arXiv subjects

Bhaskar Bagchi

Publications and source records attributed to Bhaskar Bagchi.

At least 19 recordsLinked to original sources

arXiv2026

Triple arrays from ovals in finite projective planes

In this paper we prove that whenever a projective plane of odd order n contains an oval, it may be used to construct a triple array with n + 1 rows, n.n columns and n(n + 1) symbols. In particular, for any odd prime power s, we use the projective plane over the field of order s to construct an explicitly given (s + 1) times s.s triple array on s(s + 1) symbols. This is only the third infinite series of triple arrays (up to transposition) to be found. Our construction proves a recent conjecture of Gordeev and Ohman (2025) in the case of odd prime powers.

math.CO↗

arXiv2023

On the structure of cellular pseudomanifolds

In this paper we study the structure of cellular pseudomanifolds (aka abstract polytopes). These are natural combinatorial generalisations of polytopal spheres (i.e., boundary complexes of convex polytopes). This class is closed under natural notions of duality and product. We show that they are also closed under an operation of direct product. Any cellular pseudomanifold and it's dual have homeomorphic geometric carriers, while the geometric carrier of the product of two of them is homeomorphic to the product of the carriers of the factors. The excess of a cellular pseudomanifold is defined as the non-negative integer $n - d - 2$ where $d$ is the dimension and $n$ is the number of vertices. We completely classify the cellular pseudo manifolds of excess $< 2$, and make some progress towards classifying those of excess 2.

math.CO↗

arXiv2023

Dembowski's Theorem on Finite Inversive Planes of Even Order

A remarkable theorem due to Peter Dembowski states that if $I$ is an inversive plane of even order $q$ then $q$ must be a power of two and $I$ must be the incidence system of points versus plane ovals in an ovoid in the projective $3$-space over the field of order $q$. In this paper we present a short and self-contained proof of this result. Our proof depends on the classification due to Benson of the symmetric and regular finite generalized quadrangles. Included here is a deduction of Benson's Theorem from the Dembowski-Wagner combinatorial characterization of finite projective geometries.

math.CO↗

arXiv2021

Parametric restrictions on quasi-symmetric designs

In this paper, we attach several new invariants to connected strongly regular graphs (excepting conference graphs on non-square number of vertices) : one invariant called the discriminant, and a p-adic invariant corresponding to each prime number p. We prove parametric restrictions on quasi-symmetric 2-designs with a given connected block graph $G$ and a given defect (absolute difference of the two intersection numbers) solely in terms of the defect and the parameters of $G$, including these new invariants. This is a natural analogue of Schutzenberger's Theorem and the Shrikhande-Chowla-Ryser theorem. This theorem is effective when these graph invariants can be explicitly computed. We do this for complete multipartite graphs, co-triangular graphs, symplectic non-orthogonality graphs (over the field of order $2$) and the Steiner graphs, yielding explicit restrictions on the parameters of quasi-symmetric 2-designs whose block graphs belong to any of these four classes.

math.CO↗

arXiv2020

Aspects of optimality of plans orthogonal through other factors

The concept of orthogonality through the block factor (OTB), defined in Bagchi (2010), is extended here to orthogonality through a set (say S) of other factors. We discuss the impact of such an orthogonality on the precision of the estimates as well as on the inference procedure. Concentrating on the case when $S$ is of size two, we construct a series of plans in each of which every pair of other factors is orthogonal through a given pair of factors. Next we concentrate on plans through the block factors (POTB). We construct POTBs for symmetrical experiments with two and three-level factors. The plans for two factors are E-optimal, while those for three-level factors are universally optimal. Finally, we construct POTBs for $s^t(s+1)$ experiments, where $s \equiv 3 \pmod 4$ is a prime power. The plan is universally optimal.

math.ST↗

arXiv2020

Optimality of multi-way designs

In this paper we study optimality aspects of a certain type of designs in a multi-way heterogeneity setting. These are ``duals" of plans orthogonal through the block factor (POTB). Here by the dual of a main effect plan (say $ρ$) we mean a design in a multi-way heterogeneity setting obtained from $ρ$ by interchanging the roles of the block factors and the treatment factors. Specifically, we take up two series of universally optimal POTBs for symmetrical experiments constructed in Morgan and Uddin (1996). We show that the duals of these plans, as multi-way designs, satisfy M-optimality. Next, we construct another series of multiway designs and proved their M-optimality, thereby generalising the result of Bagchi and Shah (1989). It may be noted that M-optimality includes all commonly used optimality criteria like A-, D- and E-optimality.

math.ST↗

arXiv2019

A product formula for homogeneous characteristic functions

A bounded linear operator $T$ on a Hilbert space is said to be homogeneous if $φ(T)$ is unitarily equivalent to $T$ for all $φ$ in the group Möb of bi-holomorphic automorphisms of the unit disc. A projective unitary representation $σ$ of Möb is said to be associated with an operator T if $φ(T)= σ(φ)^\star T σ(φ)$ for all $φ$ in Möb. In this paper, we develop a Möbius equivariant version of the Sz.-Nagy--Foias model theory for completely non-unitary (cnu) contractions. As an application, we prove that if T is a cnu contraction with associated (projective unitary) representation $σ$, then there is a unique projective unitary representation $\hatσ$, extending $σ$, associated with the minimal unitary dilation of $T$. The representation $\hatσ$ is given in terms of $σ$ by the formula $$ \hatσ = (π\otimes D_1^+) \oplus σ\oplus (π_\star \otimes D_1^-), $$ where $D_1^\pm$ are the two Discrete series representations (one holomorphic and the other anti-holomorphic) living on the Hardy space $H^2(\mathbb D)$, and $π, π_\star$ are representations of Möb living on the two defect spaces of $T$ defined explicitly in terms of $σ$. Moreover, a cnu contraction $T$ has an associated representation if and only if its Sz.-Nagy--Foias characteristic function $θ_T$ has the product form $θ_T(z) = π_\star(φ_z)^* θ_T(0) π(φ_z),$ $z\in \mathbb D$, where $φ_z$ is the involution in Möb mapping $z$ to $0.$ We obtain a concrete realization of this product formula %the two representations $π_\star$ and $π$ for a large subclass of homogeneous cnu contractions from the Cowen-Douglas class.

math.FA↗

arXiv2018

A coding theoretic approach to the uniqueness conjecture for projective planes of prime order

An outstanding folklore conjecture asserts that, for any prime $p$, up to isomorphism the projective plane $PG(2,\mathbb{F}_p)$ over the field $\mathbb{F}_p := \mathbb{Z}/p\mathbb{Z}$ is the unique projective plane of order $p$. Let $π$ be any projective plane of order $p$. For any partial linear space ${\cal X}$, define the inclusion number $i({\cal X},π)$ to be the number of isomorphic copies of ${\cal X}$ in $π$. In this paper we prove that if ${\cal X}$ has at most $\log_2 p$ lines, then $i({\cal X},π)$ can be written as an explicit rational linear combination (depending only on ${\cal X}$ and $p$) of the coefficients of the complete weight enumerator (c.w.e.) of the $p$-ary code of $π$. Thus, the c.w.e. of this code carries an enormous amount of structural information about $π$. In consequence, it is shown that if $p > 2^ 9=512$, and $π$ has the same c.w.e. as $PG(2,\mathbb{F}_p)$, then $π$ must be isomorphic to $PG(2,\mathbb{F}_p)$. Thus, the uniqueness conjecture can be approached via a thorough study of the possible c.w.e. of the codes of putative projective planes of prime order.

math.CO↗

arXiv2017

The fourth smallest Hamming weight in the code of the projective plane over $\mathbb{Z}/p \mathbb{Z}$

Let $p$ be a prime and let $C_p$ denote the $p$-ary code of the projective plane over ${\mathbb Z}/p\mathbb{Z}$. It is well known that the minimum weight of non-zero words in $C_p$ is $p+1$, and Chouinard proved that, for $p \geq 3$, the second and third minimum weights are $2p$ and $2p+1$. In 2007, Fack et. al. determined, for $p\geq 5$, all words of $C_p$ of these three weights. In this paper we recover all these results and also prove that, for $p \geq 5$, the fourth minimum weight of $C_p$ is $3p-3$. The problem of determining all words of weight $3p-3$ remains open.

math.CO↗

arXiv2016

A characterization of tightly triangulated 3-manifolds

For a field $\mathbb{F}$, the notion of $\mathbb{F}$-tightness of simplicial complexes was introduced by Kühnel. Kühnel and Lutz conjectured that any $\mathbb{F}$-tight triangulation of a closed manifold is the most economic of all possible triangulations of the manifold. The boundary of a triangle is the only $\mathbb{F}$-tight triangulation of a closed 1-manifold. A triangulation of a closed 2-manifold is $\mathbb{F}$-tight if and only if it is $\mathbb{F}$-orientable and neighbourly. In this paper we prove that a triangulation of a closed 3-manifold is $\mathbb{F}$-tight if and only if it is $\mathbb{F}$-orientable, neighbourly and stacked. In consequence, the Kühnel-Lutz conjecture is valid in dimension $\leq 3$.

math.GT↗

arXiv2016

Tight triangulations of closed 3-manifolds

It is well known that a triangulation of a closed 2-manifold is tight with respect to a field of characteristic two if and only if it is neighbourly; and it is tight with respect to a field of odd characteristic if and only if it is neighbourly and orientable. No such characterization of tightness was previously known for higher dimensional manifolds. In this paper, we prove that a triangulation of a closed 3-manifold is tight with respect to a field of odd characteristic if and only if it is neighbourly, orientable and stacked. In consequence, the Kühnel-Lutz conjecture is valid in dimension three for fields of odd characteristic. Next let $\mathbb{F}$ be a field of characteristic two. It is known that, in this case, any neighbourly and stacked triangulation of a closed 3-manifold is $\mathbb{F}$-tight. For triangulated closed 3-manifolds with at most 71 vertices or with first Betti number at most 188, we show that the converse is true. But the possibility of an $\mathbb{F}$-tight non-stacked triangulation on a larger number of vertices remains open. We prove the following upper bound theorem on such triangulations. If an $\mathbb{F}$-tight triangulation of a closed 3-manifold has $n$ vertices and first Betti number $β_1$, then $(n-4)(617n- 3861) \leq 15444β_1$. Equality holds here if and only if all the vertex links of the triangulation are connected sums of boundary complexes of icosahedra.

math.GT↗

arXiv2014

Efficient algorithms to decide tightness

Tightness is a generalisation of the notion of convexity: a space is tight if and only if it is "as convex as possible", given its topological constraints. For a simplicial complex, deciding tightness has a straightforward exponential time algorithm, but efficient methods to decide tightness are only known in the trivial setting of triangulated surfaces. In this article, we present a new polynomial time procedure to decide tightness for triangulations of $3$-manifolds -- a problem which previously was thought to be hard. Furthermore, we describe an algorithm to decide general tightness in the case of $4$-dimensional combinatorial manifolds which is fixed parameter tractable in the treewidth of the $1$-skeletons of their vertex links, and we present an algorithm to decide $\mathbb{F}_2$-tightness for weak pseudomanifolds $M$ of arbitrary but fixed dimension which is fixed parameter tractable in the treewidth of the dual graph of $M$.

cs.CG↗

arXiv2014

A tightness criterion for homology manifolds with or without boundary

A simplicial complex $X$ is said to be tight with respect to a field $\mathbb{F}$ if $X$ is connected and, for every induced subcomplex $Y$ of $X$, the linear map $H_\ast (Y; \mathbb{F}) \rightarrow H_\ast (X; \mathbb{F})$ (induced by the inclusion map) is injective. This notion was introduced by Kühnel in [10]. In this paper we prove the following two combinatorial criteria for tightness. (a) Any $(k+1)$-neighbourly $k$-stacked $\mathbb{F}$-homology manifold with boundary is $\mathbb{F}$-tight. Also, (b) any $\mathbb{F}$-orientable $(k+1)$-neighbourly $k$-stacked $\mathbb{F}$-homology manifold without boundary is $\mathbb{F}$-tight, at least if its dimension is not equal to $2k+1$. The result (a) appears to be the first criterion to be found for tightness of (homology) manifolds with boundary. Since every $(k+1)$-neighbourly $k$-stacked manifold without boundary is, by definition, the boundary of a $(k+1)$-neighbourly $k$-stacked manifold with boundary - and since we now know several examples (including two infinite families) of triangulations from the former class - theorem (a) provides us with many examples of tight triangulated manifolds with boundary. The second result (b) generalizes a similar result from [2] which was proved for a class of combinatorial manifolds without boundary. We believe that theorem (b) is valid for dimension $2k+1$ as well. Except for this lacuna, this result answers a recent question of Effenberger [8] affirmatively.

math.AT↗

arXiv2014

The mu vector, Morse inequalities and a generalized lower bound theorem for locally tame combinatorial manifolds

In a recent work [2] with Datta, we introduced the mu vector (with respect to a given field) of simplicial complexes and used it to study tightness and lower bounds. In this paper, we modify the definition of mu vectors. With the new definition, most results of [2] become correct without the hypothesis of 2-neighbourliness. In particular, the combinatorial Morse inequalities of [2] are now true of all simplicial complexes. As an application, we prove the following generalized lower bound theorem (GLBT) for connected locally tame combinatorial manifolds. If $M$ is such a manifold of dimension $d$, then for $1 \leq \ell \leq \frac{d-1}{2}$ and any field $\mathbb{F}, ~ g_{\ell+1} (M) \geq \binom{d+2}{\ell+1} \sum\limits_{i=1}^\ell (-1)^{\ell-i} β_i (M;\mathbb{F})$. Equality holds here if and only if $M$ is $\ell$-stacked. We conjecture that, more generally, this theorem is true of all triangulated connected and closed homology manifolds. A conjecture on the sigma vectors of triangulated homology spheres is proposed, whose validity will imply this GLB Conjecture for homology manifolds. We also prove the GLBC for all connected and closed combinatorial 3-manifolds. Thus, any connected closed combinatorial manifold $M$ of dimension three satisfies $g_2 (M) \geq 10 β_1 (M;\mathbb{F})$, with equality iff $M$ is 1-stacked. This result settles a question of Novik and Swartz [6] in the affirmative.

math.GT↗

arXiv2013

On stellated spheres and a tightness criterion for combinatorial manifolds

We introduce the $k$-stellated spheres and consider the class ${\cal W}_k(d)$ of triangulated $d$-manifolds all whose vertex links are $k$-stellated, and its subclass ${\cal W}^{\ast}_k(d)$ consisting of the $(k+1)$-neighbourly members of ${\cal W}_k(d)$. We introduce the mu-vector of any simplicial complex and show that, in the case of 2-neighbourly simplicial complexes, the mu-vector dominates the vector of its Betti numbers componentwise; the two vectors are equal precisely for tight simplicial complexes. We are able to estimate/compute certain alternating sums of the components of the mu-vector of any 2-neighbourly member of ${\cal W}_k(d)$ for $d\geq 2k$. As one consequence of this theory, we prove a lower bound theorem for such triangulated manifolds, as well as determine the integral homology type of members of ${\cal W}^{\ast}_k(d)$ for $d \geq 2k+2$. As another application, we prove that, when $d \neq 2k+1$, all members of ${\cal W}^{\ast}_k(d)$ are tight. We also characterize the tight members of ${\cal W}^{\ast}_k(2k + 1)$ in terms of their $k^{\rm th}$ Betti numbers. These results more or less answer a recent question of Effenberger, and also provide a uniform and conceptual tightness proof for all except two of the known tight triangulated manifolds. We also prove a lower bound theorem for triangulated manifolds in which the members of ${\cal W}_1(d)$ provide the equality case. This generalises a result (the $d=4$ case) due to Walkup and Kuehnel. As a consequence, it is shown that every tight member of ${\cal W}_1(d)$ is strongly minimal, thus providing substantial evidence in favour of a conjecture of Kuehnel and Lutz asserting that tight triangulated manifolds should be strongly minimal.

math.GT↗

arXiv2013

On $k$-stellated and $k$-stacked spheres

We introduce the class $Σ_k(d)$ of $k$-stellated (combinatorial) spheres of dimension $d$ ($0 \leq k \leq d + 1$) and compare and contrast it with the class ${\cal S}_k(d)$ ($0 \leq k \leq d$) of $k$-stacked homology $d$-spheres. We have $Σ_1(d) = {\cal S}_1(d)$, and $Σ_k(d) \subseteq {\cal S}_k(d)$ for $d \geq 2k - 1$. However, for each $k \geq 2$ there are $k$-stacked spheres which are not $k$-stellated. The existence of $k$-stellated spheres which are not $k$-stacked remains an open question. We also consider the class ${\cal W}_k(d)$ (and ${\cal K}_k(d)$) of simplicial complexes all whose vertex-links belong to $Σ_k(d - 1)$ (respectively, ${\cal S}_k(d - 1)$). Thus, ${\cal W}_k(d) \subseteq {\cal K}_k(d)$ for $d \geq 2k$, while ${\cal W}_1(d) = {\cal K}_1(d)$. Let $\bar{\cal K}_k(d)$ denote the class of $d$-dimensional complexes all whose vertex-links are $k$-stacked balls. We show that for $d\geq 2k + 2$, there is a natural bijection $M \mapsto \bar{M}$ from ${\cal K}_k(d)$ onto $\bar{\cal K}_k(d + 1)$ which is the inverse to the boundary map $\partial \colon \bar{\cal K}_k(d + 1) \to {\cal K}_k(d)$.

math.GT↗

arXiv2012

On polytopal upper bound spheres

Generalizing a result (the case $k = 1$) due to M. A. Perles, we show that any polytopal upper bound sphere of odd dimension $2k + 1$ belongs to the generalized Walkup class ${\cal K}_k(2k + 1)$, i.e., all its vertex links are $k$-stacked spheres. This is surprising since the $k$-stacked spheres minimize the face-vector (among all polytopal spheres with given $f_0,..., f_{k - 1}$) while the upper bound spheres maximize the face vector (among spheres with a given $f_0$). It has been conjectured that for $d\neq 2k + 1$, all $(k + 1)$-neighborly members of the class ${\cal K}_k(d)$ are tight. The result of this paper shows that, for every $k$, the case $d = 2k +1$ is a true exception to this conjecture.

math.GT↗

arXiv2012

Combinatorial triangulations of homology spheres

Let $M$ be an $n$-vertex combinatorial triangulation of a $\ZZ_2$-homology $d$-sphere. In this paper we prove that if $n \leq d + 8$ then $M$ must be a combinatorial sphere. Further, if $n = d + 9$ and $M$ is not a combinatorial sphere then $M$ can not admit any proper bistellar move. Existence of a 12-vertex triangulation of the lens space $L(3, 1)$ shows that the first result is sharp in dimension three. In the course of the proof we also show that any $\ZZ_2$-acyclic simplicial complex on $\leq 7$ vertices is necessarily collapsible. This result is best possible since there exist 8-vertex triangulations of the Dunce Hat which are not collapsible.

math.GT↗