dinesh sarvate | College of Charleston (original) (raw)

Papers by dinesh sarvate

Discrete Mathematics , 2016

We prove that the necessary condition, n ≡ 0(mod 3), is sufficient for the existence of GDD(n, 2,... more We prove that the necessary condition, n ≡ 0(mod 3), is sufficient for the existence of GDD(n, 2, 4; 3, 4) except possibly for n = 18. We prove that necessary conditions for the existence of group divisible designs GDD(n, 2, 4; λ 1 , λ 2) with equal number of even and odd blocks are sufficient for GDD(n, 2, 4; 5n, 7(n−1)) for all n ≥ 2, GDD(7s, 2, 4; 5s, 7s−1) for all s, GDD(5t +1, 2, 4; 5t +1, 7t) for t ≡ 0(mod 2) and GDD(5t +1, 2, 4; 2(5t +1), 14t) for all t. To complete the existence of such GDDs, one needs to construct two more families: GDD(5t + 1, 2, 4; 5t + 1, 7t) for all odd t, and GDD(35s + 21, 2, 4; 5s + 3, 7s + 4) for all positive integers s.

Discrete Mathematics, Jul 1, 2008

Under the right conditions it is possible for the ordered blocks of a path design PATH(v,k,@m) to... more Under the right conditions it is possible for the ordered blocks of a path design PATH(v,k,@m) to be considered as unordered blocks and thereby create a BIBD(v,k,@l). We call this a tight embedding. We show here that, for any triple system TS(v,3), there is always such an embedding and that the problem is equivalent to the existence of a (-1)-BRD(v,3,3), i.e., a c-Bhaskar Rao Design. That is, we also prove the incidence matrix of any triple system TS(v,3) can always be signed to create a (-1)-BRD(v,3,3) and, moreover, the signing determines a natural partition of the blocks of the triple system making it a nested design.

Discrete Mathematics, Jul 1, 2008

We give some constructions of new infinite families of group divisible designs, GDD(n, 2, 4; 1 , ... more We give some constructions of new infinite families of group divisible designs, GDD(n, 2, 4; 1 , 2), including one which uses the existence of Bhaskar Rao designs. We show the necessary conditions are sufficient for 3 n 8. For n = 10 there is one missing critical design. If 1 > 2 , then the necessary conditions are sufficient for n ≡ 4, 5, 8 (mod 12). For each of n=10, 15, 16, 17, 18, 19, and 20 we indicate a small minimal set of critical designs which, if they exist, would allow construction of all possible designs for that n. The indices of each of these designs are also among those critical indices for every n in the same congruence class mod 12.

Journal of Algebra Combinatorics Discrete Structures and Applications

The present note is motivated by two papers on group divisible designs (GDDs) with the same block... more The present note is motivated by two papers on group divisible designs (GDDs) with the same block size three but different number of groups: three and four where one group is of size 111 and the others are of the same size nnn. Here we present some interesting constructions of GDDs with block size 4 and three groups: one of size 111 and other two of the same size nnn. We also obtain necessary conditions for the existence of such GDDs and prove that they are sufficient in several cases. For example, we show that the necessary conditions are sufficient for the existence of a GDD$(1,n,n,4;\lambda_1,\lambda_2)$ for nequiv0,1,4,5,8,9pmod12n\equiv0,1,4,5,8,9\pmod{12}nequiv0,1,4,5,8,9pmod12 when lambda1gelambda2\lambda_1\ge \lambda_2lambda_1gelambda_2.

Ars Comb., 2018

This paper is motivated by a survey on the existence of G-designs by Adams, Bryant and Buchanan, ... more This paper is motivated by a survey on the existence of G-designs by Adams, Bryant and Buchanan, where they gave the spectrum of the decomposition of complete graphs into graphs with small numbers of vertices. We give difference family-type constructions to decompose λ copies, where λ ≥ 2, of the complete graph Kv into several multigraphs on four vertices and five edges including so called 2-petal graphs and box-edge graphs.

A note on small defining sets for some SBIBD(4t-1, 2t-1, t-1) A note on small defining sets for s... more A note on small defining sets for some SBIBD(4t-1, 2t-1, t-1) A note on small defining sets for some SBIBD(4t-1, 2t-1, t-1) Dinesh Sarvate

Utilitas Mathematica Publishing Inc., Nov 1, 1984

Sarvate, D.G. and R.C. Hamm, Row-column directed block designs, Discrete Mathematics 92 (1991) 32... more Sarvate, D.G. and R.C. Hamm, Row-column directed block designs, Discrete Mathematics 92 (1991) 321-328. A balanced incomplete block design (BIBD) is called a row-column directed BIBD (RCDBIBD) if: (i) it is directed in the usual sense,, i.e., each ordered pair of points occurs an equal number of times in the blocks (directed column wise), and (ii) the blocks are arranged in such a way that (a) each point occurs an equal number of times in each row, and (b) each ordered pair of distinct points occurs an almost equal number of times in the rows. The present paper gives construction techniques for RCDBIBDs and proves that the necessary conditions are sufficient for the existence of RCDBIBDs with block size 2. Existence of RCDBlBDs with block size 3 and u = I mod 6 is shown and it is proved that an RCDBIBD(7,7,4,4,1*) does not exist.

We give a new construction for a known family of weighing matrices using the 2-adjugate method of... more We give a new construction for a known family of weighing matrices using the 2-adjugate method of Vartak and Patwardhan. We review the existence of Wen, kl), k = 1, ... ,12, giving new results for k '" 8, ... 12.

We extend a method of Kharaghani and obtain some new constructions for weighing matrices and orth... more We extend a method of Kharaghani and obtain some new constructions for weighing matrices and orthogonal designs. In particular we show that if there exists an OD(s1,...,sr), where w = ∑si, of order n, then there exists an OD(s1w,s2w,...,8rw) of order n(n+k) for k ≥ 0 an integer. If there is an OD(t,t,t,t) in order n, then there exists an OD(12t,12t,12t,12t) in order 12n. If there exists an OD(s,s,s,s) in order 4s and an OD(t,t,t,t) in order 4t there exists an OD(12s²t,12s²t,12s²t,12s²t) in order 48s²t and an OD(20s²t,20s²t,20s²t20s²) in order 80s²t.

Generalized Bhaskar Rao n-ary are defined. This paper studies with elements from abelian groups o... more Generalized Bhaskar Rao n-ary are defined. This paper studies with elements from abelian groups of Generalized Bhaskar Rao n ary called Bhaskar Rao Bhaskar Rao a v b matrix of ±1 and such that the inner product of any two rows 0 and the matrix obtained of X by its absolute value the incidence matrix of the construction of infinite families of Balanced Balanced are Some construction methods and necessary conditions for the existence of Bhaskar Rao are A necessary condition for the existence of balanced with even A and block size 4t is given. 1.

Bull. ICA, 2018

We define a 3-GDD(n, 2, k, λ1, λ2) by extending the definitions of a group divisible design and a... more We define a 3-GDD(n, 2, k, λ1, λ2) by extending the definitions of a group divisible design and a t-design and give some necessary conditions for its existence. We prove that these necessary conditions are sufficient for the existence of a 3-GDD(n, 2, 4, λ1, λ2) except possibly when n ≡ 1, 3 (mod 6), n 6= 3, 7, 13 and λ1 > λ2. It is known that a partition of all 3-subsets of a 7-set into 5 Steiner triple systems (a large set for 7) does not exist, but we show that the collection of all 3-sets of a 7-set along with a Steiner triple system on the 7-set can be partitioned into 6 Steiner triple systems. Such a partition is then used to prove the existence of all possible 3-GDDs for n = 7. ∗The authors thank the referees for their comments which improved the paper. They also thank Dr. Bob Mignone of the College of Charleston and Bishop Stuart University Administration, especially to the Vice Chancellor Dr. Maud Kamatenesi Mugisha, for their continued support and the Council for Intern...

T emory designs with replication numbers all or all but one of the values in the range 1\(V-1J/(K... more T emory designs with replication numbers all or all but one of the values in the range 1\(V-1J/(K-l} to 1\(V-l)/(K-2) are studied. It is known that for K=3 and replication numbers all values in the range 1\{V-l) /(K-l J to 1\(V-l }/(K-2J, designs exist if and only if 1\=2. Here we show the uniqueness of such designs. When K=3 and all values in the range 1\(V-l J/(K-l) to 1\(V-l )/(K-2) but one are used, designs are constructed for 1\=2 and all V except for one case. Nonexistence is shown for the missing case. Ternary designs with even block size and replication numbers all or all but Ofle of the values in the range A(V-l }/(K-l) to 1\(V-l)/(K-2) and temary designs with block size an odd number greater than three, 1\=2, and A(V-l )/(K-l) 1\(V-l )/(K-2):;;2 are also describe::d.

T emory designs with replication numbers all or all but one of the values in the range 1\\(V-1J/(... more T emory designs with replication numbers all or all but one of the values in the range 1\\(V-1J/(K-l} to 1\\(V-l)/(K-2) are studied. It is known that for K=3 and replication numbers all values in the range 1\\{V-l) /(K-l J to 1\\(V-l }/(K-2J, designs exist if and only if 1\\=2. Here we show the uniqueness of such designs. When K=3 and all values in the range 1\\(V-l J/(K-l) to 1\\(V-l )/(K-2) but one are used, designs are constructed for 1\\=2 and all V except for one case. Nonexistence is shown for the missing case. Ternary designs with even block size and replication numbers all or all but Ofle of the values in the range A(V-l }/(K-l) to 1\\(V-l)/(K-2) and temary designs with block size an odd number greater than three, 1\\=2, and A(V-l )/(K-l) 1\\(V-l )/(K-2):;;2 are also describe::d.

The subject matter for this paper is GDDs with three groups and block size five in which each blo... more The subject matter for this paper is GDDs with three groups and block size five in which each block has configuration (1, 2, 2); that is, each block has exactly one point from one of the three groups and two points from each of the other two groups. We provide necessary and sufficient conditions of the existence of a GDD (n, 3, 5;λ1, λ2) with configuration (1, 2, 2). A highlight of this paper is a technique which uses two and then three idempotent MOLS consecutively to construct a required family of GDDs.

The subject matter for this paper is GDDs with three groups and block size five in which each blo... more The subject matter for this paper is GDDs with three groups and block size five in which each block has configuration (1, 2, 2); that is, each block has exactly one point from one of the three groups and two points from each of the other two groups. We provide necessary and sufficient conditions of the existence of a GDD (n, 3, 5;λ1, λ2) with configuration (1, 2, 2). A highlight of this paper is a technique which uses two and then three idempotent MOLS consecutively to construct a required family of GDDs.

African Journal of Teacher Education, 2020

The embrace of diversity and multiculturalism in education facilitates the broadening of students... more The embrace of diversity and multiculturalism in education facilitates the broadening of students’ experiences as they engage with teachers and classmates from backgrounds different than their own. However, while the positive effects of diversity on students are apparent, few studies have examined possible negative challenges that diversity might have on students. Where most subject matter is taught via classroom lectures and the lecture material is presented by a speaker with a different accent than the student is used to hearing, does it make the material harder for the student to understand? On the other hand, could it increase the focus and engagement required by the students in the classroom, and in the process increase their understanding? In this vein, our research sought understand whether students’ learning of the subject matter hindered when they are taught material by a teacher with a different accent. To this end, we designed a simple experiment with a small group of u...

Discrete Mathematics , 2016

We prove that the necessary condition, n ≡ 0(mod 3), is sufficient for the existence of GDD(n, 2,... more We prove that the necessary condition, n ≡ 0(mod 3), is sufficient for the existence of GDD(n, 2, 4; 3, 4) except possibly for n = 18. We prove that necessary conditions for the existence of group divisible designs GDD(n, 2, 4; λ 1 , λ 2) with equal number of even and odd blocks are sufficient for GDD(n, 2, 4; 5n, 7(n−1)) for all n ≥ 2, GDD(7s, 2, 4; 5s, 7s−1) for all s, GDD(5t +1, 2, 4; 5t +1, 7t) for t ≡ 0(mod 2) and GDD(5t +1, 2, 4; 2(5t +1), 14t) for all t. To complete the existence of such GDDs, one needs to construct two more families: GDD(5t + 1, 2, 4; 5t + 1, 7t) for all odd t, and GDD(35s + 21, 2, 4; 5s + 3, 7s + 4) for all positive integers s.

Discrete Mathematics, Jul 1, 2008

Under the right conditions it is possible for the ordered blocks of a path design PATH(v,k,@m) to... more Under the right conditions it is possible for the ordered blocks of a path design PATH(v,k,@m) to be considered as unordered blocks and thereby create a BIBD(v,k,@l). We call this a tight embedding. We show here that, for any triple system TS(v,3), there is always such an embedding and that the problem is equivalent to the existence of a (-1)-BRD(v,3,3), i.e., a c-Bhaskar Rao Design. That is, we also prove the incidence matrix of any triple system TS(v,3) can always be signed to create a (-1)-BRD(v,3,3) and, moreover, the signing determines a natural partition of the blocks of the triple system making it a nested design.

Discrete Mathematics, Jul 1, 2008

We give some constructions of new infinite families of group divisible designs, GDD(n, 2, 4; 1 , ... more We give some constructions of new infinite families of group divisible designs, GDD(n, 2, 4; 1 , 2), including one which uses the existence of Bhaskar Rao designs. We show the necessary conditions are sufficient for 3 n 8. For n = 10 there is one missing critical design. If 1 > 2 , then the necessary conditions are sufficient for n ≡ 4, 5, 8 (mod 12). For each of n=10, 15, 16, 17, 18, 19, and 20 we indicate a small minimal set of critical designs which, if they exist, would allow construction of all possible designs for that n. The indices of each of these designs are also among those critical indices for every n in the same congruence class mod 12.

Journal of Algebra Combinatorics Discrete Structures and Applications

The present note is motivated by two papers on group divisible designs (GDDs) with the same block... more The present note is motivated by two papers on group divisible designs (GDDs) with the same block size three but different number of groups: three and four where one group is of size 111 and the others are of the same size nnn. Here we present some interesting constructions of GDDs with block size 4 and three groups: one of size 111 and other two of the same size nnn. We also obtain necessary conditions for the existence of such GDDs and prove that they are sufficient in several cases. For example, we show that the necessary conditions are sufficient for the existence of a GDD$(1,n,n,4;\lambda_1,\lambda_2)$ for nequiv0,1,4,5,8,9pmod12n\equiv0,1,4,5,8,9\pmod{12}nequiv0,1,4,5,8,9pmod12 when lambda1gelambda2\lambda_1\ge \lambda_2lambda_1gelambda_2.

Ars Comb., 2018

This paper is motivated by a survey on the existence of G-designs by Adams, Bryant and Buchanan, ... more This paper is motivated by a survey on the existence of G-designs by Adams, Bryant and Buchanan, where they gave the spectrum of the decomposition of complete graphs into graphs with small numbers of vertices. We give difference family-type constructions to decompose λ copies, where λ ≥ 2, of the complete graph Kv into several multigraphs on four vertices and five edges including so called 2-petal graphs and box-edge graphs.

A note on small defining sets for some SBIBD(4t-1, 2t-1, t-1) A note on small defining sets for s... more A note on small defining sets for some SBIBD(4t-1, 2t-1, t-1) A note on small defining sets for some SBIBD(4t-1, 2t-1, t-1) Dinesh Sarvate

Utilitas Mathematica Publishing Inc., Nov 1, 1984

Sarvate, D.G. and R.C. Hamm, Row-column directed block designs, Discrete Mathematics 92 (1991) 32... more Sarvate, D.G. and R.C. Hamm, Row-column directed block designs, Discrete Mathematics 92 (1991) 321-328. A balanced incomplete block design (BIBD) is called a row-column directed BIBD (RCDBIBD) if: (i) it is directed in the usual sense,, i.e., each ordered pair of points occurs an equal number of times in the blocks (directed column wise), and (ii) the blocks are arranged in such a way that (a) each point occurs an equal number of times in each row, and (b) each ordered pair of distinct points occurs an almost equal number of times in the rows. The present paper gives construction techniques for RCDBIBDs and proves that the necessary conditions are sufficient for the existence of RCDBIBDs with block size 2. Existence of RCDBlBDs with block size 3 and u = I mod 6 is shown and it is proved that an RCDBIBD(7,7,4,4,1*) does not exist.

We give a new construction for a known family of weighing matrices using the 2-adjugate method of... more We give a new construction for a known family of weighing matrices using the 2-adjugate method of Vartak and Patwardhan. We review the existence of Wen, kl), k = 1, ... ,12, giving new results for k '" 8, ... 12.

We extend a method of Kharaghani and obtain some new constructions for weighing matrices and orth... more We extend a method of Kharaghani and obtain some new constructions for weighing matrices and orthogonal designs. In particular we show that if there exists an OD(s1,...,sr), where w = ∑si, of order n, then there exists an OD(s1w,s2w,...,8rw) of order n(n+k) for k ≥ 0 an integer. If there is an OD(t,t,t,t) in order n, then there exists an OD(12t,12t,12t,12t) in order 12n. If there exists an OD(s,s,s,s) in order 4s and an OD(t,t,t,t) in order 4t there exists an OD(12s²t,12s²t,12s²t,12s²t) in order 48s²t and an OD(20s²t,20s²t,20s²t20s²) in order 80s²t.

Generalized Bhaskar Rao n-ary are defined. This paper studies with elements from abelian groups o... more Generalized Bhaskar Rao n-ary are defined. This paper studies with elements from abelian groups of Generalized Bhaskar Rao n ary called Bhaskar Rao Bhaskar Rao a v b matrix of ±1 and such that the inner product of any two rows 0 and the matrix obtained of X by its absolute value the incidence matrix of the construction of infinite families of Balanced Balanced are Some construction methods and necessary conditions for the existence of Bhaskar Rao are A necessary condition for the existence of balanced with even A and block size 4t is given. 1.

Bull. ICA, 2018

We define a 3-GDD(n, 2, k, λ1, λ2) by extending the definitions of a group divisible design and a... more We define a 3-GDD(n, 2, k, λ1, λ2) by extending the definitions of a group divisible design and a t-design and give some necessary conditions for its existence. We prove that these necessary conditions are sufficient for the existence of a 3-GDD(n, 2, 4, λ1, λ2) except possibly when n ≡ 1, 3 (mod 6), n 6= 3, 7, 13 and λ1 > λ2. It is known that a partition of all 3-subsets of a 7-set into 5 Steiner triple systems (a large set for 7) does not exist, but we show that the collection of all 3-sets of a 7-set along with a Steiner triple system on the 7-set can be partitioned into 6 Steiner triple systems. Such a partition is then used to prove the existence of all possible 3-GDDs for n = 7. ∗The authors thank the referees for their comments which improved the paper. They also thank Dr. Bob Mignone of the College of Charleston and Bishop Stuart University Administration, especially to the Vice Chancellor Dr. Maud Kamatenesi Mugisha, for their continued support and the Council for Intern...

T emory designs with replication numbers all or all but one of the values in the range 1\(V-1J/(K... more T emory designs with replication numbers all or all but one of the values in the range 1\(V-1J/(K-l} to 1\(V-l)/(K-2) are studied. It is known that for K=3 and replication numbers all values in the range 1\{V-l) /(K-l J to 1\(V-l }/(K-2J, designs exist if and only if 1\=2. Here we show the uniqueness of such designs. When K=3 and all values in the range 1\(V-l J/(K-l) to 1\(V-l )/(K-2) but one are used, designs are constructed for 1\=2 and all V except for one case. Nonexistence is shown for the missing case. Ternary designs with even block size and replication numbers all or all but Ofle of the values in the range A(V-l }/(K-l) to 1\(V-l)/(K-2) and temary designs with block size an odd number greater than three, 1\=2, and A(V-l )/(K-l) 1\(V-l )/(K-2):;;2 are also describe::d.

T emory designs with replication numbers all or all but one of the values in the range 1\\(V-1J/(... more T emory designs with replication numbers all or all but one of the values in the range 1\\(V-1J/(K-l} to 1\\(V-l)/(K-2) are studied. It is known that for K=3 and replication numbers all values in the range 1\\{V-l) /(K-l J to 1\\(V-l }/(K-2J, designs exist if and only if 1\\=2. Here we show the uniqueness of such designs. When K=3 and all values in the range 1\\(V-l J/(K-l) to 1\\(V-l )/(K-2) but one are used, designs are constructed for 1\\=2 and all V except for one case. Nonexistence is shown for the missing case. Ternary designs with even block size and replication numbers all or all but Ofle of the values in the range A(V-l }/(K-l) to 1\\(V-l)/(K-2) and temary designs with block size an odd number greater than three, 1\\=2, and A(V-l )/(K-l) 1\\(V-l )/(K-2):;;2 are also describe::d.

The subject matter for this paper is GDDs with three groups and block size five in which each blo... more The subject matter for this paper is GDDs with three groups and block size five in which each block has configuration (1, 2, 2); that is, each block has exactly one point from one of the three groups and two points from each of the other two groups. We provide necessary and sufficient conditions of the existence of a GDD (n, 3, 5;λ1, λ2) with configuration (1, 2, 2). A highlight of this paper is a technique which uses two and then three idempotent MOLS consecutively to construct a required family of GDDs.

The subject matter for this paper is GDDs with three groups and block size five in which each blo... more The subject matter for this paper is GDDs with three groups and block size five in which each block has configuration (1, 2, 2); that is, each block has exactly one point from one of the three groups and two points from each of the other two groups. We provide necessary and sufficient conditions of the existence of a GDD (n, 3, 5;λ1, λ2) with configuration (1, 2, 2). A highlight of this paper is a technique which uses two and then three idempotent MOLS consecutively to construct a required family of GDDs.

African Journal of Teacher Education, 2020

The embrace of diversity and multiculturalism in education facilitates the broadening of students... more The embrace of diversity and multiculturalism in education facilitates the broadening of students’ experiences as they engage with teachers and classmates from backgrounds different than their own. However, while the positive effects of diversity on students are apparent, few studies have examined possible negative challenges that diversity might have on students. Where most subject matter is taught via classroom lectures and the lecture material is presented by a speaker with a different accent than the student is used to hearing, does it make the material harder for the student to understand? On the other hand, could it increase the focus and engagement required by the students in the classroom, and in the process increase their understanding? In this vein, our research sought understand whether students’ learning of the subject matter hindered when they are taught material by a teacher with a different accent. To this end, we designed a simple experiment with a small group of u...