Minimum Number Of Points Research Articles

On the acyclic point‐connectivity of the n‐cube

The acyclic point‐connectivity of a graph G, denoted α(G), is the minimum number of points whose removal from G results in an acyclic graph. In a 1975 paper, Harary stated erroneously that α(Qn) = 2n−1 − 1 where Qn denotes the n‐cube. We prove that for n > 4, 7 · 2n−4 ≤ α(Qn) ≤ 2n−1 − 2n−y−2, where y = [log2(n − 1)]. We show that the upper bound is obtained for n ≤ 8 and conjecture that it is obtained for all n.

Incidence Structures whose Planes are Nets

A d-net is a connected semilinear incidence structure π such that (D1) every plane is a net, (D2) the intersection of two subspaces is connected, (D3) if two planes in a 3-space have a point in common then they have a second point in common, and (D4) the minimum number of points which generate π is d . Let V be a vector space over a skew field F , and W a subspace of finite codimension d . Let P , L be the set of d -, ( d - 1)-dimensional subspaces respectively of V whose intersection with W is the zero vector. The incidence structure ( P , L ⊇ ) is called an attenuated space . We show every d -net for finite d ⩾ 3 is an attenuated space. We also characterize d -nets (together with AG ( d , 2)) as those incidence structures belonging to the diagram where signifies a projective plane and signifies a net.

Regular separable graphs of minimum order with given diameter

A ( d, c, v)-graph G is one which is regular of degree v and has diameter d and connectivity c. G is said to be minimum if it is of minimum order, i.e. has the minimumnumber of points; G is separable if c=1. In this paper, the minimum order of a ( d, 1, v)-graph is determined and the construction of all minimum ( d, 1, v)-graphs is described.

The minimum-order three-connected cubic graphs with specified diameters

A minimum <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(d,c, \upsilon )</tex> -graph is one which is regular of degree <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\upsilon</tex> , has diameter <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</tex> and connectivity <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</tex> , and is of minimum order, i.e., has the minimum number of points. Such graphs have application in the design of survivable communications networks. The minimum order of a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(d, 3,3)</tex> -graph is determined, and the construction of all minimum <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(d, 3,3)</tex> -graphs is described.

A method of construction of optimal experimental designs for the circle

An optimal design concentrated at a minimum number of points is considered for a polynomial regression experiment on the sphere. For the case of a polynomial experiment on the circle it is possible to explicitly find the parameters of an optimal minimal design if any optimal design is known.

F-Sets in graphs

A subset S of the vertex set of a graph G is called an F-set if every α ϵ Γ( G), the automorphism group of G, is completely specified by specifying the images under α of all the points of S, and S has a minimum number of points. The number of points, k( G), in an F-set is an invariant of G, whose properties are studied in this paper. For a finite group Γ we define k( Γ) = max{ k( G) | Γ( G) = Γ}. Graphs with a given Abelian group and given k-value ( k ≤ k( Γ)) have been constructed. Graphs with a given group and k-value 1 are constructed which give simple proofs to the theorems of Frucht and Bouwer on the existence of graphs with given abstract/permutation groups.

Comparison of various techniques for the determination of the random incidence response of microphones and sound level meters

The measurements necessary for the determination of the random incidence responses of microphones and sound level meters were made from 500–20 000 Hz in an anechoic chamber using both single-frequency tenes and 1/3-octave bands of noise. The random incidence responses were computed from these data using formulas given in the relevant IEC and ANSI standards documents and according to a Gaussian-type numerical integration formula. This Gaussian formula appears to introduce less numerical error due to the approximation procedure used to compute the random incidence response than either the IEC or ANSI formulas. The random incidence responses of the microphones and sound level meters tested in the anechoic room were also measured in two reverberation rooms using both discrete sampling of the reverberant field and two types of continuous sampling: linear and circular. The results from the various methods and procedures are compared to the exact values (determined by a numerical procedure that used at least 30 points) and the minimum number of points (angles) necessary to achieve a prescribed level of accuracy in the random incidence responses are determined.

Computing the Finite-Range Distorted-Wave Born Approximation forB11(O16,N15)C12

Two-dimensional quadrature formulas with a minimal number of points are applied to the computation of the double radial integral in the finite-range distorted-wave Born approximation for nuclear reactions. The resulting code kundry is fast, especially in cluster transfer reactions where the integrated area is largest. The code was used to investigate the reaction $^{11}\mathrm{B}$($^{16}\mathrm{O}$, $^{15}\mathrm{N}$)$^{12}\mathrm{C}$ and the coherent interference between the proton and alpha-cluster transfer reaction mechanisms.

On Minimum-Point Second-Order Designs

In this note, we discuss k-factor, second order designs with minimum number of points ½(k + l)(k + 2), in particular, those which are extensions of designs that give minimum generalized variance for k = 2 and 3. The experimental region is the unit cuboid. Minimum point designs of this type are unknown for k ≥ 4, and these designs are the best found to date except for k = 4, where a better design is known. Kiefer has shown that these designs cannot be the best for k ≥ 7, via an existence result but, even here, specific better designs are not known and appear difficult to obtain. We also discuss some difficulties of using, in practice, designs that, are D-optimal (that is give minimum generalized variance when the number of points is not restricted).

On Minimum-Point Second-Order Designs

In this note, we discuss k-factor, second order designs with minimum number of points ½(k + l)(k + 2), in particular, those which are extensions of designs that give minimum generalized variance for k = 2 and 3. The experimental region is the unit cuboid. Minimum point designs of this type are unknown for k ≥ 4, and these designs are the best found to date except for k = 4, where a better design is known. Kiefer has shown that these designs cannot be the best for k ≥ 7, via an existence result but, even here, specific better designs are not known and appear difficult to obtain. We also discuss some difficulties of using, in practice, designs that, are D-optimal (that is give minimum generalized variance when the number of points is not restricted).

Heawood's theorem and connectivity

Let G be a finite connected graph with no loops or multiple edges. The point-connectivity K ( G ) of G is the minimum number of points whose removal results in a disconnected or trivial graph. Similarly, the line-connectivity λ( G ) of G is the minimum number of lines whose removal results in a disconnected or trivial graph. For the complete graph K p we have

Minimal Point Cubatures of Precision Seven for Symmetric Planar Regions

A method of constructing 12 point cubature formulas with polynomial precision seven is given for planar regions and weight functions which are symmetric in each variable. If the nodes are real, the weights are positive. For any fully symmetric region, or any region which is the product of symmetric intervals, it is shown that infinitely many 12 point formulas exist, and that these formulas use the minimum number of points.

On Self-Contained Numerical Integration Formulas for Symmetric Regions

Approximate integration formulas for compact regions in n-dimensional Euclidean space are considered. For planar regions which are invariant under the symmetries of a square, it is shown that there exist 4-point equally-weighted formulas of (algebraic) degree three with all of the points in the domain of integration. However, for $n \geqq 3$ and odd, there exist regions which have the symmetries of an n-cube, such that none of the $2n$-point formulas of degree three are self-contained. An example is given of such a region. For planar regions which are invariant under the symmetries of a triangle, it is shown that there exist 3-point equally-weighted formulas of degree two with all points in the domain of integration. It is again shown by example that this property does not extend to $n=3$. The planar results are the first to guarantee self-contained formulas with the minimal number of points for reasonably large classes of regions.

Antler Morphometry in a Colorado Mule Deer Population

Symmetry was demonstrated between left and right antler beam diameters, lengths, and number of points in over 1,200 Cache la Poudre mule deer (Odocoileus hemionus) segregated into two age-classes. Decreasing annual precipitation, soil moisture, and three- to five-fold decreases in mean yields of two major winter range browse species over a 4-year period did not appear to affect antler development. Simple correlation coefficients (r) of antler beam weight (Y) versus antler beam mea- surements (X) of 25 yearling deer were: diameter, 0.62; length, 0.82; and number of points, 0.49. Similar values for 30 deer about 27 months and older were: diameter, 0.82; length, 0.74; and number of points, 0.52. Multiple correlation coefficients derived from antler weight versus these variables were: 0.87 (yearlings) and 0.82 (adults). Antler weight was linearly related to estimated age (0.94), and nonlinearly to eviscerated carcass weight (0.85) in 23 mule deer ranging in age from about 15-77 months. The limitations and possible management applications of these data are described. Abstract: Symmetry was demonstrated between left and right antler beam diameters, lengths, and number of points in over 1,200 Cache la Poudre mule deer (Odocoileus hemionus) segregated into two age-classes. Decreasing annual precipitation, soil moisture, and three- to five-fold decreases in mean yields of two major winter range browse species over a 4-year period did not appear to affect antler development. Simple correlation coefficients (r) of antler beam weight (Y) versus antler beam mea- surements (X) of 25 yearling deer were: diameter, 0.62; length, 0.82; and number of points, 0.49. Similar values for 30 deer about 27 months and older were: diameter, 0.82; length, 0.74; and number of points, 0.52. Multiple correlation coefficients derived from antler weight versus these variables were: 0.87 (yearlings) and 0.82 (adults). Antler weight was linearly related to estimated age (0.94), and nonlinearly to eviscerated carcass weight (0.85) in 23 mule deer ranging in age from about 15-77 months. The limitations and possible management applications of these data are described. There are no published statistical descrip- tions of antler morphometry in Rocky Moun- tain mule deer based on large, age-related samples from a single population. Cowan (1936:208-209) described the antlers and gave detailed measurements of the left beam from five Rocky Mountain mule deer of un- stated age from the Pacific Coast. Hunter (n.d.:9) listed the means and extremes of several measurements from both beams and empirically related these to age-class, car- cass weight, and several carcass measure- ments sampled from four Rocky Mountain mule deer populations in Colorado. His sample sizes ranged from 1-158 when segre- gated into five age-classes. Russo (1964: 110) gave the minimum number of points

Perfectly symmetric two-dimensional integration formulas with minimal numbers of points

Perfectly symmetric integration formula of degrees 9–15 with a minimal number of points are computed for the square, the circle and the entire plane with weight functions exp ( − ( x 2 + y 2 ) ) ( - ({x^2} + {y^2})) and exp ( − ( x 2 + y 2 ) 1 / 2 ) ( - {({x^2} + {y^2})^{1/2}}) . These rules were computed by solving a large system of nonlinear algebraic equations having a special structure. In most cases where the minimal formula has a point exterior to the region or where some of the weights are negative, ’good’ formulas, which consist only of interior points and have only positive weights, are given which contain more than the minimal number of points.

Extremal Point and Edge Sets in n-Graphs

A set of points (edges) of a graph is independent if no two distinct members of the set are adjacent. Gallai (1) observed that, if A0 (B0) is the minimum number of points (edges) of a finite graph covering all the edges (points) and A1 (B1) is the maximum number of independent points (edges), then:holds, where m is the number of points of the graph.The concepts of independence and covering are generalized in various ways for n-graphs. In this paper we establish certain connections between the corresponding extreme numbers analogous to the above result of Gallai.Ray-Chaudhuri considered (2) independence and covering problems in n-graphs and determined algorithms for finding the minimal cover and some associated numbers.

On Indecomposable Graphs

A set of points M of a graph G is a point cover if each line of G is incident with at least one point of M. A minimum cover (abbreviated m.c.) for G is a point cover with a minimum number of points. The point covering number α(G) is the number of points in any minimum cover of G. Let [V1, V2, … , Vr], r &gt; 1 be a partition of V(G), the set of points of G. Let Gi be the subgraph of G spanned by Vi for i = 1, 2, … , r.