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Relativistically induced divergence of electric field
Thesis Proposal
The proliferation of data in the modern age has spawned an entire industry; the goal of which is to examine, understand, and ultimately transform the data into useful information. Statistical learning is a framework built upon mathematical theory that attempts to make rigorous this exploration by making explicit the assumptions, proofs, and limitations of the methods upon which we rely. This is in contrast to its more publicly recognizable cousin, machine learning, whose focus is directed firmly towards the synthesis of data into information. Notably, the singular focus of machine learning allows for the use of tools for which the theoretical underpinnings are not fully understood. While both paths have merit, we err here towards the more measured path of statistical learning with hopes that the tools developed herein will be of use to both camps; indeed, we have drawn heavily from machine learning literature which is both prolific and of extraordinary practical value. We remain convinced that the fields are symbiotic, that each informs the other, and that while the approaches may be different, the goals are the same.
The internet, smartphones, drones, and the ’internet of things’ have spawned an interesting problem for statistical learning that few people had foreseen when the primary tools of the field were being developed. Data is being created at such an extraordinary rate that traditional methods are wholly incapable of keeping up. Generations in advance of their time, Robbins and Monro\cite{Robbins_1951} developed what is widely considered to be the first iterative probabilistic optimization algorithm and jump-started an entire new field of research called Stochastic Approximation. It was decades before statistical learning would have need of their work, and many \cite{Fabian_1971,Fabian_1968,Fabian_1969,Wolfowitz_1952,Wolfowitz_1956} improvements were made in the interim. Our goal in this work is to expand the current repertoire of stochastic iterative optimization algorithms to reflect the current requirements imposed by big data. In particular, we propose alternatives to more recent work \cite{Byrd_2016,Mokhtari_2014} in function optimization. In addition, we provide proofs of convergence with probability one, of asymptotic normality, and of the rates of convergence of the proposed methods.
Before we introduce our proposed methods, we provide a brief background in statistical learning in section [StatisticalLearning] so that the contribution to optimization can be motivated placed in context. Additionally, some background in probability and functional analysis is required for the proofs of convergence; this is provided in section [Probability]. In section [Overview] we explore the path that has led to the state of the art, and provide an overview of the recent relative literature. With the requisite tools in place, and having surveyed the most current alternatives we introduce our method in section [RIM] along with the aforementioned proofs. Finally, in section [Future] we detail the future of this work and propose new, complementary areas of research. All but the shortest or most important proofs are placed in the appendices, along with much detailed discussion that is imperative for a holistic understanding of the paper.
Quantum Physics at the Pre-University Level in Singapore
Macro HW #3 (2.14, 2.16. and 2.17)
i) Adding the Social Security tax, the constraint changes to:\(C_{1, t} + S_t = A w_t - T\)\(C_{2, t} = (1 + r_{t + 1})S_t + (1 + n)T\)Solving for savings gives:\(S_t = \frac{C_{2, t + 1}}{1 + r_{t + 1}} - \frac{(1 + n)}{(1 + r_{t + 1})} T\)Substitute savings into the first equation:\(C_{1, t} + \frac{C_{2, t + 1}}{1 + r_{t + 1}} = Aw_t -T \frac{(1 + n)}{(1 + r_{t + 1})} T\)Finding the intertemporal budget constraint:\(C_{1, t} + \frac{C_{2, t + 1}}{1 + r_{t + 1}} = Aw_t - \frac{(r_{t + 1}- n)}{(1 + r_{t + 1})} T\)Since with logarithmic utility the individual will consume \(\frac{(1 + \rho)}{(2 + \rho)}\) of lifetime wealth, we have:\(C_{1, t} = (\frac{1 + \rho}{2 + \rho}) [A w_t - (\frac{(r_{t + 1}- n)}{(1 + r_{t + 1})}) T\)Solving for saving per person with substitution:\(S_t = A w_t - (\frac{1 + \rho}{2 + \rho}) [A w_t - (\frac{r_{t + 1} - n}{1 + r_{t + 1}}) T] - T\)\(S_t = [1 - (\frac{1 + \rho}{2 + \rho})]A w_t - [1 - (\frac{1 + \rho}{2 + \rho}) (\frac{r_{t + 1} - n}{1 + r_{t + 1}})] - T\)If we replace the last argument in the brackets with \(Z_t\), we will have:\(S_t = [(\frac{1}{2 + \rho})]A w_t - Z_t T\)And we know that the capital stock in the second period is equal to saving of the young in the first, thus:\(k_{t + 1} = [\frac{1}{1 + n}] [(\frac{1}{2 + \rho})w_t - \frac{Z_t T}{A}]\)Substituting in what we know from Cobb-Douglas about the real wage:\(k_{t + 1} = [\frac{1}{1 + n}] [(\frac{1}{2 + \rho})(1 - \alpha) k_t ^\alpha - \frac{Z_t T}{A}]\)ii) To see the effect this has on the balanced growth path value of capital, we have to observe the sign of \(Z_t\). If it is positive, the introduction of the tax will shift down the k curve and reduce the balanced growth path value. We can simplify to see this:\(Z_t = \frac{(1 + r_{t + 1}) + (1 + \rho)(1 + n)}{(2 + \rho)(1 + r_{t + 1})} > 0\)iii) If the economy was initially dynamically efficient, a marginal increase in T would result in a gain to the old generation that would receive the extra benefits. However it would reduce capital further and leave posterity worse off. If the original level of capital was above the Golden Rule the older generation would again gain. But now, it would be welfare-improving for posterity as well. The tax could alleviate inefficiency from the over-accumulation of capital.
i) \(C_{2, t+ 1} = (1 + r_{t + 1})S_t + (1 + n)T\)Since the rate of return on social security is the same as that on saving, we can derive the intertemporal budget constraint:\(C_{1, t} + \frac{C_{2, t + 1}}{1 + r_{t + 1}} = A w_t\)Solving this yields the usual Euler equation:\(C_{2, t + 1} = [\frac{1}{1 + \rho}](1 + r_{t + 1}) C_{1, t}\)Substituting this into the budget constraint and solving for saving per person we have:\(S_t = [\frac{1}{2 + \rho}] Aw_t - T\)Thus the social security tax cases a one-for-one reduction in private saving. Examining the capital stock:\(K_{t + 1} = S_t L_t + T L_t\)Converting per person and simplifying gives us:\(k_{t + 1} = [\frac{1}{1 + n}][\frac{1}{2 + \rho}] (1 - \alpha) k_t ^\alpha\)Therefore, the fully-funded social security system has no effect on the relationship between capital stock in successive periods.ii) Since there is no effect on the relationship between the capital stock in the successive periods, the balanced growth path value is the same as before the fully funded social security system. Essentially, the total investment and saving is still the same, but government is doing some of it for the young.
Coevolving Cancer Hallmarks: Notes
Spatial Stats Notes
Correcting annotation noise in automatically labelled data
and 1 collaborator
RCK Notes
\(\mathcal {L} = \log(C_1) + \frac{1}{1 + \rho} \log (C_2) + \lambda (Y_1 + \frac{Y_2}{1 + r} - C_1 - \frac{C_2}{1 + r})\)
\(\frac{\partial \mathcal{L}}{\partial C_1} = 0 \rightarrow \frac{1}{C_1} = \lambda\)\(\frac{\partial \mathcal{L}}{\partial C_2} = 0 \rightarrow \frac{1}{C_2(1 + \rho)} = \frac{\lambda}{1 + r}\)\(\frac{\partial \mathcal{L}}{\partial \lambda} = 0 \rightarrow Y_1 + \frac{Y_2}{1 + r} = C_1 + \frac{C_2}{1 + r}\)
\(\frac{\frac{1}{C_1}}{1 + r} = \frac{1 + r}{C_1}\)\(\frac{1}{C_2(1 + \rho)} = \frac{1 + r}{C_1}\)
\(\frac{C_2}{C_1} = \frac{1 + r}{1 + \rho}\)
\(\frac{C_1 + \Delta C }{C_1} = \frac{1 + r}{1 + \rho}\)
\(\log (1+ \frac{\Delta C}{C_1}) \approx \log (1 + r) - \log (1 + \rho)\)\(\frac{\Delta C}{C_1} \approx r - \rho \rightarrow \frac{\dot C}{C} \approx r - \rho\)\(\therefore \frac{\dot C}{C} = \frac{r - \rho}{\theta}\)
\(\frac{\dot c}{c} = \) the growth rate of consumption per worker\(r =\) interest rate\(\rho = \) subjective rate of time preference (a.k.a. discount rate). If this falls, households discount their future consumption less, and they will save more. And vice versa.\(\theta = \) coefficient of relative risk aversion (a.k.a. elasticity of intertemporal substitution). If this falls, the less households' marginal utility changes as consumption changes. And vice versa.
If \(r = \rho \rightarrow \frac{\dot C}{C} = 0 \), then there is a constant amount of consumptionIf \(r > \rho \rightarrow \frac{\dot C}{C} > 0\), then there is greater present saving and less present consumption, but greater future consumptionIf \(r < \rho \rightarrow \frac{\dot C}{C} < 0\), then there is less present saving and greater present consumption, but less future consumption
\(\dot k = \frac{d k_t}{dt} = \frac{AL (\frac{d K}{dt}) - K (\frac{d AL}{dt})}{AL^2} = \frac{AL \dot K - K(A \dot L + \dot A L)}{AL^2}\)\(= \frac{AL \dot K}{(AL)(AL)} - \frac{K(A \dot L + \dot A L)}{(AL)(AL)} = \frac{\dot K}{AL} - k[\frac{A \dot L}{AL} + \frac{\dot A L}{AL}]\)\(= \frac{sY}{AL} - k(\frac{\dot L}{L} + \frac{\dot A}{A}) = \frac{Y - C}{AL} - k(n + g)\)\(\therefore \dot k_t = f(k_t) - c_t - (n + g)k\)
\(\dot c = 0 \rightarrow f'(k*) = \rho + \theta g\)\(\dot k = 0 \rightarrow c = f(k) - (n + g)k_t\)\(f'(k_{GR}) = (n + g) \Longrightarrow \rho + \theta g > n + g \neq f'(k_{GR})\)
\(f'(k*) = \rho + \theta g\) and\(f'(k_{GR}) = n + g\)\(\therefore \rho > n\), or the discount rate is greater than the population rate.
Mast Cell Imaging Lit Review
and 2 collaborators
- conjugate a pH-sensitive green fluorescent protein (pHluorin) to Vesicle Associated Membrane Protein 8 (VAMP-8) at the C-terminal, creating a protein called immuno-pHluorin (impH). It only fluoresces at a neutral pH – when acidic secretory granules fuse with plasma membrane, pH increases due to exposure to neutral external environment.
- Transfect Bone marrow-derived mast cells (BMMCs) with retrovirus, transfectants sensitised with 2,4,6-trinitrophenol TNP-IgE, then activated with 300 ng/ml of TNP-conjugated ovalbumin (TNP-OVA) or control OVA.
- Transfected cells show no impairment in degranulation process measured by b-hexosamidase release assay. histamine release (shown by comparable histamine release between untranfested and impH mast cells treated with TNP-OVA after TNP-IgE stimulation). Figure 1 E
- Flow cytometric analysis showed transfection did not affect level of IgE receptors FcERI – FIgure 1D
- Functional evidence of impH via confocal microscopy:
- Increasing pH of transfected cells via NH4Cl and by blocking acidification with Bafilomycin A1 leads to presence of green fluorescent patches that are absent in PBS-treated cells.
- This pH switch is illustrated when sensitised cells were stimulated with TNP-OVA where impH-expressing cells showed green fluorescent patches near edge of cells within 10 minutes after treatment. Cells stimulated with control ova showed no green patches.
- The green patches co-localise with newly formed CD63-a degranulation marker-indicating that impH enables visualisation of mast cell degranulation in vitro.
- The impH system showed that the degranulation process is polarised –his is seen when they used a TNP-OVA conjugated beads with a 3 microm diamete. They proposed that it only occurs at certain sites where the antibody-allergen binds to and cross-links the FcERI (Figure 3). T To test this in vivo, --
- Able to show degranulation process in vivo using mast cell deficient mice reconstituted with either wild-type of FcRgamma-deficient BMMC. Inject the ear skin of mice interdermally with TNP-IgE prior to IV injection with TNP-OVA. Green patches only seen in mice reconstituted with wild-type BMMC, illustrating that the impH system utilises the engagement of the allergen bound-IgE receptors prior to degranulation.
- Advantages:
Research Project: Project Ideas (Iteration 1)
Agregación de enlaces
MACHINE LEARNING FOR MEDICAL DIAGNOSIS AND HEALTH ANALYSIS
quotes for colsan
Knowledge discovery and data cycle.
Abstract
Introduction
Colloids And Surfaces B Biointerfaces Template
Title
and 3 collaborators
Notes on Solow
\(Y_t = F(K_t, A_t L_t)\), where output equals the function of physical capital multiplied by the effectiveness of labor.
\(cY_t = F(cK_t, cA_t cL_t) \Longrightarrow cF(K_t, A_t L_t)\)
\(\frac{\partial F}{\partial K_t} > 0, MPK > 0\)\(\frac{\partial ^2 F}{\partial ^2 K_t} > 0, \frac{\partial MPK}{\partial K_t} > 0\)
\(\lim _{K \rightarrow 0} \frac{\partial F}{\partial K} = + \infty\)\(\lim _{K \rightarrow \infty} \frac{\partial F}{\partial K} = 0\)
\(\frac{\dot L_t}{L_t} = n > 0\) where \(\frac{dL_t}{dt} \equiv \dot L_t = n L_t\)
\(\frac{\dot A_t}{A_t} = g > 0\) where \(\frac{dA_t}{dt} \equiv \dot A_t = g A_t\)
\(\dot K_t = I_t - \delta K_t\), meaning the change in capital equals investment minus the deprecation rate times the amount of capital (depreciated capital)\(K_{t + 1} = K_t (1 - \delta) + I_t\), meaning the amount of capital in the next period equals the previous period's capital minus depreciation plus investment
\(I_t = Y_t - C_t = S_t\)\(S_t = sY_t\), where \(s\) is the fraction of output not consumed,\(C_t = (1 - s) Y_t\), thus,\(I_t = S_t = sY_t\), and now we can substitute into capital accumulation:\(\dot K_t = sF(K, AL) - \delta K_t\)
\(Y = F(K, AL) = K^{\alpha} (AL)^{1 - \alpha}\)\(cY = F(cK, cAL) = c^{\alpha} c^{1 - \alpha} K^{\alpha} (AL)^{1 - \alpha}\), using constant returns to scale
\(y = F(\frac{K}{AL}^\alpha) \equiv f(k) ^\alpha\)
\(\frac{d_t}{dt} = \frac{d \frac{K_t}{A_t L_t}}{dt} \) using the quotient rule we have,\(\frac{(A_t L_t) (\frac{d K_t}{dt}) - K_t (\frac{dA_t L_t}{dt})}{(A_t L_t)^2}\)Taking the definitions of \(\frac{d K_t}{dt} = \dot K_t; \frac{d L_t}{dt} = \dot L_t\) and using the product rule:\(\frac{(A_t L_t) \dot K_t - K_t (A_t \dot L_t + \dot A_t L_t)}{(A_t L_t)^2}\)Then separating terms and canceling:\(\frac{(A_t L_t) \dot K_t}{A_t L_t ^2} - \frac{K_t (A_t \dot L_t + \dot A_t L_t)}{(A_t L_t)^2} = \frac{\dot K_t}{A_t L_t} - \frac{K_t}{A_t L_t} \cdot (\frac{A_t \dot L_t}{A_t L_t} + \frac{\dot A_t L_t}{A_t L_t})\)Substituting in the definition of \(\dot K_t = sY_t - \delta K_t\) and canceling:\(= \frac{sY_t - \delta K_t}{A_t L_t} - (\frac{K_t}{A_t L_t})(\frac{\dot L_t}{L_t} + \frac{\dot A_t}{A_t})\)Thus we have the definitions of the growth rates:\(= sy - \delta k_t - k_t (n + g)\)And finally, the capital accumulation (change in capital) per unit of effective labor:\(\dot k_t = sf(k_t) - (n +g + \delta)k_t\)
\(f'(k) > 0 ; f''(k) < 0\)\(y_t = f(k_t)\)\(i_t = sf(k_t)\)\(c_t = (1 - s)f(k_t)\)\(\dot k_t = sf(k_t) - (n + g + \delta) k_t\)
\(c_t(s) = (1 - s)f(k_t(s))\)Multiplying this out gives,\(c_t(s) = f(k_t(s)) - sf(k_t(s))\)And since \(\dot k_t = sf(k_t) - (n + g + \delta) k_t\) and \(\dot k* = 0\), thus \(sf(k*) = (n + g + \delta)k\)\(c*_t(s) = f(k_t(s)) - (n + g + \delta)k*(s)\)
\(\frac{\partial c*(s)}{\partial s} = \frac{\partial f}{\partial k} \frac{\partial k}{\partial s} - (n + g + \delta) \frac{\partial k}{\partial s} = 0\)\(f'(k) \frac{\partial k}{\partial s} - (n + g + \delta) \frac{\partial k}{\partial s} = 0\)\((f'(k) - (n + g + \delta)) \frac{\partial k}{\partial s} = 0\)\(f'(k) - (n + g + \delta)= 0\)\(\therefore c* = f'(k) = n + g + \delta\)
\(\frac{d Y}{d K} = MPK = r\)\(\frac{d Y}{d L} = MPL = w\)
UWCCC Grant
Scientific Paper