Energy-Efficient Threshold Circuits Computing Mod Functions
|
|
- Asher Lane
- 5 years ago
- Views:
Transcription
1 Energy-Efficient Threshold Circuits Coputing Mod Functions Akira Suzuki Kei Uchizawa Xiao Zhou Graduate School of Inforation Sciences, Tohoku University Araaki-aza Aoba , Aoba-ku, Sendai, , Japan. Eail: {a.suzuki, uchizawa, Abstract We prove that the odulus function MOD of n variables can be coputed by a threshold circuit C of energy e size s = O(e(n/ 1/(e 1 for any integer e 2, where the energy e is defined to be the axiu nuber of gates outputting 1 over all inputs to C, the size s to be the nuber of gates in C. Our upper bound on the size s alost atches the known lower bound s = Ω(e(n/ 1/e. 1 Introduction Neuronal signals play fundaental role in inforation processing of the brain. A neuron eitting a signal is said to be firing. Recent biological studies report the following fact about the energy consuption of the neuronal firing: the energy cost of a neuronal firing is high while energy supplied to the brain is liited, hence neural networks ust have low firing activity (Attwell & Laughlin, 2001; Lennie, Consequently, any neuroscientists consider that the etabolic liit ust influence the way in which inforation is processed, the brain has countered this etabolic constraint by adopting energy-efficient circuit designs (Földiak, 2003; Laughlin & Sejnowski, 2003; Olshausen & Field, 2004; Vinje & Gallant, Uchizawa, Douglas Maass consider the proble posed above fro the view point of theoretical coputer science, introduce a new coplexity easure called the energy coplexity of threshold circuits (Uchizawa et al., 2006, where a threshold circuit, is a cobinatorial circuit consisting of threshold gates, is a theoretical odel of neural circuit (Minsky & Papert, 1988; Parberry, 1994; Sia & Orponen, 2003; Siu et al., Based on the biological fact above, the energy e of a threshold circuit C is defined as the axiu nuber of threshold gates outputting 1 over all inputs to C. In previous research, several facts are known on the coputational power of threshold circuits with sall energy (Uchizawa et al., 2006, 2009a; Uchizawa & Takioto, 2008; Uchizawa et al., 2009b. Particularly, Uchizawa et al. (2006 find a non-trivial circuit structure that benefits energy-efficiency, provide threshold circuits of polynoial size energy O(log n for a fairly large class of Boolean functions Supported by MEXT Grant-in-Aid for Young Scientists (B No Copyright c 2011, Australian Coputer Society, Inc. This paper appeared at the 17th Coputing: The Australasian Theory Syposiu (CATS 2011, Perth, Australia, January Conferences in Research Practice in Inforation Technology (CR- PIT, Vol Alex Potanin Taso Viglas, Ed. Reproduction for acadeic, not-for-profit purposes peritted provided this text is included. of n variables. However, their construction is not specialized for a particular task, hence it soeties gives a redundant circuit. In this paper, we consider one of the fundaental well-studied Boolean functions in the theory of circuit coplexity, the odulus function, as a particular task. The odulus function MOD of n variables for two positive integers n is defined as follows: MOD (x = 0 if the nuber of 1 s in an input x {0, 1} n is a ultiple of, otherwise, MOD (x = 1. Although the odulus function ay be far fro real tasks that neural networks in the brain perfor, we believe that considering such a siple fundaental task akes an iportant step for understing what circuit structure benefits the energy-efficiency of threshold circuits. Uchizawa et al. (2009b proved that size energy of a threshold circuit coputing the odulus function cannot be siultaneously sall: Any threshold circuit C of energy e coputing MOD of n variables has size ( ( n 1/e s = Ω e. (1 We prove in this paper that MOD of n variables can be coputed by a threshold circuit of energy e size ( ( n 1/(e 1 s = O e (2 for every integer e 2. Coparing the right-h side of Eq. (1 with one of Eq. (2, we can find the difference between the ters only in the exponent of n/. our upper bound alost atches the lower bound, iplies that there exists a tight tradeoff between size energy of threshold circuits coputing odulus function. We obtain the result by construction of the desired threshold circuits, hence it exhibit a circuit design of energy-efficient threshold circuits. In addition, we consider an extree case where threshold circuits have energy e = 1. In this case, we prove that any threshold circuit C coputing the PARITY of n variables ust have an exponential nuber of gates in n. On the other h, Eq. (2 iplies that PARITY (i.e., MOD 2 can be coputed by a threshold circuit of size s = O(n energy e = 2. we know fro these facts that there exists a significant gap of coputational power between threshold circuits of e = 1 ones of e = 2. The rest of the paper is organized as follows. In Section 2, we define soe ters on threshold circuits the odulus function. In Section 3, we first provide our ain theore. We then give a technical lea, prove the theore using the lea. In
2 Section 4, we prove the technical lea given in Section 3. In Section 5, we give the lower bound for threshold circuits of energy one. In Section 6, we conclude with soe rearks. 2 Preliinaries A threshold circuit C is a cobinatorial circuit of threshold gates, is expressed by a directed acyclic graph. Let n be the nuber of input variables to C. Then each node of in-degree 0 in C corresponds to one of the n input variables x 1, x 2,..., x n, the other nodes correspond to threshold gates. We define the size s(c of a threshold circuit C as the nuber of threshold gates in C. Let g1 C, g2 C,..., gs(c C be the gates in a threshold circuit C. For each gate gi C, 1 i s(c, let z(x = (z 1 (x, z 2 (x,..., z ki (x {0, 1} ki be the k i inputs of gi C with weights w 1, w 2,..., w ki a threshold t i for x {0, 1} n. Then the output gi C (z(x of the gate gi C is defined as follows: k i gi C (z(x = sign w j z j (x t i, where sign(z = 1 if z 0 sign(z = 0 if z < 0. Siply, gi C(z(x is denoted by gc i [x]. Let f : {0, 1} n {0, 1} be a Boolean function of n variables. Let gs C be a gate of out-degree 0 in C, let the output gs C [x] of g s be the output C(x of C. C(x = g s [x] for every input x {0, 1} n. The gate gs C is called the top gate of C. A threshold circuit C coputes a Boolean function f : {0, 1} n {0, 1} if C(x = f(x for every input x {0, 1} n. We define the energy e(c of a threshold circuit C as e(c = gi C [x]. s(c ax x {0,1} n the energy e(c is the axiu nuber of gates outputting 1 over all inputs x {0, 1} n to C. Trivially, we have 0 e(c s(c. For an input variable x = (x 1, x 2,..., x n {0, 1} n, we define x as the haing weight of the inputs x, that is, x = x i. Then, for an integer 2, the odulus function MOD of n variables is defined as follows: For every input variable x = (x 1, x 2,..., x n {0, 1} n, { 1 if x is not a ultiple of ; MOD (x = 0 otherwise. If n 1, then MOD (x = 1 for all inputs x except x = (0, 0,..., 0. a circuit consisting of a single threshold gate with weight ones for all the inputs threshold one coputes MOD. One ay thus assue that n in the reainder of the paper. 3 Energy-Efficient Circuits Our ain result is the following theore that yields an energy-efficient threshold circuit coputing the odulus function. Theore 1. Let, n be any two positive integers, let e 2 be any integer. Then there is a threshold circuit coputing MOD of n variables such that its energy is at ost e its size is at ost (n 1/(e (e 1 ( ( n 1/(e 1 = O e. (3 Uchizawa et al. (2009b prove that the size s energy e of a threshold circuit C coputing MOD of n variables cannot be siultaneously sall, as described in the following theore. Theore 2 (Uchizawa et al. 2009b. Let C be a threshold circuit coputing the function MOD of n variables. Then the size s energy e of C satisfy n πe where c = is a constant. ( e 2c s (4 By a siple odification of Eq. (4, we can obtain fro Theore 2 the following lower bound on the size of threshold circuits coputing MOD of n variables. Corollary 1. Let C be any threshold circuit of energy e coputing MOD of n variables. Then ( ( n 1/e s(c = Ω e. (5 Observe the asyptotic ters in the right-h side of Eqs. (3 (5. We can find the difference between the ters only in the exponent of n/: the ter in Eq. (3 has 1/(e 1, while the ter in Eq. (5 has 1/e. Hence, the upper bound in Theore 1 alost atches the lower bound in Corollary 1. In the rest of the section, we prove Theore 1. We say that a threshold circuit C is regular if the inputs of every gate in C includes all the inputs x 1, x 2,..., x n with weight ones. In other words, every gate in C receives all the unweighted inputs x 1, x 2,..., x n. The following technical lea plays key role in our proof. Lea 1. Let, n, n be positive integers such that n n + 1. Let C be a regular threshold circuit coputing MOD of n variables. Then, there is a regular threshold circuit C coputing MOD of n variables such that e(c e(c + 1 s(c s(c + e n n +1 We will prove the lea in the next section. Using the lea, we prove Theore 1 below. Proof of Theore 1. Let e be an arbitrary integer at least 2. We prove the theore by constructing a regular threshold circuit C coputing MOD of n variables such that e(c e (n 1/(e s(c (e 1. (6
3 We provide our construction by induction on e 2. That is, we construct a threshold circuit of energy e + 1 fro a threshold circuit of energy e. We start fro the case of e = 2 as the basis. Basis: e = 2. Consider a regular threshold circuit C consisting of a single threshold gate g with threshold t = 1 1 input variables. Clearly, e(c = 1, s(c = 1, C coputes MOD of n = 1 variables. Therefore, Lea 1 iplies that there is a regular threshold circuit C coputing MOD of n variables such that e(c e(c + 1 ( n + 1 s(c s(c + 1. Since e(c = 1, we have e(c e(c + 1 = 2 = e. Since s(c = 1, we have ( n + 1 s(c s(c + 1 = (n 1/(e = (e 1. Inductive Step: e 3. By the induction hypothesis, there is a regular threshold circuit C coputing MOD of n = γ e 1 1 variables for each positive integer γ where e(c e (n s(c 1/(e (e 1 ((γ e 1 1/(e (e 1 (e 1γ. (7 We will construct a regular threshold circuit C coputing MOD of n variables, show that C has the energy e(c e + 1 (8 the size (n 1/e + 1 s(c e. (9 Since C coputes MOD of n = γ e 1 1 variables, by Lea 1 there is a regular threshold circuit C coputing MOD of n variables such that We choose e(c e(c + 1 = e + 1 s(c s(c + γ e 1 1. (10 (n 1/e + 1 γ =, then γ e (n + 1/, hence γ e 1 1 γ. (11 Therefore, by Eqs. (7, (10 (11, we have s(c s(c + γ e Proof of Lea 1 (e 1γ + γ eγ (n 1/e + 1 = e. In the section, we prove Lea 1. Let, n, n be positive integers such that n n +1. Let C be a regular threshold circuit coputing MOD of n variables, s = s(c. We denote by g 1, g 2,..., g s the threshold gates in C. One ay assue without loss of generality that g 1, g 2,..., g s are topologically ordered with respect to the underlying directed acyclic graph of C, that each gate g i, 1 i s, receives exactly (i 1+n inputs fro the outputs of the gates g 1, g 2,..., g i 1 the n inputs x 1, x 2,..., x n. If there is soe gate g i, 1 j i 1, such that g i has no input fro the output of g j, then one connects input of g i with weight 0 for the output of g j. Therefore, for each index i, 1 i s, let w i,1, w i,2,..., w i,i 1 be the weights of the gate g i for the outputs of the gates g 1, g 2,..., g i 1, respectively, denote by t i the threshold of g i. Since C is regular, each of the gates g 1, g 2,..., g s has weight ones for the n input variables. the output of the gate g i for each input x {0, 1} n can be recursively coputed by the following threshold function: sign ( x t 1 if i = 1; ( g i[x ] = sign x + i 1 w i,j g j [x ] t (12 i otherwise. We show that, for any positive integer n n + 1, MOD of n variables can be coputed by a regular threshold circuit C of energy e(c e(c + 1 size s(c s(c + β, where β = n + 1 α 1 n + 1 α =. We construct the desired threshold circuit C fro C, as described below. To obtain C, we add new input variables x n +1, x n +2,..., x n to C, connect each of the new input variables to each of the gates g 1, g 2,..., g s with weight one. Besides, for each index i, 1 i β, we add a new threshold gate ĝ i with weight ones
4 for the inputs x 1, x 2,..., x n a threshold αi to C, connect the output of the gate ĝ i to the gates g 1, g 2,..., g s with weight αi. For each index i, 1 i s, we denote by g i the gate in C that corresponds to the gate g i in C, denote by x = (x 1, x 2,..., x n {0, 1} n an input to C. Then, the output of the gate g i for an input x {0, 1} n is now represented as ( sign x β αj ĝ j [x] t 1 if i = 1; ( g i [x] = sign x + i 1 w i,j g j[x] β αj ĝ j [x] t i otherwise. (13 Moreover, for each index i, 2 i β, we connect the output of the gate ĝ i to the gates ĝ 1, ĝ 2,..., ĝ i 1 with weight αi. we have ( sign x αj ĝ j [x] αi ĝ i [x] = if 1 i β 1; sign ( x αβ if i = β. β for x {0, 1} n. Clearly, C is a regular circuit, s(c s(c + β = s(c n n +1 (14 Below we prove that C coputes MOD of n variables, e(c e(c + 1. Let x {0, 1} n be an arbitrary input to C. Note that 0 x x n α 1 α α(β for any input x {0, 1} n. Let x i =, α then trivially αi x α(i (15 We prove the following clai. Clai 1. The following (i, (ii (iii hold. (i ĝ i [x] = 0 for each i, i + 1 i β; (ii ĝ i [x] = 1 if i = i ; (iii ĝ i [x] = 0 for each i, 1 i i 1. In other words, if 0 x α 1, none of ĝ 1, ĝ 2,..., ĝ β outputs one; otherwise, only the gate ĝ i outputs one. Proof of Clai. For each index i, 1 i β, let β x αj ĝ j [x] αi p i (x = if 1 i β 1, x αβ if i = β. (16 Clearly, p i [x] is the value in the sign function of the right h side of Eq. (14 for x {0, 1} n, that is, ĝ i [x] = sign(p i (x. We evaluate p i (x, prove (i, (ii (iii. (i ĝ i [x] = 0 for each i, i + 1 i β. If i β 1, then by Eqs. (15 (16 p i (x α(i αj ĝ j [x] αi α(i + 1 i 1 1. (17 If i = β, then by Eqs. (15 (16 we siilarly have p i (x = x αβ α(i αβ 1. (18 Since i + 1 β, Eqs. (14, (17 (18 iply that ĝ i [x] = sign(p i (x = 0. (ii ĝ i [x] = 1 if i = i. In this case, we have 1 i β. By (i above, if i β 1, j=i +1 αj ĝ j [x] = 0, hence we have by Eqs. (15 (16 p i (x = x αi αi αi = 0. Thus Eq. (14 iplies that ĝ i [x] = sign(p i (x = 1. (iii ĝ i [x] = 0 for each i, 1 i i 1. In this case, we have 2 i+1 i β, hence ĝ j [x] ĝ i [x]. By (ii above, we have g i [x] = 1, hence αj ĝ j [x] αi ĝ i [x] = αi. (19 Since i + 1 β, we have i β 1. Therefore, by Eqs. (16 (19 p i (x x αi αi. (20
5 By Eqs. (15 (20, we have p i (x α(i αi αi α 1 αi 1 hence Eq. (14 iplies that ĝ i [x] = sign(p i (x = 0. We are now ready to prove the lea by the clai. There are the following two cases to consider. Case 1: 0 x α 1. In this case, the clai iplies that none of ĝ 1, ĝ 2,..., ĝ β output one. Besides, we have n + 1 α 1 = 1 n. Therefore, Eqs. (12 (13 iply that, for every index i, 1 i s, the output of g i for x {0, 1} n equals to the output of g i for an input x {0, 1} n such that x = x. Thus the nuber of gates outputting one is at ost e. Since C coputes MOD, C(x equals to MOD (x. Case 2: α x α(β In this case, the clai iplies that only the gate ĝ i of ĝ 1, ĝ 2,..., ĝ β outputs one, hence Eq. (13 iplies that, for every index i, 1 i s, the output of g i can be represented as i 1 g i [x] = sign x + w i,jg j [x] αi t i. (21 Eq. (15 iplies that 0 x αi α 1, hence, for every index i, 1 i s, we have that g i [x] for x {0, 1} n equals to the output of g i for an input x {0, 1} n such that x αi = x. at ost e gates of the gates g 1, g 2,..., g s output one, consequently the nuber of gates outputting one in C is at ost e+1. The circuit C coputes MOD of n variables, x αi is a ultiple of if only if x is a ultiple of. Hence, C(x equals to MOD (x. 5 Circuits of Energy One In this section, we consider an extree case where threshold circuits have energy e = 1. While we know fro Theore 1 that PARITY (i.e., MOD 2 of n variables can be coputed by a threshold circuit of size s = O(n energy e = 2, we can prove that any threshold circuit of energy e = 1 coputing PARITY of n variables ust have an exponential nuber of gates in n, as follows. Theore 3. If a threshold circuit C of energy one coputes PARITY of n variables, then the size s of C is at least 2 n 1. Proof. Let C be a threshold circuit of size s energy e = 1 that coputes PARITY of n variables. We denote by g 1, g 2,..., g s the threshold gates in C, let g s be the top gate of C. Let X 0 = {z {0, 1} n z is even}, n 0 be the cardinality of X 0. Clearly, n 0 = 2 n 1. We prove that s n 0 = 2 n 1 as follows. For the sake of contradiction, assue that s n 0 1. Since the top gate g s outputs zero for any input z X 0, we have, for each input z X 0, either exactly one of g 1, g 2,..., g s 1 outputs one or none of the gates outputs one. Since s n 0 1, the pigeon hole principle iplies that there exists a pair of inputs x = (x 1, x 2,..., x n, y = (y 1, y 2,..., y n X 0 that satisfies one of the following conditions: (i there exists only an index k, 1 k s 1, such that the gate g k outputs one for each of the inputs x y; (ii none of the gates g 1, g 2,..., g s outputs one for each of the inputs x y. For each of (i (ii, we derive a contradiction as follows. We first consider (i. Let the gate g k have weights w 1, w 2,..., w n for the n input variables a threshold t. Since only the gate g k outputs one for each of x y, we clearly have w i x i t 0 w i y i t 0. w i y i 2t 0. (22 Since x y, there exists an index j such that x j y j. Consider a pair of inputs x y obtained fro x y by exchanging the jth coponents of x for that of y: x = (x 1, x 2,..., x j 1, y j, x j+1,..., x n (23 y = (y 1, y 2,..., y j 1, x j, y j+1,..., y n (24 Both x y are even we have either 0 = x j y j = 1 or 1 = x j y j = 0, hence x y are odd. only the top gate g s outputs one for each of x y, consequently g k outputs zero for each of x y, which iplies that w i x i w j x j + w j y j t < 0 w i y i w j y j + w j x j t < 0. w i y i 2t < 0. (25 We obtain a contradiction fro Eqs. (22 (25 We next consider (ii, derive a contradiction in a siilar way to (i. Let the top gate g s have weights w 1, w 2,..., w n for the n input variables
6 a threshold t. Since none of the gates outputs one for x y, we clearly have w i x i t < 0 w i y i t < 0. w i y i 2t < 0. (26 Let j be an index such that x j y j, then consider a pair of inputs x y obtained fro x y by switching the jth coponents of the inputs as in Eqs (23 (24. Clearly, x y are both odd. the top gate g s outputs one for each of x y, which iplies that w i x i w j x j + w j y j t 0 w i y i w j y j + w j x j t 0. w i y i 2t 0. (27 We obtain a contradiction fro Eqs. (26 (27 Theores 1 3 iply that there exists a significant gap of coputational power between threshold circuits of e = 1 ones of e = 2. 6 Conclusions In the paper, we design energy-efficient threshold circuits coputing the odulus function MOD, show that MOD of n variables can be coputed by a threshold circuit of size s = O(e(n/ 1/(e 1 energy e for any integer e 2. The upper bound on the size s = O(e(n/ 1/(e 1 alost atches the known lower bound Ω(e(n/ 1/e. We also show that any threshold circuit of energy e = 1 needs at least 2 n 1 threshold gates to copute PARITY of n variables. A Boolean function f : {0, 1} n {0, 1} is called syetric if f(x depends only on the nuber of 1s in x for every input x {0, 1} n. the odulus function is syetric. A generalization of the result to syetric functions reains open. Földiak, P. (2003, Sparse coding in the priate cortex, The Hbook of Brain Theory Neural Networks 1, Laughlin, S. B. & Sejnowski, T. J. (2003, Counication in neuronal networks, Science 301(5641, Lennie, P. (2003, The cost of cortical coputation, Current Biology 13, Minsky, M. & Papert, S. (1988, Perceptrons: An Introduction to Coputational Geoetry, MIT Press, Cabridge, MA. Olshausen, B. A. & Field, D. J. (2004, Sparse coding of sensory inputs, Current Opinion in Neuro Biology 14, Parberry, I. (1994, Circuit Coplexity Neural Networks, MIT Press, Cabridge, MA. Sia, J. & Orponen, P. (2003, General-purpose coputation with neural networks: A survey of coplexity theoretic results, Neural Coputation 15, Siu, K. Y., Roychowdhury, V. & Kailath, T. (1995, Discrete Neural Coputation; A Theoretical Foundation, Prentice-Hall, Inc., Upper Saddle River, NJ. Uchizawa, K. & Takioto, E. (2008, Exponential lower bounds on the size of constant-depth threshold circuits with sall energy coplexity, Theoretical Coputer Science 407(1-3, Uchizawa, K., Douglas, R. & Maass, W. (2006, On the coputational power of threshold circuits with sparse activity, Neural Coputation 18(12, Uchizawa, K., Nishizeki, T. & Takioto, E. (2009a, Energy coplexity depth of threshold circuits, in Proceedings of the 17th International Syposiu on Fundaentals of Coputation Theory, Springer Lect. Notes in Coputer Science 5699, pp Uchizawa, K., Takioto, E. & Nishizeki, T. (2009b, Size energy of threshold circuits coputing od functions, in Proceedings of the 34th Int. Syp. on Matheatical Foundations of Coputer Science, Aug , 2009, High Tatras, Slovakia, Springer Lect. Notes in Coputer Science 5734, pp Vinje, W. E. & Gallant, J. L. (2000, Sparse coding decorrelation in priary visual cortex during natural vision, Science 287(5456, Acknowledgents We would like to thank Takao Nishizeki for his fruitful discussion the referees for their helpful coents. References Attwell, D. & Laughlin, S. B. (2001, An energy budget for signaling in the gray atter of the brain, Journal of Cerebral Blood Flow Metabolis 21,
Complexity of Counting Output Patterns of Logic Circuits
Proceedings of the Nineteenth Computing: The Australasian Theory Symposium (CATS 2013), Adelaide, Australia Complexity of Counting Output Patterns of Logic Circuits Kei Uchizawa 1 Zhenghong Wang 2 Hiroki
More informationPolygonal Designs: Existence and Construction
Polygonal Designs: Existence and Construction John Hegean Departent of Matheatics, Stanford University, Stanford, CA 9405 Jeff Langford Departent of Matheatics, Drake University, Des Moines, IA 5011 G
More informationOn Poset Merging. 1 Introduction. Peter Chen Guoli Ding Steve Seiden. Keywords: Merging, Partial Order, Lower Bounds. AMS Classification: 68W40
On Poset Merging Peter Chen Guoli Ding Steve Seiden Abstract We consider the follow poset erging proble: Let X and Y be two subsets of a partially ordered set S. Given coplete inforation about the ordering
More informationOn Conditions for Linearity of Optimal Estimation
On Conditions for Linearity of Optial Estiation Erah Akyol, Kuar Viswanatha and Kenneth Rose {eakyol, kuar, rose}@ece.ucsb.edu Departent of Electrical and Coputer Engineering University of California at
More informationA new type of lower bound for the largest eigenvalue of a symmetric matrix
Linear Algebra and its Applications 47 7 9 9 www.elsevier.co/locate/laa A new type of lower bound for the largest eigenvalue of a syetric atrix Piet Van Mieghe Delft University of Technology, P.O. Box
More informationMANY physical structures can conveniently be modelled
Proceedings of the World Congress on Engineering Coputer Science 2017 Vol II Roly r-orthogonal (g, f)-factorizations in Networks Sizhong Zhou Abstract Let G (V (G), E(G)) be a graph, where V (G) E(G) denote
More informationAnalysis of Polynomial & Rational Functions ( summary )
Analysis of Polynoial & Rational Functions ( suary ) The standard for of a polynoial function is ( ) where each of the nubers are called the coefficients. The polynoial of is said to have degree n, where
More informationOptimal Jamming Over Additive Noise: Vector Source-Channel Case
Fifty-first Annual Allerton Conference Allerton House, UIUC, Illinois, USA October 2-3, 2013 Optial Jaing Over Additive Noise: Vector Source-Channel Case Erah Akyol and Kenneth Rose Abstract This paper
More informationA Note on Scheduling Tall/Small Multiprocessor Tasks with Unit Processing Time to Minimize Maximum Tardiness
A Note on Scheduling Tall/Sall Multiprocessor Tasks with Unit Processing Tie to Miniize Maxiu Tardiness Philippe Baptiste and Baruch Schieber IBM T.J. Watson Research Center P.O. Box 218, Yorktown Heights,
More informationTight Bounds for Maximal Identifiability of Failure Nodes in Boolean Network Tomography
Tight Bounds for axial Identifiability of Failure Nodes in Boolean Network Toography Nicola Galesi Sapienza Università di Roa nicola.galesi@uniroa1.it Fariba Ranjbar Sapienza Università di Roa fariba.ranjbar@uniroa1.it
More informationCharacterization of the Line Complexity of Cellular Automata Generated by Polynomial Transition Rules. Bertrand Stone
Characterization of the Line Coplexity of Cellular Autoata Generated by Polynoial Transition Rules Bertrand Stone Abstract Cellular autoata are discrete dynaical systes which consist of changing patterns
More informationLecture 21. Interior Point Methods Setup and Algorithm
Lecture 21 Interior Point Methods In 1984, Kararkar introduced a new weakly polynoial tie algorith for solving LPs [Kar84a], [Kar84b]. His algorith was theoretically faster than the ellipsoid ethod and
More informationUniform Approximation and Bernstein Polynomials with Coefficients in the Unit Interval
Unifor Approxiation and Bernstein Polynoials with Coefficients in the Unit Interval Weiang Qian and Marc D. Riedel Electrical and Coputer Engineering, University of Minnesota 200 Union St. S.E. Minneapolis,
More informationarxiv: v1 [math.nt] 14 Sep 2014
ROTATION REMAINDERS P. JAMESON GRABER, WASHINGTON AND LEE UNIVERSITY 08 arxiv:1409.411v1 [ath.nt] 14 Sep 014 Abstract. We study properties of an array of nubers, called the triangle, in which each row
More informationLow complexity bit parallel multiplier for GF(2 m ) generated by equally-spaced trinomials
Inforation Processing Letters 107 008 11 15 www.elsevier.co/locate/ipl Low coplexity bit parallel ultiplier for GF generated by equally-spaced trinoials Haibin Shen a,, Yier Jin a,b a Institute of VLSI
More informationGeneralized Alignment Chain: Improved Converse Results for Index Coding
Generalized Alignent Chain: Iproved Converse Results for Index Coding Yucheng Liu and Parastoo Sadeghi Research School of Electrical, Energy and Materials Engineering Australian National University, Canberra,
More informationMODULAR HYPERBOLAS AND THE CONGRUENCE ax 1 x 2 x k + bx k+1 x k+2 x 2k c (mod m)
#A37 INTEGERS 8 (208) MODULAR HYPERBOLAS AND THE CONGRUENCE ax x 2 x k + bx k+ x k+2 x 2k c (od ) Anwar Ayyad Departent of Matheatics, Al Azhar University, Gaza Strip, Palestine anwarayyad@yahoo.co Todd
More informationOn Process Complexity
On Process Coplexity Ada R. Day School of Matheatics, Statistics and Coputer Science, Victoria University of Wellington, PO Box 600, Wellington 6140, New Zealand, Eail: ada.day@cs.vuw.ac.nz Abstract Process
More informationE0 370 Statistical Learning Theory Lecture 5 (Aug 25, 2011)
E0 370 Statistical Learning Theory Lecture 5 Aug 5, 0 Covering Nubers, Pseudo-Diension, and Fat-Shattering Diension Lecturer: Shivani Agarwal Scribe: Shivani Agarwal Introduction So far we have seen how
More informationResearch Article Some Formulae of Products of the Apostol-Bernoulli and Apostol-Euler Polynomials
Discrete Dynaics in Nature and Society Volue 202, Article ID 927953, pages doi:055/202/927953 Research Article Soe Forulae of Products of the Apostol-Bernoulli and Apostol-Euler Polynoials Yuan He and
More informationLinear recurrences and asymptotic behavior of exponential sums of symmetric boolean functions
Linear recurrences and asyptotic behavior of exponential sus of syetric boolean functions Francis N. Castro Departent of Matheatics University of Puerto Rico, San Juan, PR 00931 francis.castro@upr.edu
More informationarxiv: v1 [cs.ds] 3 Feb 2014
arxiv:40.043v [cs.ds] 3 Feb 04 A Bound on the Expected Optiality of Rando Feasible Solutions to Cobinatorial Optiization Probles Evan A. Sultani The Johns Hopins University APL evan@sultani.co http://www.sultani.co/
More informationDivisibility of Polynomials over Finite Fields and Combinatorial Applications
Designs, Codes and Cryptography anuscript No. (will be inserted by the editor) Divisibility of Polynoials over Finite Fields and Cobinatorial Applications Daniel Panario Olga Sosnovski Brett Stevens Qiang
More informationFast Montgomery-like Square Root Computation over GF(2 m ) for All Trinomials
Fast Montgoery-like Square Root Coputation over GF( ) for All Trinoials Yin Li a, Yu Zhang a, a Departent of Coputer Science and Technology, Xinyang Noral University, Henan, P.R.China Abstract This letter
More informationA note on the multiplication of sparse matrices
Cent. Eur. J. Cop. Sci. 41) 2014 1-11 DOI: 10.2478/s13537-014-0201-x Central European Journal of Coputer Science A note on the ultiplication of sparse atrices Research Article Keivan Borna 12, Sohrab Aboozarkhani
More informationEstimation of the Population Mean Based on Extremes Ranked Set Sampling
Aerican Journal of Matheatics Statistics 05, 5(: 3-3 DOI: 0.593/j.ajs.05050.05 Estiation of the Population Mean Based on Extrees Ranked Set Sapling B. S. Biradar,*, Santosha C. D. Departent of Studies
More informationBipartite subgraphs and the smallest eigenvalue
Bipartite subgraphs and the sallest eigenvalue Noga Alon Benny Sudaov Abstract Two results dealing with the relation between the sallest eigenvalue of a graph and its bipartite subgraphs are obtained.
More informationVC Dimension and Sauer s Lemma
CMSC 35900 (Spring 2008) Learning Theory Lecture: VC Diension and Sauer s Lea Instructors: Sha Kakade and Abuj Tewari Radeacher Averages and Growth Function Theore Let F be a class of ±-valued functions
More informationCurious Bounds for Floor Function Sums
1 47 6 11 Journal of Integer Sequences, Vol. 1 (018), Article 18.1.8 Curious Bounds for Floor Function Sus Thotsaporn Thanatipanonda and Elaine Wong 1 Science Division Mahidol University International
More informationComputational and Statistical Learning Theory
Coputational and Statistical Learning Theory Proble sets 5 and 6 Due: Noveber th Please send your solutions to learning-subissions@ttic.edu Notations/Definitions Recall the definition of saple based Radeacher
More informationlecture 36: Linear Multistep Mehods: Zero Stability
95 lecture 36: Linear Multistep Mehods: Zero Stability 5.6 Linear ultistep ethods: zero stability Does consistency iply convergence for linear ultistep ethods? This is always the case for one-step ethods,
More informationPattern Recognition and Machine Learning. Artificial Neural networks
Pattern Recognition and Machine Learning Jaes L. Crowley ENSIMAG 3 - MMIS Fall Seester 2017 Lessons 7 20 Dec 2017 Outline Artificial Neural networks Notation...2 Introduction...3 Key Equations... 3 Artificial
More informationA1. Find all ordered pairs (a, b) of positive integers for which 1 a + 1 b = 3
A. Find all ordered pairs a, b) of positive integers for which a + b = 3 08. Answer. The six ordered pairs are 009, 08), 08, 009), 009 337, 674) = 35043, 674), 009 346, 673) = 3584, 673), 674, 009 337)
More information3.8 Three Types of Convergence
3.8 Three Types of Convergence 3.8 Three Types of Convergence 93 Suppose that we are given a sequence functions {f k } k N on a set X and another function f on X. What does it ean for f k to converge to
More informationDSPACE(n)? = NSPACE(n): A Degree Theoretic Characterization
DSPACE(n)? = NSPACE(n): A Degree Theoretic Characterization Manindra Agrawal Departent of Coputer Science and Engineering Indian Institute of Technology, Kanpur 208016, INDIA eail: anindra@iitk.ernet.in
More informationVulnerability of MRD-Code-Based Universal Secure Error-Correcting Network Codes under Time-Varying Jamming Links
Vulnerability of MRD-Code-Based Universal Secure Error-Correcting Network Codes under Tie-Varying Jaing Links Jun Kurihara KDDI R&D Laboratories, Inc 2 5 Ohara, Fujiino, Saitaa, 356 8502 Japan Eail: kurihara@kddilabsjp
More informationDescent polynomials. Mohamed Omar Department of Mathematics, Harvey Mudd College, 301 Platt Boulevard, Claremont, CA , USA,
Descent polynoials arxiv:1710.11033v2 [ath.co] 13 Nov 2017 Alexander Diaz-Lopez Departent of Matheatics and Statistics, Villanova University, 800 Lancaster Avenue, Villanova, PA 19085, USA, alexander.diaz-lopez@villanova.edu
More informationAn EGZ generalization for 5 colors
An EGZ generalization for 5 colors David Grynkiewicz and Andrew Schultz July 6, 00 Abstract Let g zs(, k) (g zs(, k + 1)) be the inial integer such that any coloring of the integers fro U U k 1,..., g
More informationarxiv: v1 [cs.ds] 29 Jan 2012
A parallel approxiation algorith for ixed packing covering seidefinite progras arxiv:1201.6090v1 [cs.ds] 29 Jan 2012 Rahul Jain National U. Singapore January 28, 2012 Abstract Penghui Yao National U. Singapore
More informationEXPLICIT CONGRUENCES FOR EULER POLYNOMIALS
EXPLICIT CONGRUENCES FOR EULER POLYNOMIALS Zhi-Wei Sun Departent of Matheatics, Nanjing University Nanjing 10093, People s Republic of China zwsun@nju.edu.cn Abstract In this paper we establish soe explicit
More informationComputable Shell Decomposition Bounds
Coputable Shell Decoposition Bounds John Langford TTI-Chicago jcl@cs.cu.edu David McAllester TTI-Chicago dac@autoreason.co Editor: Leslie Pack Kaelbling and David Cohn Abstract Haussler, Kearns, Seung
More informationFPTAS for optimizing polynomials over the mixed-integer points of polytopes in fixed dimension
Matheatical Prograing anuscript No. (will be inserted by the editor) Jesús A. De Loera Rayond Heecke Matthias Köppe Robert Weisantel FPTAS for optiizing polynoials over the ixed-integer points of polytopes
More informationCourse Notes for EE227C (Spring 2018): Convex Optimization and Approximation
Course Notes for EE227C (Spring 2018): Convex Optiization and Approxiation Instructor: Moritz Hardt Eail: hardt+ee227c@berkeley.edu Graduate Instructor: Max Sichowitz Eail: sichow+ee227c@berkeley.edu October
More informationIn this chapter, we consider several graph-theoretic and probabilistic models
THREE ONE GRAPH-THEORETIC AND STATISTICAL MODELS 3.1 INTRODUCTION In this chapter, we consider several graph-theoretic and probabilistic odels for a social network, which we do under different assuptions
More informationThe concavity and convexity of the Boros Moll sequences
The concavity and convexity of the Boros Moll sequences Ernest X.W. Xia Departent of Matheatics Jiangsu University Zhenjiang, Jiangsu 1013, P.R. China ernestxwxia@163.co Subitted: Oct 1, 013; Accepted:
More informationGeneralized eigenfunctions and a Borel Theorem on the Sierpinski Gasket.
Generalized eigenfunctions and a Borel Theore on the Sierpinski Gasket. Kasso A. Okoudjou, Luke G. Rogers, and Robert S. Strichartz May 26, 2006 1 Introduction There is a well developed theory (see [5,
More informationM ath. Res. Lett. 15 (2008), no. 2, c International Press 2008 SUM-PRODUCT ESTIMATES VIA DIRECTED EXPANDERS. Van H. Vu. 1.
M ath. Res. Lett. 15 (2008), no. 2, 375 388 c International Press 2008 SUM-PRODUCT ESTIMATES VIA DIRECTED EXPANDERS Van H. Vu Abstract. Let F q be a finite field of order q and P be a polynoial in F q[x
More informationFigure 1: Equivalent electric (RC) circuit of a neurons membrane
Exercise: Leaky integrate and fire odel of neural spike generation This exercise investigates a siplified odel of how neurons spike in response to current inputs, one of the ost fundaental properties of
More informationThe Frequent Paucity of Trivial Strings
The Frequent Paucity of Trivial Strings Jack H. Lutz Departent of Coputer Science Iowa State University Aes, IA 50011, USA lutz@cs.iastate.edu Abstract A 1976 theore of Chaitin can be used to show that
More informationIntelligent Systems: Reasoning and Recognition. Artificial Neural Networks
Intelligent Systes: Reasoning and Recognition Jaes L. Crowley MOSIG M1 Winter Seester 2018 Lesson 7 1 March 2018 Outline Artificial Neural Networks Notation...2 Introduction...3 Key Equations... 3 Artificial
More informationAcyclic Colorings of Directed Graphs
Acyclic Colorings of Directed Graphs Noah Golowich Septeber 9, 014 arxiv:1409.7535v1 [ath.co] 6 Sep 014 Abstract The acyclic chroatic nuber of a directed graph D, denoted χ A (D), is the iniu positive
More informationE0 370 Statistical Learning Theory Lecture 6 (Aug 30, 2011) Margin Analysis
E0 370 tatistical Learning Theory Lecture 6 (Aug 30, 20) Margin Analysis Lecturer: hivani Agarwal cribe: Narasihan R Introduction In the last few lectures we have seen how to obtain high confidence bounds
More informationORIGAMI CONSTRUCTIONS OF RINGS OF INTEGERS OF IMAGINARY QUADRATIC FIELDS
#A34 INTEGERS 17 (017) ORIGAMI CONSTRUCTIONS OF RINGS OF INTEGERS OF IMAGINARY QUADRATIC FIELDS Jürgen Kritschgau Departent of Matheatics, Iowa State University, Aes, Iowa jkritsch@iastateedu Adriana Salerno
More informationFourier Series Summary (From Salivahanan et al, 2002)
Fourier Series Suary (Fro Salivahanan et al, ) A periodic continuous signal f(t), - < t
More informationPrediction by random-walk perturbation
Prediction by rando-walk perturbation Luc Devroye School of Coputer Science McGill University Gábor Lugosi ICREA and Departent of Econoics Universitat Popeu Fabra lucdevroye@gail.co gabor.lugosi@gail.co
More information13.2 Fully Polynomial Randomized Approximation Scheme for Permanent of Random 0-1 Matrices
CS71 Randoness & Coputation Spring 018 Instructor: Alistair Sinclair Lecture 13: February 7 Disclaier: These notes have not been subjected to the usual scrutiny accorded to foral publications. They ay
More informationFeature Extraction Techniques
Feature Extraction Techniques Unsupervised Learning II Feature Extraction Unsupervised ethods can also be used to find features which can be useful for categorization. There are unsupervised ethods that
More informationPRELIMINARIES This section lists for later sections the necessary preliminaries, which include definitions, notations and lemmas.
MORE O SQUARE AD SQUARE ROOT OF A ODE O T TREE Xingbo Wang Departent of Mechatronic Engineering, Foshan University, PRC Guangdong Engineering Center of Inforation Security for Intelligent Manufacturing
More informationTHE AVERAGE NORM OF POLYNOMIALS OF FIXED HEIGHT
THE AVERAGE NORM OF POLYNOMIALS OF FIXED HEIGHT PETER BORWEIN AND KWOK-KWONG STEPHEN CHOI Abstract. Let n be any integer and ( n ) X F n : a i z i : a i, ± i be the set of all polynoials of height and
More informationOn the Inapproximability of Vertex Cover on k-partite k-uniform Hypergraphs
On the Inapproxiability of Vertex Cover on k-partite k-unifor Hypergraphs Venkatesan Guruswai and Rishi Saket Coputer Science Departent Carnegie Mellon University Pittsburgh, PA 1513. Abstract. Coputing
More information4 = (0.02) 3 13, = 0.25 because = 25. Simi-
Theore. Let b and be integers greater than. If = (. a a 2 a i ) b,then for any t N, in base (b + t), the fraction has the digital representation = (. a a 2 a i ) b+t, where a i = a i + tk i with k i =
More informationIntelligent Systems: Reasoning and Recognition. Perceptrons and Support Vector Machines
Intelligent Systes: Reasoning and Recognition Jaes L. Crowley osig 1 Winter Seester 2018 Lesson 6 27 February 2018 Outline Perceptrons and Support Vector achines Notation...2 Linear odels...3 Lines, Planes
More informationDesign of Spatially Coupled LDPC Codes over GF(q) for Windowed Decoding
IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED PAPER) 1 Design of Spatially Coupled LDPC Codes over GF(q) for Windowed Decoding Lai Wei, Student Meber, IEEE, David G. M. Mitchell, Meber, IEEE, Thoas
More information1 Proof of learning bounds
COS 511: Theoretical Machine Learning Lecturer: Rob Schapire Lecture #4 Scribe: Akshay Mittal February 13, 2013 1 Proof of learning bounds For intuition of the following theore, suppose there exists a
More informationarxiv: v2 [math.co] 3 Dec 2008
arxiv:0805.2814v2 [ath.co] 3 Dec 2008 Connectivity of the Unifor Rando Intersection Graph Sion R. Blacburn and Stefanie Gere Departent of Matheatics Royal Holloway, University of London Egha, Surrey TW20
More informationarxiv:math/ v1 [math.nt] 15 Jul 2003
arxiv:ath/0307203v [ath.nt] 5 Jul 2003 A quantitative version of the Roth-Ridout theore Toohiro Yaada, 606-8502, Faculty of Science, Kyoto University, Kitashirakawaoiwakecho, Sakyoku, Kyoto-City, Kyoto,
More informationBernoulli Numbers. Junior Number Theory Seminar University of Texas at Austin September 6th, 2005 Matilde N. Lalín. m 1 ( ) m + 1 k. B m.
Bernoulli Nubers Junior Nuber Theory Seinar University of Texas at Austin Septeber 6th, 5 Matilde N. Lalín I will ostly follow []. Definition and soe identities Definition 1 Bernoulli nubers are defined
More informationChicago, Chicago, Illinois, USA b Department of Mathematics NDSU, Fargo, North Dakota, USA
This article was downloaded by: [Sean Sather-Wagstaff] On: 25 August 2013, At: 09:06 Publisher: Taylor & Francis Infora Ltd Registered in England and Wales Registered Nuber: 1072954 Registered office:
More informationNew upper bound for the B-spline basis condition number II. K. Scherer. Institut fur Angewandte Mathematik, Universitat Bonn, Bonn, Germany.
New upper bound for the B-spline basis condition nuber II. A proof of de Boor's 2 -conjecture K. Scherer Institut fur Angewandte Matheati, Universitat Bonn, 535 Bonn, Gerany and A. Yu. Shadrin Coputing
More informationTesting Properties of Collections of Distributions
Testing Properties of Collections of Distributions Reut Levi Dana Ron Ronitt Rubinfeld April 9, 0 Abstract We propose a fraework for studying property testing of collections of distributions, where the
More informationarxiv: v1 [cs.ds] 17 Mar 2016
Tight Bounds for Single-Pass Streaing Coplexity of the Set Cover Proble Sepehr Assadi Sanjeev Khanna Yang Li Abstract arxiv:1603.05715v1 [cs.ds] 17 Mar 2016 We resolve the space coplexity of single-pass
More informationSolving initial value problems by residual power series method
Theoretical Matheatics & Applications, vol.3, no.1, 13, 199-1 ISSN: 179-9687 (print), 179-979 (online) Scienpress Ltd, 13 Solving initial value probles by residual power series ethod Mohaed H. Al-Sadi
More informationKernel Methods and Support Vector Machines
Intelligent Systes: Reasoning and Recognition Jaes L. Crowley ENSIAG 2 / osig 1 Second Seester 2012/2013 Lesson 20 2 ay 2013 Kernel ethods and Support Vector achines Contents Kernel Functions...2 Quadratic
More informationA := A i : {A i } S. is an algebra. The same object is obtained when the union in required to be disjoint.
59 6. ABSTRACT MEASURE THEORY Having developed the Lebesgue integral with respect to the general easures, we now have a general concept with few specific exaples to actually test it on. Indeed, so far
More informationTight Information-Theoretic Lower Bounds for Welfare Maximization in Combinatorial Auctions
Tight Inforation-Theoretic Lower Bounds for Welfare Maxiization in Cobinatorial Auctions Vahab Mirrokni Jan Vondrák Theory Group, Microsoft Dept of Matheatics Research Princeton University Redond, WA 9805
More informationSolutions of some selected problems of Homework 4
Solutions of soe selected probles of Hoework 4 Sangchul Lee May 7, 2018 Proble 1 Let there be light A professor has two light bulbs in his garage. When both are burned out, they are replaced, and the next
More informationThe Simplex Method is Strongly Polynomial for the Markov Decision Problem with a Fixed Discount Rate
The Siplex Method is Strongly Polynoial for the Markov Decision Proble with a Fixed Discount Rate Yinyu Ye April 20, 2010 Abstract In this note we prove that the classic siplex ethod with the ost-negativereduced-cost
More informationComputable Shell Decomposition Bounds
Journal of Machine Learning Research 5 (2004) 529-547 Subitted 1/03; Revised 8/03; Published 5/04 Coputable Shell Decoposition Bounds John Langford David McAllester Toyota Technology Institute at Chicago
More informationOn the Existence of Pure Nash Equilibria in Weighted Congestion Games
MATHEMATICS OF OPERATIONS RESEARCH Vol. 37, No. 3, August 2012, pp. 419 436 ISSN 0364-765X (print) ISSN 1526-5471 (online) http://dx.doi.org/10.1287/oor.1120.0543 2012 INFORMS On the Existence of Pure
More informationError Exponents in Asynchronous Communication
IEEE International Syposiu on Inforation Theory Proceedings Error Exponents in Asynchronous Counication Da Wang EECS Dept., MIT Cabridge, MA, USA Eail: dawang@it.edu Venkat Chandar Lincoln Laboratory,
More informationResearch Article On the Isolated Vertices and Connectivity in Random Intersection Graphs
International Cobinatorics Volue 2011, Article ID 872703, 9 pages doi:10.1155/2011/872703 Research Article On the Isolated Vertices and Connectivity in Rando Intersection Graphs Yilun Shang Institute for
More informationA Note on the Applied Use of MDL Approximations
A Note on the Applied Use of MDL Approxiations Daniel J. Navarro Departent of Psychology Ohio State University Abstract An applied proble is discussed in which two nested psychological odels of retention
More informationarxiv: v2 [math.co] 8 Mar 2018
Restricted lonesu atrices arxiv:1711.10178v2 [ath.co] 8 Mar 2018 Beáta Bényi Faculty of Water Sciences, National University of Public Service, Budapest beata.benyi@gail.co March 9, 2018 Keywords: enueration,
More informationThe Weierstrass Approximation Theorem
36 The Weierstrass Approxiation Theore Recall that the fundaental idea underlying the construction of the real nubers is approxiation by the sipler rational nubers. Firstly, nubers are often deterined
More informationThis article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and
This article appeared in a ournal published by Elsevier. The attached copy is furnished to the author for internal non-coercial research and education use, including for instruction at the authors institution
More informationA Bernstein-Markov Theorem for Normed Spaces
A Bernstein-Markov Theore for Nored Spaces Lawrence A. Harris Departent of Matheatics, University of Kentucky Lexington, Kentucky 40506-0027 Abstract Let X and Y be real nored linear spaces and let φ :
More informationQualitative Modelling of Time Series Using Self-Organizing Maps: Application to Animal Science
Proceedings of the 6th WSEAS International Conference on Applied Coputer Science, Tenerife, Canary Islands, Spain, Deceber 16-18, 2006 183 Qualitative Modelling of Tie Series Using Self-Organizing Maps:
More informationThe Wilson Model of Cortical Neurons Richard B. Wells
The Wilson Model of Cortical Neurons Richard B. Wells I. Refineents on the odgkin-uxley Model The years since odgkin s and uxley s pioneering work have produced a nuber of derivative odgkin-uxley-like
More informationTHE SUPER CATALAN NUMBERS S(m, m + s) FOR s 3 AND SOME INTEGER FACTORIAL RATIOS. 1. Introduction. = (2n)!
THE SUPER CATALAN NUMBERS S(, + s FOR s 3 AND SOME INTEGER FACTORIAL RATIOS XIN CHEN AND JANE WANG Abstract. We give a cobinatorial interpretation for the super Catalan nuber S(, + s for s 3 using lattice
More informationConvex Programming for Scheduling Unrelated Parallel Machines
Convex Prograing for Scheduling Unrelated Parallel Machines Yossi Azar Air Epstein Abstract We consider the classical proble of scheduling parallel unrelated achines. Each job is to be processed by exactly
More informationPerturbation on Polynomials
Perturbation on Polynoials Isaila Diouf 1, Babacar Diakhaté 1 & Abdoul O Watt 2 1 Départeent Maths-Infos, Université Cheikh Anta Diop, Dakar, Senegal Journal of Matheatics Research; Vol 5, No 3; 2013 ISSN
More informationAlireza Kamel Mirmostafaee
Bull. Korean Math. Soc. 47 (2010), No. 4, pp. 777 785 DOI 10.4134/BKMS.2010.47.4.777 STABILITY OF THE MONOMIAL FUNCTIONAL EQUATION IN QUASI NORMED SPACES Alireza Kael Mirostafaee Abstract. Let X be a linear
More informationPolytopes and arrangements: Diameter and curvature
Operations Research Letters 36 2008 2 222 Operations Research Letters wwwelsevierco/locate/orl Polytopes and arrangeents: Diaeter and curvature Antoine Deza, Taás Terlaky, Yuriy Zinchenko McMaster University,
More informationOn Concurrent Detection of Errors in Polynomial Basis Multiplication
1 On Concurrent Detection of Errors in Polynoial Basis Multiplication Siavash Bayat-Saradi and M. Anwar Hasan Abstract The detection of errors in arithetic operations is an iportant issue. This paper discusses
More information. The univariate situation. It is well-known for a long tie that denoinators of Pade approxiants can be considered as orthogonal polynoials with respe
PROPERTIES OF MULTIVARIATE HOMOGENEOUS ORTHOGONAL POLYNOMIALS Brahi Benouahane y Annie Cuyt? Keywords Abstract It is well-known that the denoinators of Pade approxiants can be considered as orthogonal
More informationIEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 60, NO. 2, FEBRUARY ETSP stands for the Euclidean traveling salesman problem.
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 60, NO., FEBRUARY 015 37 Target Assignent in Robotic Networks: Distance Optiality Guarantees and Hierarchical Strategies Jingjin Yu, Meber, IEEE, Soon-Jo Chung,
More informationList Scheduling and LPT Oliver Braun (09/05/2017)
List Scheduling and LPT Oliver Braun (09/05/207) We investigate the classical scheduling proble P ax where a set of n independent jobs has to be processed on 2 parallel and identical processors (achines)
More informationAlgebraic Montgomery-Yang problem: the log del Pezzo surface case
c 2014 The Matheatical Society of Japan J. Math. Soc. Japan Vol. 66, No. 4 (2014) pp. 1073 1089 doi: 10.2969/jsj/06641073 Algebraic Montgoery-Yang proble: the log del Pezzo surface case By DongSeon Hwang
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 11 10/15/2008 ABSTRACT INTEGRATION I
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 11 10/15/2008 ABSTRACT INTEGRATION I Contents 1. Preliinaries 2. The ain result 3. The Rieann integral 4. The integral of a nonnegative
More informationRobustness and Regularization of Support Vector Machines
Robustness and Regularization of Support Vector Machines Huan Xu ECE, McGill University Montreal, QC, Canada xuhuan@ci.cgill.ca Constantine Caraanis ECE, The University of Texas at Austin Austin, TX, USA
More information