent subgroups

1998 ◽  
Vol 40 (2) ◽  
pp. 257-262 ◽  
Author(s):  
Howard Smith ◽  
James Wiegold
Keyword(s):  
Finite Groups ◽  
Proper Subgroup ◽  
Complete Classification ◽  
Finitely Generated ◽  
Infinite Groups ◽  
Finite Case ◽  
Locally Nilpotent ◽  
Infinite Case ◽  
Natural Generalisation ◽  
Abelian Subgroups

We were concerned in [12] with groups G that are isomorphic to all of their non-abelian subgroups. In order to exclude groups with all proper subgroups abelian, which are well understood in the finite case [7] and which include Tarski groups in the infinite case, we restricted attention to the class X of groups G that are isomorphic to their nonabelian subgroups and that contain proper subgroups of this type; such groups are easily seen to be 2-generator, and a complete classification was given in [12, Theorem 2] for the case G soluble. In the insoluble case, G/Z(G) is infinite simple [12; Theorem 1], though not much else was said in [12] about such groups. Here we examine a property which represents a natural generalisation of that discussed above. Let us say that a group G belongs to the class W if G is isomorphic to each of its non-nilpotent subgroups and not every proper subgroup of G is nilpotent. Firstly, note that finite groups in which all proper subgroups are nilpotent are (again) well understood [9]. In addition, much is known about infinite groups with all proper subgroups nilpotent (see, in particular, [8] and [13] for further discussion) although, even in the locally nilpotent case, there are still some gaps in our understanding of such groups. We content ourselvesin the present paper with discussing finitely generated W-groups— note that a W-group is certainly finitely generated or locally nilpotent. We shall have a little more to say about the locally nilpotent case below.

scholarly journals Nonlinear Young Differential Equations: A Review

2021 ◽  
Author(s):  
Lucio Galeati
Keyword(s):  
Differential Equations ◽  
Banach Spaces ◽  
Numerical Schemes ◽  
Infinite Dimensional ◽  
Versatile Tool ◽  
Natural Generalisation

AbstractNonlinear Young integrals have been first introduced in Catellier and Gubinelli (Stoch Process Appl 126(8):2323–2366, 2016) and provide a natural generalisation of classical Young ones, but also a versatile tool in the pathwise study of regularisation by noise phenomena. We present here a self-contained account of the theory, focusing on wellposedness results for abstract nonlinear Young differential equations, together with some new extensions; convergence of numerical schemes and nonlinear Young PDEs are also treated. Most results are presented for general (possibly infinite dimensional) Banach spaces and without using compactness assumptions, unless explicitly stated.

scholarly journals Safety in s-t Paths, Trails and Walks

Algorithmica ◽  
2021 ◽  
Author(s):  
Massimo Cairo ◽  
Shahbaz Khan ◽  
Romeo Rizzi ◽  
Sebastian Schmidt ◽  
Alexandru I. Tomescu
Keyword(s):  
Directed Graph ◽  
Data Structures ◽  
Linear Time ◽  
Compact Representation ◽  
Complex Data ◽  
Graph Problems ◽  
The Third ◽  
Natural Generalisation ◽  
Basic Graph ◽  
Time Linear

AbstractGiven a directed graph G and a pair of nodes s and t, an s-tbridge of G is an edge whose removal breaks all s-t paths of G (and thus appears in all s-t paths). Computing all s-t bridges of G is a basic graph problem, solvable in linear time. In this paper, we consider a natural generalisation of this problem, with the notion of “safety” from bioinformatics. We say that a walk W is safe with respect to a set $${\mathcal {W}}$$ W of s-t walks, if W is a subwalk of all walks in $${\mathcal {W}}$$ W . We start by considering the maximal safe walks when $${\mathcal {W}}$$ W consists of: all s-t paths, all s-t trails, or all s-t walks of G. We show that the solutions for the first two problems immediately follow from finding all s-t bridges after incorporating simple characterisations. However, solving the third problem requires non-trivial techniques for incorporating its characterisation. In particular, we show that there exists a compact representation computable in linear time, that allows outputting all maximal safe walks in time linear in their length. Our solutions also directly extend to multigraphs, except for the second problem, which requires a more involved approach. We further generalise these problems, by assuming that safety is defined only with respect to a subset of visible edges. Here we prove a dichotomy between the s-t paths and s-t trails cases, and the s-t walks case: the former two are NP-hard, while the latter is solvable with the same complexity as when all edges are visible. We also show that the same complexity results hold for the analogous generalisations of s-tarticulation points (nodes appearing in all s-t paths). We thus obtain the best possible results for natural “safety”-generalisations of these two fundamental graph problems. Moreover, our algorithms are simple and do not employ any complex data structures, making them ideal for use in practice.

scholarly journals Free Fibonacci algebras

1989 ◽  
Vol 40 (2) ◽  
pp. 235-241 ◽  
Author(s):  
D.L. Jonnson ◽  
A.C. Kim
Keyword(s):  
Abelian Groups ◽  
Free Objects ◽  
One Stage ◽  
Natural Generalisation

Fibonacci varieties were introduced by one of us in 1978 and a natural generalisation was studied shortly afterwards. We carry this investigation one stage further by giving a description of the free objects in these varieties. This is done in terms of the n-abelian groups of Levi.

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