On relative counterpart of Auslander’s conditions

It is now well known that the conditions used by Auslander to define the Gorenstein projective modules on Noetherian rings are independent. Recently, Ringel and Zhang adopted a new approach in investigating Auslander’s conditions. Instead of looking for examples, they investigated rings on which certain implications between Auslander’s conditions hold. In this paper, we investigate the relative counterpart of Auslander’s conditions. So, we extend Ringel and Zhang’s work and introduce other concepts. Namely, for a semidualizing module [Formula: see text], we introduce weakly [Formula: see text]-Gorenstein and partially [Formula: see text]-Gorenstein rings as rings representing relations between the relative counterpart of Auslander’s conditions. Moreover, we introduce a relative notion of the well-known Frobenius category. We show how useful are [Formula: see text]-Frobenius categories in characterizing weakly [Formula: see text]-Gorenstein and partially [Formula: see text]-Gorenstein rings.

In quest of higher dimensions – Superstring Theory and the Calabi – Yau manifolds

Higher dimensions are impossible to visualize as the size of dimension varies inversely proportional to its level. The more the dimension ranges, the least its size. We are a set of points living in a particular point of space and a particular frame of time. i.e, we live in space-time. The space has more dimensions that meets the human eye. We are living in a world of hyper-space. Our world being a smaller dimension is floating in higher dimensions. The quest for the visually of higher dimensions has been a fantasy to mankind but this aspect of nature is completely locked. We can transform dimensions i.e., from higher to lower dimensions, or from lower to higher dimensions, but only through mathematics. The relative notion of mathematics helps us to do the thing, which is perhaps impossible in the experimental part of physical reality. Humans being an element of 3 Dimensions – length, breath, height can only perceive one higher dimensions, that is space-time. but beyond that the notion of dimension itself changes. The dimensions got curled up in every intersection of the coordinates of space in such a way that the higher dimensions remain stable to us. But in reality it is highly unstable. In the higher dimensions, above 4, the space is tearing apart and joining again spontaneously, but the tearing portion itself covered by 2 dimensional Branes which acts as a stabilizer for the unstable dimensions. Dimensions will get smaller and smaller with the space-time interwoven in it. But at Planks length that is 10^-33 meter, the notion of space-time itself breaks down thereby making impossible for the higher dimensions to coexist along with space. Without space, there will be no identity of any dimension. The space itself is the fabric for the milestone of residing higher dimensions. Imagine our room, which is 3 dimensional. But what is there inside the room. The space and of course the time. Space-time being a totally separate entity is not quite separate when compared with other dimensions because it makes the residing place for the higher dimensions or the hyperspace itself. We all are confined within a lower dimensional world within a randomness of higher dimensions. Time being alike like space is an arrow which has the capability of slicing space into different forms. Thereby taking a snapshot of our every nano-second we vibrate within space-time. As each slice of time represents each slice of space, similarly each slice of space represents each slice of time. The nature of space-time is beyond human consciousness. It is the identity by which we breathe, we play, we survive. It is the whole localization of species that encompasses itself with space thereby making space-time a relative quantity depending upon the reference frame. The only thing that can encompass space-time or even change the relative definition of space-time is the speed, the speed far beyond the speed of light. The more the speed, the less the array of time flows. Space-time being an invisible entity makes the other dimensions visible residing in it only into the level of 3, that is l, b, h. After that there is a infamous structure formed by the curling of higher dimensions called CALABI-YAU manifold. This manifold depicts the usual nature of the dimensional quadrants of the higher order by containing a number of small spherical spheres inside it. The mathematics of string theory is still unable to solve the genus and the containing spheres of the manifold which can be the ultimate quest for the hidden dimensions. Hidden, as, the higher dimensions are hidden from human perspective of macro level but if we probe deeper into the fabric of the space-time of General Relativity then we will find the 5th dimension according to the Kaluza-Klein theory. And if we probe even deeper into it at the perspective of string theory we will be amazed to see the real nature of quantum world. They are so marvelously beautiful, they contain so many forms of higher dimensions ranging from 6 to 10. And even many more of that, but we are still not sure about it where they may exist in a ghost state. After all, the quantum nature is far more beautiful that one can even imagine with a full faze of weirdness.

In quest of higher dimensions – Superstring Theory and the Calabi – Yau manifolds

Higher dimensions are impossible to visualize as the size of dimension varies inversely proportional to its level. The more the dimension ranges, the least its size. We are a set of points living in a particular point of space and a particular frame of time. i.e, we live in space-time. The space has more dimensions that meets the human eye. We are living in a world of hyper-space. Our world being a smaller dimension is floating in higher dimensions. The quest for the visually of higher dimensions has been a fantasy to mankind but this aspect of nature is completely locked. We can transform dimensions i.e., from higher to lower dimensions, or from lower to higher dimensions, but only through mathematics. The relative notion of mathematics helps us to do the thing, which is perhaps impossible in the experimental part of physical reality. Humans being an element of 3 Dimensions – length, breath, height can only perceive one higher dimensions, that is space-time. but beyond that the notion of dimension itself changes. The dimensions got curled up in every intersection of the coordinates of space in such a way that the higher dimensions remain stable to us. But in reality it is highly unstable. In the higher dimensions, above 4, the space is tearing apart and joining again spontaneously, but the tearing portion itself covered by 2 dimensional Branes which acts as a stabilizer for the unstable dimensions. Dimensions will get smaller and smaller with the space-time interwoven in it. But at Planks length that is 10^-33 meter, the notion of space-time itself breaks down thereby making impossible for the higher dimensions to coexist along with space. Without space, there will be no identity of any dimension. The space itself is the fabric for the milestone of residing higher dimensions. Imagine our room, which is 3 dimensional. But what is there inside the room. The space and of course the time. Space-time being a totally separate entity is not quite separate when compared with other dimensions because it makes the residing place for the higher dimensions or the hyperspace itself. We all are confined within a lower dimensional world within a randomness of higher dimensions. Time being alike like space is an arrow which has the capability of slicing space into different forms. Thereby taking a snapshot of our every nano-second we vibrate within space-time. As each slice of time represents each slice of space, similarly each slice of space represents each slice of time. The nature of space-time is beyond human consciousness. It is the identity by which we breathe, we play, we survive. It is the whole localization of species that encompasses itself with space thereby making space-time a relative quantity depending upon the reference frame. The only thing that can encompass space-time or even change the relative definition of space-time is the speed, the speed far beyond the speed of light. The more the speed, the less the array of time flows. Space-time being an invisible entity makes the other dimensions visible residing in it only into the level of 3, that is l, b, h. After that there is a infamous structure formed by the curling of higher dimensions called CALABI-YAU manifold. This manifold depicts the usual nature of the dimensional quadrants of the higher order by containing a number of small spherical spheres inside it. The mathematics of string theory is still unable to solve the genus and the containing spheres of the manifold which can be the ultimate quest for the hidden dimensions. Hidden, as, the higher dimensions are hidden from human perspective of macro level but if we probe deeper into the fabric of the space-time of General Relativity then we will find the 5th dimension according to the Kaluza-Klein theory. And if we probe even deeper into it at the perspective of string theory we will be amazed to see the real nature of quantum world. They are so marvelously beautiful, they contain so many forms of higher dimensions ranging from 6 to 10. And even many more of that, but we are still not sure about it where they may exist in a ghost state. After all, the quantum nature is far more beautiful that one can even imagine with a full faze of weirdness.

An analytical and experimental comparison of maximal lottery schemes

Maximal lottery ($$ ML $$ ML ) schemes constitute an interesting class of randomized voting rules that were proposed by Peter Fishburn in 1984 and have been repeatedly recommended for practical use. However, the subtle differences between different $$ ML $$ ML schemes are often overlooked. Two canonical subsets of $$ ML $$ ML schemes are "Image missing" schemes (which only depend on unweighted majority comparisons) and "Image missing" schemes (which only depend on weighted majority comparisons). We prove that "Image missing" schemes are the only homogeneous $$ ML $$ ML schemes that satisfy $$ SD $$ SD -efficiency and $$ SD $$ SD -participation, but are also among the most manipulable $$ ML $$ ML schemes. While all $$ ML $$ ML schemes are manipulable and even violate monotonicity, they are never manipulable when a Condorcet winner exists and satisfy a relative notion of monotonicity. We also evaluate the frequency of manipulable preference profiles and the degree of randomization of $$ ML $$ ML schemes via extensive computer simulations. In summary, $$ ML $$ ML schemes are rarely manipulable and often do not randomize at all, especially for few alternatives. The average degree of randomization of "Image missing" schemes is consistently lower than that of "Image missing" schemes.

Shake slice and shake concordant links

We can construct a [Formula: see text]-manifold by attaching [Formula: see text]-handles to a [Formula: see text]-ball with framing [Formula: see text] along the components of a link in the boundary of the [Formula: see text]-ball. We define a link as [Formula: see text]-shake slice if there exists embedded spheres that represent the generators of the second homology of the [Formula: see text]-manifold. This naturally extends [Formula: see text]-shake slice, a generalization of slice that has previously only been studied for knots, to links of more than one component. We also define a relative notion of shake[Formula: see text]-concordance for links and versions with stricter conditions on the embedded spheres that we call strongly[Formula: see text]-shake slice and strongly[Formula: see text]-shake concordance. We provide infinite families of links that distinguish concordance, shake concordance, and strong shake concordance. Moreover, for [Formula: see text] we completely characterize shake slice and shake concordant links in terms of concordance and string link infection. This characterization allows us to prove that the first non-vanishing Milnor [Formula: see text] invariants are invariants of shake concordance. We also argue that shake concordance does not imply link homotopy.

Skills, Efficiency, and Timing in a Simple Attack and Defense Model

In this paper, we consider a simple two-player attack and defense model, focusing on the role of players’ abilities and choice timing. Abilities are divided into skills and efficiency, where the former is an absolute notion and the latter a relative notion of ability. Timing is investigated by comparing players’ investments in a simultaneous and a Stackelberg game. In the simultaneous game, the Nash Equilibrium investment level in attack and defense resources is symmetric, increasing in the skills but non-monotonic in the relative efficiency. In the Stackelberg game, the equilibrium investment levels are asymmetric, increasing in the skills, but with their ranking affected only by the relative efficiency. Therefore, interestingly, players’ choice is mostly characterized by players’ relative efficiency rather than by their skills, in regards to timing.

Furstenberg boundaries for pairs of groups

Furstenberg has associated to every topological group $G$ a universal boundary $\unicode[STIX]{x2202}(G)$ . If we consider in addition a subgroup $H<G$ , the relative notion of $(G,H)$ -boundaries admits again a maximal object $\unicode[STIX]{x2202}(G,H)$ . In the case of discrete groups, an equivalent notion was introduced by Bearden and Kalantar (Topological boundaries of unitary representations. Preprint, 2019, arXiv:1901.10937v1) as a very special instance of their constructions. However, the analogous universality does not always hold, even for discrete groups. On the other hand, it does hold in the affine reformulation in terms of convex compact sets, which admits a universal simplex $\unicode[STIX]{x1D6E5}(G,H)$ , namely the simplex of measures on $\unicode[STIX]{x2202}(G,H)$ . We determine the boundary $\unicode[STIX]{x2202}(G,H)$ in a number of cases, highlighting properties that might appear unexpected.

„Much book, little know“ – Enzyklopädie als Lederstrumpf

Though James Fenimore Cooper’s five Leatherstocking-Tales (1823–1841) seem unlikely candidates for an encyclopaedic poetics, they encompass frontier life in an extensive scope, integrating a wide range of sources as well as using immense descriptive detail that often enough overgrows their plot. Even more to the point is the often voiced scepticism towards academic knowledge. By taking the tales as a test case for the ‘encyclopaedic novel‘, I explore how encyclopaedic reference shifts in the genesis of these five novels. In addition, a survey of contemporary publishing shows a similar fluidity in the various bearings that encyclopaedic forms could take on at the time. Given the double flexibility of the status and formation of encyclopaedic knowledge, both in the Leatherstocking tales and in contemporary discourse, the article suggests a less strict, more profane and popularly grounded, as well as more historically relative notion of what constitutes an encyclopaedic narrative. As to the Leatherstocking tales, I suggest that they occupy an epistemic space that was then only recently given up by academic encyclopaedic thought, and that they do so by claiming an encyclopaedic function of their own.