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The purpose of this paper is to study the relation between the |$C^0$| -topology and the topology induced by the spectral norm on the group of Hamiltonian diffeomorphisms of a closed symplectic manifold. Following the approach of Buhovsky–Humilière–Seyfaddini, we prove the |$C^0$| -continuity of the spectral norm for complex projective spaces and negative monotone symplectic manifolds. The case of complex projective spaces provides an alternative approach to the |$C^0$| -continuity of the spectral norm proven by Shelukhin. We also prove a partial |$C^0$| -continuity of the spectral norm for rational symplectic manifolds. Some applications such as the Arnold conjecture in the context of |$C^0$| -symplectic topology are also discussed. [ABSTRACT FROM AUTHOR]

Mustafin varieties are well-studied degenerations of projective spaces induced by a choice of integral points in a Bruhat–Tits building. In recent work, Annette Werner and the author initiated the study of degenerations of plane curves obtained by Mustafin varieties by means of arithmetic geometry. Moreover, we applied these techniques to construct models of vector bundles on plane curves with strongly semistable reduction. In this work, we take a Groebner basis approach to the more general problem of studying degenerations of projective varieties. Our methods include determining the behaviour of Groebner bases under substitution over unique factorisation rings. Finally, we outline applications to the adic Simpson correspondence, when the respective projective variety is a curve. [ABSTRACT FROM AUTHOR]

In this paper, we study the rigidity theorem of closed minimally immersed Legendrian submanifolds in the unit sphere. Utilizing the maximum principle, we obtain a new characterization of the Calabi torus in the unit sphere which is the minimal Calabi product Legendrian immersion of a point and the totally geodesic Legendrian sphere. We also establish an optimal Simons' type integral inequality in terms of the second fundamental form of three-dimensional closed minimal Legendrian submanifolds in the unit sphere. Our optimal rigidity results for minimal Legendrian submanifolds in the unit sphere are new and also can be applied to minimal Lagrangian submanifolds in the complex projective space. [ABSTRACT FROM AUTHOR]

Copyright of Al-Mustansiriyah Journal of Science is the property of Republic of Iraq Ministry of Higher Education & Scientific Research (MOHESR) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We consider weak Fano manifolds with small contractions obtained by blowing up successively curves and subvarieties of codimension 2 in the product of a projective space and a projective line. We give a classification result for a special case. We determine the nef cones and describe the extremal contractions for related smooth projective varieties. [ABSTRACT FROM AUTHOR]

The major goals of this workshop are to lay paths for a systematic study of geproci (and related, e.g., projecting to almost complete intersections or full intersections) sets of points in projective spaces, study algebraic properties of their ideals (e.g. in the spirit of the Cayley-Bacharach properties), and to identify the most promising new directions for study. [ABSTRACT FROM AUTHOR]

Some cohomology elements, called ν -classes, as a supergeneralization of universal Chern classes, are introduced for canonical super line bundles over ν -projective spaces, a novel supergeometric generalization of projective spaces. It is shown that these classes may be described by analytic representatives of elements of generalized de Rham cohomology. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to prove the finiteness theorems for meromorphic mappings of a complete connected Kähler manifold into a projective space sharing few hyperplanes in subgeneral position without counting multiplicity, where all zeros with multiplicities more than a certain number are omitted. Our results are extensions and generalizations of some recent ones. [ABSTRACT FROM AUTHOR]

Let M¯0,n(G(r,V),d)$\overline{M}_{0,n}(G(r,V), d)$ be the coarse moduli space of stable degree d maps from n‐pointed genus 0 curves to a Grassmann variety G(r,V)$G(r,V)$. We provide a recursive method for the computation of the Hodge numbers and the Betti numbers of M¯0,n(G(r,V),d)$\overline{M}_{0,n}(G(r,V), d)$ for all n and d. Our method is a generalization of Getzler and Pandharipande's work for maps to projective spaces. [ABSTRACT FROM AUTHOR]

We provide new answers about the distribution of mass on spheres so as to minimize energies of pairwise interactions. We find optimal measures for the pframe energies, i.e., energies with the kernel given by the absolute value of the inner product raised to a positive power p. Application of linear programming methods in the setting of projective spaces allows for describing the minimizing measures in full in several cases: we show optimality of tight designs and of the 600-cell for several ranges of p in different dimensions. Our methods apply to a much broader class of potential functions, namely, those which are absolutely monotonic up to a particular order [ABSTRACT FROM AUTHOR]

In this study, all embeddings that map each line of projective plane order 4 to an oval of Projective Space of dimensional 4 will be investigated and it was shown that the image of these maps generate projective spaces P G(4, 4). [ABSTRACT FROM AUTHOR]

We define a Euler characteristic χ (X , G) for a finite cell complex X with a finite group G acting cellularly on it. Then, each K i (X) (a complex vector space with basis the i-cells of X) is a representation of G, and we define χ (X , G) to be the alternating sum of the representations K i (X) , as elements of the representation ring R (G) of G. By adapting the ordinary proof that the alternating sum of the dimensions of the chain complexes is equal to the alternating sum of the dimensions of the homology groups, we prove that there is another definition of χ (X , G) with the alternating sum of the representations H i (X) , again as elements of the representation ring R (G) . We also show that the character of this virtual representation χ (X , G) , with respect to a given element g, is just the ordinary Euler characteristic of the fixed-point set by this element. Finally, we give a topological proof of a version of Artin's induction theorem. More precisely, we show that, if G is a group with an irreducible representation of dimension greater than 1, then each character of G is a linear combination with rational coefficients of characters induced up from characters of proper subgroups of G. [ABSTRACT FROM AUTHOR]

Spun normal surfaces are a useful way of representing proper essential surfaces, using ideal triangulations for 3-manifolds with tori and Klein bottle boundaries. In this paper, we consider spinning essential surfaces in an irreducible P^2-irreducible, anannular, atoroidal 3-manifold with tori and Klein bottle boundary components. We can assume that such a 3-manifold is equipped with an ideal 1-efficient triangulation. In particular, we prove that for a given choice of a set of boundary slopes for a proper essential surface, there is a set of essential vertex solutions for the projective solution space at these boundary slopes, answering a question of Dunfield and Garoufalidis [Trans. Amer. Math. Soc. 364 (2012), pp. 6109–6137], so long as the slopes are not of a fiber of a bundle structure. [ABSTRACT FROM AUTHOR]

We provide a classification theorem for compact stable minimal immersions (CSMI) of codimension 1 or dimension 1 (codimension 1 and 2 or dimension 1 and 2) in the product of a complex (quaternionic) projective space with any other Riemannian manifold. We characterize the complex minimal immersions of codimension 2 or dimension 2 as the only CSMI in the product of two complex projective spaces. As an application, we characterize the CSMI of codimension 1 or dimension 1 (codimension 1 and 2 or dimension 1 and 2) in the product of a complex (quaternionic) projective space with any compact rank one symmetric space. [ABSTRACT FROM AUTHOR]

The extended coset leader weight enumerator of the generalized Reed–Solomon [ q + 1 , q - 3 , 5 ] q code is computed. In this computation methods in finite geometry, combinatorics and algebraic geometry are used. For this we need the classification of the points, lines and planes in the projective three space under projectivities that leave the twisted cubic invariant. A line in three space determines a rational function of degree at most three and vice versa. Furthermore, the double point scheme of a rational function is studied. The pencil of a true passant of the twisted cubic, not in an osculation plane gives a curve of genus one as double point scheme. With the Hasse–Weil bound on F q -rational points we show that there is a 3-plane containing the passant. [ABSTRACT FROM AUTHOR]

In an earlier paper by three of the present authors and Csaba Schneider, it was shown that, for m ≥ 2 , a set of m + 1 partitions of a set Ω , any m of which are the minimal non-trivial elements of a Cartesian lattice, either form a Latin square (if m = 2 ), or generate a join-semilattice of dimension m associated with a diagonal group over a base group G. In this paper we investigate what happens if we have m + r partitions with r ≥ 2 , any m of which are minimal elements of a Cartesian lattice. If m = 2 , this is just a set of mutually orthogonal Latin squares. We consider the case where all these squares are isotopic to Cayley tables of groups, and give an example to show the groups need not be all isomorphic. For m > 2 , things are more restricted. Any m + 1 of the partitions generate a join-semilattice admitting a diagonal group over a group G. It may be that the groups are all isomorphic, though we cannot prove this. Under an extra hypothesis, we show that G must be abelian and must have three fixed-point-free automorphisms whose product is the identity. (We describe explicitly all abelian groups having such automorphisms.) Under this hypothesis, the structure gives an orthogonal array, and conversely in some cases. If the group is cyclic of prime order p, then the structure corresponds exactly to an arc of cardinality m + r in the (m - 1) -dimensional projective space over the field with p elements, so all known results about arcs are applicable. More generally, arcs over a finite field of order q give examples where G is the elementary abelian group of order q. These examples can be lifted to non-elementary abelian groups using p-adic techniques. [ABSTRACT FROM AUTHOR]

A ϱ -saturating set of PG (N , q) is a point set S such that any point of PG (N , q) lies in a subspace of dimension at most ϱ spanned by points of S . It is generally known that a ϱ -saturating set of PG (N , q) has size at least c · ϱ q N - ϱ ϱ + 1 , with c > 1 3 a constant. Our main result is the discovery of a ϱ -saturating set of size roughly (ϱ + 1) (ϱ + 2) 2 q N - ϱ ϱ + 1 if q = (q ′) ϱ + 1 , with q ′ an arbitrary prime power. The existence of such a set improves most known upper bounds on the smallest possible size of ϱ -saturating sets if ϱ < 2 N - 1 3 . As saturating sets have a one-to-one correspondence to linear covering codes, this result improves existing upper bounds on the length and covering density of such codes. To prove that this construction is a ϱ -saturating set, we observe that the affine parts of q ′ -subgeometries of PG (N , q) having a hyperplane in common, behave as certain lines of AG (ϱ + 1 , (q ′) N) . More precisely, these affine lines are the lines of the linear representation of a q ′ -subgeometry PG (ϱ , q ′) embedded in PG (ϱ + 1 , (q ′) N) . [ABSTRACT FROM AUTHOR]

A t-intersecting constant dimension subspace code C is a set of k-dimensional subspaces in a projective space PG (n , q) , where distinct subspaces intersect in exactly a t-dimensional subspace. A classical example of such a code is the sunflower, where all subspaces pass through the same t-space. The sunflower bound states that such a code is a sunflower if | C | > q k + 1 - q t + 1 q - 1 2 + q k + 1 - q t + 1 q - 1 + 1 . In this article we will look at the case t = 0 and we will improve this bound for q ≥ 9 : a set S of k-spaces in PG (n , q) , q ≥ 9 , pairwise intersecting in a point is a sunflower if | S | > 2 q 6 + 4 q 3 - 5 q q k + 1 - 1 q - 1 2 . [ABSTRACT FROM AUTHOR]

We characterize the smallest codimension components of the Hodge locus of smooth degree d hypersurfaces of the projective space ℙ n + 1 of even dimension n , passing through the Fermat variety (with d ≠ 3 , 4 , 6). They correspond to the locus of hypersurfaces containing a linear algebraic cycle of dimension n 2 . Furthermore, we prove that among all the local Hodge loci associated to a nonlinear cycle passing through Fermat, the ones associated to a complete intersection cycle of type (1 , 1 , ... , 1 , 2) attain the minimal possible codimension of their Zariski tangent spaces. This answers a conjecture of Movasati, and generalizes a result of Voisin about the first gap between the codimension of the components of the Noether–Lefschetz locus to arbitrary dimension, provided that they contain the Fermat variety. [ABSTRACT FROM AUTHOR]

Consider the real, complex and quaternionic n-dimensional projective spaces, R P n , C P n and H P n ; to unify notation, write K d P n for the real ( d = 1 ), complex ( d = 2 ) and quaternionic ( d = 4 ) n-dimensional projective space. Consider a pair (M, T), where M is a closed smooth manifold and T is a smooth involution defined on M. Write F for the fixed point set of T. In this paper we obtain the equivariant cobordism classification of the pairs (M, T) when F is of the form F = K d P m ∪ K e P n , with d < e , in the cases (m , n) = (o d d , o d d) and (m , n) = (o d d , e v e n) . As will be seen in the history described in Sect. 1, the case F = a disjoint union of projective spaces has an intense and still unfinished history in the literature; in particular, there are several results in the literature in which F is the union of two projective spaces, but with d = e ; our paper is the first to consider unions of projective spaces under different rings. For example, if d = e , in the case (m , n) = (o d d , o d d) all involution bounds, while for different rings there are nonbounding involutions. [ABSTRACT FROM AUTHOR]

We describe a method to construct completions of affine spaces into total spaces of Q$\mathbb {Q}$‐factorial terminal Mori fiber spaces over the projective line. As an application, we provide families of examples with non‐rational, birationally rigid and non‐stably rational general fibers. [ABSTRACT FROM AUTHOR]

Let G be a group acting properly and by isometries on a metric space X; it follows that the quotient or orbit space X/G is also a metric space. We study the Vietoris–Rips and Čech complexes of X/G. Whereas (co) homology theories for metric spaces let the scale parameter of a Vietoris–Rips or Čech complex go to zero, and whereas geometric group theory requires the scale parameter to be sufficiently large, we instead consider intermediate scale parameters (neither tending to zero nor to infinity). As a particular case, we study the Vietoris–Rips and Čech thickenings of projective spaces at the first scale parameter where the homotopy type changes. [ABSTRACT FROM AUTHOR]

By modeling the space of projective measured laminations in the cotangent space to Teichmller space via hyperbolic length functions and extremal length functions, we associate two classes of linear structures to the space of measured laminations. We prove that both of these two linear structures are rigid, the induced linear structures on different Riemann surfaces are different. [ABSTRACT FROM AUTHOR]

We address several extremal problems concerning the spreading property of point sets of Steiner triple systems. This property is closely related to the structure of subsystems, as a set is spreading if and only if there is no proper subsystem which contains it. We give sharp upper bounds on the size of a minimal spreading set in a Steiner triple system and show that if all the minimal spreading sets are large then the examined triple system must be a projective space. We also show that the size of a minimal spreading set is not an invariant of a Steiner triple system. [ABSTRACT FROM AUTHOR]

Let |$H$| be a Hilbert space and |$P(H)$| be the projective space of all quantum pure states. Wigner's theorem states that every bijection |$\phi \colon P(H)\to P(H)$| that preserves the quantum angle between pure states is automatically induced by either a unitary or an antiunitary operator |$U\colon H\to H$|⁠. Uhlhorn's theorem generalizes this result for bijective maps |$\phi $| that are only assumed to preserve the quantum angle |$\frac{\pi }{2}$| (orthogonality) in both directions. Recently, two papers, written by Li–Plevnik–Šemrl and Gehér, solved the corresponding structural problem for bijections that preserve only one fixed quantum angle |$\alpha $| in both directions, provided that |$0 < \alpha \leq \frac{\pi }{4}$| holds. In this paper we solve the remaining structural problem for quantum angles |$\alpha $| that satisfy |$\frac{\pi }{4} < \alpha < \frac{\pi }{2}$|⁠ , hence complete a programme started by Uhlhorn. In particular, it turns out that these maps are always induced by unitary or antiunitary operators, however, our assumption is much weaker than Wigner's. [ABSTRACT FROM AUTHOR]

In this paper, we give essential concepts of geometry of three-dimensional spaces in vector formulation in an affine vector space An. [ABSTRACT FROM AUTHOR]

Copyright of Al-Mustansiriyah Journal of Science is the property of Republic of Iraq Ministry of Higher Education & Scientific Research (MOHESR) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The purpose of this paper is to study holomorphic curves f from C to C³ avoiding four complex hyperplanes and a real subspace of real dimension four in C³ . We show that the projection of f into the complex projective space CP2 does not remain constant as in the complex case studied by Green, which indicates that the complex structure of the avoided hyperplanes is a necessary condition in the Green theorem. [ABSTRACT FROM AUTHOR]

We prove that total spaces of CP2-bundles generate the oriented cobordism ring Ω*. Combined with the surgery lemma [8, 22], this yields a somewhat different proof of Gromov's and Lawson's theorem [8] that all simply connected non-spin manifolds of dimension at least 5 admit metrics of positive scalar curvature. [ABSTRACT FROM AUTHOR]

D. Davis introduced projective product spaces in 2010 as a generalization of real projective spaces and discussed some of their topological properties. On the other hand, Dold manifolds were introduced by A. Dold in 1956 to study the generators of the non-oriented cobordism ring. Recently, in 2019, A. Nath and P. Sankaran made a modest generalization of Dold manifolds. In this paper we simultaneously generalize both the notions of projective product spaces and Dold manifolds, leading to infinitely many different classes of new smooth manifolds. Our main goal will be to study the integral homology groups, cohomology rings, stable tangent bundles, and vector field problems, on certain generalized projective product spaces and Dold manifolds. [ABSTRACT FROM AUTHOR]

We discuss inequalities between the values of homotopical and cohomological Poincar'e polynomials of the self-products of rationally elliptic spaces. For rationally elliptic quasi-projective varieties, we prove inequalities between the values of generating functions for the ranks of the graded pieces of the weight and Hodge filtrations of the canonical mixed Hodge structures on homotopy and cohomology groups. Several examples of such mixed Hodge polynomials and related inequalities for rationally elliptic quasi-projective algebraic varieties are presented. One of the consequences is that the homotopical (resp. cohomological) mixed Hodge polynomial of a rationally elliptic toric manifold is a sum (resp. a product) of polynomials of projective spaces. We introduce an invariant called stabilization threshold pp(X; e) for a simply connected rationally elliptic space X and a positive real number e, and we show that the Hilali conjecture implies that pp(X; 1) ⩽ 3. [ABSTRACT FROM AUTHOR]

We define two generalizations of the totally nonnegative Grassmannian and determine their topology in the case of real projective space.We find the spaces to be PL manifoldswith boundary which are homotopy equivalent to another real projective space of smaller dimension. One generalization makes use of sign variation while the other uses boundary measurement. Spaces arising from boundary measurement are shown to admit Cohen--Macaulay triangulations. [ABSTRACT FROM AUTHOR]

Non-simply connected Calabi-Yau threefolds play a central role in the study of string compactifications. Such manifolds are usually described by quotienting a simply connected Calabi-Yau variety by a freely acting discrete symmetry. For the Calabi-Yau threefolds described as complete intersections in products of projective spaces, a classification of such symmetries descending from linear actions on the ambient spaces of the varieties has been given in [16]. However, which symmetries can be described in this manner depends upon the description that is being used to represent the manifold. In [24] new, favorable, descriptions were given of this data set of Calabi-Yau threefolds. In this paper, we perform a classification of cyclic symmetries that descend from linear actions on the ambient spaces of these new favorable descriptions. We present a list of 129 symmetries/non-simply connected Calabi-Yau threefolds. Of these, at least 33, and potentially many more, are topologically new varieties. [ABSTRACT FROM AUTHOR]

A vector space partition P of the projective space PG(v - 1; q) is a set of subspaces in PG(v - 1; q) which partitions the set of points. We say that a vector space partition P has type ... if precisely mi of its elements have dimension i, where 1 ≤ i ≤ v - 1. Here we determine all possible types of vector space partitions in PG(7; 2). [ABSTRACT FROM AUTHOR]

The generalized Miller–Morita–Mumford classes of a manifold bundle with fiber M depend only on the underlying τ M -fibration, meaning the family of vector bundles formed by the tangent bundles of the fibers. This motivates a closer study of the classifying space for τ M -fibrations, B a u t (τ M) , and its cohomology ring, i.e., the ring of characteristic classes of τ M -fibrations. For a bundle ξ over a simply connected Poincaré duality space, we construct a relative Sullivan model for the universal ξ -fibration with holonomy in a given connected monoid, together with explicit cocycle representatives for the characteristic classes of the canonical bundle over its total space. This yields tools for computing the rational cohomology ring of B a u t (ξ) as well as the subring generated by the generalized Miller–Morita–Mumford classes. To illustrate, we carry out sample computations for spheres and complex projective spaces. We discuss applications to tautological rings of simply connected manifolds and to the problem of deciding whether a given τ M -fibration comes from a manifold bundle. [ABSTRACT FROM AUTHOR]

Quantum games represent the really twenty-first century branch of game theory, tightly linked to the modern development of quantum computing and quantum technologies. The main accent in these developments so far was made on stationary or repeated games. In this paper, we aim at initiating the truly dynamic theory with strategies chosen by players in real time. Since direct continuous observations are known to destroy quantum evolutions (so-called quantum Zeno paradox), the necessary new ingredient for quantum dynamic games must be the theory of non-direct observations and the corresponding quantum filtering. Apart from the technical problems in organizing feedback quantum control in real time, the difficulty in applying this theory for obtaining mathematically amenable control systems is due partially to the fact that it leads usually to rather non-trivial jump-type Markov processes and/or degenerate diffusions on manifolds, for which the corresponding control is very difficult to handle. The starting point for the present research is the remarkable discovery (quite unexpected, at least to the author) that there exists a very natural class of homodyne detections such that the diffusion processes on projective spaces resulting by filtering under such arrangements coincide exactly with the standard Brownian motions (BM) on these spaces. In some cases, one can even reduce the process to the plain BM on Euclidean spaces or tori. The theory of such motions is well studied making it possible to develop a tractable theory of related control and games, which can be at the same time practically implemented on quantum optical devices. [ABSTRACT FROM AUTHOR]

We relate the geometry of curves to the notion of hyperbolicity in real algebraic geometry. A hyperbolic variety is a real algebraic variety that (in particular) admits a real fibered morphism to a projective space whose dimension is equal to the dimension of the variety. We study hyperbolic varieties with a special interest in the case of hypersurfaces that admit a real algebraic ruling. The central part of the paper is concerned with secant varieties of real algebraic curves where the real locus has the maximal number of connected components, which is determined by the genus of the curve. For elliptic normal curves, we further obtain definite symmetric determinantal representations for the hyperbolic secant hypersurfaces, which implies the existence of symmetric Ulrich sheaves of rank one on these hypersurfaces. We also use this to derive better bounds on the size of semidefinite representations for convex hulls of real algebraic curves of genus 1. [ABSTRACT FROM AUTHOR]

In the first months of life, babies develop visual perception. The notions of space evolve in the everyday of experiences, the recognition of the self through your body, and relationships with others. The topological notions developed by babies correspond to closeness, proximity, continuity and separation. As babies grow, their skills are developed both in the projective space and in the geometric space. These even influence the baby's development in an integral way. This article intends to present results of the topological notions of closure, proximity separation and projections in the baby's space. This qualitative research is developed under a descriptive perspective with interdisciplinary contributions. Data collection was made from cartography, photographic and filmic records of babies in different cities in Brazil and Colombia. The reflections developed point to the development of perception from the offer of multiple experiences since the first months. In addition, it is evident that the understanding of important mathematical concepts happens since the beginning of life, from everyday experiences of exploration and relationship with spaces and regardless of the formal school learning of geometry and its concepts. [ABSTRACT FROM AUTHOR]

For a nonnegative weakly irreducible tensor, its spectral radius is an eigenvalue corresponding to a unique positive eigenvector up to a scalar, called Perron vector. But including the Perron vector, it may have more than one eigenvector corresponding to the spectral radius. The projective eigenvariety associated with the spectral radius is the set of the eigenvectors corresponding to the spectral radius considered in the complex projective space. We prove that the dimension of the above projective eigenvariety is zero, i.e. there are finitely many eigenvectors associated with the spectral radius up to a scalar. We characterize a nonnegative combinatorially symmetric tensor for which the dimension of projective eigenvariety associated with spectral radius is greater than zero. Finally we apply those results to the adjacency tensors of uniform hypergraphs. [ABSTRACT FROM AUTHOR]

For a graph $\Gamma$ , let $K(H_{\Gamma },\,1)$ denote the Eilenberg–Mac Lane space associated with the right-angled Artin (RAA) group $H_{\Gamma }$ defined by $\Gamma$. We use the relationship between the combinatorics of $\Gamma$ and the topological complexity of $K(H_{\Gamma },\,1)$ to explain, and generalize to the higher TC realm, Dranishnikov's observation that the topological complexity of a covering space can be larger than that of the base space. In the process, for any positive integer $n$ , we construct a graph $\mathcal {O}_n$ whose TC-generating function has polynomial numerator of degree $n$. Additionally, motivated by the fact that $K(H_{\Gamma },\,1)$ can be realized as a polyhedral product, we study the LS category and topological complexity of more general polyhedral product spaces. In particular, we use the concept of a strong axial map in order to give an estimate, sharp in a number of cases, of the topological complexity of a polyhedral product whose factors are real projective spaces. Our estimate exhibits a mixed cat-TC phenomenon not present in the case of RAA groups. [ABSTRACT FROM AUTHOR]

In this article, we will show some algebraic dependence theorems for meromorphic mappings into a projective space sharing few hyperplanes. Our result is an improvement of many previous results in this topic. [ABSTRACT FROM AUTHOR]

We prove that for a Banach algebra A having a bounded $\mathcal {Z}(A)$ -approximate identity and for every $\mathbf {[IN]}$ group G with a weight w which is either constant on conjugacy classes or satisfies $w \geq 1$ , $\mathcal {Z}(L^{1}_{w}(G) \otimes ^{\gamma } A) \cong \mathcal {Z}(L^{1}_{w}(G)) \otimes ^{\gamma } \mathcal {Z}(A)$. As an application, we discuss the conditions under which $\mathcal {Z}(L^{1}_{\omega }(G,A))$ enjoys certain Banach algebraic properties, such as weak amenability or semisimplicity. [ABSTRACT FROM AUTHOR]

We investigate containment statements between symbolic and ordinary powers and bounds on the Waldschmidt constant of defining ideals of points in projective spaces. We establish the stable Harbourne conjecture for the defining ideal of a general set of points. We also prove Chudnovsky's Conjecture and the stable version of the Harbourne–Huneke containment conjectures for a general set of sufficiently many points. [ABSTRACT FROM AUTHOR]

Linear codes have widespread applications in data storage systems. There are two major contributions in this paper. We first propose infinite families of optimal or distance-optimal linear codes over F p constructed from projective spaces. Moreover, a necessary and sufficient condition for such linear codes to be Griesmer codes is presented. Secondly, as an application in data storage systems, we investigate the locality of the linear codes constructed. Furthermore, we show that these linear codes are alphabet-optimal locally repairable codes with locality 2. [ABSTRACT FROM AUTHOR]

In this paper, the authors discuss a generalization of Lappan's theorem to higher dimensional complex projective space and get the following result: Let f be a holomorphic mapping of Δ into ℙn(ℂ), and let H1, ⋯, Hq be hyperplanes in general position in ℙn(ℂ). Assume that sup { (1 − | z | 2) f ♯ (z) : z ∈ ∪ j = 1 q f − 1 (H j) } < ∞ , if q ≥ 2n2 + 3, then f is normal. [ABSTRACT FROM AUTHOR]

In this article, we present a classification for the divisor topologies of the projective complete intersection Calabi-Yau (pCICY) 3-folds realized as hypersurfaces in the product of complex projective spaces. There are 7890 such pCICYs of which 7820 are favorable, and can be subsequently useful for phenomenological purposes. To our surprise we find that the whole pCICY database results in only 11 (so-called coordinate) divisors (D) of distinct topology and we classify those surfaces with their possible deformations inside the pCICY 3-fold, which turn out to be satisfying 1 ≤ h2,0(D) ≤ 7. We also present a classification of the so-called ample divisors for all the favorable pCICYs which can be useful for fixing all the (saxionic) Kähler moduli through a single non-perturbative term in the superpotential. We argue that this relatively unexplored pCICY dataset equipped with the necessary model building ingredients, can be used for a systematic search of physical vacua. To illustrate this for model building in the context of type IIB CY orientifold compactifications, we present moduli stabilization with some preliminary analysis of searching possible vacua in simple models, as a template to be adopted for analyzing models with a larger number of Kähler moduli. [ABSTRACT FROM AUTHOR]

Copyright of Erzincan University Journal of Science & Technology is the property of Erzincan Binali Yildirim Universitesi and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)