Trivial Elements Research Articles

On homology planes and contractible 4‐manifolds

AbstractWe call a non‐trivial homology sphere a Kirby–Ramanujam sphere if it bounds both a homology plane and a Mazur or Poénaru manifold. In 1980, Kirby found the first example by proving that the boundary of the Ramanujam surface bounds a Mazur manifold and it has remained a single example since then. By tracing their initial step, we provide the first additional examples and we present three infinite families of Kirby–Ramanujam spheres. Also, we show that one of our families of Kirby–Ramanujam spheres is diffeomorphic to the splice of two certain families of Brieskorn spheres. Since this family of Kirby–Ramanujam spheres bound contractible 4‐manifolds, they lie in the class of the trivial element in the homology cobordism group; however, both splice components are separately linearly independent in that group.

A SOCIAL SEMIOTIC STUDY OF IMAGES IN FOOTBALL GAMES

This study delvesinto the realm of funny posters and images displayed by fans during the World Cup and Gulf Cup games, employing a social semiotic approach to interpret and understand their content, imagery, and cultural significance. The investigation focuses on how these images function as communicative tools that not only entertain but also convey deeper social and cultural messages to a global audience.The findings underscore the significance of visual literacy in contemporary media and offer a new perspective on the social dynamics of major sporting events. This research contributes to the fields of visual communication, cultural studies, and sports sociology by highlighting how seemingly trivial elements like fan posters can have profound social implications.

IDEMPOTENT ELEMENTS IN MATRIX RING OF ORDER 2 OVER POLYNOMIAL RING $\mathbb{Z}_{p^2q}[x

An idempotent element in the algebraic structure of a ring is an element that, when multiplied by itself, yields an outcome that remains unchanged and identical to the original element. Any ring with a unity element generally has two idempotent elements, 0 and 1, these particular idempotent elements are commonly referred to as the trivial idempotent elements However, in the case of rings $\mathbb{Z}_n$ and $\mathbb{Z}_n[x]$ it is possible to have non-trivial idempotent elements. In this paper, we will investigate the idempotent elements in the polynomial ring $\mathbb{Z}_{p^2q}[x]$ with $p,q$ different primes. Furthermore, the form and characteristics of non-trivial idempotent elements in $M_2(\mathbb{Z}_{p^2q}[x])$ will be investigated. The results showed that there are 4 idempotent elements in $\mathbb{Z}_{p^2q}[x]$ and 7 idempotent elements in $M_2(\mathbb{Z}_{p^2q}[x])$.

Free limits of free algebras

Consider a diagram ⋯→F3→F2→F1 of algebraic systems, where Fn denotes the free object on n generators and the connecting maps send the extra generator to some distinguished trivial element. We prove that (a) if the Fi are free associative algebras over a fixed field then the limit in the category of graded algebras is again free on a set of homogeneous generators; (b) on the other hand, the limit in the category of associative (ungraded) algebras is a free formal power series algebra on a set of homogeneous elements, and (c) if the Fi are free Lie algebras then the limit in the category of graded Lie algebras is again free.

Generalized torsion for hyperbolic 3‐manifold groups with arbitrary large rank

AbstractLet be a group and a non‐trivial element in . If some non‐empty finite product of conjugates of equals to the trivial element, then is called a generalized torsion element. To the best of our knowledge, we have no hyperbolic 3‐manifold groups with generalized torsion elements whose rank is explicitly known to be greater than two. The aim of this short note is to demonstrate that for a given integer there are infinitely many closed hyperbolic 3‐manifolds which enjoy the property: (i) the Heegaard genus of is , (ii) the rank of is , and (ii) has a generalized torsion element. Furthermore, we may choose as homology lens spaces and so that the order of the generalized torsion element is arbitrarily large.

Caustics of Lagrangian homotopy spheres with stably trivial Gauss map

For each positive integer $n$, we give a geometric description of the stably trivial elements of the group $\pi_n U_n/O_n$. In particular, we show that all such elements admit representatives whose tangencies with respect to a fixed Lagrangian plane consist only of folds. By the h-principle for the simplification of caustics, this has the following consequence: if a Lagrangian distribution is stably trivial from the viewpoint of a Lagrangian homotopy sphere, then by an ambient Hamiltonian isotopy one may deform the Lagrangian homotopy sphere so that its tangencies with respect to the Lagrangian distribution are only of fold type. Thus the stable triviality of the Lagrangian distribution, which is a necessary condition for the simplification of caustics to be possible, is also sufficient. We give applications of this result to the arborealization program and to the study of nearby Lagrangian homotopy spheres.

Deontología profesional de la abogacía, una aproximación comparada. Implicaciones del uso de los nuevos medios de comunicación digitales

It is not uncommon that the proper operation of a complex system depends on an element seemingly irrelevant. Private law practice is one of these apparently trivial elements that, actually, are essential for proper legal protection and rule of law. Hence, the importance of deontology as a touchstone of good professional practice. This paper studies from a comparative perspective how this issue is addressed at the international level and within both main surrounding legal systems: European Union and United States with regard to Spanish Legal System. Likewise, deontological treatment of digitization in the professional practice of law will be analysed.

Soft-self and Hard-cross Graph Attention Network for Knowledge Graph Entity Alignment

Knowledge Graph (KG) entity alignment aims to identify entities across different KGs that refer to the same real world object, and it is the key step towards KG integration and KG complement. Recently, Graph Attention Network (GAT) based models become a popular paradigm in entity alignment community owing to its ability in modeling structural data. But current GAT based models either ignore relation semantics and edge directions when learning entity neighbor representations or make no distinction between incoming neighbors and outgoing neighbors when calculating their attention scores. Furthermore, softmax functions utilized in soft attention mechanisms of current models always assign small but nonzero probabilities to trivial elements, which is unsuitable for learning alignment oriented entity embeddings. Taking these issues into account, this paper proposes a novel GAT based entity alignment model SHEA (Soft-self and Hard-cross Graph Attention Networks for Knowledge Graph Entity Alignment), which takes both relation semantics and edge directions into consideration when modeling single KG, and distinguishes prior aligned neighbors from the general ones to take full advantage of prior aligned information. Specifically, a type of four-channels graph attention layer is conceived to aggregate information from entity neighbors in different cases. The first two channels teach entities to aggregate information from their neighbors with soft-self attention, where both neighboring entities and the linked relations are used to obtain attention values. The other two channels teach entities to aggregate information from their neighbors with hard-cross graph attention, where tf_idf11tf_idf (term frequency–inverse document frequency) is a commonly used weighting technology in statistics based information retrieval. It measures the importance of a word with regard to a document. If we think of a pre-aligned neighbor as a word and look all pre-aligned neighbors for a center entity as a document, tf_idf is suitable for measuring the importance of each pre-aligned neighborhood. is utilized to measure the importance of entity neighbors. Extensive experiments on five publicly available datasets demonstrate our superior performances.

Generalized small cancellation conditions, non-positive curvature and diagrammatic reducibility

We present a metric condition $\TTMetric$ which describes the geometry of classical small cancellation groups and applies also to other known classes of groups such as two-dimensional Artin groups. We prove that presentations satisfying condition $\TTMetric$ are diagrammatically reducible in the sense of Sieradski and Gersten. In particular, we deduce that the standard presentation of an Artin group is aspherical if and only if it is diagrammatically reducible. We show that, under some extra hypotheses, $\TTMetric$-groups have quadratic Dehn functions and solvable conjugacy problem. In the spirit of Greendlinger's lemma, we prove that if a presentation P = 〈X| R〉 of group G satisfies conditions $\TTMetric -C'(\frac {1}{2})$, the length of any nontrivial word in the free group generated by X representing the trivial element in G is at least that of the shortest relator. We also introduce a strict metric condition $\TTMetricStrict$, which implies hyperbolicity.

Deep Neural Network Acceleration With Sparse Prediction Layers

The ever-increasing computation cost of Convolutional Neural Network (CNN) makes it imperative for real-world applications to accelerate the key steps especially the inference. In this work, we propose an efficient yet general scheme called Sparse Prediction Layer (SPL) which can predict and skip the trivial elements in the CNN layer. Pruned weights are used to predict the locations of maximum values in max-pooling kernels and those of positive values before Rectified Linear Units (ReLUs). Thereafter, the precise values of these predicted important elements are calculated selectively and the complete outputs are restored from them. Our experiments on ImageNet Large Scale Visual Recognition Competition (ILSVRC) 2012 show that SPL can reduce 68.3%, 58.6% and 59.5% Floating-point Operations (FLOPs) on AlexNet, VGG-16 and ResNet-50, respectively, within an accuracy loss of less than 1% without retraining. The proposed SPL scheme can further accelerate these networks pruned by other pruning-based methods, such as a FLOP reduction of 50.2% on the ResNet-50 which has been pruned by Channel Pruning (CP) before being applied with SPLs. A special matrix multiplication called Sparse Result Matrix Multiplication (SRMM) is proposed to support the implementation of SPL, and its acceleration effect is in line with expectations.

Mixed threefolds isogenous to a product

In this paper, we study threefolds isogenous to a product of mixed type i.e. quotients of a product of three compact Riemann surfaces [Formula: see text] of genus at least two by the action of a finite group [Formula: see text], which is free, but not diagonal. In particular, we are interested in the systematic construction and classification of these varieties. Our main result is the full classification of threefolds isogenous to a product of mixed type with [Formula: see text] in the absolutely faithful case, i.e. any automorphism in [Formula: see text], which restricts to the trivial element in [Formula: see text] for some [Formula: see text], is the identity on the product. Since the holomorphic Euler–Poincaré-characteristic of a smooth threefold of general type with ample canonical class is always negative, these examples lie on the boundary, in the sense of threefold geography. To achieve our result we use techniques from computational group theory. Indeed, we develop a MAGMA algorithm to classify these threefolds for any given value of [Formula: see text] in the absolutely faithful case.

Four dimensional topological quantum field theories from $G$-crossed braided categories

We construct a state-sum type invariant of smooth closed oriented 4-manifolds out of a G -crossed braided spherical fusion category ( G -BSFC) for G a finite group. The construction can be extended to obtain a (3+1) -dimensional topological quantum field theory (TQFT). The invariant of 4-manifolds generalizes several known invariants in literature such as the Crane–Yetter invariant from a ribbon fusion category and Yetter's invariant from homotopy 2-types. If the G -BSFC is concentrated only at the sector indexed by the trivial group element, a cohomology class in H^4(G,U(1)) can be introduced to produce a different invariant, which reduces to the twisted Dijkgraaf–Witten theory in a special case. Although not proven, it is believed that our invariants are strictly different from other known invariants. It remains to be seen if the invariants are sensitive to smooth structures. It is expected that the most general input to the state-sum type construction of (3+1) -TQFTs is a spherical fusion 2-category. We show that a G -BSFC corresponds to a monoidal 2-category with certain extra structure, but that structure does not satisfy all the axioms of a spherical fusion 2-category given by M. Mackaay. Thus the question of what axioms properly define a spherical fusion 2-category is open.

‘Bitter Enemies of the Sporting Fraternity’: The War Precautions Act and the Restriction of Horse-Racing in Australia during World War One

The First World War as a form of total warfare required belligerent governments to take centralized control of society in order to better prosecute their respective war efforts. In Australia, this was enforced through the War Precautions Act, an item of legislation which greatly expanded executive power and restricted civil liberties. Among the various regulations emplaced on society came restrictions on entertainments, including sport. This paper focuses on the curtailment of sport on the Australian home front, principally horse-racing, as an example of government authoritarianism in wartime and the turmoil it fuelled. These regulations illuminate the government’s willingness to control even the trivial elements of Australian life, as well as its deteriorating relationship with factions of the Australian populace. Unlike other studies, however, this paper uncovers a broad level of discontent among racing institutions at either end of the social spectrum. Though the middle classes are said to have supported the government in its strict execution of the war effort, socially prestigious clubs expressed resentment toward the repressive methods of government that matched working class discontent. Such evidence brings into question the clear dividing lines used to explain social conflict on the home front during the war.

Próba wyczerpania pamięci. Funkcja listy w pisarstwie Georges’a Pereca

The aim of the article is to analyse the function of the list in the works of Georges Perec, a French writer of the second half of the twentieth century. Writing by enumeration was one of the most important literary strategies practiced by the author and it took various forms depending on the specific text. Enumeration in Perec’s work can thus be perceived as a mnemontechnical tool, thanks to which it becomes a way to force one's memory to remember what is forgotten. This mne-motechnical aspect will be particularly important in the literary project called Lieux où j’ai dormi. Simultaneously, the creation of literary lists and enumerations can be linked to author’s need to control his surrounding reality. From this perspective, the list gives the illusion of double control. On the one hand, it fights the obliteration of traces of the past, and on the other, by recording even the most trivial elements of the reality, it seems to be a way to consolidate it.

Detecting trivial elements of periodic quotient of hyperbolic groups

Detecting trivial elements of periodic quotient of hyperbolic groups

Detecting trivial elements of periodic groups

In this article we give a sufficient and necessary condition to determine wether or not an element of the free group induces a non-trivial element of the free Burnside group of sufficiently large odd exponent. This criterion can be stated without any knowledge about Burnside groups, in particular about the proof of its infiniteness. Therefore it provides a useful tool that we will use later to study outer automorphisms of Burnside groups. We also state an analogue result for periodic quotients of torsion-free hyperbolic groups.

Poetics and Pâtisserie

In this paper, I explore how the sensory experiences provided -- or, at least, promised -- by the croissant inflect a range of performances. My investigation draws on personal experience, public performances, and fieldwork interviews with both pâtissiers and consumers.Brazilian anthropologist Roberto Da Matta stressed that the study of ritual should address how trivial elements of one’s social world can be transformed into symbols which, in certain contexts, construct moments of special intensity. I draw on Da Matta’s work to investigate how and why croissant consumption can form the centrepiece of ritualised processes. My analysis of pâtissiers’ narratives about their croissant-making reveals the importance of a gustatory taste imaginary. Descriptions of their learning experiences and motivations allows for parallels to be made between croissant preparation and participation in the more conventionally-conceived performance form of music. I also consider how gustatory taste vis-à-vis the croissant can inform notions of discriminatory taste, and then interrogate ideas concerning class and connoisseurship which can result in (admittedly idiosyncratic) manifestations of Pierre Bourdieu’s concept of “cultural capital”.My paper is book-ended by accounts of two performances of croissant-eating given in very different circumstances – one was a personal experience and illuminates complex familial relations; the other took place in highly charged circumstances before the international media and was disseminated around the world.Overall, I present the richness of the croissant as a performative entity and demonstrate how distinct performances bring different colours to the pastry.

Level sets of $$(p,e-p)$$(p,e-p) outer generalized pseudo spectrum

Let A be a complex Banach algebra with unit e. Let p be a non trivial idempotent element in A and e>0. For a∈A, it is proved that the interior of the level set of (p,e−p)−e pseudo spectrum of a is empty in the unbounded component of (p,e−p) resolvent set of a. An example is constructed to show that the condition ‘unbounded component’ can not be dropped. Further, it is proved this ‘unbounded component’ can be dropped in the case when A is B(X) where X is a complex uniformly convex Banach space. That is, if T∈B(X) then interior of the level set of (p,I−p)−e pseudo spectrum is empty in (p,I−p) resolvent set of T.

Balanced finite presentations of the trivial group

We construct a sequence of balanced finite presentations of the trivial group with two generators and two relators with the following property: The minimal number of relations required to demonstrate that a generator represents the trivial element grows faster than the tower of exponentials of any fixed height of the length of the finite presentation.

Trivial center element and Coulombic potential of the thick center vortex model

The thick center vortex potentials in the SU(3) gauge group have been calculated by means of the modified inter-quark potential which consists of two terms. One term is the result of the area law fall-off for the large Wilson loop which leads to the linear potentials. The second term represents vacuum fluctuations leading to the perimeter law fall-off believed to contain the trivial center element. We introduce a new Gaussian flux limited to vary in a finite region of space which causes the corresponding group factor to have only some small deviations from the trivial center element. So, this flux increases the role of the trivial center element and W0 is enhanced in the induced potential of the model at small quark separations. Using both trivial and non-trivial center elements in the potential between static color sources, results in the correct 3-ality dependence at large quark separations and a very good agreement with Casimir scaling at short and intermediate distances. In fact, the ratios of the potential of each representation to that of the fundamental one have been improved — in comparison with the previous work on the short distance potentials, remarkably. So, one might use the thick center vortex model to describe the inter-quark potential of every regime.