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Open Physics

formerly Central European Journal of Physics

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Physical-chemical properties studying of molecular structures via topological index calculating

Jianzhang Wu
/ Xiao Yu
/ Wei Gao
Published Online: 2017-05-05 | DOI: https://doi.org/10.1515/phys-2017-0029

Abstract

It’s revealed from the earlier researches that many physical-chemical properties depend heavily on the structure of corresponding moleculars. This fact provides us an approach to measure the physical-chemical characteristics of substances and materials. In our article, we report the eccentricity related indices of certain important molecular structures from mathematical standpoint. The eccentricity version indices of nanostar dendrimers are determined and the reverse eccentric connectivity index for V-phenylenic nanotorus is discussed. The conclusions we obtained mainly use the trick of distance computation and mathematical derivation, and the results can be applied in physics engineering.

PACS: 02.10.0x

1 Introduction

Many experiments reveal the evidence that the physical-chemical properties of pounds and materials are closely related to their molecular structures. For example, the structure-dependency of total π-electron energy Eπ heavily relies on the sum of squares of the vertex degrees of the molecular graph. Topological indices, as a result, are introduced as numerical parameters of molecular structures, and they play a key role in realizing the physical-chemical properties of substances.

In theoretical computational model, a molecular structure can be represented as a molecular graph G in which each vertex is expressed as an atom and each bound between atoms is denoted as an edge. Then, a topological index can be regarded as a score operator f : G → ℝ+ which maps each molecular graph to a positive real number.

In the past four decades, scientists introduced many indices from the application perspective, such as Zagreb index, Wiener index, sum connectivity index and harmonic index which reflect certain physical-chemical characteristics of molecular structure. There were many contributions to report these distance-based or degree-based indices of special molecular structures (See Farahani et al. [1], Jamil et al. [2], Gao and Wang [3], Gao et al. [47] and Gao and Wang [8, 9] for more details). The notations and terminologies that were used (but undefined in our paper) can be found in Bondy and Murty [10].

For a fixed vertex uV(G), the eccentricity ec(u) of vertex u is defined as the largest distance between u and any other vertex v in G. There are several eccentricity related indices introduced for the engineering purpose.

The first atom-bond connectivity index (in short, ABC index) is introduced by Estrada et al. [11] as $ABC(G)=∑uv∈E(G)d(u)+d(v)−2d(u)d(v),$

where d(v) is the degree of vertex v in the molecular graph G. Correspondingly, the fifth atom bond connectivity index is denoted as $ABC5(G)=∑uv∈E(G)ec(u)+ec(v)−2ec(u)ec(v).$

The first geometric-arithmetic index (shortly, GA index) rased by Vukić ević and Furtula [12] as $GA(G)=∑uv∈E(G)2d(u)d(v)d(u)+d(v).$

Several results on GA index can refer to Zhou et al. [13], Rodríguez and Sigarreta [14], [15] and [16], Husin et al. [17], Bahrami and Alaeiyan [18], Sigarreta [19], Divnic et al. [20], Das et al. [21], Mahmiani et al. [22], Fath-Tabar et al. [23] and [24], Das et al. [25], Gutman and Furtula [26], Furtula and Gutman [27], and Shabani et al. [28]. The fourth geometric-arithmetic index is defined by Lee et al. [29] which is stated as $GA4(G)=∑uv∈E(G)2ec(u)ec(v)ec(u)+ec(v).$

Moreover, the first and the second multiplicative eccentricity index are defined as $Π1∗(G)=Πuv∈E(G)(ec(u)+ec(v))$ and $Π2∗(G)=Πuv∈E(G)(ec(u)ec(v)),$ respectively. And, the fourth and sixth Zagreb polynomials are denoted as $Zg4(G,x)=∑uv∈E(G)xec(u)+ec(v)$ and $Zg6(G,x)=∑uv∈E(G)xec(u)ec(v),$ respectively.

As an important index, the eccentric connectivity index (ECI) of molecular graph G is introduced as $ξc(G)=∑v∈V(G)ec(v)d(v).$

Ranjini and Lokesha [30] determined the eccentric connectivity index of the subdivision graph of tadpole graphs, complete graphs and wheel graphs. Morgan et al. [31] yielded the sharp lower bound on ξc(G) for a given graph order. Furthermore, the tight upper and lower bounds are manifested for trees with given vertex number and diameter. Hua and Das [32] presented the relationship between the eccentric connectivity index and Zagreb indices. De [33] deduced the eccentric connectivity index and polynomial of the thorn graphs. Eskender and Vumar [34] studied the eccentric distance sum and eccentric connectivity index of generalized hierarchical product graphs. In addition, the eccentric connectivity index of F-sum graphs by virtue of certain invariants of the factors are presented as well. Ilić and Gutman [35] proposed that the broom reaches the maximum ξc(G) among trees with fixed maximum vertex degree, and the trees with minimum ξc(G) are also characterized. Iranmanesh and Hafezieh [36] computed the eccentric connectivity index of some special graph families. Dankelmann et al. [37] raised the sharp upper bound for eccentric connectivity index and presented the molecular graphs which asymptotically attain this bound. Morgan et al. [38] presented the tight lower bound of eccentric connectivity index for a tree with given order and diameter. Rao and Lakshmi [39] determined the eccentric connectivity index of phenylenic nanotubes.

Ediz [40] introduced a new distance-based topological index called reverse eccentric connectivity index which is stated as $REξc(G)=∑v∈V(G)ec(v)S(v),$

where $S\left(v\right)=\sum _{u\in N\left(v\right)}d\left(u\right).$ Nejati and Mehdi [41] obtained the reverse eccentric connectivity index of tetragonal carbon nanocones.

Although there have been several contributions in distance-based and degree-based indices of molecular structures, the study of eccentricity related indices for certain special compounds is still largely limited. On the other hand, as critical and widespread molecular structures, nanostar dendrimers and V-phenylenic nanotorus are widely used in physics, chemistry, biology, medical and material science. For these reasons, we study the exact expressions of eccentricity related indices for nanostar dendrimers and V-phenylenic nanotorus.

2 Main Results and Proofs

In this paper, we mainly study the eccentricity related indices for nanostar dendrimers (Subsection 2.1) and the reverse eccentric connectivity index of V-phenylenic nanotorus (Subsection 2.2).

2.1 Eccentricity related index of nanostar dendrimers

D1[n] is the first class of nanostar dendrimer family in which |V(D1[n])| = 36n – 12 and |E(D1[n])| = 42n – 15, and D2[n] is the second class of nanostar dendrimer family in which |V(D2[n])| = 120 · 2n – 108 and |E(D2[n])| = 140 · 2n – 127. The example of basic structure of D1[n] and D2[n] can refer to Figure 1 and Figure 2, respectively.

Figure 1

D1[1] and D1[2]

Figure 2

D2[1] and D2[2]

Theorem 1

For n = 1, we have $GA4(D1[1])=184213+241415+72217+361019,ABC5(D1[1])=91142+31314+152+21710,Π1∗(D1[1])=139156176196,Π2∗(D1[1])=429566726906,Zg4(D1[1],x)=9x13+6x15+6x17+6x19,Zg6(D1[1],x)=9x42+6x56+6x72+6x90.$

For n ≥ 2, we get $GA4(D1[n])=18(3n+3)⋅(3n+4)6n+7+∑i=1n(3⋅2i+1(3n+3i+1)⋅(3n+3i+2)6n+6i+3+3⋅2i+1(3n+3i+2)⋅(3n+3i+3)6n+6i+5+3⋅2i+1(3n+3i+3)⋅(3n+3i+4)6n+6i+7)+∑i=1n−13⋅2i+1(3n+3i+3)⋅(3n+3i+4)6n+6i+7,$ $ABC5(D1[n])=96n+5(3n+3)⋅(3n+4)+∑i=1n(3⋅2i6n+6i+1(3n+3i+1)⋅(3n+3i+2)+3⋅2i6n+6i+3(3n+3i+2)⋅(3n+3i+3)+3⋅2i6n+6i+5(3n+3i+3)⋅(3n+3i+4))+∑i=1n−13⋅2i6n+6i+5(3n+3i+3)⋅(3n+3i+4),$ $Π1∗(D1[n])=(6n+7)9⋅∏i=1n((6n+6i+3)3⋅2i⋅(6n+6i+5)3⋅2i⋅(6n+6i+7)3⋅2i)⋅∏i=1n−1(6n+6i+7)3⋅2i,$ $Π2∗(D1[n])=((3n+3)⋅(3n+4))9⋅∏i=1n(((3n+3i+1)⋅(3n+3i+2))3⋅2i⋅((3n+3i+2)⋅(3n+3i+3))3⋅2i⋅((3n+3i+3)⋅(3n+3i+4))3⋅2i)⋅∏i=1n−1((3n+3i+3)⋅(3n+3i+4))3⋅2i,$ $Zg4(D1[n],x)=96n+7+∑i=1n((3⋅2i)6n+6i+3+(3⋅2i)6n+6i+5+(3⋅2i)6n+6i+7)+∑i=1n−1(3⋅2i)6n+6i+7,$ $Zg6(D1[n],x)=9(3n+3)⋅(3n+4)+∑i=1n((3⋅2i)(3n+3i+1)⋅(3n+3i+2)+(3⋅2i)(3n+3i+2)⋅(3n+3i+3)+(3⋅2i)(3n+3i+3)⋅(3n+3i+4)+∑i=1n−1(3⋅2i)(3n+3i+3)⋅(3n+3i+4.$

Proof

Since D1[n] is symmetrical, we can mark the vertices some representative symbols which are showed in Figure 1. Next, we only present the detailed proof of GA4 index, and other parts of result can be obtained in the similar way.

By the analysis of molecular structure, the edge set of D1[n] can be divided into six subsets which are described as follows:

• (u, v): with eccentricities 3n+4 and 3n+3, and there are six edges in this class;

• (a1, v): with eccentricities 3n + 4 and 3n + 3, and there are three edges in this class;

• (ai, bi): with eccentricities 3n + 3i+1 and 3n+3i+2, and there are 3 · 2i edges in i-th generation of this class;

• (bi, ci): with eccentricities 3n+3i+2 and 3n+3i+3, and there are 3 · 2i edges in i-th generation of this class;

• (ci, di) : with eccentricities 3n+3i+3 and 3n+3i+4, and there are 3 · 2i edges in i-th generation of this class;

• (ci, ai+1): with eccentricities 3n+3i+3 and 3n+3i+4, and there are 3 · 2i edges in i-th generation of this class.

Thus, by the definition of the fourth GA index, we infer $GA4(D1[1])=∑uv∈E(D1[1])2ec(u)ec(v)ec(u)+ec(v)=627⋅67+6+327⋅67+6+627⋅87+8+629⋅89+8+629⋅109+10=184213+241415+72217+361019,$ and for n ≥ 2 $GA4(D1[n])=∑uv∈E(D1[n])2ec(u)ec(v)ec(u)+ec(v)=62(3n+3)⋅(3n+4)(3n+3)+(3n+4)+32(3n+3)⋅(3n+4)(3n+3)+(3n+4)+∑i=1n(3⋅2i2(3n+3i+1)⋅(3n+3i+2)(3n+3i+1)+(3n+3i+2)+3⋅2i2(3n+3i+2)⋅(3n+3i+3)(3n+3i+2)+(3n+3i+3)+3⋅2i2(3n+3i+3)⋅(3n+3i+4)(3n+3i+3)+(3n+3i+4))+∑i=1n−13⋅2i2(3n+3i+3)⋅(3n+3i+4)(3n+3i+3)+(3n+3i+4)=18(3n+3)⋅(3n+4)6n+7+∑i=1n(3⋅2i2(3n+3i+1)⋅(3n+3i+2)6n+6i+3+3⋅2i2(3n+3i+2)⋅(3n+3i+3)6n+6i+5+3⋅2i2(3n+3i+3)⋅(3n+3i+4)6n+6i+7)+∑i=1n−13⋅2i2(3n+3i+3)⋅(3n+3i+4)6n+6i+7.$

Therefore, we get the expected result.  □

Theorem 2

For n = 1, we have $GA4(D2[1])=4+161415+24217+241019+811021+163323+323925,$ $ABC5(D2[1]=473+81314+13152+431710+419110+2711+226+42339,$ $Π1∗(D2[1])=142154172194214234242258,Π2∗(D2[1])=4925647229041104132414421568,$ $Zg4(D2[1],x)=2x14+4x15+2x17+4x19+4x21+4x23+2x24+8x25,$ $Zg6(D2[1],x)=2x49+4x56+2x72+4x90+4x110+4x132+2x144+8x156.$

For n ≥ 2, we get $GA4(D2[n])=∑uv∈E(D2[n])2ec(u)ec(v)ec(u)+ec(v)=2+8(10n−3)(10n−2)20n−5+4(10n−1)(10n−2)20n−3+∑i=1n(2i+2(10n+10i−10)(10n+10i−11)20n−20i−21+2i+2(10n+10i−10)(10n+10i−9)20n−20i−19+2i+2(10n+10i−8)(10n+10i−9)20n−20i−17+(2i+2−2n+2+2n+3)(10n+10i−8)(10n+10i−7)20n−20i−15+2i)+∑i=1n(2i+3(10n+10i−5)(10n+10i−6)20n−20i−11+2i+3(10n+10i−5)(10n+10i−4)20n−20i−9+2i+3(10n+10i−3)(10n+10i−4)20n−20i−7+2i+3(10n+10i−3)(10n+10i−2)20n−20i−5+2i+2(10n+10i−7)(10n+10i−6)20n−20i−13+2i+2(10n+10i−1)(10n+10i−2)20n−20i−3.$ $ABC5(D2[n])=210n−320n−8+420n−7(10n−3)(10n−2)+220n−5(10n−1)(10n−2)+∑i=1n(2i+120n+20i−23(10n+10i−10)(10n+10i−11)+2i+120n+20i−21(10n+10i−10)(10n+10i−9)+2i+120n+20i−19(10n+10i−8)(10n+10i−9)+(2i+1−2n+1−2n+2)20n+20i−17(10n+10i−8)(10n+10i−7)+2i10n+10i−820n+20i−18)+∑i=1n(2i+120n+20i−13(10n+10i−5)(10n+10i−6)+2i+120n+20i−11(10n+10i−5)(10n+10i−4)+2i+220n+20i−9(10n+10i−3)(10n+10i−4)+2i+220n+20i−7(10n+10i−3)(10n+10i−2)+2i+120n+20i−15(10n+10i−7)(10n+10i−6)+2i+120n+20i−5(10n+10i−1)(10n+10i−2)),$ $Π1∗(D2[n])=(20n−6)2⋅(20n−5)4⋅(20n−3)2⋅∏i=1n((20n+20i−21)2i+1⋅(20n+20i−19)2i+1⋅(20n+20i−17)2i+1⋅(20n+20i−15)(2i+1−2n+1−2n+2)⋅(20n+20i−16)2i+1)⋅∏i=1n((20n+20i−11)2i+2⋅(20n+20i−9)2i+2⋅(20n+20i−7)2i+2⋅(20n+20i−5)2i+2⋅(20n+20i−13)2i+1⋅(20n+20i−3)2i+1),$ $Π2∗(D2[n])=(10n−3)4⋅((10n−3)(10n−2))4⋅((10n−1)(10n−2))2⋅∏i=1n(((10n+10i−10)(10n+10i−11))2i⋅((10n+10i−10)(10n+10i−9))2i⋅((10n+10i−8)(10n+10i−9))2i+1⋅((10n+10i−8)(10n+10i−7))(2i+1−2n+1−2n+2)⋅(10n+10i−8)2i+1)⋅∏i=1n(((10n+10i−5)(10n+10i−6))2i+2⋅((10n+10i−5)(10n+10i−4))2i+2⋅((10n+10i−3)(10n+10i−4))2i+2⋅((10n+10i−3)(10n+10i−2))2i+2⋅((10n+10i−7)(10n+10i−6))2i+1⋅((10n+10i−1)(10n+10i−2))2i+1),$ $Zg4(D2[n],x)=2x20n−6+4x20n−5+2x20n−3+∑i=1n(2i+1x20n+20i−21+2i+1x20n+20i−19+2i+1x20n+20i−17+(2i+1−2n+1−2n+2)x20n+20i−15+2i+1x20n+20i−16+∑i=1n(2i+2x20n+20i−11+2i+2x20n+20i−9+2i+2x20n+20i−7+2i+2x20n+20i−5+2i+1x20n+20i−13+2i+1x20n+20i−3),$ $Zg6(D2[n],x)=2x(10n−3)2+4x(10n−3)(10n−2)+2x(10n−1)(10n−2)+∑i=1n(2i+1x(10n+10i−10)(10n+10i−11)+2i+1x(10n+10i−10)(10n+10i−9)+2i+1x(10n+10i−8)(10n+10i−9)+(2i+1−2n+1−2n+2)x(10n+10i−8)(10n+10i−7)+2i+1x(10n+10i−8)(10n+10i−8)+∑i=1n(2i+2x(10n+10i−5)(10n+10i−6)+2i+2x(10n+10i−5)(10n+10i−4)+2i+2x(10n+10i−3)(10n+10i−4)+2i+2x(10n+10i−3)(10n+10i−2)+2i+1x(10n+10i−7)(10n+10i−6)+2i+1x(10n+10i−1)(10n+10i−2)).$

Proof

Since D2[n] is also symmetrical, we can mark the vertices some representative symbols which are showed in Figure 2. We only present the detailed proof of GA4 index, and other parts of result can be obtained in the similar way.

By the analysis of molecular structure, the edge set of D2[n] can be divided into 17 subsets which are described as follows:

• (u, v):with eccentricities 10n – 3 and 10n – 3, and there are two edges in this class;

• (v, w): with eccentricities 10n – 3 and 10n – 2, and there are four edges in this class;

• (w, b1):with eccentricities 10n – 2 and 10n – 1, and there are two edges in this class;

• (ai, bi): with eccentricities 10n+10i–10 and 10n + 10i– 11, and there are 2i edges in i-th generation of this class;

• (bi, ci): with eccentricities 10n + 10i–11 and 10n +10i– 10, and there are 2i edges in i-th generation of this class;

• (ci, di): with eccentricities 10n+10i–10 and 10n+10i–9, and there are 2i+1 edges in i-th generation of this class;

• (di, ei): with eccentricities 10n+10i–9 and 10n+10i–8, and there are 2i+1 edges in i-th generation of this class;

• (ei, fi): with eccentricities 10n+10i–8 and 10n+10i–7, and there are 2i+1 edges in i-th (in)generation of this class;

• (en, fn): with eccentricities 10n+10i–8 and 10n+10i–7, and there are 2n+2 edges in this class;

• (ei, gi): with eccentricities 10n+10i–8 and 10n +10i–8, and there are 2i edges in i-th generation of this class;

• (ai′, bi′):with eccentricities 10n+10i–5 and 10n+10i– 6, and there are 2i+1 edges in i-th (1 ≤ in –1 and n ≥ 2) generation of this class;

• (bi′, ci′): with eccentricities 10n+10i–6 and 10n+10i–5, and there are 2i+1 edges in i-th (1 ≤ in –1 and n ≥ 2) generation of this class;

• (ci′, di′): with eccentricities 10n+10i–5 and 10n+10i–4, and there are 2i+2 edges in i-th (1 ≤ in –1 and n ≥ 2) generation of this class;

• (di′, ei′): with eccentricities 10n +10i–4 and 10n + 10i– 3, and there are 2i+2 edges in i-th (1 ≤ in – 1 and n ≥ 2) generation of this class;

• (ei′, fi′): with eccentricities 10n+10i–3 and 10n+10i–2, and there are 2i+2 edges in i-th (1 ≤ in – 1 and n ≥ 2) generation of this class;

• (fi, bi′):with eccentricities 10n+10i–7 and 10n+10i–6, and there are 2i+1 edges in i-th (1 ≤ in –1 and n ≥ 2) generation of this class;

• (fi′, bi+1): with eccentricities 10n + 10i–2 and 10n+ 10i – 1, and there are 2i+1 edges in i-th (1 ≤ in – 1 and n ≥ 2) generation of this class.

Thus, in view of the definition of the fourth GA index, we have $GA4(D2[1])=∑uv∈E(D2[1])2ec(u)ec(v)ec(u)+ec(v)=4+427⋅87+8+229⋅89+8+429⋅109+10+4211⋅1011+10+4211⋅1211+12+8213⋅1213+12=4+161415+24217+241019+811021+163323+323925,$ and for n ≥ 2 $GA4(D2[n])=∑uv∈E(D2[n])2ec(u)ec(v)ec(u)+ec(v)=2+8(10n−3)(10n−2)20n−5+4(10n−1)(10n−2)20n−3+∑i=1n(2i+2(10n+10i−10)(10n+10i−11)20n−20i−21+2i+2(10n+10i−10)(10n+10i−9)20n−20i−19+2i+2(10n+10i−8)(10n+10i−9)20n−20i−17+(2i+2−2n+2+2n+3)(10n+10i−8)(10n+10i−7)20n−20i−15+2i)+∑i=1n(2i+3(10n+10i−5)(10n+10i−6)20n−20i−11+2i+3(10n+10i−5)(10n+10i−4)20n−20i−9+2i+3(10n+10i−3)(10n+10i−4)20n−20i−7+2i+3(10n+10i−3)(10n+10i−2)20n−20i−5+2i+2(10n+10i−7)(10n+10i−6)20n−20i−13+2i+2(10n+10i−1)(10n+10i−2)20n−20i−3.$

Thus, we yield the expected result.  □

2.2 Reverse eccentric connectivity index for V-phenylenic nanotorus

In this subsection, we consider the V-phenylenic nanotorus TO[p, q] (here p is the number of hexagons in each row and q is the number of hexagons in each column) which is a widely used nano structure. For the example of TO[4, 5], please refer to Figure 3.

Figure 3

V-phenylenic nanotorus TO[4, 5]

By the molecular structure analysis, we check that |V(TO[p, q])| = 6pq and |E(TO[p, q])| = 9pq. Furthermore, TO[p, q] is a cubic such that d(v) = 3 for any vV(G) and ec(v) = ec(u) for any two vertices u and v.

Now, we present the main result of this part.

Theorem 3

The reverse eccentric connectivity index for V-phenylenic nanotorus is sated as follows:

1. If p ≡ 0(mod2), then $REξc(TO[p,q])=(4q+p)pq3,ifq≥p(3p+2q)pq3,ifq≤p,$

2. If p ≡ 1(mod2), then $REξc(TO[p,q])=(4q+p−1)pq3,ifq≥p(3p+2q−1)pq3,ifq≤p,$

Proof

Since TO[p, q] is a cubic, we have S(v) = 9 for any vV(G).

The whole proof process can be divided into four parts.

1. p ≡ 0(mod2) and q ≡ 0(mod2).

• If qp, then $ec\left(v\right)=\frac{4q+p}{2}$ for any vV(G).

Then $REξc(TO[p,q])=(4q+p)pq3.$

• If qp, then $ec\left(v\right)=\frac{3p+2q}{2}$ for any vV(G).

Then $REξc(TO[p,q])=(3p+2q)pq3.$

2. p ≡ 1(mod2) and q ≡ 1(mod2).

• If qp, then $ec\left(v\right)=\frac{4q+p-1}{2}$ for any vV(G).

Then $REξc(TO[p,q])=(4q+p−1)pq3.$

• If qp, then $ec\left(v\right)=\frac{3p+2q-1}{2}$ for any vV(G).

Then $REξc(TO[p,q])=(3p+2q−1)pq3.$

3. p ≡ 0(mod2) and q ≡ 1(mod2).

• If qp, then $ec\left(v\right)=\frac{4q+p}{2}$ for any vV(G).

Then $REξc(TO[p,q])=(4q+p)pq3.$

• If qp, then $ec\left(v\right)=\frac{3p+2q}{2}$ for any vV(G).

Then $REξc(TO[p,q])=(3p+2q)pq3.$

4. p ≡ 1(mod2) and q ≡ 0(mod2).

• If qp, then $ec\left(v\right)=\frac{4q+p-1}{2}$ for any vV(G).

Then $REξc(TO[p,q])=(4q+p−1)pq3.$

• If qp, then $ec\left(v\right)=\frac{3p+2q-1}{2}$ for any vV(G).

Then $REξc(TO[p,q])=(3p+2q−1)pq3.$

Hence, we get the desired conclusion.  □

3 Conclusion

In this paper, we mainly report the eccentricity related indices for nanostar dendrimers and the reverse eccentric connectivity index of V-phenylenic nanotorus. Since these indices are widely used in the analysis of physical-chemical properties, they possess a promising prospect of application in physical, chemical, medical and material engineering.

Acknowledgement

The authors thank the reviewers for their constructive comments in improving the quality of this paper. This work was supported in part by the National Natural Science Foundation of China [grant number 60903131]; Science and Technology of Jiangsu Province [grant number BE2011173]; and Key Laboratory of Computer Network and Information Integration Founding in Southeast University.

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Accepted: 2016-09-23

Published Online: 2017-05-05

Conflicts of interest: The authors declare that there is no conflict of interests regarding the publication of this paper.

Citation Information: Open Physics, Volume 15, Issue 1, Pages 261–269, ISSN (Online) 2391-5471,

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## Content

### Some derivatives

So far in the examples and exercises we have found the following derivatives.

 $$f(x)$$ $$f'(x)$$ constant $$c$$ 0 $$ax+b$$ $$a$$ $$x^2$$ $$2x$$ $$x^3$$ $$3x^2$$

Note that the example $$f(x) = ax+b$$ includes the case $$f(x) = x$$, which has derivative $$f'(x) = 1$$. Since we have seen that

$\dfrac{d}{dx} (x) = 1, \qquad \dfrac{d}{dx} (x^2) = 2x \qquad\text{and}\qquad \dfrac{d}{dx} (x^3) = 3x^2,$

it is natural to conjecture that the derivative of $$x^n$$ is $$nx^{n-1}$$.

We will now compute the derivative of $$f(x) = x^n$$, for any positive integer $$n$$. To do so, we need the binomial expansion

$(a+b)^n = a^n + \dbinom{n}{1} a^{n-1} b + \dbinom{n}{2} a^{n-2} b^2 + \dots + \dbinom{n}{n-2} a^2 b^{n-2} + \dbinom{n}{n-1} a b^{n-1} + b^n.$

See the module The binomial theorem for details.

We begin by using the binomial theorem to expand

\begin{align*} f(x + \Delta x) &= (x + \Delta x)^n \\ &= x^n + \dbinom{n}{1} x^{n-1} (\Delta x) + \dbinom{n}{2} x^{n-2} (\Delta x)^2 + \dots + \dbinom{n}{n-1} x (\Delta x)^{n-1} + (\Delta x)^n. \end{align*}

We can then compute

\begin{align*} \dfrac{f(x+\Delta x) - f(x)}{\Delta x} &= \dfrac{(x+\Delta x)^n - x^n}{\Delta x} \\ &= \dbinom{n}{1} x^{n-1} + \dbinom{n}{2} x^{n-2} (\Delta x) + \dots + \dbinom{n}{n-1} x (\Delta x)^{n-2} + (\Delta x)^{n-1}. \end{align*}

The last line has a $$\Delta x$$ in every term except the first. Since $$\binom{n}{1} = n$$, we have

\begin{align*} f'(x) &= \lim_{\Delta x \to 0} \dfrac{f(x + \Delta x) - f(\Delta x)}{\Delta x} \\ &= nx^{n-1}. \end{align*}

The fact that the derivative of $$x^n$$ is $$nx^{n-1}$$ holds more generally than just when $$n$$ is a positive integer. The next exercise shows it is also true when $$n=-1$$. exercise 14, later in this module, shows how to find the derivative of negative integer powers of $$x$$ in general.

##### Exercise 5

Prove that the derivative of $$f(x) = \dfrac{1}{x}$$ is $$f'(x) = - \dfrac{1}{x^2}$$. That is, prove that

$\dfrac{d}{dx}(x^{-1}) = -x^{-2}.$

In fact, it's also true that for any non-zero rational number (i.e., fraction) $$n$$, the derivative of $$f(x) = x^n$$ is $$f'(x) = nx^{n-1}$$. See exercises 20 and 21 later in this module.

Even more generally, for any real number $$a$$, including irrational $$a$$, the derivative of

$f(x) = x^a \qquad\text{is}\qquad f'(x) = a x^{a-1}.$

It is not obvious how to even define what it means to raise a number to the power of an irrational number. For instance, $$2^3$$ just means $$2 \times 2 \times 2$$, and $$2^{\frac{7}{5}}$$ just means $$\sqrt[5]{2^7}$$, but what does $$2^{\sqrt{3}}$$ mean? In the module Exponential and logarithmic functions, we explore these issues, show how to define $$x^a$$ precisely for any real number $$a$$, and show that the derivative of $$x^a$$ is $$a x^{a-1}$$.

In summary, the following theorem is true.

###### Theorem

For any real number $$a$$, the derivative of $$f(x) = x^a$$ is $$f'(x) = a x^{a-1}$$, wherever $$f(x)$$ is defined.

Next page - Content - Properties of the derivative

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Home / Mechanical Engineering / Engineering Mechanics :: Discussion

### Discussion :: Engineering Mechanics

1. The range of a projectile is maximum, when the angle of projection is

2.  A. 30° B. 45° C. 60° D. 90°


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# Triple integral. Spherical coordinates

Find the volume of the solid that lies within the sphere $x^2 + y^2 + z^2 =25$, above the $xy$-plane, and outside the cone $z=3\sqrt{x^2+y^2}$.

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$z=\sqrt{3(x^2+y^2)}$ or $z=3\sqrt{(x^2+y^2)}$? –  Sigur Dec 1 '12 at 23:44
z= 3 * sqrt(x^2+y^2) –  AAbc Dec 1 '12 at 23:49
your second one, sorry I don't know how to type sqrt. –  AAbc Dec 1 '12 at 23:50
Please, try to write in TeX. Use dollar signs $ to enclose your formulas and backslash for commands. For example, \sqrt{}. – Sigur Dec 1 '12 at 23:50 For some basic information about writing math at this site see e.g. here or here. – Américo Tavares Dec 1 '12 at 23:54 ## 2 Answers Hints: $$z=3\sqrt{x^2+y^2}\Longrightarrow 25=x^2+y^2+9x^2+9y^2\Longrightarrow x^2+y^2=2.5\Longrightarrow$$ the cone interesects the sphere on the above rightmost circle. Since everything symetric with respect all the axis and the origin, you can try to calcualte the volumet in the first octant and the multiply by 4 (as we're interested only in what happens above the$\,xy-$plane. Finally, you can try to calculate the volume between the$\,xy-$plane and inside the cone (that is inside the sphere): $$4\int_0^{\sqrt{2.5}}\int_0^{\sqrt{2.5-x^2}}\int_0^{3\sqrt{x^2+y^2}}dzdydx\,\,\,\\\text{Very strongly adviced to change this to cylindrical coordinates}$$ and then substract this from the half sphere's volume. $${}$$ Disclaimer: The above are only general hints. Check it thoroughly. - Using the following substitutions for spherical coordinates: $$z = \rho \cos(\phi)$$ $$x = \rho \sin(\phi)\cos(\theta)$$ $$y = \rho \sin(\phi)\sin(\theta)$$ We can substitute these into your equations: $$\rho ^2 < 25 \implies \rho < 5 \\ \text{Above the xy -plane} \implies \phi > \pi / 2 \\ z < 3 \sqrt{x^2 + y^2} \implies z^2 < 9(x^2 + y^2) \implies (\rho \cos(\phi))^2 < 9(\rho \sin(\phi))^2 \implies \tan(\phi) < 1/3$$$\phi < \tan^{-1}(1/3) \approx 20.48^o\$

Now we can set up our triple integral:

$$\int_0^{2\pi} \int_{20.48}^{90} \int_0^5 \rho^2 \sin(\phi) d\rho d\phi d\theta$$

Inner: $$1/3 \rho^3 \sin(\phi) ]_0^5 = 125/3 sin(\phi)$$ Outer: $$-125/3\cos(\phi)]_{20.48}^{90} = -125/3(0-0.9487) = 39.529$$ Outer(last): $$39.529]_0^{2 \pi} = 248.369$$

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Hi this is what I was trying to do. but would't this be the cone only? since phi is from 0 to 20.48. but i want area under the cone . thank you –  AAbc Dec 2 '12 at 0:28
Yep, my mistake, editing now... –  Ben Dec 2 '12 at 0:29
hi your answer is correct. thank you –  AAbc Dec 2 '12 at 0:43
but i don't understand why is it 20.48 to 90. this would be the volume of the semi sphere cut out the cone form 20.48 to 90. but on top the cone, wouldn't there be a little part as well. since the cone is flat at the top and sphere is round. by going form 20.48-90 we ignore the little part from 0 to 20.48 –  AAbc Dec 2 '12 at 0:46
The cone extends out infinitely, not stopping once it hits the sphere, so it eventually "surrounds" the top part too. This may help visualize. –  Ben Dec 2 '12 at 1:17


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# So, this girl wants to be my roommate

But there is still this issue of $450 a month...maybe a psycho is worth it A prostitute is cheaper and won't land you having to pay tens of thousands in jail bonds or worse whenever she ends up in ER or croaks or goes (more) nuts and decides to ruin your life, something which she is in a very good position to do already. Maybe it won't stroke your ego as much, but then again, neither will your buff jail cell inmate. Last edited: #### Borg Science Advisor Gold Member In all honesty she is financially stable. She teaches at a technical school, writes for a few companies, has her own dog watching business. She even gave me$30 for gas money yesterday for driving her to a clients house.

And shes not slutty, if she was she would have slept with me when I first met her months ago.
So, she needed to be driven to a client after taking every drug and alcohol combination she could get? Why do you make excuses for something that everyone clearly sees as a train wreck in the making? Even if you get $450 a month in cash and 'services', you will regret it. The problems will be greater than you can apparently imagine. Everyone sees this clearly. #### jackmell Weird night. I doubt I'll let her be my roommate because I don't want someone digging through my pills in secret trying to get high off anti psychotics. I personally believe living with a woman in an environment of psychosis will only make matters worst. Would your doctor agree or disagree? Actually it often makes things worst even if you're not psychotic but I digress. My opinion in this matter is to more recognize your mental health and to treat it in a way that makes you better: seek treatment, take your meds, socialize and engage, and at least for the time being, live alone and find happiness in solitude. This is the wise solution to this problem in my opinion. And when you get older, you will look back at some of your failings and wish very, very much that you could go back and make wiser choices in your youth. So there you go: we're back. :) Last edited: #### Mépris In all honesty she is financially stable. She teaches at a technical school, writes for a few companies, has her own dog watching business. She even gave me$30 for gas money yesterday for driving her to a clients house.

And shes not slutty, if she was she would have slept with me when I first met her months ago.
Why are you even trying to rationalise this? This guy below, he speaks the truth:

A prostitute is cheaper and won't land you having to pay tens of thousands in jail bonds or worse whenever she ends up in ER or croaks or goes (more) nuts and decides to ruin your life, something which she is in a very good position to do already. Maybe it won't stroke your ego as much, but then again, neither will your buff jail cell inmate.
Things could get that messed up. Or worse...a false rape charge or something.

Aero, I am willing to bet that if you just went out and spoke to more *young* girls (college bar, streets, coffee shop, movies, anywhere - nothing creepy about it), you will eventually get better at talking to strangers/women, and soon enough, you will find a woman that is attractive, pleasant, interesting, and not "five years older than you" and "insane"...and by then, you will realise how insane the very thought of letting this woman into your house is again.

#### reenmachine

Gold Member
You're still not convinced it's not a good idea? I'm thinking you actually enjoy the excitement a crazy psycho girl could bring into your life.Are you bored?? In need of some adrenaline rushes? Try roller coasters , they are much cheaper in the long run.

#### Aero51

1) In all truthfulness I am attracted to chaos as this is how I grew up.

2) I need the money badly. I know its a bad idea but right now I have $400 a month spending money, which is barely enough for food and gas. #### Lavabug 1) In all truthfulness I am attracted to chaos as this is how I grew up. 2) I need the money badly. I know its a bad idea but right now I have$400 a month spending money, which is barely enough for food and gas.
Seriously man, you're an aerospace engineer right? I'm sure you can figure out a way to cut back on costs and earn some extra money on the side in a much safer fashion.

Is 400$a month really worth the potential drama, court costs and months or years of income because of some bogus conviction she could easily concoct against you? Let me be blunt(er). Let's assume the positive scenario in which she doesn't deliberately go out to screw your life up at some point as we have been hinting. Your DNA is all over her body and probably will be regularly if you keep this up. She could trip down some stairs while tripping balls. Good luck convincing her family or a judge that it wasn't a case of date rape or passionate homicide. Last edited: #### cristo Staff Emeritus Science Advisor I think people are overreacting somewhat. Just because someone (a college kid) was drunk/high does not make her a psycho who's going to stitch you up, or is going to kill herself such that you would get the blame. That's the sort of thing that happens in movies. Still, while the situation may be exciting, she's probably not someone you want to live with! If you really need the money, you could get a part time job? #### Drakkith Staff Emeritus Science Advisor 2018 Award 1) In all truthfulness I am attracted to chaos as this is how I grew up. Or maybe you just consider this 'normal' behavior for people and think it's acceptable. It's not. And the sooner you realize that the better off you'll be. 2) I need the money badly. I know its a bad idea but right now I have$400 a month spending money, which is barely enough for food and gas.
You have other options. Go put out an ad somewhere or ask around and get another roommate if you need money that badly. Move into a smaller, cheaper place. Move closer to work/school so you're not spending so much on gas.

#### zoobyshoe

In all honesty she is financially stable. She teaches at a technical school, writes for a few companies, has her own dog watching business. She even gave me $30 for gas money yesterday for driving her to a clients house. And shes not slutty, if she was she would have slept with me when I first met her months ago. If she stops paying her rent, you'll want to get rid of her, right? If she refuses to leave you'll have to start a legal action in motion. This ends with sheriff's deputies arriving at the apt. who will force her to move her stuff out onto the sidewalk if she claims she has nowhere to put it. Regardless, here in San Diego is COSTS something like$250.00 to file an unlawful detainer against someone, and it takes as much as three months before the deputies come to kick them out.

You had better check into the way it works there, because here it's very hard to get rid of a roommate who refuses to move.

#### Mépris

I think people are overreacting somewhat. Just because someone (a college kid) was drunk/high does not make her a psycho who's going to stitch you up, or is going to kill herself such that you would get the blame. That's the sort of thing that happens in movies.
I wish...I made a google search for "4 drunk girls false rape cab driver", and I saw 3 cases. I was looking for one instance, where the driver was about to be arrested, but his car had a camera, so he was let off the hook. He tried to sue, but I don't recall seeing anything positive coming out of it. The girls were drunk and smoking in his cab, he said they couldn't smoke. They said they'd get down, and they wouldn't pay their fare, and then said they'd call the cops and say he tried to (or actually? I can't remember) raped them. His life would have gone down the drain if it weren't for the video footage! [source]

Another more recent example (UK this time) is one where a girl's relationship was coming to an end. She couldn't deal with that. So, in a desperate attempt to get some attention from her boyfriend, she hits herself, tears off her clothes and says she was raped. Then, here's the worse part, she *randomly* picks two guys on Facebook and says it was them. Luckily, the court ruled in their favour, but they were not compensated and all she got was some community service iirc. [source]

Also this.

All sound like something that could be in a bad soap like Girls or Desperate Housewives. Those are just 3 examples, but if one were to spend a few minutes on a men's rights forum, one would easily find more examples. They usually have sub forums with numerous discussions on the subject.

Note how the men in these scenarios were treated more poorly than the women would have. In any of those cases, if those men did not have valid evidence, everything would be biased in favour of the women. Accusing a man of rape means he guilty of rape until otherwise proven. Recently, an international student was expelled from Vassar College just for being accused of having raped a fellow student (a professor's daughter, it seems). While we don't know if he actually raped her, we also don't know if she's lying. Why not suspend both? Or not allow them to take classes together until evidence is found?

I'm not well versed in the law, but this doesn't sound very right to me. That said, I choose to not complain about it, and instead, stay away from crazy people in general.

I will echo what others have said elsewhere: this is just disrespectful to women (and men!) who have actually been raped. The above examples are just pathetic. I read something a while back, which put this nicely: "I didn't wanna pay my cab fare...no, I was raped" or "I just want some affection...oh honey, look! I just got raped!"

The bottom line is: watch for the red flags. A drunken woman wandering into your home, looking to get high from your pills, drinking your booze, having sex with you, and then actually taking some of your pills after drinking your whisky...well, those all look like red flags to me. Chances are nothing will happen to the OP, but if something does happen, and he has no evidence to prove otherwise (say, video footage - go see a lawyer, and see if you can get her to agree to have cameras in your house, haha), he will likely be considered guilty.

Let me put it this way: is this girl worth your career and your life?

Still, while the situation may be exciting, she's probably not someone you want to live with! If you really need the money, you could get a part time job?
AND he could cut down his expenses like someone else said. Chicken thighs instead of chicken breasts, ground beef, regular peanut butter/banana sandwiches, eggs (scrambled, boiled, fried, poached), oats, milk, rice...that's a healthy foundation for a diet, that's not too expensive. Buy whatever veggies and fruits that are on sale, and you're golden. Add some spices to the mix and use YouTube, and you can do a lot with that.

On that note, this is more or less my diet plan for the next few months.

1) In all truthfulness I am attracted to chaos as this is how I grew up.
Oh lord.

---

On another note, does anyone feel sorry for the dog?

#### 1MileCrash

Sounds like she'd only be a problem, not a solution.

#### Lavabug

I think people are overreacting somewhat.
Not at all. My sister works in law and works with a few criminal defense lawyers. The amount of false domestic violence accusations on the part of greedy/crazy wives, ex-wifes or girlfriends she encounters are astounding. One of the more basic values of democratic law is the presumption of innocence, but it is rarely upheld in favor of men accused of committing violence against women. A lot of people DO get their lives ruined like this.

A more recent case she dealt with was that of a woman who suffered narcolepsy, she passed out and fell, bruising herself (something bound to happen to someone with a drinking problem that looks to combine alcohol with narcotics). She tried to accuse her husband (who he himself had asked for a divorce months earlier) of hitting her and tried to bring him to court. The medical examinations are jokes, nothing like in a CSI episode, just a bold-faced assertion, a quick look at the bruise, and you get a document from the doc indicating you were beaten. Changed her story around several times during the trial and she later gave up on it after the judge gave her a layman's run-down of the consequences of perjury, since he smelled something was fishy. Way to waste the court's time and taxpayer money, that kind of abuse of the legal system should carry some punishment.

Back on topic: please heed our advice Aero. The risks far outweigh any benefits.

Last edited:

#### chemisttree

Homework Helper
Gold Member
On another note, does anyone feel sorry for the dog?
He has already determined that the dog is a slut.

#### Aero51

Ok, I said this already but, apparently it didnt resonate with some of you: she is not slutty...at all. And her dog is very nice and well taken care of. I'm not gonna let her room with me, but I'll try to get to know her and maybe that night she visited me was just particularly bad. Who knows? Anyone whos gone out drinking has had good and bad nights.

#### jedishrfu

Mentor
Ok, I said this already but, apparently it didnt resonate with some of you: she is not slutty...at all. And her dog is very nice and well taken care of. I'm not gonna let her room with me, but I'll try to get to know her and maybe that night she visited me was just particularly bad. Who knows? Anyone whos gone out drinking has had good and bad nights.
I think the key point is to be careful in this relationship, don't let your laisse-faire attitude cloud your commonsense and the warnings from your intuition. Clearly, you were bothered enough to even ask on this forum so for me that means just don't do it, just don't get too tied up into this situation. Be friends with her and then decide later on when things settle down.

Two-bits is worth more than one.

#### chemisttree

Homework Helper
Gold Member
Ok, I said this already but, apparently it didnt resonate with some of you: she is not slutty...at all. And her dog is very nice and well taken care of. I'm not gonna let her room with me, but I'll try to get to know her and maybe that night she visited me was just particularly bad. Who knows? Anyone whos gone out drinking has had good and bad nights.
Aero, I hope you know that I was kidding about the dog. My Kindle doesn't do easily. I was just noting that the dog warmed up to you immediately, followed you into your bedroom and...

#### Aero51

I wasnt sure. I just wrote your comment off as an obnoxious 16 year old who happened to rank very high in PF. Sorry. Vocal inflections and body language do not propagate well though the internet medium.

#### Naty1

Your job is to take care of yourself. You cannot 'rescue' anybody else...they will be who they will be.

Chicks be crazy sometimes...and some of those are almost impossible to resist. You must resist the crazies.

Having said that, if miss crazy takes good care of her dog, that says a lot about who she would like to be. You job is to figure out if she can get there on her own.

Another test: if you see her again, sober, tell her your impressions and why you are hesitant in a helpful, supportive way. See how she reacts. Set some boundaries for her and see how she responds. Be wary of false promises.

Good luck. You will need it. Let us know what happens!!

#### Aero51

Id love to tell her how I felt about the night but shes a pain to get in contact with. Ill text her and hear nothing back, the one day weeks later she will ask me to hang out then claim she never got them. Last time she asked me I just was a snide jerk - you get like that when you have to deal with bs all the time. Well, if she ever decides im worth her time shell let me know.

#### Naty1

Last time she asked me I just was a snide jerk - you get like that when you have to deal with bs all the time.
not really. YOU get that way because that's YOU.
we all revert to such behavior sometimes, but that doesn't make it nice.

#### franklinwu

It sounds like you already made up your mind that you need this cash badly, and that she's not that bad (from all the replies you've posted). Then why the heck are you asking the rest of us a question you already know the answer to?

#### Enigman

Maybe if I lock all the cabinets an put child safety warnings on my pills she will behave.

In all seriousness though its sad because she could be a nice person if she didn't make dumb decisions.
Well, as we know she DOES make dumb decisions. For Example:
1) She has hung out with you only 3 times and wants to be your roommate.
2) You talked about becoming room mates today and she wanted to move in...tomorrow.
3) She hasnt even seen your place
4) Didn't care to make any agreements on anything before hand.

My diagnosis is she is a controlling psycho or a plain iDiot.
Sorry to be a pessimist, but am in a bad mood right now.

#### Chronos

Gold Member
Nobody asks to move in with you after 3 dates - unless desperately broke, or nuts. Who needs that?

#### Mk

I don't see anything wrong with this girl. As long as she's not violent and loud and doesn't steal your stuff (feeding your chicken wings to her dog doesn't count), what's the problem?

You should try hanging out with her. Maybe you'd learn a little about the other side? Loosen up dude. She thinks you're ok enough to pay you money to live with you. And you said she looks good. Expand your boundaries! You might find something you like.

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text:
# sklearn.metrics.classification_report¶

sklearn.metrics.classification_report(y_true, y_pred, labels=None, target_names=None, sample_weight=None, digits=2, output_dict=False)[source]

Build a text report showing the main classification metrics

Read more in the User Guide.

Parameters: y_true : 1d array-like, or label indicator array / sparse matrix Ground truth (correct) target values. y_pred : 1d array-like, or label indicator array / sparse matrix Estimated targets as returned by a classifier. labels : array, shape = [n_labels] Optional list of label indices to include in the report. target_names : list of strings Optional display names matching the labels (same order). sample_weight : array-like of shape = [n_samples], optional Sample weights. digits : int Number of digits for formatting output floating point values. When output_dict is True, this will be ignored and the returned values will not be rounded. output_dict : bool (default = False) If True, return output as dict report : string / dict Text summary of the precision, recall, F1 score for each class. Dictionary returned if output_dict is True. Dictionary has the following structure: {'label 1': {'precision':0.5, 'recall':1.0, 'f1-score':0.67, 'support':1}, 'label 2': { ... }, ... }  The reported averages include micro average (averaging the total true positives, false negatives and false positives), macro average (averaging the unweighted mean per label), weighted average (averaging the support-weighted mean per label) and sample average (only for multilabel classification). See also precision_recall_fscore_support for more details on averages. Note that in binary classification, recall of the positive class is also known as “sensitivity”; recall of the negative class is “specificity”.

Examples

>>> from sklearn.metrics import classification_report
>>> y_true = [0, 1, 2, 2, 2]
>>> y_pred = [0, 0, 2, 2, 1]
>>> target_names = ['class 0', 'class 1', 'class 2']
>>> print(classification_report(y_true, y_pred, target_names=target_names))
precision    recall  f1-score   support

class 0       0.50      1.00      0.67         1
class 1       0.00      0.00      0.00         1
class 2       1.00      0.67      0.80         3

micro avg       0.60      0.60      0.60         5
macro avg       0.50      0.56      0.49         5
weighted avg       0.70      0.60      0.61         5


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# Shizmoo Games, 2003

By Drew Sikora | Published Feb 15 2003 05:10 AM in Interviews

dan joshua games game word year ninja new last
 If you find this article contains errors or problems rendering it unreadable (missing images or files, mangled code, improper text formatting, etc) please contact the editor so corrections can be made. Thank you for helping us improve this resource

"Be all you can be. Come play our snazzy web games. Shizmoo - it ownz joo."

- Haiku by Dan Goldstein

The Independent Games Festival was established in 1998 as a forum for independent developers to exhibit their work, receive recognition, and meet with commercial publishers. Finalists to this event attend the GDC and compete for several awards and cash prizes. This year's Seumas McNally grand prize is $15,000. I once again had the pleasure of sitting down and interviewing the two Goldstein brothers, Dan and Joshua about their IGF finalist entry. This year it's Word Ninja, a fast-paced action-crazy crossword game that pits you against multiple opponents to create words faster than the rest. It's the tried-and-true Shizmoo formula at work. If you thought they were off the hook last year, fasten your safety belts and strap in tight (and keep your eyes open for an Easter Egg link) - These guys may have been Kings before, but now they're Ninjas!! Who are you and what do you do? Dan: We are two crazy brothers who decided to write the world's greatest network multiplayer game engine. Joshua: We are two crazy brothers and we plan to take over the world via mind control and trained apes who pick tea leaves! Dan: We make great web games that leverage our cutting-edge network technology. Games that have action, strategy and everything in between I want to hold this off but I can't - how's it feel coming back for another round at the IGF? Dan: It feels warm and fuzzy! The IGF was an incredibly fun and hectic time last year, and I fully expect it to be the same or better this time around! Joshua: We couldn't wait for another chance to wear funny clothes and attend the prestigious DirectX geek party! Last year we got at least 40 free drinks! Microsoft = Owned! Dan: I anticipate owning a lot more this year, too. For those not in the know, you guys walked away last year with an Audience Choice award for Kung Fu Chess. Word Ninja seems another contender thanks to it being easy to get into. Yay or nay? Any other higher aspirations? Dan: Well we'd always love to win the grand prize, but yes I think people will be able to pick up and get a really good feel for Word Ninja right away. Maybe that will translate into another Audience Choice - who knows! Joshua: No... this year we don't think we have any chance of winning. However we're almost sure to win anyway, mostly thanks to our good looks and Dan's girlish figure. Dan: Yes I plan to be a letter this year - the letter Z or maybe X. We are, after all, sponsored by the letter Q and the number 3. Ok, ok - we have to hear it again :-) Where did the name Shizmoo come from? Joshua: OMG - Wasn't the answer bad enough last time? Dan: Not a very interesting answer, I'm afraid. Our father thinks it sounds like a cow crapping. But that's not the origin. Joshua: Okay, you want to hear the real origin? I'll give u the scoop. Joshua: I was having sex with my now ex-girlfriend and we're both VERY weird people, so we like to make strange noises that sound cute when we do it. And so she started to go "Shizmoooo" in this very cute animal-like voice… Dan: OMG I wish I had known that before agreeing to use the name. Too late now… Joshua: …and I got so turned on. Dan: Dude that's so not cool! Joshua: I decided to name the company after it. Dan is stunned Dan: Please don't ever tell me whether you just made that up. So wait a minute... wasn't Word Ninja originally called Word 'em Up? What happened? Dan: Well some developer friends of ours that we met last year asked us to change it, because they have a game called Word Up. So since we didn't care about the name so much and they seemed to enough that they threatened us with legal battles we changed it. Joshua: Also, our site is already too ghetto Dan: Lol, too ghetto for what? Oh well… So why Ninja? I mean I love Ninjas as much as the next guy and they are extremely cool but does it have anything to do with the gameplay? Joshua: Yes. There are lots of ninjas… they come out and enter your house and kill your wives and kids and have lots of cool weapons. Dan: But only if you can't spell correctly. Joshua: Come play the game. It's just a click away at shiizmoo.com Dan: SPELL THE SITE RIGHT!! Joshua: Oh - minus one i Dozens of Ninjas break through the windows - both brothers scream Ok so what exactly is Word Ninja? Joshua: It is a game so intense it defies description! Dan: A fast-paced word game where you compete with an opponent or three to make words on a crossword board! Joshua: That too Dan: Unlike most crossword games you never have to wait your turn, so your friend can't take an hour to figure out his next word. This, as we all know, can get REALLY annoying. Joshua: Yes, especially if you have sucky friends. Hmmm - real-time Chess, real-time Scrabble... a trend perhaps? Taking slow, sometimes boring games and whipping them up to pace? Dan: Sounds good to us. Taking the turned-basedness out of Chess and Crossword games definitely makes for a fun, new competitive experience. We have more in the works too, as well as other styles of games that you might see for next year's competition. Joshua: Super-Size it! So was there really any decision between Word Ninja and Sumo Volleyball for the competition? Dan: Yes, actually we didn't even quite have Word Ninja done at the time so it was a big debate. Last year we actually entered two games, but we decided that Sumo Volleyball and Word Ninja have such similar interfaces that we didn't want to enter both. Joshua: We also considered writing a completely different game for the competition Dan: At this point writing games takes us a matter of days, so it was tough to decide. I'm glad that in the end we went with Word Ninja. So how has your technology evolved over the past year? And is it a total game framework or mostly a networking framework? Joshua: It is a total game framework, but the key component is the networking engine. Dan: The tech is coming fabulously. In fact, I'm working on version 2.0 right now which should be even better than the original. Joshua: You can prototype online games with full-featured lobbies in it in hours. Dan: Writing an online multiplayer game in hours is not something any other system I know of allows you to do. Joshua: Yeah. Pretty much you write the game -- with no special networking code, and the system figures out the rest for you. It's packaged with an extremely simple to use graphics and sound library which make making games with it a breeze. On top of this, you can easily add configurability support to allow players to redesign the game you programmed. Along those lines are you guys considering any plans to license the engine? Dan: We're testing with external developers now and anticipate creating a lot more games with it in the months to come. Eventually we'll be licensing the version 2.0 technology to other game companies, who can use it to create web games, downloadable games, retail games, or even massively multiplayer environments. The engine is very flexible, and by using it developers can avoid almost all of the hassles involved with network programming. Not only that, but they can use the same code and output both an in-browser game and a downloadable 'full' version of the same game. With all this and only three games it sounds like you guys have been working really hard on the engine side. How long has it been in development total so far? Joshua: I think we've been in development around 2 years... though some of the code dates back even further than that. Dan: Yeah we've been playing with the network side of things for years at this point. We didn't decide to try to make a business of it until maybe two years ago or so though. Our original product was a massively multiplayer game called StarThief, which has been sort of put on hold for a bit while we get the technology really right. So I see the site has been redone as well - just better-looking or is there anything new server-side for the games? Dan: There's a lot of new stuff! Joshua: Haha, this is a long answer question Dan: We just launched the "Shizmoo Subscription", which comes with a boatload of new games and features. Joshua: It includes the ability to redesign any of our games to your choosing. For example, you can make new sports games in Sumo Volleyball, new chess variants with up to 8 players in Kung Fu Chess, new crossword games in Word Ninja with whatever rules you like, new graphics, scheduled tournaments,.. Dan: Every month we add new 'official' game types to the games as well - like we just added Crazyhouse, Bughouse, and Team 4-way Kung Fu Chess. We're about to do the same for Sumo Volleyball and Word Ninja as well. For the last week or so there's been new stuff for subscribers practically every day, and we don't intend to stop. Is there any cost for the subscription? Is it indefinite or renewable? Dan: Right now the cost is$25 for six months of play. That gets you all the features on all the games, as well as the new subscriber-only games that are going to be coming out very soon.

I understand that you guys are quite a distance from each other now - any troubles working together over the Pacific?

Joshua: We work together quite well remotely. However, I've also gone to visit Dan in Hawaii a couple times over the past six months.

Dan: Yeah I don't think there's ever a downside to your co-worker living in Hawaii and taking working vacations.

Joshua: Needless to say, Christmas in Hawaii is pretty sweet =)

You know, I think you guys should definitely come to the GDC dressed as Ninjas this year :-)

Joshua: We don't like to disappoint =)

Dan: I think we just might take you up on that offer. The PR from the Karate outfit and Crown was amazing last year. There was practically nobody who didn't know who I was. Of course, I had a silly crown on at the time so it's not that surprising.

Great. Guys, thanks again for taking the time to do the interview, best of luck and I'll see you at the GDC... or will I?

Joshua: It probably won't be hard =)

Dan: I'll bring my shuriken. And remember, seeing in one's mind's eye the attacker brings peace and tranquility to the defender. Do not underestimate the power of overwhelmingly tacky clothing and bravado.


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# How to choose prior means?

Hello,

I am replicating a paper similar to Smets and Wouters (2007) with my own data. I have measurement equations and the data. I guess the prior distributions can just be set to standard ones. But I would like to know how to specify prior means? I have read the dynare manual and the guide to observation equations, but did not find much on that topic. I would be looking forward to your answer.

Best

Jing

This is pretty well documented in the reference manual - which you suggest you have read - as well as in many of the tutorials, guides, and examples. Can I ask what you have tried? It’s hard to know exactly the source of confusion here is. Are you asking the positive question “how do I implement a given prior for a particular parameter in dynare”? Or are you asking the normative question “how should I go about deciding on the prior I want to implement”?

Hi bdombeck,

Thanks very much for your prompt reply. I am just wondering if the prior mean should be related to my data, and how. For instance, suppose I have a measurement equation for output. dy = y - y(-1) + ctrend. And I have a time series for dy. Is there some rule that I need to stick to when I set the prior mean for “ctrend”?

My problem is that the estimation code works for the original data (for US), but not for my data (for UK). The error message is that the Hessian is not positive. So I am trying to find out why.

And I have another question: Does it matter if one mode check plot has some red dots? If so, how should I fix the problem?

Best

Jing

Based on this my suspicion is your issues are to do with the data you are feeding in rather than the specification of priors. As a first step I would recommend calculating and comparing the means and standard deviations of the corresponding observables from each (e.g. “consumption series for US” and “consumption series for UK”).

Issues with mode-finding frequently arise from issues of scaling or trying to match demeaned data to a positive-meaned model variable. In your example above, it looks like dy has a steady state value of zero, assuming ctrend is a mean-zero exogenous stochastic process. In that case you should make sure your empiric counterpart for output growth has also been demeaned.

See remarks 13 and 10 in A Guide to Specifying Observation Equations for the Estimation of DSGE Models, which deal with the issues of forgetting the mean and scaling, respectively.

These should be simple to diagnose: the theoretic moments are calculated from the stoch_simul command and contained in “oo_.mean” and “oo_.var” respectively. If these values for your model-observables are quite different than those of your actual data, you have likely found your problem.

This depends on what ctrend is supposed to measure. Is it measurement error? In that case you would use your understanding of the actual data gathering process for guidance. Do you have a reason to suspect that there are systematic mistakes made in measuring the data? My sense is generally we assume measurement error shocks are mean zero. But there are cases in which a reasonable person could probably make an argument for non-zero mean shocks. For example it is well known that measuring inflation using the CPI biases the inflation rate upwards.

I will leave it to others on the forum to comment on this idea.

Thank you very much for your detailed answer! I’ll try out these thoughts soon.

I had another thought about this: you have not described what ctrend actually is. My answers above assumed it was some kind of exogenous stochastic shock such as measurement error. But the name you have given it suggests it may be an endogenous variable related to the trend growth rate of non-demeaned data used to detrend non-stationary model variables, such as the growth rate of output term \mu^{y}_{t} in e.g. Schmitt-Grohe and Uribe (2012) and Born, Peter and Pfeifer (2013). As you can see in these papers, the steady state value for this variable is calibrated to the sample average of the corresponding data series.

I see! Thanks a lot! I will try this out immediately

Hi bdombeck,

Would you like to take a look at my code, when you have some free time, please?

Following your thoughts, I have checked the oo._ sections and changed some of the initialized parameters in the parameter block so that they are close to my sample means, but the code still doesn’t work. You could just give me a brief impression what you think the problem is, and how I could fix it, if time is tight.

Posterior_Generation.mod (22.3 KB) data.mat (5.4 KB)

There are significant differences in the means given by your data as compared to the means of the corresponding observable model variables. For example, the mean of the variable rspreadobs is 0; the mean of the corresponding data is 0.7427. The problem is reversed for the labobs variable, where you’ve tried to match a model variable with positive mean (0.10) to a data series with a mean of zero. There are also orders of magnitude differences in the volatility between some of your data and the corresponding model observables.

Essentially you are forcing the estimation routine to try to account for means and volatilities which are very different. This frequently pushes parameters to their boundaries which results in a non-pd Hessian.

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Thanks very much, bdombeck! I am currently trying to match those values carefully.

May I ask you another question? Suppose that I have such time series data. What are the differences between the values to be entered in the 1. parameter block (initialize parameters which are to be estimated); 2. initval block (initial values for the observables); and 3. the estimation block (initial values)? Do they all have to be the same with the sample means of the time series? I have seen some discrepancies in the original paper, which is why I am wondering about this.

Best

Jing

The initval block is about finding the steady state for the endogenous variables and has nothing to do with the estimation of parameters.
The parameter initialization is for simulating the model. The estimated_params-block is about the starting value of the parameter for mode-finding. Those two are often the same, but there is no strict necessity. Sometimes you are interested in simulating the model before estimation with a parameter value different from what you think the mode might be.

Hi bdombeck,

Thanks to your help–after changing the initialized values of the parameters in the model, the code could work, but there are still red dots in some mode check plots. Just now you mentioned that the volatility between my data and the model observables are also different. And I can also see this from using the oo._var command. But how can I match them? I mean–how to change the theoretical volatility once I know the volatility of the data? I guess the theoretical volatility comes from the shocks, but the link with that of the data doesn’t seem very straightforward, or does it?

Best

Jing

If it’s orders of magnitude, then typically there is some scaling error (usually a factor 100)

Hi jpfeifer,

Thanks for your reply. What I am wondering is that I have changed a different dataset for the model, and whether I need to make changes to the standard deviations of the shocks in the model, according to my own data too, and how?

Best

Jing

Everything should be explained there. Data and model need to be consistent.

Hi jpfeifer,

I am very sorry to bother you again. I have compared the theoretical standard deviation of the model variables and that of my data. I think there is a difference of 5 times at most. Would you please show me just one observable which has “a difference in orders of magnitude”, when you have some time to look at my code? (I think a scaling error of 100 is unlikely to arise because I had compared my measurement equations and data with those of the author before I put them in.)

And in general I am very curious about how to match the standard deviation of the model shocks to my data.

And the theoretical mean of the model variable robs is 1.17 compared to 0.58 in my data. Do you think problems could arise from there?

The problem is that I still have red dots in three of the mode check plots.

Posterior_Generation.mod (22.7 KB)
data.mat (5.3 KB)

Best

Jing

1. When I use mode_compute=5 the model works.
2. When adding options_.debug=1 you will get information where the red dots in the mode_check-plots come from. In your case, the model does not satisfy the Blanchard-Kahn conditions for those parameter values.

Hi JPfeifer,

Thanks very much for your time!

Does that mean that the boundaries for the prior distribution are not appropriate?

If I am unable to erase the red dots in the very end, are my estimation results ineligible?

Best

Jing

1. The prior bounds seem unproblematic to me.
2. It is completely normal that not the whole parameter space leads to unique stable solutions. So some red dots are normal.

Hi JPfeifer,

I see. So my estimation is successful. Thanks very much for your patience!

Best

Jing


text:
# Data Sources

This page describes several places where you might look for interesting data to use in your SPIS projects.

Here is an overview, followed by more specific information about each one.

# Corgis datasets for Python

When working with the Corgis datasets for Python, be sure to read the part about the test=False parameter.

For many of these datasets, you only get a small sample of the data when you use this code:

import cars
list_of_car = cars.get_cars()
list_of_car = cars.get_cars_by_year("2001")
list_of_car = cars.get_cars_by_make("'Pontiac'")

If instead, you set the test parameter to False, you get a much larger data set that could be considered “big data”:

import cars
# These may be slow!
import cars
# These may be slow!
list_of_car = cars.get_cars(test=False)
list_of_car = cars.get_cars_by_year("2001", test=False)
list_of_car = cars.get_cars_by_make("'Pontiac'", test=False)


text:
Problem proving property quasi-convexity (quasi-concavity) & optima

Let D $\subset \mathbb{R^n}$ be an open convex domain and let f : D $\rightarrow \mathbb{R}$ be a map such that f has a locally strict maximum and a locally strict minimum.
Prove: The function f is neither quasi-convex nor quasi-concave.

I am trying to prove this property but my textbook gives very little information about quasi-convexity (quasi-concavity) at all to come up with an intelligent proof, so I have no idea where to start (or which properties to use).

• Could you put down a definition of quasi-convexity you're using? Oct 19, 2016 at 18:34
• @xyzzyz Of course. The map f is called quasi-convex if and only if the level set {x is an element of D : f(x) is less than or equal to alpha} is convex for every alpha in R.
– Anna
Oct 19, 2016 at 18:37
• The property I want to prove provides a criterion to exclude quasi-convexity (this is a comment by the author of my textbook).
– Anna
Oct 19, 2016 at 18:40

If $p$ is a locally strict maximum with $f(p) = r$, then for $\epsilon > 0$ sufficiently small $\{x: f(x) \le r-\epsilon\}$ contains a sphere around $p$ but not $p$ itself, so is not convex, and $f$ is not quasi-convex. Similarly...

EDIT:

• Thanks. Is the idea f is not convex (and not quasi-convex) since the open ball does not contain p (there is a 'hole' in the set)? How should I visualize this statement?
– Anna
Oct 19, 2016 at 19:21
• I do not fully understand why this would be the case (open ball).
– Anna
Oct 19, 2016 at 20:12
• If $\delta$ is small enough, the sphere $S_\delta = \{x: \|x - p\| = \delta\}$ is contained in $D$ and the maximum value of $f$ on $S_\delta$ is less than $r$. If $0 < \epsilon < r - \max_{S_\delta} f$, the level set $L = \{x: f(x) \le r - \epsilon\}$ contains $S_\delta$ but not $p$. If $x \in S_\delta$, so is $2 p - x$, and $(x + (2p-x))/2 = p \notin L$ while $p$ and $2p-x$ are in $L$, so $L$ is not convex. If $L$ is not convex, the function is not quasi-convex. Oct 20, 2016 at 0:11
• I'll draw a picture, with the level set $L$ in pink and the sphere in blue. Oct 20, 2016 at 0:25
• Thanks for the clarification and the visualization. It is a lot more clear now.
– Anna
Oct 22, 2016 at 20:23