scholarly journals A Simple Sum of Products Formula to Compute the Reliability of the KooN System

IEEE Access ◽  
2021 ◽  
Vol 9 ◽  
pp. 31161-31169
Author(s):  
Saeid Pashazadeh ◽  
Leila Namvari Tazeh-Kand
Keyword(s):  
Sum Of Products

scholarly journals On the Ternary Exponential Diophantine Equation Equating a Perfect Power and Sum of Products of Consecutive Integers

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1813
Author(s):  
S. Subburam ◽  
Lewis Nkenyereye ◽  
N. Anbazhagan ◽  
S. Amutha ◽  
M. Kameswari ◽  
...  
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Upper Bound ◽  
Exponential Diophantine Equation ◽  
Research Article ◽  
All Solutions ◽  
Sum Of Products ◽  
Consecutive Integers ◽  
Sums Of Products ◽  
Positive Integers

Consider the Diophantine equation yn=x+x(x+1)+⋯+x(x+1)⋯(x+k), where x, y, n, and k are integers. In 2016, a research article, entitled – ’power values of sums of products of consecutive integers’, primarily proved the inequality n= 19,736 to obtain all solutions (x,y,n) of the equation for the fixed positive integers k≤10. In this paper, we improve the bound as n≤ 10,000 for the same case k≤10, and for any fixed general positive integer k, we give an upper bound depending only on k for n.

Lower bounds for Sum and Sum of Products of Read-once Formulas

2019 ◽  
Vol 11 (2) ◽  
pp. 1-27 ◽  
Author(s):  
C. Ramya ◽  
B. V. Raghavendra Rao
Keyword(s):  
Lower Bounds ◽  
Sum Of Products

scholarly journals SUMS OF PRODUCTS OF CONGRUENCE CLASSES AND OF ARITHMETIC PROGRESSIONS

2009 ◽  
Vol 05 (04) ◽  
pp. 625-634
Author(s):  
SERGEI V. KONYAGIN ◽  
MELVYN B. NATHANSON
Keyword(s):  
Arithmetic Progression ◽  
Congruence Class ◽  
Arithmetic Progressions ◽  
Sum Of Products ◽  
Sums Of Products ◽  
Positive Integers ◽  
Congruence Classes

Consider the congruence class Rm(a) = {a + im : i ∈ Z} and the infinite arithmetic progression Pm(a) = {a + im : i ∈ N0}. For positive integers a,b,c,d,m the sum of products set Rm(a)Rm(b) + Rm(c)Rm(d) consists of all integers of the form (a+im) · (b+jm)+(c+km)(d+ℓm) for some i,j,k,ℓ ∈ Z. It is proved that if gcd (a,b,c,d,m) = 1, then Rm(a)Rm(b) + Rm(c)Rm(d) is equal to the congruence class Rm(ab+cd), and that the sum of products set Pm(a)Pm(b)+Pm(c)Pm eventually coincides with the infinite arithmetic progression Pm(ab+cd).

CONTINUOUS-TIME FINANCE AND THE WAITING TIME DISTRIBUTION: MULTIPLE CHARACTERISTIC TIMES

2012 ◽  
Vol 26 (23) ◽  
pp. 1250151 ◽  
Author(s):  
KWOK SAU FA
Keyword(s):  
Distribution Function ◽  
Random Walk ◽  
Waiting Time ◽  
Continuous Time ◽  
Probability Distribution Function ◽  
Time Distribution ◽  
Waiting Time Distribution ◽  
Exponential Functions ◽  
Multiple Characteristic ◽  
Sum Of Products

In this paper, we model the tick-by-tick dynamics of markets by using the continuous-time random walk (CTRW) model. We employ a sum of products of power law and stretched exponential functions for the waiting time probability distribution function; this function can fit well the waiting time distribution for BUND futures traded at LIFFE in 1997.

scholarly journals A Short Path Quantum Algorithm for Exact Optimization

Quantum ◽  
2018 ◽  
Vol 2 ◽  
pp. 78 ◽  
Author(s):  
M. B. Hastings
Keyword(s):  
Hybrid Algorithm ◽  
Optimal Solution ◽  
Quantum Algorithm ◽  
Low Energy ◽  
Weighted Sum ◽  
Natural Hybrid ◽  
Sum Of Products ◽  
Time Required ◽  
The Cost ◽  
Exact Optimization

We give a quantum algorithm to exactly solve certain problems in combinatorial optimization, including weighted MAX-2-SAT as well as problems where the objective function is a weighted sum of products of Ising variables, all terms of the same degree D; this problem is called weighted MAX-ED-LIN2. We require that the optimal solution be unique for odd D and doubly degenerate for even D; however, we expect that the algorithm still works without this condition and we show how to reduce to the case without this assumption at the cost of an additional overhead. While the time required is still exponential, the algorithm provably outperforms Grover's algorithm assuming a mild condition on the number of low energy states of the target Hamiltonian. The detailed analysis of the runtime dependence on a tradeoff between the number of such states and algorithm speed: fewer such states allows a greater speedup. This leads to a natural hybrid algorithm that finds either an exact or approximate solution.

Synaptic integration in an excitable dendritic tree

1993 ◽  
Vol 70 (3) ◽  
pp. 1086-1101 ◽  
Author(s):  
B. W. Mel
Keyword(s):  
Pyramidal Cell ◽  
Neuronal Response ◽  
Dendritic Tree ◽  
Pattern Discrimination ◽  
Synaptic Activity ◽  
Visual Neurons ◽  
Voltage Dependent ◽  
Sum Of Products ◽  
Very High ◽  
Synaptic Drive

1. Compartmental modeling experiments were carried out in an anatomically characterized neocortical pyramidal cell to study the integrative behavior of a complex dendritic tree containing active membrane mechanisms. Building on a previously presented hypothesis, this work provides further support for a novel principle of dendritic information processing that could underlie a capacity for nonlinear pattern discrimination and/or sensory processing within the dendritic trees of individual nerve cells. 2. It was previously demonstrated that when excitatory synaptic input to a pyramidal cell is dominated by voltage-dependent N-methyl-D-aspartate (NMDA)-type channels, the cell responds more strongly when synaptic drive is concentrated within several dendritic regions than when it is delivered diffusely across the dendritic arbor. This effect, called dendritic "cluster sensitivity," persisted under wide-ranging parameter variations and directly implicated the spatial ordering of afferent synaptic connections onto the dendritic tree as an important determinant of neuronal response selectivity. 3. In this work, the sensitivity of neocortical dendrites to spatially clustered synaptic drive has been further studied with fast sodium and slow calcium spiking mechanisms present in the dendritic membrane. Several spatial distributions of the dendritic spiking mechanisms were tested with and without NMDA synapses. Results of numerous simulations reveal that dendritic cluster sensitivity is a highly robust phenomenon in dendrites containing a sufficiency of excitatory membrane mechanisms and is only weakly dependent on their detailed spatial distribution, peak conductances, or kinetics. Factors that either work against or make irrelevant the dendritic cluster sensitivity effect include 1) very high-resistance spine necks, 2) very large synaptic conductances, 3) very high baseline levels of synaptic activity, and 4) large fluctuations in level of synaptic activity on short time scales. 4. The functional significance of dendritic cluster sensitivity has been previously discussed in the context of associative learning and memory. Here it is demonstrated that the dendritic tree of a cluster-sensitive neuron implements an approximative spatial correlation, or sum of products operation, such as that which could underlie nonlinear disparity tuning in binocular visual neurons.

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