Sharp lower bounds involving circuit layout system
© Wen et al.; licensee Springer. 2013
Received: 12 September 2013
Accepted: 5 December 2013
Published: 30 December 2013
is the minimal one.
The above problem can easily be generalized. To this end, we need to recall somebasic concepts as follows.
Let be a Euclideanspace, and . The inner product of α andβ is denoted by and the norm of α is denoted by. The dimension of satisfies if and only if there exist n linearlyindependent vectors in (see ).
Now we give the definition of the circuit layout system in a Euclidean space asfollows.
is a circuit layout system (or CLS for short) if the following conditions aresatisfied:
(H1.1) , .
(H1.2) for and .
(H1.3) If , then for and .
(H1.4) If and for and , then .
(H1.5) For any , there exists such that .
In this paper, we are concerned with the sharp lower bound (see [2–5]) of , its purpose is to estimate the installation costs ofthe circuit layout problem. In other words, we will mainly be concerned with thefollowing problem.
Problem 1.1 (Circuit layout problem)
Let be a CLS. How can we determine the lower bound of by means of n, N, δand ?
In this paper, by means of algebraic, analytic, geometric and inequality theories,several sharp lower bounds of in Problem 1.1 are obtained. As applications of ourresults, in Section 4, we calculate that for the special circuit layout system by means of three effective examples.
We provide in this section some basic terminologies and results which are necessaryfor the investigation of Problem 1.1.
We first recall the concept of parallel vectors for later use. Two vectors xand y in are said to be inthe same (opposite) direction if (i) or , or (ii) and and x is a positive (respectively negative)constant multiple of y. Two vectors x and y in the same(opposite) direction are indicated by (respectively ) (see ).
In order to study Problem 1.1, we need six lemmas as follows.
Lemma 2.1 (Minkowski’s inequality )
Furthermore, the equality holds if and only if.
According to Lemma 2.1 and the algebraic theory, we easily get the followinglemma.
Lemma 2.2 (Minkowski-type inequality )
Furthermore, if, then the equality in (2) holds if andonly if.
where, . Then the function φ is nondecreasing. If in addition, , then φ is increasing.
Thus, is nondecreasing. Furthermore, if, then is a strictly increasing function. This ends theproof. □
The result of Lemma 2.4 is well known.
By our assumptions (H1.2)-(H1.5), we may easily get the following result.
whereandare defined in Lemma 2.5.
This is contrary to the minimality of . The proof is completed. □
3 Study of Problem 1.1
3.1 The case where n is an odd number
We first study the case of Problem 1.1 where n is an odd number. In thissituation we have the following result.
This means that inequality (12) holds.
In addition, from the above analysis we may easily see that the equality in (12)holds if
(H3.1) is an equiangular n-gon;
(H3.2) for any , , equality (5) holds;
(H3.3) equality (4) holds; and
The proof is completed. □
3.2 The case where n is an even number
Now, we consider the case of Problem 1.1 where n is an even number.
Thus, inequality (16) is proved.
Thus, inequality (16) is proved.
Finally, the conditions for the equality in (16) to hold are as follows:
(H3.5) The n-gon is an equiangular n-gon.
(H3.6) Equality (5) holds for any , .
(H3.7) Equality (4) holds.
(H3.8) , where is defined by (23).
While the conditions for the equality in (17) to hold are as follows:
(H3.10) The conditions (H3.5)-(H3.7) and (H3.9) hold.
This completes the proof of this theorem. □
3.3 The case is an equiangular n-gon
We now turn to the calculation of .
Theorem 3.3 Letbe a CLS with n odd, and letbe an equiangular n-gon. Suppose that there existfor each, such that:
(H3.14) , where.
Proof By the assumptions in Theorem 3.3, conditions (H3.1) and(H3.3) hold. Since , there exists such that condition (H3.2) holds. Thus, we justneed to consider condition (H3.4).
According to the assumption in Theorem 3.3, and (33)-(35), condition (H3.4)holds. Consequently, by Theorem 3.1, Theorem 3.3 holds.
This completes the proof of Theorem 3.3. □
Suppose that there existfor eachand free variablesuch that:
(H3.15) Condition (H3.12) holds.
where ω is defined by (22).
Proof We first look for the conditions for equality in (16)-(17) tohold. The conditions are either (H3.5)-(H3.9) or (H3.10)-(H3.11). By theassumptions in Theorem 3.4 and , conditions (H3.5)-(H3.8) and (H3.11) hold. If(H3.5)-(H3.9) hold, then (H3.10) hold. Therefore we just need to show that(H3.9) holds.
This means that condition (H3.9) can be deduced from conditions (H3.15)-(H3.17).Thus Theorem 3.4 holds by applying Theorem 3.2.
This completes the proof of Theorem 3.4. □
4 Three effective examples
For a general , the equalities in (12), (16) and (17) may not hold,this is most probably because conditions (H3.1)-(H3.4) or conditions (H3.5)-(H3.11)cannot be met at the same time. We will discuss Problem 1.1 of a special CLS in.
and a, b, c is the length of the sides of the triangle. We will calculate that .
where A, B, C are three inner angles of the triangle. According to conditions (H3.1)-(H3.4), the equalityin (46) holds if and only if is a normal triangle, and , , are the midpoints of line segment, , , respectively. Consequently, the equality ininequality (12) does not hold in general.
We will calculate that .
We can also give another intuitive proof of equation (52) as follows.
where w is a free variable. This means that (51) holds if and only if(53)-(54) hold. This proves equation (52).
In addition, we can also prove (52) by Theorem 3.4.
Example 4.3 is a geometry problem. However, we can see this example as a circuitlayout problem of a family. In addition, this example also means that the equalitiesin (12), (16) and (17) can hold.
Remark 4.1 Since a Euclidean space is an abstract space, we will find theapplications of CLS in theoretical fields such as statistics (see [1, 7]), matrix theory (see ), geometry (see [6, 9–12]) and space science (see [1, 9]), etc.
The angle between two nonzero vectors B and C is defined to be.
This work was supported in part by the Natural Science Foundation of China (No.61309015) and in part by the Foundation of Scientific Research Project of FujianProvince Education Department of China (No. JK2012049). The authors are deeplyindebted to Professor Sui Sun Cheng, Tsing Hua University, Taiwan, for manyuseful comments and keen observations which led to the present improved versionof the paper as it stands.
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