projective space

The plane degree g K (2) of a subset K of PG(3, q) is the greatest integer such that at least one plane intersecting K in exactly g K (2) points exists. In this note, (q+1)-arcs of PG(3, q) (that is, twisted cubics when q is odd) are characterized as (q + 1)-sets of type (0, 1, s) 1 of PG(3, q) of minimal plane degree.

Recently, in Innamorati and Zuanni (J. Geom 111:45, 2020. 10.1007/s00022-020-00557-0) the authors give a characterization of a Baer cone of $$\mathrm {PG}(3, q^2)$$ PG ( 3 , q 2 ) , q a prime power, as a subset of points of the projective space intersected by any line in at least one point and by every plane in $$q^2+1$$ q 2 + 1 , $$q^2+q+1$$ q 2 + q + 1 or $$q^3+q^2+1$$ q 3 + q 2 + 1 points. In this paper, we show that a similar characterization holds even without assuming that the order of the projective space is a square, and weakening the assumptions on the three intersection numbers with respect to the planes.

The plane degree g_K(2) of a subset K of PG(3, q) is the greatest integer such that at least one plane intersecting K in exactly g_K(2) points exists. In this note, (q+1)-arcs of PG(3, q) (that is, twisted cubics when q is odd) are characterized as (q + 1)-sets of type (0, 1, s)1 of PG(3, q) of minimal plane degree.

the authors give a characterization of a Baer cone of PG(3, q 2 ), q a prime power, as a subset of points of the projective space intersected by any line in at least one point and by every plane in q 2 + 1, q 2 + q + 1 or q 3 + q 2 + 1 points. In this paper, we show that a similar characterization holds even without assuming that the order of the projective space is a square, and weakening the assumptions on the three intersection numbers with respect to the planes.

Let X be an anisotropic projective quadric over a field F of characteristic not 2. The essential dimension dim es (X) of X, as defined by Oleg Izhboldin, is where i(X) is the first Witt index of X (i.e., the Witt index of X over its own function field). Let Y be a complete (possibly singular) algebraic variety over F with all closed points of even degree and such that Y has a closed point of odd degree over F (X). Our main theorem states that dim es (X) ≤ dim(Y ) and that in the case dim es (X) = dim(Y ) the quadric X F (Y ) is isotropic. Applying the main theorem to a projective quadric Y , we get a proof of Izhboldin's conjecture stated as follows: if an anisotropic quadric Y becomes isotropic over F (X), then dim es (X) ≤ dim es (Y ), and the equality holds if and only if X is isotropic over F (Y ).

A complete classification of 2-arc-transitive dihedrants, that is, Cayley graphs of dihedral groups is given, thus completing the study of these graphs initiated by the third author in [D. Marušič, On 2-arc-transitivity of Cayley graphs, J. Combin. Theory Ser. B 87 (2003) 162-196]. The list consists of the following graphs: (i) cycles C 2n , n 3; (ii) complete graphs K 2n , n 3; (iii) complete bipartite graphs K n,n , n 3; (iv) complete bipartite graphs minus a matching K n,n -nK 2 , n 3; (v) incidence and nonincidence graphs B(H 11 ) and B (H 11 ) of the Hadamard design on 11 points; (vi) incidence and nonincidence graphs B(PG(d, q)) and B (PG(d, q)), with d 2 and q a prime power, of projective spaces; (vii) and an infinite family of regular Z d -covers K 2d q+1 of K q+1,q+1 -(q + 1)K 2 , where q 3 is an odd prime power and d is a divisor of q-1 2 and q -1, respectively, depending on whether q ≡ 1 (mod 4) or q ≡ 3 (mod 4), obtained by identifying the vertex set of the base graph with two copies of the projective line PG(1, q), where the missing matching consists of all pairs of the form [i, i ], i ∈ PG(1, q), and the edge [i, j ] carries trivial voltage if i = ∞ or j = ∞, and carries voltage h ∈ Z d , the residue class of h ∈ Z, if and only if ij = θ h , where θ generates the multiplicative group F * q of the Galois field F q .

We distribute the points and lines ofPG(2, 2n+1) according to a special structure that we call the daisy structure. This distribution is intimately related to a special block design which turns out to be isomorphic toPG(n, 2).We show a blocking set of 3qpoints inPG(2, 2n+1)that intersects each line in at least two points and we apply this to find a lower bound for the heterochromatic number of the projective plane.

We construct linear codes from scrolls over curves of high genus and study the higher support weights di of these codes. We embed the scroll into projective space P k-1 and calculate bounds for the di by considering the maximal number of Fq-rational points that are contained in a codimension h subspace of P k-1 . We nd lower bounds of the di and for the cases of large i calculate the exact values of the di. This work follows the natural generalisation of Goppa codes to higher-dimensional varieties as studied by S.H. Hansen, C. Lomont and T. Nakashima.

Let X be a completely regular Hausdorff space and let C(X) be the ring of all continuous real valued functions defined on X. The complement graph for the zero-divisors in C(X) is a simple graph in which two zero-divisor functions are adjacent if their product is non-zero. In this article, the complement graph for the zero-divisor graph of C(X) and its line graph are studied. It is shown that if X has more than 2 points, then these graphs are connected with radius 2, and diameter less than or equal to 3. The girth is also calculated for them to be 3, and it is shown that they are always triangulated and hypertriangulated. Bounds for the dominating number and clique number are also found for them in terms of the density number of X.

Let (X, d) be a metric space. A subset A of X resolves X if each point x in X is uniquely determined by the distances d(x, a), where a ∈ A. The metric dimension of (X, d) is the smallest integer k such that there is a set A of cardinality k that resolves X. Much is known about the metric dimension when X is the vertex set of a graph, but very little seems to be known for a general metric space. Here we provide some basic results for general metric spaces.

This paper examines the effective representation of the punctual Hilbert scheme. We give new equations, which are simpler than Bayer and Iarrobino-Kanev equations. These new Plücker-like equations define the Hilbert scheme as a subscheme of a single Grassmannian and are of degree two in the Plücker coordinates. This explicit complete set of defining equations for Hilb µ (P n ) are deduced from the commutation relations characterising border bases and from generating equations. We also prove that the punctual Hilbert functor Hilb µ P n can be represented by the scheme Hilb µ (P n ) defined by these relations and the well-known Plücker relations on the Grassmanian. A new description of the tangent space at a point of the Hilbert scheme, seen as a subvariety of the Grassmannian, is also given in terms of projections with respect to the underlying border basis.

This paper is devoted to the study of affine quaternionic manifolds and to a possible classification of all compact affine quaternionic curves and surfaces. It is established that on an affine quaternionic manifold there is one and only one affine quaternionic structure. A direct result, based on the celebrated Kodaira Theorem that studies compact complex manifolds in complex dimension 2, states that the only compact affine quaternionic curves are the quaternionic tori and the primary Hopf surface S 3 × S 1 . As for compact affine quaternionic surfaces, we restrict to the complete ones: the study of their fundamental groups, together with the inspection of all nilpotent hypercomplex simply connected 8-dimensional Lie Groups, identifies a path towards their classification.

First and foremost, I would like to express my gratitude to my supervisor, Pr. Karim Mounirh. Without his scrupulous attention and unwavering patience, this project would have been an impossibility. Furthermore, his guidance, encouragements and friendship have been an invaluable part of my experience at The Faculty of Sciences Dhar El Mehrez, in Mathematical and Humanity levels. My thanks also go to the jury team, Pr. Mohamed Tahar Kadaoui Abbassi and Pr. Hakima Mouanis for their availability and their presence to evaluate this work. I would also like to thank all my friends who offered me a hand with the accomplishment of this thesis. Finally, I must thank my family. My mother, and my father, are a constant source of strength. Thank you for your support in all of my endeavors. * A field k is algebraically closed if every non-constant polynomial (on one indeterminate and with coefficients in k) has a root in k. It follows that every polynomial of degree n can be uniquely factorized (up to permutation of the factors) as where c and the a i are elements of k. ii) If I is the ideal generated by the polynomials in S, then we have Z(I) = Z(S). So algebraic sets can be defined Z(I) for ideals I ⊆ k[T 1 , . . . , T n ]. Recall that all ideals in k[T 1 , . . . , T n ] are finitely generated by the Hilbert Basis Theorem. Examples 1.1.1 1) Affine n-space itself is an algebraic set, since A n = Z(0). Similarly, the empty set ∅ = Z(1) is an algebraic set. 2) Any single point in A n is an algebraic set. Indeed, we have {(a 1 , . . . , a n )} = Z(T 1a 1 , . . . , T na n ). 3) The special linear group, SL(n, k) which is the set of all matrices A = (a ij ) 1≤i,j≤n with entries in k and such that det(A) = 1, can be viewed as a subset of A n 2 by the correspondence b) X ⊆ Z(I(X)) and S ⊆ I(Z(S)). c) The Zariski closure of X is exactly Z(I(X)). So, if X is an algebraic set, then Z(I(X)) = X. c) By b), we have X ⊆ Z(I(X)) and so X ⊆ Z(I(X)). Conversely, let W ⊆ A n be an algebraic set containing X and write W = Z(S) for some S ⊆ k [T 1 , . . . , T n ]. Then, again by b), we have S ⊆ I(Z(S)) = I(W) ⊆ I(X) and so Z(I(X)) ⊆ Z(S) = W, as required. By this lemma, the only thing left that would be needed for a bijective correspondence between ideals of k[T 1 , . . . , T n ] and algebraic sets A n would be I(Z(J)) ⊂ J for any ideal J (so that then I(Z(J)) = J by part b). Unfortunately, the following example shows that why this is not true in general. for some n ∈ N, distinct elements b 1 , . . . , b n ∈ C, and m 1 , . . . , m n ∈ N. Obviously, the zero locus of this ideal in A 1 is Z(J) = {b 1 , . . . , b n } . The polynomials vanishing on this set are precisely those that contain each factor Xb i for i = 1, . . . , n at least once, i. e. we have If at least one of the numbers m 1 , . . . , m n is greater than 1, this is a bigger ideal than J. In what follows we will see that a bijective correspondence does however exist between algebraic sets in A n and some special ideals (radical ideals) of k[T 1 , . . . , 2) Similarly, it is easily verified that O x is a k-algebra when equipped by the following operations : Definition 1.3.5 Let X be a variety, we define the function field k(X) of X as follows : an element of k(X) is an equivalence class of pairs (U, f ) where U is a nonempty open subset of X, f is a regular function on U, and where we identify two pairs (U, f ) and (V, g) when f = g on U ∩ V. Remark 1.3.3 Note that k(X) is indeed a field, for : * Let < U, f > and < V, g > two elements of k(X). Since X is irreducible, any two nonempty open subsets have a nonempty intersection (see proposition 1.1.3). We define : We show that this defines an abelian group structure on k(X). In the same way we define the product of two elements of k(X) and the product of an element of k(X) by a scalar of k. We can easily see that this gives a (commutative) ring structure on k(X). * If < U, f >∈ k(X) with f ̸ = 0, we can restrict f to the open set W = U \ Z( f ) it does not vanish, so that To see that δ( f ) = 0, we using the injectivity of α p+1 and we show that ). So we have β p (g) ∈ Z p+1 (Ω, G) and by definition of ∆ Ω , we have ∆ β p (g) + B p (Ω, G) = f + B p+1 (Ω, F ). Finally, f + B p+1 (Ω, F ) ∈ Im(∆ Ω ). * Im(∆ Ω ) ⊆ ker( αΩ ) : Let f ∈ Z P-1 (Ω, F ) and f + B p+1 (Ω, F ) ∈ Im(∆ Ω ) by definition we have h ∈ Z p (Ω, H) and g ∈ C p (Ω, G) such that α p+1 ( f ) = δ(g) and β p (g) = h. So we have αΩ (Ω, G). Hence f + B p+1 (Ω, F ) ∈ Ker( αΩ ). * William Rowan Hamilton (4 August 1805-2 September 1865) was an Irish mathematician, Andrews Professor of Astronomy at Trinity College Dublin, and Royal Astronomer of Ireland at Dunsink Observatory. He made major contributions to optics, classical mechanics and abstract algebra. His work was of importance to theoretical physics, particularly his reformulation of Newtonian mechanics, now called Hamiltonian mechanics. It is now central both to electromagnetism and to quantum mechanics. In pure mathematics, he is best known as the inventor of quaternions.

It is conjectured that the question of the existence of a set of d + 1 mutually unbiased bases in a ddimensional Hilbert space if d differs from a power of prime is intimatelly linked with the problem whether there exist projective planes whose order d is not a power of prime. Recently, there has been a considerable resurgence of interest in the concept of the so-called mutually unbiased bases [see, e.g., 1-7], especially in the context of quantum state determination, cryptography, quantum information theory and the King's problem. We recall that two different orthonormal bases A and B of a d-dimensional Hilbert space H d are called mutually unbiased if and only if | a|b | = 1/ √ d for all a∈A and all b∈B. An aggregate of mutually unbiased bases is a set of orthonormal bases which are pairwise mutually unbiased. It has been found that the maximum number of such bases cannot be greater than d + 1 . It is also known that this limit is reached if d is a power of prime. Yet, a still unanswered question is if there are non-prime-power values of d for which this bound is attained. The purpose of this short note is to draw the reader's attention to the fact that the answer to this question may well be related with the (non-)existence of finite projective planes of certain orders. A finite projective plane is an incidence structure consisting of points and lines such that any two points lie on just one line, any two lines pass through just one point, and there exist four points, no three of them on a line . From these properties it readily follows that for any finite projective plane there exists an integer d with the properties that any line contains exactly d + 1 points, any point is the meet of exactly d + 1 lines, and the number of points is the same as the number of lines, namely d 2 + d + 1. This integer d is called the order of the projective plane. The most striking issue here is that the order of known finite projective planes is a power of prime . The question of which other integers occur as orders of finite projective planes remains one of the most challenging problems of contemporary mathematics. The only "no-go" theorem known so far in this respect is the Bruck-Ryser theorem saying that there is no projective plane of order d if d -1 or d -2 is divisible by 4 and d is not the sum of two squares. Out of the first few non-prime-power numbers, this theorem rules out finite projective planes of order 6, 14, 21, 22, 30 and 33. Moreover, using massive computer calculations, it was proved by Lam [12] that there is no projective plane of order ten. It is surmised that the order of any projective plane is a power of a prime. ¿From what has already been said it is quite tempting to hypothesize that the above described two problems are nothing but different aspects of one and the same problem. That is, we conjecture that non-existence of a projective plane of the given order d implies that there are less than d + 1 mutually unbiased bases (MUBs) in the corresponding H d , and vice versa. Or, slightly rephrased, we say that if the dimension d of Hilbert space is such that the maximum of MUBs is less than d + 1, then there does not exist any projective plane of this particular order d. An important observation speaking in favour of our claim is the following one. Let us find the minimum number of different measurements we need to determine uniquely the state of an ensemble of identical d-state systems. The density matrix of such an ensemble, being Hermitian and of unit trace, is specified by (2d 2 /2) -1 = d 2 -1 real parameters. As a given non-degenerate measurement applied to a sub-ensemble gives d -1 real numbers (the probabilities of all but one of the d possible outcomes), the minimum number of different measurements needed to determine the state uniquely is (d 2 -1)/(d -1) = d + 1 . On the other hand, it is a well-known fact [see,

More than thirty new upper bounds on the smallest size t2(2, q) of a complete arc in the plane PG(2, q) are obtained for 169 ≤ q ≤ 839. New upper bounds on the smallest size t2(n, q) of the complete cap in the space PG(n, q) are given for n = 3 and 25 ≤ q ≤ 97, q odd; n = 4 and q = 7, 8, 11, 13, 17; n = 5 and q = 5, 7, 8, 9; n = 6 and q = 4, 8. The bounds are obtained by computer search for new small complete arcs and caps. New upper bounds on the largest size m2(n, q) of a complete cap in PG(n, q) are given for q = 4, n = 5, 6, and q = 3, n = 7, 8, 9. The new lower bound 534 ≤ m2(8, 3) is obtained by finding a complete 534-cap in PG(8, 3). Many new sizes of complete arcs and caps are obtained. The updated tables of upper bounds for t2(n, q), n ≥ 2, and of the spectrum of known sizes for complete caps are given. Interesting complete caps in PG(3, q) of large size are described. A proof of the construction of complete caps in PG(3, 2 h ) announced in previous papers is given; this is modified from a construction of Segre. In PG(2, q), for q = 17, δ = 4, and q = 19, 27, δ = 3, we give complete ( 1 2 (q + 3) + δ)-arcs other than conics that share 1 2 (q + 3) points with an irreducible conic. It is shown that they are unique up to collineation. In PG(2, q), q ≡ 2 (mod 3) odd, we propose new constructions of 1 2 (q + 7)-arcs and show that they are complete for q ≤ 3701.

In this paper we consider the problem of optimizing a quadratic pseudo-Boolean function subject to the cardinality constraint 1 i n x i = k with a polyhedral method. More precisely we propose a study of the convex hull of feasible points included in the Padberg's Boolean quadric polytope and satisfying the cardinality constraint. Specifically, we investigate the connection with the Boolean quadric polytope and study a facet family. The relationship with two other polytopes of the literature is also explored.

By the method of synthetic geometry, we define a seemingly new transformation of a three-dimensional projective space where the corresponding points lie on the rays of the first order, nth class congruence C 1 n and are conjugate with respect to a proper quadric Ψ. We prove that this transformation maps a straight line onto an n + 2 order space curve and a plane onto an n + 2 order surface which contains an n-ple (i.e. n-multiple) straight line. It is shown that in Euclidean space the pedal surfaces of the congruences C 1 n can be obtained by this transformation. The analytical approach enables new visualizations of the resulting curves and surfaces with the program Mathematica. They are shown in four examples.

For a smooth surface S⊂ PK there are well known classical formulas giving the numberρ(S) of secants ofS passing through a generic point of P5. In this paper, for possibly singular surfaces T, a computer assisted computation of ρ(T) from the defining ideal I(T) ⊂ K[x0, . . . ,x5] is proposed. It is based on the Stückrad-Vogel self-interse ction cycle ofT and requires the computation of the normal cone of the ruled joi n J(T,T) along the diagonal. It is shown that in the case when T ⊂ P5 arises as the linear projection with center L of a surfaceS⊂ PK (N &gt; 5) (which satisfies some mild assumptions), the computational complexity can be reduced considerably by using the normal con e of SecS alongL∩SecS instead of the former normal cone. Many examples and the relativ code for the computer algebra systems R EDUCE, CoCoA, Macaulay2 and Singular are given.

The notion of a projective system, defined as a set X of n-points in a projective space over a finite field, was introduced by Tsfasman and Vlaˇdut. By this notion, the weight distribution of a nondegenerate linear code can be computed by the configuration of X and hyperplanes in the dual projective space of C: In this paper, we construct a projective system X defined as a set of n-linear subspaces on the dual projective space of a poset code. This gives a natural oneto-one correspondence between the set of equivalence classes of nondegenerate poset projective systems and the set of nondegenerate poset codes. By this correspondence, the weight distribution of a nondegenerate poset code can be also computed by the configuration of X and hyperplanes in the dual projective space of C: Finally, we provide an algorithm for finding perfect poset codes.

An Alexander self-dual complex gives rise to a compactification of $M_{0,n}$, called ASD compactification, which is a smooth algebraic variety. ASD compactifications include (but are not exhausted by) the polygon spaces, or the moduli spaces of flexible polygons. We present an explicit description of the Chow rings of ASD compactifications. We study the analogues of Kontsevich tautological bundles, compute their Chern classes, compute top intersections of the Chern classes, and derive a recursion for the intersection numbers.

First, I would like to express my profound gratitude to my PhD supervisors, Prof. Slim Chaabane and Prof. Houssem Haddar. I sincerely thank you both for your trust, granting me research freedom, your invaluable guidance and mentorship, and most importantly, for standing by my side during all the difficult moments. I would also like to extend my thanks to Professors Amel Ben Abda and Maher Moakher for agreeing to review this work and for the time they dedicated to reading and assessing this thesis, as well as for preparing their reports. I am deeply grateful to Professors Afif Masmoudi, Mondher Benjemaa, and Mohamed Hbaib for agreeing to be part of the jury. My appreciation extends to all the members of the Mathematics Department at the Faculty of Sciences of Sfax and the LAMHA laboratory for providing me with such a friendly and supportive environment. Special thanks to Bilel Charfi and Zouhour Jlali for our enjoyable discussions, and to Assistant Fatma Abid for her valuable administrative assistance. I also want to acknowledge my doctoral friends, Mariem Ammar, Yosra Barkaoui, Khouloud Dammak, and Chaima Hmani, with whom I shared many cherished moments during my thesis. Additionally, I want to express my gratitude to all the members of the IDEFIX team for their warmth and friendliness. A special thanks goes to Marcella, who provided invaluable assistance during my internship when we shared the same office, and to my doctoral and post-doctoral friends, Amal, Nouha, and Mohamed Aziz, for the fantastic moments we shared during my internships. A huge thank you to my friends Yassmine, Vanessa and Walid, who provided invaluable assistance during my stays in France and with whom I shared wonderful moments. Before I wrap up, I want to thank my friends whom I met in Albania during my participation in the CIMPA summer school and who have remained in contact with me, especially Ina Hila, Gazi Ozdemir and Destiny S. Lutero. Finally, I want to express my deepest gratitude to my family, with a special mention of my sister Hanen, for their unwavering support during these years of my thesis.

Let M n be a compact Riemannian manifold isometrically immersed in the Euclidean space R n+m . Then a modification of a beautiful method first used by Lawson and Simons [22] is used to give a pointwise algebraic condition on the second fundamental form of M in R n+m which implies that M has no complete stable submanifolds (or integral currents) in some given dimension p (1 ≤ p ≤ n -1). It is then shown that this condition is preserved under small deformation of the metric on M in the C 2 topology. Some results of these are (1) There is a C 2 neighborhood of the standard metric on the Euclidean sphere S n (and other sufficiently convex hypersurfaces in R n+1 ), such that for any g in this neighborhood (S n , g) has no stable submanifolds (or integral currents). (2) A characterization (and in some cases a classification ) of stable submanifolds and integral currents of all the rank one symmetric spaces (extending the work of Lawson and Simons on the spheres and complex projective spaces ), and some information about what happens in this case under a small C 2 deformation of the metric. (3) There is a C 2 neighborhood U of the standard metric on the complete simply connected manifold R n+m (c) of constant sectional curvature c ≥ 0 such that if M n is a compact immersed submanifold of (R n+m (c), g) with mean curvature vector and the second fundamental form satisfying (5.17) for some g ∈ U , 1 ≤ p ≤ n 2 , and q = n -1. Then (a) M has no stable p-integral or (n -p)-integral currents over any finitely abelian group G, (b) Hp(M, G) = Hq(M, G) = 0 for any G and if p = 1 or p = n-1 then M is simply connected, and (c) If (5.17) holds for some g ∈ U , and p = 1, then M is diffeomorphic to S n for all n ≥ 2. Indeed, by a Theorem of Federer and Fleming [10] every non-zero integral homology class in a compact manifold N can be represented by a stable integral current; thus, the method can also be used to give pointwise conditions on the second fundamental form of a compact submanifold which forces some of the integral homology groups of M to vanish.

Lisema Victor Rammea uena le ntate Mohaila, le moo a ithobaletseng teng, le nkholisitse ka thata, ha ke tsebe nka u leboha joang, 'mé oa ka. Uena le bana beso le ntŝehelitse ka mokhoa o ikhethang le ha ke ntse ke le fosetsa, ha lea ka la mphuralla. Molimo oa ka o tla mpaballela lona batebang. Finally, I thank my wife 'Makeletso Rammea, for her love and devotion and her tremendous support and encouragement. Kea u rata lehakoe la pelo ea ka, ke leboha lerato le boitelo ba hau maratuoa. To my son Moati Rammea, I hope that he did not miss home too much and that his experience in England will bear fruits for him later in life. ii Decker and Schreyer [19] 2000 ≤ 52 *in response to errors they found above b X = 7, irregularity q = 0 and geometric genus p g = 6. Moreover, P 4 has an involution under which X is invariant, giving a quotient X ∈ P 4 (w). Chapter 5 contains the third main result of this thesis: Proposition. There exists a smooth general type surface X in P 4 , of degree d = 9, sectional genus π = 10, topological Euler characteristic c 2 ( X) = 63, first Chern number c 2 1 ( X) = 9, Euler characteristic of the structure sheaf χ O b X = 6, irregularity q = 0 and geometric genus p g = 5. We also give some evidence for

If X ⊂ P n is a reduced subscheme, we say that X admits an unexpected hypersurface of degree t for multiplicity m if the imposition of having multiplicity m at a general point P fails to impose the expected number of conditions on the linear system of hypersurfaces of degree t containing X. Conditions which either guarantee the occurrence of unexpected hypersurfaces, or which ensure that they cannot occur, are not well understand. We introduce new methods for studying unexpectedness, such as the use of generic initial ideals and partial elimination ideals to clarify when it can and when it cannot occur. We also exhibit algebraic and geometric properties of X which in some cases guarantee and in other cases preclude X having certain kinds of unexpectedness. In addition, we formulate a new way of quantifying unexpectedness (our AV sequence), which allows us detect the extent to which unexpectedness persists as t increases but tm remains constant. Finally, we study to what extent we can detect unexpectedness from the Hilbert function of X.

We extend Oprea's result that the Gottlieb group G 1 (S 2n+1 /H) is ZH (the center of H) and show that for a map f : A → S 2n+1 /H, under some conditions on A, we have , the centralizer of the image f * (π 1 (A)) in H. Then, we compute or estimate the groups G f m (S 2n+1 /H) and P f m (S 2n+1 /H) for certain m > 1.

We show that points in specific degree 2 hypersurfaces in the Grassmannian Gr(3, n) correspond to generic arrangements of n hyperplanes in C 3 with associated discriminantal arrangement having intersections of multiplicity three in codimension two.

The aim of this paper is to establish unified integrals that encompass a diverse set of elements, including multivariable general class of polynomials, Jacobi polynomials, and the I-function of two variables. Initially, we computed integrals involving multivariable general class of polynomials and the I-function of two variables using the Oberhettinger formula. Subsequently, we derived integrals involving Jacobi polynomials and the I-function of two variables. Due to the general nature of I-function of two variables, we also derived specific integrals involving the I-function and the H-function of one variable as special cases based on our primary findings.

In 1824, Abel showed that there is no general algebraic solution for the roots of a quintic equation, or any polynomial equation of degree greater than four, using explicit algebraic operations, as stated in the Abel-Ruffini theorem (1799). In doing so, he independently developed a branch of mathematics known as group theory, which has proven invaluable not only in many areas of mathematics but also in much of physics. Abel sent a paper on the unsolvability of the quintic equation to Carl Friedrich Gauss, who dismissed it without a glance, believing it to be the worthless work of a crank. Later, in 1957, Kolmogorov and Arnold proved that functions of three variables are reducible to superpositions of functions of two variables, thus providing an affirmative answer on this problem. In this work, our focus is to clarify the concept of solving equations of degree greater than six using continuous functions or hypergeometric functions, and to provide another proof of the nonexistence of algebraic solutions for equations of degree greater than four. According to the Kolmogorov-Arnold theorem, we will demonstrate that equations of degree greater than five are not solvable without specific conditions between their coefficients, using hypergeometric functions. However, trinomial equations can generally be solved using hypergeometric functions. Additionally, it is well-known that a general polynomial equation has a solution in the complex plane using iterative approximation methods, GRIM(arXiv:2205.04241), which can also solve the transcendental equations. We develop in the first part the polynomial equations and will continue in the second part (new version of the publication) with the solution of transcendental equations that can be solved mainly by hypergeometric functions. With these two articles we conclude a large section of polynomial and transcendental equations that can be solved by these modern methods.

In 1824, Abel showed that there is no general algebraic solution for the roots of a quintic equation, or any polynomial equation of degree greater than four, using explicit algebraic operations, as stated in the Abel-Ruffini theorem (1799). In doing so, he independently developed a branch of mathematics known as group theory, which has proven invaluable not only in many areas of mathematics but also in much of physics. Abel sent a paper on the unsolvability of the quintic equation to Carl Friedrich Gauss, who dismissed it without a glance, believing it to be the worthless work of a crank. Later, in 1957, Kolmogorov and Arnold proved that functions of three variables are reducible to superpositions of functions of two variables, thus providing an affirmative answer on this problem. In this work, our focus is to clarify the concept of solving equations of degree greater than six using continuous functions or hypergeometric functions, and to provide another proof of the nonexistence of algebraic solutions for equations of degree greater than four. According to the Kolmogorov-Arnold theorem, we will demonstrate that equations of degree greater than five are not solvable without specific conditions between their coefficients, using hypergeometric functions. However, trinomial equations can generally be solved using hypergeometric functions. Additionally, it is well-known that a general polynomial equation has a solution in the complex plane using iterative approximation methods, GRIM(arXiv:2205.04241), which can also solve the transcendental equations. We develop in the first part the polynomial equations and will continue in the second part (new version of the publication) with the solution of transcendental equations that can be solved mainly by hypergeometric functions. With these two articles we conclude a large section of polynomial and transcendental equations that can be solved by these modern methods.

This paper links two previously more or less unrelated important examples at the intersection of the fields of transformation groups, exotic spheres, and nonnegative curvature. These two examples are the exotic Gromoll-Meyer sphere 7 and the Brieskorn sphere W 5 3 . Several applications are drawn from the interplay between the Riemannian geometry of a 2-parameter family of metrics on 7 and the equivariant geometry of W 5 3 , which, surprisingly, determines the equivariant geometry of 7 much more than the equivariant geometry of the exotic Brieskorn spheres W 7 6j -1,3 , although the latter contain W 5 3 in a much more obvious way. In 1974, Gromoll and Meyer [GrMy] constructed 7 as a biquotient of the compact group Sp(2) and thereby the first exotic sphere with nonnegative sectional curvature. Note that 7 can be regarded as the basic example of a biquotient in Riemannian geometry and, simultaneously, as the basic example of an exotic sphere. It generates the group 7 ≈ Z 28 of homotopy spheres in dimension 7, the first dimension except possibly 4 where exotic spheres can occur. Recently, it was shown that 7 is actually the only exotic sphere that can be modeled by a biquotient of a compact Lie group [KaZ; To]. Because of this exceptional status of the Gromoll-Meyer sphere, it seems natural to study the geometry of 7 in detail. Papers that do this from various viewpoints are [D; EK; PaSp; Y; Wi], for example. Here we investigate 7 through the interaction between symmetry arguments, submanifold stratifications, and geodesic constructions. It is important, however, to note that we consider not only the Gromoll-Meyer metric on 7 but also the entire 2-parameter family of metrics •, • µ,ν that are O(2) × SO(3) invariant by construction. This family includes the Gromoll-Meyer metric µ = ν = 1 2 and the pointed wiedersehen metric constructed in [D] (µ = ν = 1) but not the metrics of almost positive sectional curvature obtained in [EK] and [Wi]. Extending the constructions of [D] and [ADPR], we obtain the following structural information.

Generalizations of the classical affine Lelieuvre formula to surfaces in projective three-dimensional space and to hypersurfaces in multidimensional projective space are given. A discrete version of the projective Lelieuvre formula is presented too.

Given a collection of 2 k -1 real vector bundles εa over a closed manifold F , suppose that, for some a 0 , εa 0 is of the form ε a 0 ⊕ R, where R → F is the trivial one-dimensional bundle. In this paper we prove that if a εa → F is the fixed data of a (Z 2 ) k -action, then the same is true for the Whitney sum obtained from a εa by replacing εa 0 by ε a 0 . This stability property is well-known for involutions. Together with techniques previously developed, this result is used to describe, up to bordism, all possible (Z 2 ) kactions fixing the disjoint union of an even projective space and a point.

In this paper we obtain conditions for a Whitney sum of three vector bundles over a closed manifold, ε 1 ⊕ ε 2 ⊕ ε 3 → F \varepsilon _{1} \oplus \varepsilon _{2} \oplus \varepsilon _{3} \rightarrow F , to be the fixed data of a ( Z 2 ) 2 (Z_{2})^{2} -action; these conditions yield the fact that if ( ε 1 ⊕ R ) ⊕ ε 2 ⊕ ε 3 → F (\varepsilon _{1} \oplus R) \oplus \varepsilon _{2} \oplus \varepsilon _{3} \rightarrow F is the fixed data of a ( Z 2 ) 2 (Z_{2})^{2} -action, where R → F R \rightarrow F is the trivial one dimensional bundle, then the same is true for ε 1 ⊕ ε 2 ⊕ ε 3 → F \varepsilon _{1} \oplus \varepsilon _{2} \oplus \varepsilon _{3} \rightarrow F . The results obtained, together with techniques previously developed, are used to obtain, up to bordism, all possible ( Z 2 ) 2 (Z_{2})^{2} -actions fixing the disjoint union of an even projective space and a point.

Given a collection of 2 k -1 real vector bundles εa over a closed manifold F , suppose that, for some a 0 , εa 0 is of the form ε a 0 ⊕ R, where R → F is the trivial one-dimensional bundle. In this paper we prove that if a εa → F is the fixed data of a (Z 2 ) k -action, then the same is true for the Whitney sum obtained from a εa by replacing εa 0 by ε a 0 . This stability property is well-known for involutions. Together with techniques previously developed, this result is used to describe, up to bordism, all possible (Z 2 ) kactions fixing the disjoint union of an even projective space and a point.

Let S m be a m-sphere (or, more generally, a homology m-sphere), Y a k-dimensional CW-complex and f : S m → Y a continuous map. Let G be a finite group which acts freely on S m . Suppose that H ⊂ G is a nontrivial normal cyclic subgroup of a prime order. We estimate the cohomological dimension of the set A(f, H, G) of (H, G)-coincidence points of f .

Let A be the family of all equivtiant bordism classes of involutions containing a representative (M, T) with M connected and with the fixed point set of T being the disjoint union of a fixed connected n-dimensional manifold V" and a point. In this paper we analyse the influence of A in the determination of the equivariant bordism classes of (ZZ)~-actions which contain a representative fixing Vn U {point}. The results obtained are used to determine, up to bordism, all possible (Z2)kactions fixing RIP(n) U {point} with n odd.

Let X be a simply connected finite CW complex whose cohomology groups with coefficients in the group of integers, Z, satisfy H j (X; Z) = Z if j = 0, n, 2n or 3n, and H j (X; Z) = 0 otherwise. Let u i generate H in (X; Z) for i = 0, 1, 2 and 3. We say that X has type (a, b), for integers a and b, if u 2 1 = au 2 and u 1 u 2 = bu 3 . We show that the group G = Z 2 can not act freely on a space of type (a, b) if a is odd and b is even, and that the group G = S 1 can not act freely on a space of type (a, b) if a = 0. For the remaining pairs (a, b), we may have free actions, and thus it makes sense to ask for the possible cohomology rings of the corresponding orbit spaces. In this direction, we determine the possible Z 2 -cohomology rings of orbit spaces of free actions of Z 2 on spaces of type (a, b), where a and b are even, and of free actions of S 1 on spaces of type (0, b). As a consequence of these cohomological calculations, we also obtain some results of the Borsuk-Ulam type, concerning the existence of equivariant maps S m → X, where S m is equipped with standard G-actions (G = Z 2 or S 1 ) and X are spaces of type (a, b) equipped with arbitrary G-actions.

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We show that the Coble hypersurfaces, uniquely characterized by the remarkable property
that their singular loci are an abelian surface and a Kummer threefold, respectively, belong to a family
of hypersurfaces exhibiting similar behavior, but defined in various types of homogeneous spaces. With
the help of Jordan-Vinberg theory, we show how these hypersurfaces can be parametrized by G¨opel type
varieties inside projectivized representations of complex reflection groups.

This paper extends a recently proposed singularity analysis method to lower-mobility parallel manipulators having an articulated nacelle. Using screw theory, a twist graph is introduced in order to simplify the constraint analysis of such manipulators. Then, a wrench graph is obtained in order to represent some points at infinity on the Plücker lines of the Jacobian matrix. Using Grassmann-Cayley algebra, the rank deficiency of the Jacobian matrix amounts to the vanishing condition of the superbracket. Accordingly, the parallel singularities are expressed in three different forms involving superbrackets, meet and join operators, and vector cross and dot products, respectively. The approach is explained through the singularity analysis of the H4 robot. All the singularity conditions of this robot are enumerated and the motions associated with these singularities are characterized.