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AbstractAn additive action on an irreducible algebraic variety X is an effective action with an open orbit of the vector group . Any two additive actions on X are conjugate by a birational automorphism of X. We prove that, if X is the projective space, the conjugating element can be chosen in the affine Cremona group and it is given by so-called basic polynomials of the corresponding local algebra. [ABSTRACT FROM AUTHOR]

The hull of a linear code plays an important role in determining the complexity of algorithms for checking permutation equivalence of two linear codes and computing the automorphism group of a linear code. Regarding the quantum error correction, linear codes with determined hull are used to construct quantum codes. In this paper, we focus on the hull of Simplex codes and punctured Simplex codes. We firstly study the properties of the matrix produced by the column vectors of a projective space and determine the Euclidean and Hermitian hull of punctured Simplex codes completely. Secondly, we investigate the Euclidean and Hermitian hull of several classes of linear codes from Simplex codes. [ABSTRACT FROM AUTHOR]

A multidimensional projective space with a marked point (center) is considered. On the manifold of projective frames of the given space that are adapted to the center, the action of the stabilizer of the center of the group of projective transformations is introduced. We prove that linear frames, i.e., bases of the tangent vector space of the projective space at its center, can be identified with orbits of the adapted projective frames with respect to the action of the kernel of the epimorphism of Lie groups, which assigns to each transformation from the stabilizer its differential at the center. Using a multidimensional generalization of the Desargues theorem, we obtain a criterion for two adapted projective frames to belong to the same orbit. [ABSTRACT FROM AUTHOR]

The relationship of discrete L-algebras to projective geometry is deepened and made explicit in several ways. Firstly, a geometric lattice is associated to any discrete L-algebra. Monoids of I-type are obtained as a special case where the perspectivity relation is trivial. Secondly, the structure group of a non-degenerate discrete L-algebra X is determined and shown to be a complete invariant. It is proved that X ∖ {1} is a projective space with an orthogonality relation. A new definition of non-symmetric quantum sets, extending the recursive definition of symmetric quantum sets, is provided and shown to be equivalent to the former one. Quantum sets are characterized as complete projective spaces with an anisotropic duality, and they are also characterized in terms of their complete lattice of closed subspaces, which is one-sided orthomodular and semimodular. For quantum sets of finite cardinality n > 3, a representation as a projective space with duality over a skew-field is given. Quantum sets of cardinality 2 are classified, and the structure group of their associated L-algebra is determined. [ABSTRACT FROM AUTHOR]

It is expected that a totally invariant divisor of a non-isomorphic endomorphism of the complex projective space is a union of hyperplanes. In this paper, we compute an upper bound for the degree of such a divisor. As a consequence, we prove the linearity of totally invariant divisors with isolated singularities. [ABSTRACT FROM AUTHOR]

We study induced additive actions on projective hypersurfaces, i.e. effective regular actions of the algebraic group G a m with an open orbit that can be extended to a regular action on the ambient projective space. We prove that if a projective hypersurface admits an induced additive action, then it is unique if and only if the hypersurface is non-degenerate. We also show that for any n ≥ 2 , there exists a non-degenerate hypersurface in P n of each degree d from 2 to n. [ABSTRACT FROM AUTHOR]

A line coloring of PG(n,q) $\text{PG}(n,q)$, the n $n$‐dimensional projective space over GF(q) $(q)$, is an assignment of colors to all lines of PG(n,q) $\text{PG}(n,q)$ so that any two lines with the same color do not intersect. The chromatic index of PG(n,q) $\text{PG}(n,q)$, denoted by χ′(PG(n,q)) $\chi ^{\prime} (\text{PG}(n,q))$, is the least number of colors for which a coloring of PG(n,q) $\text{PG}(n,q)$ exists. This paper translates the problem of determining the chromatic index of PG(n,q) $\text{PG}(n,q)$ to the problem of examining the existences of PG(3,q) $\text{PG}(3,q)$ and PG(4,q) $\text{PG}(4,q)$ with certain properties. In particular, it is shown that for any odd integer n $n$ and q∈{3,4,8,16},χ′(PG(n,q))=(qn−1)∕(q−1) $q\in \{3,4,8,16\},\chi ^{\prime} (\text{PG}(n,q))=({q}^{n}-1)\unicode{x02215}(q-1)$, which implies the existence of a parallelism of PG(n,q) $\text{PG}(n,q)$ for any odd integer n $n$ and q∈{3,4,8,16} $q\in \{3,4,8,16\}$. [ABSTRACT FROM AUTHOR]

We consider the structures of the plane-line and point-line incidence matrices of the projective space PG (3 , q) connected with orbits of planes, points, and lines under the stabilizer group of the twisted cubic. In the literature, lines are partitioned into classes, each of which is a union of line orbits. In this paper, for all q, even and odd, we determine the incidence matrices connected with a family of orbits of the class named O 6 . This class contains lines external to the twisted cubic. The considered family includes an essential part of all O 6 orbits, whose complete classification is an open problem. [ABSTRACT FROM AUTHOR]

In this paper, we present a combinatorial characterization of a quasi-Hermitian surface as a set H of points of PG (3 , q) , q = p 2 h h ≥ 1 , p a prime number and q ≠ 4 , having the same size as the Hermitian surface and containing no plane, such that either a line is contained in H or intersects H in at most q + 1 points and every plane intersects H in at least q q + 1 points. Moreover, if there is no external line, the set H is a Hermitian surface. [ABSTRACT FROM AUTHOR]

Let X be a right C*‐module over a unital C*‐algebra A$ {\cal A}$. We study the Hopf fibration of X: h:XP→P1(X)=projective space ofX,$$\begin{equation*} \mathfrak {h}: {\bf X} _ {\cal P} \rightarrow P1({\bf X}) = \hbox{projective space of } {\bf X} , \end{equation*}$$where the projective space of X is the set of singly generated orthocomplemented submodules of X, XP$ {\bf X} _ {\cal P}$ is the set of elements of X, which generate such submodules, and h(x)=$\mathfrak {h}({\bf x})=$module generated by x∈XP$ {\bf x} \in {\bf X} _ {\cal P}$. The group of unitary operators of the module X acts on both spaces. We introduce a Finsler metric in XP$ {\bf X} _ {\cal P}$, which is invariant under the unitary action. Our main results establish that the map h$\mathfrak {h}$ is distance decreasing (when the projective space of X is considered with its natural unitary invariant metric), and a minimality result in XP$ {\bf X} _ {\cal P}$, characterizing metric geodesics in this space. [ABSTRACT FROM AUTHOR]

Let X be a right C*‐module over a unital C*‐algebra A$ {\cal A}$. We study the Hopf fibration of X: h:XP→P1(X)=projective space ofX,$$\begin{equation*} \mathfrak {h}: {\bf X} _ {\cal P} \rightarrow P1({\bf X}) = \hbox{projective space of } {\bf X} , \end{equation*}$$where the projective space of X is the set of singly generated orthocomplemented submodules of X, XP$ {\bf X} _ {\cal P}$ is the set of elements of X, which generate such submodules, and h(x)=$\mathfrak {h}({\bf x})=$module generated by x∈XP$ {\bf x} \in {\bf X} _ {\cal P}$. The group of unitary operators of the module X acts on both spaces. We introduce a Finsler metric in XP$ {\bf X} _ {\cal P}$, which is invariant under the unitary action. Our main results establish that the map h$\mathfrak {h}$ is distance decreasing (when the projective space of X is considered with its natural unitary invariant metric), and a minimality result in XP$ {\bf X} _ {\cal P}$, characterizing metric geodesics in this space. [ABSTRACT FROM AUTHOR]

In the projective space PG (3 , q) , we consider orbits of lines under the stabilizer group of the twisted cubic. In the literature, lines of PG (3 , q) are partitioned into classes, each of which is a union of line orbits. We propose an approach to obtain orbits of the class named O 6 , whose complete classification is an open problem. For all q, we describe a family of orbits of O 6 and their stabilizer groups. The orbits of this family include an essential part of all O 6 orbits. [ABSTRACT FROM AUTHOR]

In the projective space PG (3 , q) , we consider the orbits of lines under the stabilizer group of the twisted cubic. In the literature, lines of PG (3 , q) are partitioned into classes, each of which is a union of line orbits. In this paper, all classes of lines consisting of a unique orbit are found. For the remaining line types, with one exception, it is proved that they consist exactly of two or three orbits; sizes and structures of these orbits are determined. Also, the subgroups of the stabilizer group of the twisted cubic fixing lines of the orbits are obtained. Problems which remain open for one type of lines are formulated and, for 5 ≤ q ≤ 37 and q = 64 , a solution is provided. [ABSTRACT FROM AUTHOR]

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 C P^2 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

We continue to study the space of centered planes in projective space . In this paper, we use E. Cartan's method of external forms and the group-theoretical method of G. F. Laptev to study the space of centered planes of the same dimension. These methods are successfully applied in physics. In a generalized bundle, a bilinear connection associated with a space is given. The connection object contains two simplest subtensors and subquasi-tensors (four simplest and three simple subquasi-tensors). The object field of this connection defines the objects of torsion, curvature-torsion, and curvature. The curvature tensor contains six simplest and four simple subtensors, and curvature-torsion tensor contains three simplest and two simple subtensors. The canonical case of a generalized bilinear connection is considered.

The pure state evaluation map from M n (C) to C (C P n − 1) is a completely positive map of C * -algebras intertwining the U n symmetries on the two algebras. We show that it extends to a cochain map from the universal calculus on M n (C) to the holomorphic ∂ ¯ calculus on C P n − 1 . The method uses connections on Hilbert C * -bimodules. [ABSTRACT FROM AUTHOR]

This contribution presents a brief survey of clipping and intersection algorithms in E 2 and E 3 with a nearly complete list of relevant references. Some algorithms use the projective extension of the Euclidean space and vector-vector operations, which support GPU and SSE use. This survey is intended to help researchers, students, and practitioners dealing with intersection and clipping algorithms. [ABSTRACT FROM AUTHOR]

In this article, we analyse maximal sets of k-spaces, in PG(n, q) and AG(n, q) with n ≥ 2k + t + 3, that pairwise meet in at least a t-space. It is known that for both PG(n, q) and AG(n, q), the largest example is a t-pencil, i.e. the set of all k-spaces containing a fixed t-space. In this paper, we analyse the structure of the second largest maximal example in both PG(n, q) and AG(n, q). [ABSTRACT FROM AUTHOR]

The purpose of this article is to partition PG(3,11) into orbits. These orbits are studied from the view of caps using the subgroups of PGL(4,11) which are determined by nontrivial positive divisors of the order of PG(3,11). The τi - distribution and ci -distribution are also founded for each cap. [ABSTRACT FROM AUTHOR]

We clarify the undecided case C2 = 3 of a result of Ein, Hartshorne and Vogelaar [8] about the restriction of a stable rank 3 vector bundle with C1 = 0 on the projective 3-space to a general plane. It turns out that there are more exceptions to the stable restriction property than those conjectured by the three authors. One of them is a Schwarzenberger bundle (twisted by -1); it has C3 = 6. There are also some exceptions with C3 = 2 (plus, of course, their duals). [ABSTRACT FROM AUTHOR]

This contribution presents a brief survey of clipping and intersection algorithms in E 2 and E 3 with a nearly complete list of relevant references. Some algorithms use the projective extension of the Euclidean space and vector-vector operations, which support GPU and SSE use. This survey is intended to help researchers, students, and practitioners dealing with intersection and clipping algorithms. [ABSTRACT FROM AUTHOR]

Among probability measures on d-dimensional real projective space, one which maximizes the expected angle arccos (x | x | · y | y |) between independently drawn projective points x and y was conjectured to equidistribute its mass over the standard Euclidean basis { e 0 , e 1 , ... , e d } by Fejes Tóth (Acta Math Acad Sci Hung 10:13–19, 1959. https://doi.org/10.1007/BF02063286). If true, this conjecture evidently implies the same measure maximizes the expectation of arccos α (x | x | · y | y |) for any exponent α > 1 . The kernel arccos α (x | x | · y | y |) represents the objective of an infinite-dimensional quadratic program. We verify discrete and continuous versions of this milder conjecture in a non-empty range α > α Δ d ≥ 1 , and establish uniqueness of the resulting maximizer μ ^ up to rotation. We show μ ^ no longer maximizes when α < α Δ d . At the endpoint α = α Δ d of this range, we show another maximizer μ must also exist which is not a rotation of μ ^ . For the continuous version of the conjecture, an "Appendix A" provided by Bilyk et al in response to an earlier draft of this work combines with the present improvements to yield α Δ d < 2 . The original conjecture α Δ d = 1 remains open (unless d = 1 ). However, in the maximum possible range α > 1 , we show μ ^ and its rotations maximize the aforementioned expectation uniquely on a sufficiently small ball in the L ∞ -Kantorovich–Rubinstein–Wasserstein metric d ∞ from optimal transportation; the same is true for any measure μ which is mutually absolutely continuous with respect to μ ^ , but the size of the ball depends on α , d , and ‖ d μ ^ d μ ‖ ∞ . [ABSTRACT FROM AUTHOR]

This paper is a contribution to the classification of parallelisms in three-dimensional projective spaces over small finite fields of order q by computer. The smallest space in which parallelisms have not yet been classified is for q = 4. Partial results are available. The parallelisms admitting a nontrivial automorphism of odd prime order are known. Moreover, much is known about the case of parallelisms of PG (3 , 4) whose automorphism group is a two group. Namely, everything is known for two of the three possible groups of order two, as well as for cyclic groups of order 4. The present paper will settle the case of parallelisms whose automorphism group is elementary abelian of order 4. This leaves open the cases of parallelisms whose full automorphism groups are either trivial or a specific group of order two. [ABSTRACT FROM AUTHOR]

Echelon-Ferrers construction is one of the most powerful methods to improve the lower bounds for constant dimension codes. Although the method was proposed more than 10 years ago, it is still the best construction in a number of cases. In this paper, we remove parts of lifted FDRM codes from the echelon-Ferrers construction, and insert an SC-representation set, which finally improves the Echelon-ferrers construction, to get lower bounds for the following cases: A q (11 , 4 , 4) , A q (15 , 4 , 4) and A q (19 , 4 , 4) . [ABSTRACT FROM AUTHOR]

In this paper, studied the types of (k, r)-arcs were constructed by action of groups on the three-dimensional projective space over the Galois field of order eight. Also, determined if they form complete arcs or not.

We complete the classification of globally generated vector bundles with c1≤5$c_1 \le 5$ on projective spaces by treating the case c1=5$c_1 = 5$ on Pn$\mathbb {P}^n$, n≥4$n \ge 4$. It turns out that there are very few indecomposable bundles of this kind: besides some obvious examples there are, roughly speaking, only the (first twist of the) rank 5 vector bundle which is the middle term of the monad defining the Horrocks bundle of rank 3 on P5$\mathbb {P}^5$, and its restriction to P4$\mathbb {P}^4$. We recall, in an appendix, from one of our previous papers, the main results allowing the classification of globally generated vector bundles with c1=5$c_1 = 5$ on P3$\mathbb {P}^3$. Since there are many such bundles, a large part of the main body of the paper is occupied with the proof of the fact that, except for the simplest ones, they do not extend to P4$\mathbb {P}^4$ as globally generated vector bundles. [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]

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 main aim of this paper is to introduce the relationship between the topic of coding theory and the projective space in field three and test the code. The maximum value of size of code over finite field of order three and an incidence matrix with the parameters, n (length of code), d (minimum distance of code) and e (error-correcting of code) have been constructed. With a theorem and a result that test the code if it is perfect or not."

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]

The point-plane, the point-line, and the plane-line incidence matrices of PG (3 , q) are of interest in combinatorics, finite geometry, graph theory and group theory. Some of the properties of these matrices and their submatrices are related with the interplay of orbits of points, lines and planes under the action of subgroups of PGL (4 , q) . A remarkable particular case is the subgroup G ≅ PGL (2 , q) , viewed as the stabilizer group of the twisted cubic C . For this case, the study of the point-plane incidence matrix, initiated by D. Bartoli and the present authors, has attracted attention as being related to submatrices with useful applications in coding theory for the construction of multiple covering codes. In this paper, we extend our investigation to the plane-line incidence matrix apart from just one class of the line orbits, named O 6 in the literature. For all q ≥ 2 , in each submatrix, the numbers of lines in any plane and planes through any line are obtained. As a tool for the present investigation, we use submatrices of incidences arising from orbits of planes and unions of line orbits, including O 6 . For each such submatrix we determine the total number of lines from the union in any plane and the average number of planes from the orbit through any line of the union. [ABSTRACT FROM AUTHOR]

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]

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.)

In this note, we propose a "projective-geometric" version of quantization, based on the surjective rational self-maps of . Relations with the classical subjects via several examples and key properties of these maps are provided. [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]

We classify holomorphic isometric embeddings of the projective space into quadrics using the generalization of do Carmo–Wallach theory in Nagatomo [Harmonic maps into Grassmann manifolds, [mathDG], Holomorphic maps into Grassmann manifolds (Harmonic maps into Grassmann manifolds III), Annals of Global analysis and Geometry 60, 33–63 (2021). An explicit description of their moduli spaces up to image and gauge equivalence is provided. [ABSTRACT FROM AUTHOR]

Let 𝔽q[ɛ] := 𝔽q [X]/(X4 − X3) be a finite quotient ring where ɛ4 = ɛ3, with 𝔽q is a finite field of order q such that q is a power of a prime number p greater than or equal to 5. In this work, we will study the elliptic curve over 𝔽q[ɛ], ɛ4 = ɛ3 of characteristic p ≠ 2, 3 given by homogeneous Weierstrass equation of the form Y 2Z = X3 + aXZ2 + bZ3 where a and b are parameters taken in 𝔽q[ɛ]. Firstly, we study the arithmetic operation of this ring. In addition, we define the elliptic curve Ea,b(𝔽q[ɛ]) and we will show that Eπ0(a),π0(b)(𝔽q) and Eπ1(a),π1(b)(𝔽q) are two elliptic curves over the finite field 𝔽q, such that π0 is a canonical projection and π1 is a sum projection of coordinate of element in 𝔽q[ɛ]. Precisely, we give a classification of elements in elliptic curve over the finite ring 𝔽q[ɛ].

We continue to study of the Grassmann-like manifold of -centered planes. A special case is considered when the center de­scribes an -dimensional surface . We will denote this mani­fold by . An analogue of the strong Norden normalization of the manifold is realized. It is proved that this normalization induces a connection in the bundle associated with the manifold . A geometric characteristic of this connection is given with the help of parallel displacements. In our research we use the Cartan method of external forms and the group-theoretical method of Laptev. These methods are used by many geometers and physicists. The Grassmann-like manifold is closely related to such a well-known and popular manifold as the Grassmann manifold. The Grassmann mani­fold is an example of a homogeneous space and forms an important fun­damental class of projective manifolds, and the projective space itself can be represented as a Grassmann manifold.

In n-dimensional projective space Pn a manifold , i. e., a family of pairs of planes one of which is a hyperplane in the other, is considered. A principal bundle arises over it, . A typi­cal fiber is the stationarity subgroup of the generator of pair of planes: external plane and its multidimensional center — hyperplane. The princi­pal bundle contains four factor-bundles. A fundamental-group connection is set by the Laptev — Lumiste method in the associated fibering. It is shown that the connection object contains four subobjects that define connections in the corresponding fac­tor-bundles. It is proved that the curvature object of fundamental-group connection forms pseudotensor. It contains four subpseudotensors, which are curvature objects of the corresponding subconnections. The composite equipment of the family of hypercentered planes set by means of a point lying in the plane and not belonging to its hypercent­er and an (n – m – 1)-dimensional plane, which does not have common points with the hypercentered plane. It is proved, that composite equip­ment induces the fundamental-group connections of two types in the as­sociated fibering.

In this paper, the -caps are created by action of groups on the three-dimensional projective space over the Galois field of order eight. The types of -caps are also studied and determined either they form complete caps or not. [ABSTRACT FROM AUTHOR]

The quantum transition probability bears the symmetry of a process “evolving” around a symplectic area-bounding loop in the projective Hilbert space that essentially underlies the notion of a global geometric phase. The basic idea is that this symmetry can be associated with a synchronization procedure in the quantum phase space, which is only implicit due to the insistence of interpreting the temporal variable as a classical one, overlooking the subtle interrelation of the complex with the symplectic structure. This leads to the qualification of quantum amplitudes in terms of symplectic spinorial objects which doubly cover the corresponding complex vectors obtained through squaring. Their interrelation can be simply formulated in terms of null vectors of a 3-d Minkowski space, which proves to be instrumental for the Hermitian and unitary representations of the symplectic group. In this light, we show that the quantum transition probability assignment constitutes an equiareal transformation from the annulus of spinors to the disk of complex vectors, which makes it equivalent to the equiareal measure-preserving transformation of Archimedes. This realization ignited a process of re-evaluating the original works of Archimedes in light of their conceptual significance. It turns out that the symplectic method of equiareal projection on a disk pertains to a precise synchronization procedure that can be applied to the quantum phase space. From this viewpoint, we examine the pertinent notions of time, entangled symplectic area, objective indistinguishability, and qualify physically the non-squeezing theorem of symplectic geometry. [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.)

In this article we generalize the concepts that were used in the PhD thesis of Drudge to classify Cameron–Liebler line classes in PG (n , q) , n ≥ 3 , to Cameron–Liebler sets of k-spaces in PG (n , q) and AG (n , q) . In his PhD thesis, Drudge proved that every Cameron–Liebler line class in PG (n , q) intersects every 3-dimensional subspace in a Cameron–Liebler line class in that subspace. We are using the generalization of this result for sets of k-spaces in PG (n , q) and AG (n , q) . Together with a basic counting argument this gives a very strong non-existence condition, n ≥ 3 k + 3 . This condition can also be improved for k-sets in AG (n , q) , with n ≥ 2 k + 2 . [ABSTRACT FROM AUTHOR]

The pure state evaluation map from Mn(C) to C(CPn−1) is a completely positive map of C*-algebras intertwining the Un symmetries on the two algebras. We show that it extends to a cochain map from the universal calculus on Mn(C) to the holomorphic ∂¯ calculus on CPn−1. The method uses connections on Hilbert C*-bimodules.

In this paper, we propose an angular momentum-based advection technique that can express the turbulent foam effect. The motion of foam particles, which are strongly bound to the motion of the underlying fluid, is viscous, and sometimes clumping problems occur. This problem is a decisive factor that makes it difficult to express realistic foam effects. Since foam particles, which are secondary effects, depend on the motion of the underlying water, in order to exaggerate the foam effects or express more lively foam effects, it is inevitable to tune the motion of the underlying water and then readjust the foam particles. Because of such a cumbersome process, the readjustment of the foam effects requires a change in the motion of the underlying water, and it is not easy to produce such a scene because the water and foam effects must change at the same time. In this paper, we present a method to maintain angular momentum-based force from water particles without tuning the motion of the underlying water. We can restore the lost turbulent flow by additional advection of foam particles based on this force. In addition, our method can be integrated with screen-space projection frameworks, allowing us to fully embrace all the advantages of this approach. In this paper, the turbulence of the foam particles was improved by minimizing the viscous motion of the foam particles without tuning the motion of the underlying water, and as a result, lively foam effects can be expressed. [ABSTRACT FROM AUTHOR]