### Table 2: Computational Power of Deterministic and Probabilistic Discrete-Time Analog Neural Networks with the Saturated-Linear Activation Function.

"... In PAGE 17: ...4 de- pends on the descriptive complexity of their weights. The respective results are summarized in Table2 as presented by Siegelmann (1994), including the comparison with the probabilistic recurrent networks discussed in sec- tion 2.... In PAGE 24: ..., 2000). This implies that the results on the computational power of deterministic asymmetric networks summarized in Table2 are still valid for Hopfield nets with an external oscillator of certain type. Especially for rational weights, these devices are Turing universal.... In PAGE 26: ...4 (Siegelmann, 1999b). The results are summarized and compared to the corresponding deterministic models in Table2 . Thus, for integer weights, the results co- incide with those for deterministic networks (see section 2.... In PAGE 39: ... Figure 2). Furthermore, Table2 , summarizing the results concerning the computational power of recurrent neural networks, shows that the only difference between deterministic and probabilistic mod- els is in polynomial time computations with rational weights, which are characterized by the corresponding Turing complexity classes P and BPP. This means that from the computational power point of view, stochasticity plays a similar role in neural networks as in conventional Turing computa- tions.... ..."

### Table 3: Discretization Time in CPU seconds

2000

"... In PAGE 4: ... fyes,nog (i.e., was the student admitted to UCI). Table 2: Description of Data Sets Data Set # Features # Continuous # Examples Adult 14 5 48812 Census-Income 41 7 199523 SatImage 37 36 6435 Shuttle 10 9 48480 UCI Admissions 19 8 123028 6.1 Execution Time Table3 shows the discretization time for MVD, ME-MDL and the time taken by Apriori to perform frequent set mining on MVD apos;s discretizations. MVD apos;s time was generally com- parable to ME-MDL.... ..."

Cited by 8

### Table 25: De nitions of discrete time operators (a 2 A )

2001

"... In PAGE 38: ...3. The explicit de nitions needed are given in Table25 . Notice that the operators abs, abs and abs of ACPdatp are simply de ned as the operators abs, abs and abs of ACPsatIp restricted in their rst argument to N.... In PAGE 38: ... Notice that the operators abs, abs and abs of ACPdatp are simply de ned as the operators abs, abs and abs of ACPsatIp restricted in their rst argument to N. We will establish the existence of an embedding by proving that for closed terms the axioms of ACPdatp are derivable from the axioms of ACPsatIp and the explicit de nitions given in Table25 . However, we rst take another look at the connection between ACPsatIp and ACPdatp by introducing the notions of a discretized real time process and a discretely initialized real time process.... ..."

Cited by 30

### Table 2: Discrete Time Hazard Estimation of Age at Marriage Males with Primary

"... In PAGE 18: ...17 5. Empirical Results In this section we discuss separately the effects of each of the covariates on the age at marriage ( Table2 ), age at first birth (Table 3) and the duration of subsequent birth intervals (Tables 4 and 5). Although we have also estimated the intervals from marriage to a first birth, we do not discuss the results due to some problems.... ..."

### TABLE 2: Inherent Delays Due To Filtering And Other Processing. Loop Type Low-Pass Filter Other Processing Total Delay Discrete-Time Decision-Directed T/2 T/2 T

in Carrier Synchronization for Homodyne and Heterodyne Detection of Optical Quadriphase-Shift Keying

### Table 2: Parameters of the Discrete-Time Model

2003

### Table 1: Two pairs of a discrete time autoregression matrix A and corresponding continuous time drift matrix A

"... In PAGE 13: ...Table 1: Two pairs of a discrete time autoregression matrix A and corresponding continuous time drift matrix A represents in fact a smaller auto-effect than 0:50 over t = 1:25. Not less paradoxical differences between the discrete time models studied in behavioral sci- ence on the one hand and the underlying continuous time models on the other hand can be ob- served in Table1 . On the basis of simple simulated autoregression matrices A (both typical in the sense of having higher diagonal than nondiagonal elements), it is shown that the conclusions drawn with respect to the cross-lagged coef cients in A may differ quite fundamentally from those drawn on the basis of the corresponding cross-effects in drift matrices A.... In PAGE 14: ... Therefore, statements about direction and strength of a causal effect in discrete time are meaningless without indicating the exact time interval t the statement refers to. This is the clear message of Figures 2 and 3, where for sev- eral of the cross-lagged coef cients in Table1 not only the value at t = 1 but the development over the whole period from t = 0 until t = 2 years according to exponential form A = eA t is given. Figures 2 and 3 give the continuous time impulse-response, that is the effects of an iso- lated unit-impulse in a single independent variable over continuously increasing time intervals on the dependent variable.... In PAGE 14: ... Figures 2 and 3 give the continuous time impulse-response, that is the effects of an iso- lated unit-impulse in a single independent variable over continuously increasing time intervals on the dependent variable. The implication of Figure 2 is that the conclusion about the relative strength of the recipro- cal causal effects between x3 and x1 (pair I of Table1 ) on the basis of the discrete time model depends on the time interval chosen in the model. Researchers choosing the discrete time in- terval t between 0 and 0:66 year will come to the conclusion that x1 has a larger effect on x3 (maximum difference reached at t = 0:27), while researchers choosing t gt; 0:66 come to the opposite conclusion (maximum difference reached at t = 2:74).... In PAGE 14: ...200 0.250 Coefficient Value Figure 2: Cross-lagged coef cients a ;31 (solid line) and a ;13 (dotted line) in autoregression matrix A of pair I in Table1 for corresponding continuous time coef cients a31 = 0:50 and a13 = 0:43 in A as functions of the time interval t 2 [0; 2]... In PAGE 15: ...150 0.200 Coefficient Value Figure 3: Cross-lagged coef cient a ;21 in autoregression matrix A of pair II in Table1 for corresponding continuous time coef cient a21 = 0:11 in A as a function of the time interval t 2 [0; 2] cients, by de nition having value 0 over t = 0, will rst once or repeatedly go up or down but eventually go to 0 for this and other asymptotically stable models (all eigenvalues of A strictly negative). Such stable models also imply a maximum value for the cross-lagged effect to be reached after some nite time interval t.... In PAGE 15: ... For a ;31 the maximum values of 0:208 is reached at t = 1:42, for a ;13 it is 0:230, reached at t = 1:70. Figure 3 describes the discrete time effect from x1 on x2 (pair II of Table1 ) in models with different t. Its clear implication is that even the sign of the cross-lagged coef cient need not be the same as the one of the true underlying continuous time effect.... ..."

### Table 12: De nitions of discrete time operators (a 2 A , n 2 N, k 2 Z)

"... In PAGE 33: ...operators a, abs, abs, abs and pd (for processes) in Table12 . In [6], use is made of two lemmas that do not go through for the extension with conditionals, viz.... ..."

### Table 3: Discrete Time Hazard Estimation of Age at First Birth Males with Primary

"... In PAGE 18: ...17 5. Empirical Results In this section we discuss separately the effects of each of the covariates on the age at marriage (Table 2), age at first birth ( Table3 ) and the duration of subsequent birth intervals (Tables 4 and 5). Although we have also estimated the intervals from marriage to a first birth, we do not discuss the results due to some problems.... ..."

### Table 4 Deduction rules for MPA with relative discrete time (a 2 A).

2000

"... In PAGE 61: ... Intuitively, x 1 7! x0 means that x evolves into x0 by passing to the next time slice. We add the rules in Table4 to the rules of Table 2. Note that x 1 67! means that x cannot execute a 1 7! transition, i.... In PAGE 86: ... We detected four signi cant language features which give rise to di erent languages and make three performance-sensitive equiv- alences, performance congruence, lazy equivalence and eager equivalence, to behave di erently over these languages. Table4 summarizes our results. We would like to note that if process synchronization is allowed and eager equivalence and/or lazy equivalence do not coincide with performance con- gruence, then they are not even compositional.... ..."