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# Create a label if reference undefined, else reference it (unique label)

I'm trying to label items on a table. In my use, multiple labels may exist, so the initial location of the creation of the label is irrelevant (not trying to hyperlink). I want to use one command that will assign a label. So I thought the easy solution would be to check if the reference exists, if not, increment a counter and assign a new label, otherwise reference it.

I do notice that I get different results alternating compiles. I also notice that when testing the ifundefined part in isolation (i.e. print true or false), it takes two compiles. I believe this is because of the way TeX interacts with its .aux file ?

\documentclass{article}

\newcounter{myrefcnt}% my referene counter
\makeatletter
\newcommand\myref[1]{% my command
\@ifundefined{r@#1}{%if reference doesnt exist
\refstepcounter{myrefcnt}\themyrefcnt\label{#1}% create a label for reference
}{%
\ref{#1}% else reference it
}%
}
\makeatother

\begin{document}

\textbf{My Attempt:}

\begin{tabular}{lr}
\hline
Things & Applicable Things \\\hline
Aardvark & \myref{fourlegs}, \myref{lookslikerat} \\
Bat & \myref{lookslikerat} \\
Cheetah & \myref{fourlegs} \\\hline
\end{tabular}

\myref{fourlegs} indicates speciment has four legs.
\myref{lookslikerat} indicates speciment looks like a rat.

\textit{When I run the file, the labels are sometimes 1,2,3,4, but other times 5,6,6,5}

\textbf{I want to produce the following:}

\begin{tabular}{lr}
\hline
Things & Applicable Things \\\hline
Aardvark & 1, 2 \\
Bat & 2 \\
Cheetah & 1 \\\hline
\end{tabular}

1 indicates speciment has four legs.
2 indicates speciment looks like a rat.

\end{document}

• so that you can make forward references \label writes to the aux file which is read at begin document so all the r@.. macros are defined on the second run. you could make a global \my@'#1 macro that you don't record the the aux, but then test that in your macro in the same way. (You probably wouldn't need \label at all in that case. – David Carlisle Nov 20 '20 at 14:06

so that you can make forward references \label writes to the aux file which is read at begin document so all the r@.. macros are defined on the second run. you could make a global \my@'#1 macro that you don't record the the aux, but then test that in your macro in the same way. (You probably wouldn't need \label at all in that case.

\documentclass{article}

\newcounter{myrefcnt}% my referene counter
\makeatletter
\newcommand\myref[1]{% my command
\@ifundefined{myr@#1}{%if reference doesnt exist
\stepcounter{myrefcnt}\themyrefcnt\expandafter\xdef\csname myr@#1\endcsname{\themyrefcnt}% create a label for reference
}{%
\csname myr@#1\endcsname% else reference it
}%
}
\makeatother

\begin{document}

\textbf{My Attempt:}

\begin{tabular}{lr}
\hline
Things & Applicable Things \\\hline
Aardvark & \myref{fourlegs}, \myref{lookslikerat} \\
Bat & \myref{lookslikerat} \\
Cheetah & \myref{fourlegs} \\\hline
\end{tabular}

\myref{fourlegs} indicates speciment has four legs.
\myref{lookslikerat} indicates speciment looks like a rat.

\textit{When I run the file, the labels are sometimes 1,2,3,4, but other times 5,6,6,5}

\textbf{I want to produce the following:}

\begin{tabular}{lr}
\hline
Things & Applicable Things \\\hline
Aardvark & 1, 2 \\
Bat & 2 \\
Cheetah & 1 \\\hline
\end{tabular}

1 indicates speciment has four legs.
2 indicates speciment looks like a rat.

\end{document}

You may still have problems with environments that evaluate things more than once (amsmath alignments and tabularx for example) as you need to guard the test to so that the trial runs do not make every use seem to be not the first.

• Interesting. I noticed that it is not printing references other than the first (I put a dummy text before and it is printing), so there I believe there is an error in \csname my@r#1\endcsname. Based on your comment about some environment evaluating things more than once, then would it perhaps be wiser to use the global definiton method (not requiring label at all) you mentioned? – likethevegetable Nov 20 '20 at 14:18
• @likethevegetable that's what I did here (\xdef is global and I didn't use \label) – David Carlisle Nov 20 '20 at 14:36
• @likethevegetable but your are right I have myr@ in one place and my@r in the other can be anything but needs to be consistent, I'll fix sorry – David Carlisle Nov 20 '20 at 14:38
• That's great, thank you very much! I'm still learning here. If I wanted to re-use this frame work, I suppose I could set the counter to 0, but would have to ensure that I use new reference names. If I wanted to allow the resuse of reference names, I could make another counter and somehow tie it's value to the myr ? – likethevegetable Nov 20 '20 at 14:52
• yes myr@ is just an arbitrary prefix so \myref{zzz} gets stored in \myr@zzz so if you had a different counter and a \myrefB command just use a different prefix myrB@ if you want to be consistent but zzzzz would work as well. – David Carlisle Nov 20 '20 at 14:56


text:
Now showing items 52-71 of 193

• #### Edge-based nonlinear diffusion for finite element approximations of convection–diffusion equations and its relation to algebraic flux-correction schemes

For the case of approximation of convection–diffusion equations using piecewise affine continuous finite elements a new edge-based nonlinear diffusion operator is proposed that makes the scheme satisfy a discrete maximum principle. The diffusion operator is shown to be Lipschitz continuous and linearity preserving. Using these properties we provide a full stability and error analysis, which, in the diffusion dominated regime, shows existence, uniqueness and optimal convergence. Then the algebraic flux correction method is recalled and we show that the present method can be interpreted as an algebraic flux correction method for a particular definition of the flux limiters. The performance of the method is illustrated on some numerical test cases in two space dimensions.
• #### Error estimates of high-order numerical methods for solving time fractional partial differential equations

Error estimates of some high-order numerical methods for solving time fractional partial differential equations are studied in this paper. We first provide the detailed error estimate of a high-order numerical method proposed recently by Li et al. \cite{liwudin} for solving time fractional partial differential equation. We prove that this method has the convergence order $O(\tau^{3- \alpha})$ for all $\alpha \in (0, 1)$ when the first and second derivatives of the solution are vanish at $t=0$, where $\tau$ is the time step size and $\alpha$ is the fractional order in the Caputo sense. We then introduce a new time discretization method for solving time fractional partial differential equations, which has no requirements for the initial values as imposed in Li et al. \cite{liwudin}. We show that this new method also has the convergence order $O(\tau^{3- \alpha})$ for all $\alpha \in (0, 1)$. The proofs of the error estimates are based on the energy method developed recently by Lv and Xu \cite{lvxu}. We also consider the space discretization by using the finite element method. Error estimates with convergence order $O(\tau^{3- \alpha} + h^2)$ are proved in the fully discrete case, where $h$ is the space step size. Numerical examples in both one- and two-dimensional cases are given to show that the numerical results are consistent with the theoretical results.
• #### Error estimates of a high order numerical method for solving linear fractional differential equations

In this paper, we first introduce an alternative proof of the error estimates of the numerical methods for solving linear fractional differential equations proposed in Diethelm [6] where a first-degree compound quadrature formula was used to approximate the Hadamard finite-part integral and the convergence order of the proposed numerical method is O(∆t 2−α ), 0 < α < 1, where α is the order of the fractional derivative and ∆t is the step size. We then use the similar idea to prove the error estimates of a high order numerical method for solving linear fractional differential equations proposed in Yan et al. [37], where a second-degree compound quadrature formula was used to approximate the Hadamard finite-part integral and we show that the convergence order of the numerical method is O(∆t 3−α ), 0 < α < 1. The numerical examples are given to show that the numerical results are consistent with the theoretical results.
• #### Existence and regularity of solution for a Stochastic CahnHilliard / Allen-Cahn equation with unbounded noise diffusion

The Cahn-Hilliard/Allen-Cahn equation with noise is a simpliﬁed mean ﬁeld model of stochastic microscopic dynamics associated with adsorption and desorption-spin ﬂip mechanisms in the context of surface processes. For such an equation we consider a multiplicative space-time white noise with diﬀusion coeﬃcient of linear growth. Applying technics from semigroup theory, we prove local existence and uniqueness in dimensions d = 1,2,3. Moreover, when the diﬀusion coeﬃcient satisﬁes a sub-linear growth condition of order α bounded by 1 3, which is the inverse of the polynomial order of the nonlinearity used, we prove for d = 1 global existence of solution. Path regularity of stochastic solution, depending on that of the initial condition, is obtained a.s. up to the explosion time. The path regularity is identical to that proved for the stochastic Cahn-Hilliard equation in the case of bounded noise diﬀusion. Our results are also valid for the stochastic Cahn-Hilliard equation with unbounded noise diﬀusion, for which previous results were established only in the framework of a bounded diﬀusion coeﬃcient. As expected from the theory of parabolic operators in the sense of Petrovsk˘ıı, the bi-Laplacian operator seems to be dominant in the combined model.
• #### Existence of time periodic solutions for a class of non-resonant discrete wave equations

In this paper, a class of discrete wave equations with Dirichlet boundary conditions are obtained by using the center-difference method. For any positive integers m and T, when the existence of time mT-periodic solutions is considered, a strongly indefinite discrete system needs to be established. By using a variant generalized weak linking theorem, a non-resonant superlinear (or superquadratic) result is obtained and the Ambrosetti-Rabinowitz condition is improved. Such a method cannot be used for the corresponding continuous wave equations or the continuous Hamiltonian systems; however, it is valid for some general discrete Hamiltonian systems.
• #### Existence theory for a class of evolutionary equations with time-lag, studied via integral equation formulations

In discussions of certain neutral delay differential equations in Hale’s form, the relationship of the original problem with an integrated form (an integral equation) proves to be helpful in considering existence and uniqueness of a solution and sensitivity to initial data. Although the theory is generally based on the assumption that a solution is continuous, natural solutions of neutral delay differential equations of the type considered may be discontinuous. This difficulty is resolved by relating the discontinuous solution to its restrictions on appropriate (half-open) subintervals where they are continuous and can be regarded as solutions of related integral equations. Existence and unicity theories then follow. Furthermore, it is seen that the discontinuous solutions can be regarded as solutions in the sense of Caratheodory (where this concept is adapted from the theory of ordinary differential equations, recast as integral equations).
• #### Explosive solutions of a stochastic non-local reaction–diffusion equation arising in shear band formation

In this paper, we consider a non-local stochastic parabolic equation which actually serves as a mathematical model describing the adiabatic shear-banding formation phenomena in strained metals. We first present the derivation of the mathematical model. Then we investigate under which circumstances a finite-time explosion for this non-local SPDE, corresponding to shear-banding formation, occurs. For that purpose some results related to the maximum principle for this non-local SPDE are derived and afterwards the Kaplan's eigenfunction method is employed.
• #### Exponential stability in p-th mean of solutions, and of convergent Euler-type solutions, of stochastic delay differential equations

This article carries out an analysis which proceeds as follows: showing that an inequality of Halanay type (derivable via comparison theory) can be employed to derive conditions for p-th mean stability of a solution; producing a discrete analogue of the Halanay-type theory, that permits the development of a p-th mean stability analysis of analogous stochastic difference equations. The application of the theoretical results is illustrated by deriving mean-square stability conditions for solutions and numerical solutions of a constant-coefficient linear test equation.
• #### Finite Difference Method for Two-Sided Space-Fractional Partial Differential Equations

Finite difference methods for solving two-sided space-fractional partial differential equations are studied. The space-fractional derivatives are the left-handed and right-handed Riemann-Liouville fractional derivatives which are expressed by using Hadamard finite-part integrals. The Hadamard finite-part integrals are approximated by using piecewise quadratic interpolation polynomials and a numerical approximation scheme of the space-fractional derivative with convergence order O(Δx^(3−α )),10 , where Δt,Δx denote the time and space step sizes, respectively. Numerical examples are presented and compared with the exact analytical solution for its order of convergence.
• #### A finite element method for time fractional partial differential equations

This article considers the finite element method for time fractional differential equations.
• #### Finite-time blow-up of a non-local stochastic parabolic problem

The main aim of the current work is the study of the conditions under which (finite-time) blow-up of a non-local stochastic parabolic problem occurs. We first establish the existence and uniqueness of the local-in-time weak solution for such problem. The first part of the manuscript deals with the investigation of the conditions which guarantee the occurrence of noise-induced blow-up. In the second part we first prove the $C^{1}$-spatial regularity of the solution. Then, based on this regularity result, and using a strong positivity result we derive, for first in the literature of SPDEs, a Hopf's type boundary value point lemma. The preceding results together with Kaplan's eigenfunction method are then employed to provide a (non-local) drift term induced blow-up result. In the last part of the paper, we present a method which provides an upper bound of the probability of (non-local) drift term induced blow-up.
• #### Fixed point theroms and their application - discrete Volterra applications

The existence of solutions of nonlinear discrete Volterra equations is established. We define discrete Volterra operators on normed spaces of infinite sequences of finite-dimensional vectors, and present some of their basic properties (continuity, boundedness, and representation). The treatment relies upon the use of coordinate functions, and the existence results are obtained using fixed point theorems for discrete Volterra operators on infinite-dimensional spaces based on fixed point theorems of Schauder, Rothe, and Altman, and Banach’s contraction mapping theorem, for finite-dimensional spaces.
• #### Fourier spectral methods for some linear stochastic space-fractional partial differential equations

Fourier spectral methods for solving some linear stochastic space-fractional partial differential equations perturbed by space-time white noises in one-dimensional case are introduced and analyzed. The space-fractional derivative is defined by using the eigenvalues and eigenfunctions of Laplacian subject to some boundary conditions. We approximate the space-time white noise by using piecewise constant functions and obtain the approximated stochastic space-fractional partial differential equations. The approximated stochastic space-fractional partial differential equations are then solved by using Fourier spectral methods. Error estimates in $L^{2}$- norm are obtained. Numerical examples are given.
• #### Fourier spectral methods for stochastic space fractional partial differential equations driven by special additive noises

Fourier spectral methods for solving stochastic space fractional partial differential equations driven by special additive noises in one-dimensional case are introduced and analyzed. The space fractional derivative is defined by using the eigenvalues and eigenfunctions of Laplacian subject to some boundary conditions. The space-time noise is approximated by the piecewise constant functions in the time direction and by some appropriate approximations in the space direction. The approximated stochastic space fractional partial differential equations are then solved by using Fourier spectral methods. For the linear problem, we obtain the precise error estimates in the $L_{2}$ norm and find the relations between the error bounds and the fractional powers. For the nonlinear problem, we introduce the numerical algorithms and MATLAB codes based on the FFT transforms. Our numerical algorithms can be adapted easily to solve other stochastic space fractional partial differential equations with multiplicative noises. Numerical examples for the semilinear stochastic space fractional partial differential equations are given.
• #### Fractional boundary value problems: Analysis and numerical methods

This journal article discusses nonlinear boundary value problems.
• #### Fractional pennes' bioheat equation: Theoretical and numerical studies

In this work we provide a new mathematical model for the Pennes’ bioheat equation, assuming a fractional time derivative of single order. Alternative versions of the bioheat equation are studied and discussed, to take into account the temperature-dependent variability in the tissue perfusion, and both finite and infinite speed of heat propagation. The proposed bio heat model is solved numerically using an implicit finite difference scheme that we prove to be convergent and stable. The numerical method proposed can be applied to general reaction diffusion equations, with a variable diffusion coefficient. The results obtained with the single order fractional model, are compared with the original models that use classical derivatives.
• #### G-codes over Formal Power Series Rings and Finite Chain Rings

In this work, we define $G$-codes over the infinite ring $R_\infty$ as ideals in the group ring $R_\infty G$. We show that the dual of a $G$-code is again a $G$-code in this setting. We study the projections and lifts of $G$-codes over the finite chain rings and over the formal power series rings respectively. We extend known results of constructing $\gamma$-adic codes over $R_\infty$ to $\gamma$-adic $G$-codes over the same ring. We also study $G$-codes over principal ideal rings.
• #### Galerkin methods for a Schroedinger-type equation with a dynamical boundary condition in two dimensions

In this paper, we consider a two-dimensional Schodinger-type equation with a dynamical boundary condition. This model describes the long-range sound propagation in naval environments of variable rigid bottom topography. Our choice for a regular enough finite element approximation is motivated by the dynamical condition and therefore, consists of a cubic splines implicit Galerkin method in space. Furthermore, we apply a Crank-Nicolson time stepping for the evolutionary variable. We prove existence and stability of the semidiscrete and fully discrete solution.
• #### A genetic-algorithm approach to simulating human immunodeficiency virus evolution reveals the strong impact of multiply infected cells and recombination

It has been previously shown that the majority of human immunodeficiency virus type 1 (HIV-1)-infected splenocytes can harbour multiple, divergent proviruses with a copy number ranging from one to eight. This implies that, besides point mutations, recombination should be considered as an important mechanism in the evolution of HIV within an infected host. To explore in detail the possible contributions of multi-infection and recombination to HIV evolution, the effects of major microscopic parameters of HIV replication (i.e. the point-mutation rate, the crossover number, the recombination rate and the provirus copy number) on macroscopic characteristics (such as the Hamming distance and the abundance of n-point mutants) have been simulated in silico. Simulations predict that multiple provirus copies per infected cell and recombination act in synergy to speed up the development of sequence diversity. Point mutations can be fixed for some time without fitness selection. The time needed for the selection of multiple mutations with increased fitness is highly variable, supporting the view that stochastic processes may contribute substantially to the kinetics of HIV variation in vivo.
• #### Group Rings, G-Codes and Constructions of Self-Dual and Formally Self-Dual Codes

We describe G-codes, which are codes that are ideals in a group ring, where the ring is a finite commutative Frobenius ring and G is an arbitrary finite group. We prove that the dual of a G-code is also a G-code. We give constructions of self-dual and formally self-dual codes in this setting and we improve the existing construction given in [13] by showing that one of the conditions given in the theorem is unnecessary and, moreover, it restricts the number of self-dual codes obtained by the construction. We show that several of the standard constructions of self-dual codes are found within our general framework. We prove that our constructed codes must have an automorphism group that contains G as a subgroup. We also prove that a common construction technique for producing self-dual codes cannot produce the putative [72, 36, 16] Type II code. Additionally, we show precisely which groups can be used to construct the extremal Type II codes over length 24 and 48. We define quasi-G codes and give a construction of these codes.

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# The value of expression tan^(- 1)((sqrt(2))/2)+sin^(- 1)((sqrt(5))/5)-cos^(- 1)((sqrt(10))/10)

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cot^(-1) ((1+sqrt2)/(1-sqrt2))cot^(-1) ((sqrt2 + 1)/(sqrt2-1))- pi + cot^(-1) ((1+sqrt2)/(1 - sqrt2))pi - cot^(-1) ((1-sqrt2)/(1 + sqrt2))

Answer :
C

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# Tag Info

12

No, positive definiteness (and symmetry) are only precondition to using the Conjugate Gradient method. But there are plenty of other iterative methods such as MinRes and GMRES that can be used for indefinite and non-symmetric matrices.

7

Yes. The constants that appear in the interpolation estimates upon which finite element error estimates are based contain minimum and maximum angles of triangles/tetrahedra (or similar geometric measures for quadrilaterals/hexahedra). These constants are smallest whenever you have equilateral triangles or square quadrilaterals.

4

When dealing with conservation laws like your case, you can often make use of the divergence theorem (as you did). You can then express the fact that the total mass within your integration region is preserved by the following surface integral: $$\oint_{\partial \Omega} k \nabla T \cdot \mathbf{n} ~\partial S = 0$$ Now, as it stands, it is irrelevant which ...

3

The correct way is to average the source term which you can do easily in this case as you have a polynomial. In general you can do the average with a quadrature. For second order accuracy if the source term is smooth as in your case, you can also just evaluate it at the cell center, which is like mid-point quadrature. It should not matter much in your ...

3

Generally the step from compressible Euler equations to the Navier-Stokes equations is not that hard, at least the coding part. If you want to implement it with an explicit scheme you have to consider the severe time step restriction of the parabolic contributions. One tricky part, at least for a consistent FV implementation, is the calculation of the ...

3

From the discussions and the paper, OpenFOAM seems to have implemented a measure of skewness. This answer is not an explanation why the different definitions of skewness might be equivalent, I am just going to justify why this is a measure of skewness. Consider following two elements -for sake of simplicity- Blue arrow is the outward surface normal fAreas[...

2

Solving the Riemann problem at the interface gives you the values of the solution at this interface just after $t_n$. Then, the basic assumption is that this interface state remains constant during the whole time step, which is true for piece-wise constant reconstruction (i.e. the solution is assumed uniform in each cell), and as long as the CFL condition is ...

2

Normal direction depends on the cell that you are writing equation for. the word outward is relative to the cell under study. In order to write equation for each of cells, i.e. $\Sigma \nabla T.n S_f=0$, stick to this : $\nabla T_{face}=\frac{T_c-T_i}{r_c-r_i}$ and assume $n$ as outward pointing normal vector for that face. I think your problem is that you ...

2

Most of this was already discussed in the comments, but I would like to elaborate and put a detailed answer. There are no elementary characteristics (definiteness, symmetry, bandwidth) which can tell you whether the underlying (mixed or not) FEM/FVM is stable to solve the continuous problem. You can not tell anything about that just by looking at those ...

2

An extended answer. For more arbitrary meshes you have to consider that generally CFD/FEM solvers rely on generic data-structures with element and side lists: Element list Side list Consider the following pictures, which is the standard case for simple Cartesian meshes. Since there is a single plus and a single minus side on each face, the definition is ...

2

Upwinding is not necessary. The best known and most-used limited central scheme is Nessyahu-Tadmor Nessyahu, H. and Tadmor, E., 1990. Non-oscillatory central differencing for hyperbolic conservation laws. Journal of computational physics, 87(2), pp.408-463. However a price is paid in that the contact discontinuity is badly smeared. Google Scholar will turn ...

2

You can use discontinuous Galerkin methods also for $P_0$ elements. It's true that the gradient in the cell interior is zero, so your formulation will exclusively consist of the jump terms at cell interfaces.

2

If $(x_i,y_i)$ is the centroid of the triangle, then $$\frac{1}{\Delta_i}\int_{\Delta_i}{p(x,y)}\,dx\,dy = p(x_i,y_i) = \bar{\psi}_i$$ This is mid-point quadrature, which is exact for an affine function. The centroid of a triangle is the arithmetic average of its three vertices. Note that $$x_i = \frac{1}{\Delta_i} \int_{\Delta_i} x dx dy = \frac{1}{3}\... 2 It depends on the problem being solved and the element formulation. The orthogonality criterion you state may not be as important as the shape of each element. For example in structural mechanics, with an irregular (e.g. automatically generated and refined) mesh and isoparametric elements, an important criterion for accuracy is the largest angle between ... 2 I would leave out a few things to make it more simple. This is how we do it for our code which is capable of using polyhedral meshes: https://github.com/nikola-m/freeCappuccino-dev/blob/master/src/mesh/geometry.f90 It is so called face based data structure. We use divergence theorem to compute geometrical data like volumes and cell center coordinates. This ... 1 The equation is derived from momentum balance, so there is nothing wrong with. But you can resolve the apparent paradox like this. Take a square control volume C = [a,b] \times [c,d] and look at x-momentum balance$$ \frac{d}{dt}\int_C \rho u dx dy = -\int_c^d (\rho u^2)(b,y) dy + \int_c^d (\rho u^2)(a,y) dy + \ldots $$The first term on the right is the ... 1 The eigenvalues given correspond to the flux jacobians in the Euler equations. Physically, they represent the speed at which waves of information can travel in a space-time domain. The goal is to compute the eigenvalues at each face based on the values in each neighboring cell so there is no 'left' or 'right' face in this equation, but instead a left and ... 1 The smooth solution turned out to have BC's applied in the following way: Walls and inlet: \frac{\partial p}{\partial n}=0 Outlet: p=0 1 Look at the diffusion dispersion relations: Upwind, ENO, WENO, see Fig. 2 DG, CG, see Fig. 15 Regards 1 Use a finite volume method. Define$$ \delta_x \phi(x,y) = \frac{\phi(x+\Delta x/2,y) - \phi(x-\Delta x/2,y)}{\Delta x}  a_x \phi(x,y) = \frac{\phi(x+\Delta x/2,y) + \phi(x-\Delta x/2,y)}{2} $$etc. For example, consider \tau_{xx} which is required at (i+1/2,j).$$ (\tau_{xx})_{i+1/2,j} = \mu_{i+1/2,j} \left[ \frac{4}{3} \delta_x u_{i+1/2,j} - \...

1

Just open up the parentheses, e.g., $\partial_{x} (\alpha \partial_{x} v) = (\partial_{x} \alpha) (\partial_x v) + \alpha \partial^2_x v$, where $\alpha=\mu$ or $\alpha=\mu u$ etc., and apply your central differences: $(\partial_{x} \alpha) (\partial_x v) = (\alpha_{i+1}-\alpha_{i-1})(v_{i+1}-v_{i-1})/(4h^2)$; $\alpha \partial^2_x v = \alpha_i (v_{i+1}+v_{i-... 1 Have you plotted the solution using a visualization package or just a cut through the middle of your domain? Sometimes just looking at the solution you're generating will give you an insight into where it's arising from. Based on your description, I'd say there's a mistake in your implementation of your boundary condition. ETA: The last sentence is probably ... 1 Disclaimer: I'm not 100% sure but I thought that I should provide my working solution to the above problem, for any future visitor that might have the same questions. The answer is for a steady temperature diffusion problem. What if the face is a boundary face and insulated? how to get$\phi_f$in such case? Since a boundary face is in contact with only ... 1 To expand on Wolfgang Bangerth's answer, I think P0 DG schemes reduce to two-point cell-centered finite volume schemes. I don't know if DG convergence analysis always includes$p = 0\$, but the resulting finite volume schemes can be shown to converge under appropriate "mesh orthogonality" conditions. https://math.unice.fr/~minjeaud/Donnees/...

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# Nut - Cliff, Hex, for mounting 1/4" Jacks

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# Nut - Cliff, Hex, for mounting 1/4" Jacks

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Nut - Hex, for Cliff UK Jacks S-H902, S-H903, S-H904, & S-H905

RoHS Compliant
SKU:
S-H9NT
Item ID:
005744
UPC/EAN:
609722143561
Cliff Electronics Part Number:
CL1408
Brand:
Cliff
Type:

### Accessory Items:

 Inner Diameter 0.4 in. Outer Diameter 0.7 in. Thickness 0.19 in. U.S. Thread Major Diameter 0.4375 in. U.S. Thread Pitch 20 tpi.
 Packaging Dimensions 0.192 in. × 0.192 in. × 0.681 in. Weight (Packaging) 0.001 lbs.
 Specification Sheet All Models

## Specifications, Files, and Documents

Specification Sheet 237.94 KB

Click each question to see its answers.
Asked by Anonymous on September 21st, 2015.
September 21st, 2015
Staff Member
Top Contributor
This nut is specifically for use with Cliff brand jacks. Our compatible part numbers would be S-H902, 903, 904, & 905.
Asked by Anonymous on January 11th, 2017.
January 11th, 2017
We took a Fender input jack and the Cliff nut did fit. With this in mind, it should work with your amp.
Asked by Anonymous on January 25th, 2017.
January 25th, 2017
This jack is made to be compatible with specific Cliff brand jacks from the UK. We would not have info on whether it is compatible with Roland products but many manufacturers use Cliff jacks as OEM equipment in their designs.
Asked by Anonymous on January 25th, 2017.
January 25th, 2017
This nut has a threaded diameter of 7/16" and a TPI (Threads Per Inch) of 20.
Asked by Anonymous on July 7th, 2017.
July 7th, 2017
Yes, these are typically found on Marshall's and should fit.
Asked by Anonymous on July 18th, 2018.
July 19th, 2018
This jack is made to be compatible with specific Cliff brand jacks from the UK. We would not have info on whether it is compatible with Roland products but many manufacturers use Cliff jacks as OEM equipment in their designs.
Asked by Anonymous on August 25th, 2018.
August 28th, 2018
Staff Member
Yes, these are typically found on Marshall's and should fit.
Asked by Anonymous on December 13th, 2019.
December 17th, 2019
Staff Member
There are a number of these types of nuts on the market and many of them use different sizing, so we can not confirm these will fit your amp, but Cliff jacks are a popular choice for amp manufacturers, so there is a good chance this will fit your jack.
Asked by Anonymous on May 17th, 2020.
May 19th, 2020
Staff Member
We do not have sizing information on the jack in question, so you will need to use the technical drawing above to confirm these will be compatible with the jacks in your cab.

## Product Reviews

3.00 out of 5 based on 1 reviews
- April 24th, 2019
3 out of 5

Did not fit the Roland amp that I ordered them for. However, they did fit Peavey and Kustom amps and monitors. Not a total loss, but disappointing . . .


text:
# Triangle area on surfaces of constant curvature

I am looking for an elementary derivation of the formula for the area of a geodesic triangle lying in a surface of constant curvature $\kappa$, depending on the angles and side length.

Of course, the formula can easily be derived from the Gauss–Bonnet formula to be $$A = \frac{1}{\kappa}(\alpha + \beta + \gamma - \pi)$$ for $\kappa \neq 0$. However, I would like to have an elementary geometric proof.

Does anybody know a reference?

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For $\kappa > 0$, I believe you can use inclusion-exclusion with 3 geodesics on a sphere. –  S. Carnahan May 12 '13 at 12:39
For $\kappa<0$ Gauss had a similar inclusion-exclusion proof: It is in his collected works (letter to Wolfgang/Farkas Bolyai). –  Misha May 12 '13 at 21:31
For both cases, there are visual proofs, which can be considered elementary: mathoverflow.net/questions/8846/proofs-without-words/… and mathoverflow.net/questions/8846/proofs-without-words/… –  Igor Khavkine May 13 '13 at 0:30
There are some proofs by considering bigons (for spherical geometry) and subdivisions of an ideal triangle (in hyperbolic geometry). You may also consult J. Ratcliffe's book "Foundations of hyperbolic manifolds", some of the first chapters. However, the area of a bigon or ideal triangle is computed by "not very elementary" means there. I agree with Anton, all "elementary proofs" are cheating up to a certain extent. –  SashaKolpakov Aug 31 '13 at 23:25

M. Berger, Geometrie, vol. V. MR0536874

Edit. Let me sketch a proof for the spherical triangle. Let the sphere have area $4\pi$. First you derive the area of diangle. It is $2\alpha$, where $\alpha$ is the angle, by completely elementary reasons. Now consider a triangle. Extend its sides to three full great circles. These three circles make several diangles and two equal triangles (the second one is centrally symmetric to the original one). Make a picture showing how these three circles partition the sphere. As the areas of all diangles are known the area of a triangle is simply derived by the exclusion-inclusion formula!

Notice: this proof is truly elementary in the sense that it only uses the existence of the area for a diangle and triangle, its invariance with respect to rotations, and finite additivity. Euclid COULD give a rigorous proof of this. As rigorous as his investigation of areas of Euclidean triangles.

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Chapter 5 is "Affine-projective relationship" did you really mean this? –  Anton Petrunin May 13 '13 at 23:20
Sorry, I was using Russian edition, where this is called Chapter V. Now I checked the original, and in the original it is VOLUME V. And unfortnately I did not find an English translation:-( –  Alexandre Eremenko May 14 '13 at 21:10
It use the properties of the area which (if you look carefully) already include the original statement inside. –  Anton Petrunin May 15 '13 at 4:50

All the "elementary derivations of that type" are cheating (it may look nice but it proves nothing).

The only elementary way to introduce area is adding it as an axiom (which is already kind of cheating). You have to say that there is a additive area-functional on the set of all polygons. Then you probably want to prove that this functional is unique (or include it in the same axiom).

It remains to notice that your functional $A$ satisfies the same properties and nothing left to prove.

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You would need some normalization axiom, in order to distinguish between proportional measures. Do you have a specific one in mind? –  Sergei Ivanov May 12 '13 at 22:38
@Sergei, yes sure, all I wanted to say is that if one knows what is area and curvature then there is nothing to prove. –  Anton Petrunin May 13 '13 at 2:17
Anton: I disagree with what you say. The area of a TRIANGLE is an elementary notion. (The theory of areas of triangles in Euclid is completely rigorous, by all modern standards.) And the formula has a really elementary proof. –  Alexandre Eremenko May 14 '13 at 20:47
@BS, check this question mathoverflow.net/questions/119953/definition-of-area I would be very happy if you know a better answer. –  Anton Petrunin May 16 '13 at 18:33
@Alexandre, Euclid (and Kiselev) did not prove the existence, essentially they add the existence as an axiom, but they did not say that it is an "axiom". This axiom follows from the rest of axioms, but it takes 20 pages at least. Instead of unit square you have to use other normalization (which essentially defines curvature). A rigourous way to introduce area given in "Elementary Geometry From An Advanced Standpoint" by Moise 35 pages Euclidean plane onlyy,and in "Geometry: A Metric Approach with Models" by Millman and Parker 40 pages neutral plane and contains a gap. –  Anton Petrunin May 16 '13 at 23:37


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OBJECTIVE

Despite widespread dissemination of target values, achieving a blood pressure of <130/80 mmHg is challenging for many individuals with diabetes. The purpose of the present study was to examine temporal trends in blood pressure control in hypertensive individuals with diabetes as well as the potential for race, sex, and geographic disparities.

RESEARCH DESIGN AND METHODS

We analyzed baseline data from the REasons for Geographic And Racial Differences in Stroke (REGARDS) study, a national, population-based, longitudinal cohort study of 30,228 adults (58% European American and 42% African American), examining the causes of excess stroke mortality in the southeastern U.S. We calculated mean blood pressure and blood pressure control rates (proportion with blood pressure <130/80 mmHg) for 5,217 hypertensive diabetic participants by year of enrollment (2003–2007) using multivariable logistic regression models.

RESULTS

Only 43 and 30% of European American and African American diabetic hypertensive participants, respectively, demonstrated a target blood pressure of <130/80 mmHg (P < 0.001). However, a temporal trend of improved control was evident; the odds of having a blood pressure <130/80 mmHg among diabetic hypertensive participants of both races enrolled in 2007 (as compared with those enrolled in 2003) were ∼50% greater (P < 0.001) in multivariate models.

CONCLUSIONS

These data suggest temporal improvements in blood pressure control in diabetes that may reflect broad dissemination of tighter blood pressure control targets and improving medication access. However, control rates remain low, and significant racial disparities persist among African Americans that may contribute to an increased risk for premature cardiovascular disease.

In 2003, the Joint National Committee (JNC) on Prevention, Detection, Evaluation, and Treatment of High Blood Pressure published (1) revised recommendations (JNC 7) on the management of hypertension that recommended a lower target for patients with diabetes (130/80 mmHg) than for the general population with uncomplicated essential hypertension (140/90 mmHg). This differential target was originally based on early studies (2,3), suggesting a benefit from further blood pressure reduction in helping to offset higher renal and cardiovascular risk. Subsequent evidence from the UK Prospective Diabetes Study (UKPDS) (4), the Hypertension Optimal Treatment (HOT) (5) Study, the Action in Diabetes and Vascular Disease: Preterax and Diamicron MR Controlled Evaluation (ADVANCE) (6) Study, and the Stop Atherosclerosis in Native Diabetics Study (SANDS) (7) all demonstrated improved outcomes in patients assigned to tighter blood pressure control. The Action to Control Cardiovascular Risk in Diabetes (ACCORD) (8) Study, despite discontinuation of an intensive glucose reduction arm in February 2008 (9), is investigating the value of reducing blood pressure to ≤120 mmHg, and results will be available soon.

Published goal values for blood pressure in patients with diabetes of <130 mmHg for systolic blood pressure and <80 mmHg for diastolic blood pressure represent a differential and more aggressive target for diabetic patients when compared with uncomplicated nondiabetic patients with essential hypertension. However, the extent to which this differential goal has been effectively achieved in routine practice has not been sufficiently examined, especially in large national samples. Wang (10) examined the effect of this guideline change on hypertension control in diabetic patients during the time period 1995 through 2005 and demonstrated only a modest differential improvement in blood pressure control in diabetic subjects. However, this physician office–based study was insufficient to examine the population-level impact of the 2003 JNC 7 guideline change. Population-level data from National Health and Nutrition Examination Survey 1999 through 2004 (11) suggest more substantial improvement in blood pressure control in patients with diabetes; however, the timing of these observations does not allow examination of the longitudinal effects of the 2003 guideline change. Additionally, investigators have not sufficiently examined the potential for racial and geographic differences in the extent to which blood pressure control has improved in a population-based sample in response to these guideline changes. Because of differential risks for diabetic complications by race, such studies are of critical importance. Therefore, the purpose of the present study is to examine temporal trends in blood pressure control in hypertensive individuals with diabetes as well as the potential for race, sex, and geographic disparities. To meet these goals, the primary analysis herein is to compare mean systolic and diastolic blood pressure values and the proportion of hypertensive diabetic subjects with guideline-recommended blood pressure values by race, sex, and region across different time points over the time period 2003 through 2007.

The REasons for Geographic And Racial Disparities in Stroke (REGARDS) study is a study of health disparities and stroke incidence, the methods of which have been described in detail elsewhere (12). In short, the study is a population-based, longitudinal cohort study of 30,228 subjects aged >45 years, 55% of whom are male, 45% female, 42% African American, 58% European American, 56% from stroke belt regions, and 44% from the rest of the country. The study was designed to examine the causes of excess stroke mortality in the southeastern U.S. and among African Americans, relative to the rest of the nation. Because of the large size of this study, its national scope, its enrollment of the cohort during the implementation of new guidelines for blood pressure control in diabetic subjects, and the inclusion of large numbers of adults with diabetes, many of whom also have hypertension, this study provides a unique opportunity to examine recent trends in blood pressure control among African American and European American hypertensive adults with diabetes using baseline data assembled across 5 years in the REGARDS study.

The study included a detailed computer-assisted telephone interview with each adult subject to collect demographic characteristics, including age, race (each subject self-reported race; this study compared only African American and European American subjects), sex, annual household income (<$20,000/year,$20,000–$35,000/year, >$35,000–$75,000/year, and >$75,000/year), education level (less than high school education, high school graduate, some college, and college graduate or higher), health insurance, health behaviors such as smoking (categorized as nonsmoker, past smoker, or current smoker), exercise (how often per week they exercised enough to work up a sweat; categorized as never, one to three times per week, or four or more times per week), alcohol consumption (categorized as nondrinker, one to seven drinks per week, and seven or more drinks per week), and medical history information.

Following the telephone interview, each subject was evaluated by a trained health professional who conducted an in-home assessment that included measurement of height and weight via standard procedures, after which BMI was calculated as the weight in kilograms divided by the square of height in meters. Blood pressure was obtained with an aneroid sphygmomanometer, using the appropriate cuff size and measured after being in the seated position with both feet on the floor for at least 3 min. Blood pressure measurement was repeated twice, and the average value was used in these analyses. At the home visit, blood specimens were obtained and total and HDL cholesterol were measured in accordance with the National Cholesterol Education Program guidelines (13). All medications were carefully recorded at the home visit, and self-reported medication adherence/nonadherence was assessed based on a validated scale (14).

For the present study, all subjects were identified who reported in their telephone interview that a physician had told them that they had both hypertension/high blood pressure and diabetes, as well as those who reported having hypertension and who had a fasting blood glucose >125 mg/dl measured during the home visit. Excluded from these analyses were any subjects who failed to respond to either the question on self-reported hypertension or self-reported diabetes (n = 32) and those without self-reported hypertension (n = 12,774), resulting in an analysis cohort of 5,217 subjects. The study was approved by the institutional review board, and each subject provided informed consent.

This study included 5,217 subjects who reported having both hypertension and diabetes. Details regarding the study population are given in Table 1. The majority of the European Americans with hypertension and diabetes were male, while the majority of African Americans with hypertension and diabetes were female. As detailed in Table 2, mean blood pressure in European American diabetic hypertensive subjects was significantly lower than that in African American diabetic hypertensive subjects (130 ± 16/75 ± 10 mmHg vs. 135 ± 17/78 ± 10 mmHg) (P < 0.001). While the majority of subjects in both races were not at a target blood pressure value of <130/80 mmHg, significantly more European American than African American diabetic hypertensive subjects were at or below this threshold (43 vs. 30%, respectively; P < 0.001). In multivariate-adjusted models, the odds of African American subjects being below this target value were 39% lower than the odds of European American subjects (Table 3). Further, in multivariate-adjusted models, the odds of being below this target value were 66% higher in women than men. Self-reported medication adherence was not significantly different by race, sex, or year of enrollment; however, older age was associated with modestly lower medication adherence that was independent of education and income level. There was no clear pattern of correlates in multivariate-adjusted models that might explain the racial differences in blood pressure control. There were also no significant geographic trends in multivariate models; hypertensive diabetic subjects from the southeastern U.S. had blood pressure control rates that did not differ from those observed in subjects from across the country after other factors were controlled for.

Table 1

Demographic and cardiovascular risk characteristics by race of the REGARDS study population with hypertension and diabetes (n = 5,217)

European AmericanAfrican American
n 2,054 3,163
Mean age (years) 67.2 ± 8.4 65.7 ± 8.6
Sex (% female) 43.2 62.1
Stroke belt resident (%) 64.1 54.1
Less than high school education (%) 11.4 26.9
High school graduate (%) 29.0 29.0
Some college (%) 29.6 23.9
Health insurance (%) 95.6 91.4
Household annual income (%)
<$20,000 20.8 37.9$20,000–$35,000 30.1 32.1$35,000–$75,000 35.4 23.2 >$75,000 13.7 6.9
Proportion on antihypertensive medication reporting any nonadherence (%) 35.5 34.2
Current smoker (%) 11.7 15.2
Weekly exercise (%)
None 43.2 43.1
One to three times 30.4 34.7
Four or more times 26.5 22.3
Alcohol use (%)
None 5.7 2.2
Less than one drink per day 35.2 28.7
More than one drink per day 59.1 69.1
Mean BMI 32 ± 6 33 ± 7
Mean total cholesterol (mg/dl) 176 ± 41 181 ± 42
Mean HDL cholesterol (mg/dl) 43 ± 13 50 ± 15
European AmericanAfrican American
n 2,054 3,163
Mean age (years) 67.2 ± 8.4 65.7 ± 8.6
Sex (% female) 43.2 62.1
Stroke belt resident (%) 64.1 54.1
Less than high school education (%) 11.4 26.9
High school graduate (%) 29.0 29.0
Some college (%) 29.6 23.9
Health insurance (%) 95.6 91.4
Household annual income (%)
<$20,000 20.8 37.9$20,000–$35,000 30.1 32.1$35,000–$75,000 35.4 23.2 >$75,000 13.7 6.9
Proportion on antihypertensive medication reporting any nonadherence (%) 35.5 34.2
Current smoker (%) 11.7 15.2
Weekly exercise (%)
None 43.2 43.1
One to three times 30.4 34.7
Four or more times 26.5 22.3
Alcohol use (%)
None 5.7 2.2
Less than one drink per day 35.2 28.7
More than one drink per day 59.1 69.1
Mean BMI 32 ± 6 33 ± 7
Mean total cholesterol (mg/dl) 176 ± 41 181 ± 42
Mean HDL cholesterol (mg/dl) 43 ± 13 50 ± 15

Data are means ± SD unless otherwise indicated.

Table 2

Mean systolic and diastolic pressure values in diabetic hypertensive subjects and proportion meeting guideline targets by race

European AmericanAfrican American
n 2,054 3,163
Systolic blood pressure 130 ± 16 135 ± 17*
Diastolic blood pressure 75 ± 10 78 ± 10*
Percent meeting American Diabetes Association guidelines (<130/80 mmHg) 43% 30%
European AmericanAfrican American
n 2,054 3,163
Systolic blood pressure 130 ± 16 135 ± 17*
Diastolic blood pressure 75 ± 10 78 ± 10*
Percent meeting American Diabetes Association guidelines (<130/80 mmHg) 43% 30%

Data are means ± SD unless otherwise indicated.

*P < 0.01.

Table 3

Correlates of blood pressure <130/80 mmHg among hypertensive diabetic patients from multivariate model

DemographicFitnessSocioeconomic statusCV risk factor
n 5,214 5,064 4,299 4,054
Race 0.55 (0.49–0.62) 0.55 (0.48–0.62) 0.59 (0.51–0.68) 0.61 (0.52–0.70)
Female sex 1.27 (1.12–1.43) 1.38 (1.22–1.57) 1.45 (1.26–1.67) 1.66 (1.42–1.94)
Age (10-year difference) 1.06 (0.99–1.14) 1.00 (0.93–1.07) 0.99 (0.91–1.08) 0.97 (0.88–1.05)
2004 (relative to 2003) 0.99 (0.83–1.18) 0.99 (0.82–1.19) 1.04 (0.86–1.27) 1.01 (0.82–1.23)
2005 (relative to 2003) 1.24 (1.02–1.50) 1.23 (1.01–1.50) 1.15 (0.93–1.42) 1.08 (0.87–1.35)
2006 (relative to 2003) 1.24 (1.01–1.53) 1.28 (1.03–1.58) 1.20 (0.95–1.51) 1.11 (0.88–1.35)
2007 (relative to 2003) 1.62 (1.31–2.00) 1.64 (1.32–2.03) 1.62 (1.28–2.05) 1.50 (1.18–1.92)
Stroke belt region 0.98 (0.87–1.11) 0.97 (0.86–1.10) 1.02 (0.89–1.17) 1.04 (0.90–1.20)
BMI  0.97 (0.96–0.98) 0.97 (0.96–0.98) 0.97 (0.96–0.98)
Exercise one to three times per week (relative to none)  0.95 (0.82–1.09) 0.92 (0.79–1.06) 0.90 (0.77–1.05)
Exercise four or more times per week (relative to none)  1.02 (0.88–1.19) 1.00 (0.85–1.18) 0.99 (0.83–1.17)
Income $20,000–$35,000 (relative to <$20,000) 1.07 (0.90–1.28) 1.08 (0.90–1.29) Income$35,000– $75,000 (relative to <$20,000)   1.36 (1.12–1.65) 1.38 (1.13–1.69)
Income >$75,000 (relative to <$20,000)   1.27 (0.96–1.66) 1.30 (0.98–1.72)
Education high school graduate (relative to less than high school graduate)   1.15 (0.94–1.41) 1.19 (0.96–1.46)
Education some college (relative to less than high school graduate)   1.20 (0.97–1.48) 1.20 (0.97–1.50)
Education college graduate (relative to less than high school graduate)   1.08 (0.86–1.35) 1.09 (0.86–1.38)
Insurance   0.74 (0.56–0.98) 0.80 (0.60–1.07)
Medication adherence   1.08 (0.94–1.24) 1.07 (0.93–1.23)
Current smoking    0.92 (0.75–1.12)
Total cholesterol (10 mg/dl)    0.95 (0.93–0.97)
HDL (10 mg/dl)    0.95 (0.90–1.00)
DemographicFitnessSocioeconomic statusCV risk factor
n 5,214 5,064 4,299 4,054
Race 0.55 (0.49–0.62) 0.55 (0.48–0.62) 0.59 (0.51–0.68) 0.61 (0.52–0.70)
Female sex 1.27 (1.12–1.43) 1.38 (1.22–1.57) 1.45 (1.26–1.67) 1.66 (1.42–1.94)
Age (10-year difference) 1.06 (0.99–1.14) 1.00 (0.93–1.07) 0.99 (0.91–1.08) 0.97 (0.88–1.05)
2004 (relative to 2003) 0.99 (0.83–1.18) 0.99 (0.82–1.19) 1.04 (0.86–1.27) 1.01 (0.82–1.23)
2005 (relative to 2003) 1.24 (1.02–1.50) 1.23 (1.01–1.50) 1.15 (0.93–1.42) 1.08 (0.87–1.35)
2006 (relative to 2003) 1.24 (1.01–1.53) 1.28 (1.03–1.58) 1.20 (0.95–1.51) 1.11 (0.88–1.35)
2007 (relative to 2003) 1.62 (1.31–2.00) 1.64 (1.32–2.03) 1.62 (1.28–2.05) 1.50 (1.18–1.92)
Stroke belt region 0.98 (0.87–1.11) 0.97 (0.86–1.10) 1.02 (0.89–1.17) 1.04 (0.90–1.20)
BMI  0.97 (0.96–0.98) 0.97 (0.96–0.98) 0.97 (0.96–0.98)
Exercise one to three times per week (relative to none)  0.95 (0.82–1.09) 0.92 (0.79–1.06) 0.90 (0.77–1.05)
Exercise four or more times per week (relative to none)  1.02 (0.88–1.19) 1.00 (0.85–1.18) 0.99 (0.83–1.17)
Income $20,000–$35,000 (relative to <$20,000) 1.07 (0.90–1.28) 1.08 (0.90–1.29) Income$35,000– $75,000 (relative to <$20,000)   1.36 (1.12–1.65) 1.38 (1.13–1.69)
Income >$75,000 (relative to <$20,000)   1.27 (0.96–1.66) 1.30 (0.98–1.72)
Education high school graduate (relative to less than high school graduate)   1.15 (0.94–1.41) 1.19 (0.96–1.46)
Education some college (relative to less than high school graduate)   1.20 (0.97–1.48) 1.20 (0.97–1.50)
Education college graduate (relative to less than high school graduate)   1.08 (0.86–1.35) 1.09 (0.86–1.38)
Insurance   0.74 (0.56–0.98) 0.80 (0.60–1.07)
Medication adherence   1.08 (0.94–1.24) 1.07 (0.93–1.23)
Current smoking    0.92 (0.75–1.12)
Total cholesterol (10 mg/dl)    0.95 (0.93–0.97)
HDL (10 mg/dl)    0.95 (0.90–1.00)

Data are odds ratio (95% CI) unless otherwise indicated.

Most compelling, however, were the differences observed across the time period of enrollment. There was a significant trend for those enrolled in later years to have better blood pressure control. Mean systolic blood pressure for hypertensive diabetic African Americans enrolled in 2003 was 135.5 mmHg, while mean systolic blood pressure for those enrolled in 2007 was 130.7 mmHg. Similarly, mean systolic blood pressure for hypertensive diabetic European Americans enrolled in 2003 was 131.6 mmHg, while mean systolic blood pressure for those enrolled in 2007 was 125.9 mmHg. When examined in multivariate-adjusted models that progressively adjusted for demographics, BMI, exercise, income, education, insurance, self-reported medication adherence, smoking, total cholesterol, and HDL cholesterol, the odds of achieving a blood pressure <130/80 mmHg among those enrolled in 2007 were 50% higher than among those subjects enrolled in 2003, regardless of race (P = 0.0008 for trend) (Fig. 1). There was no significant race × time interaction observed across this time period, suggesting that improvements over time did not differ by race.

Figure 1

Odds of hypertension control among hypertensive diabetic participants. Odds ratio and 95% CIs; n = 4,054; 2003 reference year. Adjusted for race, sex, age, region, BMI, weekly exercise income, education, insurance, availability, adherence to treatment, smoking, total cholesterol, and HDL.

Figure 1

Odds of hypertension control among hypertensive diabetic participants. Odds ratio and 95% CIs; n = 4,054; 2003 reference year. Adjusted for race, sex, age, region, BMI, weekly exercise income, education, insurance, availability, adherence to treatment, smoking, total cholesterol, and HDL.

Close modal

This nationwide study of blood pressure control in African American and European American hypertensive diabetic subjects demonstrates that less than half of the subjects in both racial groups had a measured blood pressure less than the target of 130/80 mmHg recommended by the American Diabetes Association and in JNC 7, that race and sex disparities in blood pressure control continue to exist, but that between 2003 and 2007 there has been improved blood pressure control in both racial groups. The need for careful blood pressure control in hypertensive diabetic subjects is underscored by the fact that blood pressure control is one of the most important determinants of cardiovascular risk (4,15,16); it is the subject of numerous investigations, including current studies to investigate the magnitude of blood pressure lowering (6,,9), and it is a major component of published guidelines on patient management that have been widely promulgated (1,17). Our data showed that income, education, insurance, and medication adherence were not independently associated with inadequate blood pressure control when other variables were controlled for, suggesting that other factors may be influencing this outcome. The potential reasons for inadequate blood pressure control are numerous and include not only patient knowledge but also biological resistance, patient lifestyle/behavioral choices, comorbid conditions and treatments, medication nonadherence, and health system factors, including limited access to care, discrimination by race and sex, competing demands of providers, lack of knowledge or adoption of the guideline, and the lack of treatment intensification. These and other factors have been recently described in the study by Bosworth et al. (18).

Data from the REGARDS study reported previously by Safford et al. (19) demonstrated that African American hypertensive subjects were more likely to have a more intense antihypertensive regimen than European Americans across all age and income strata but still were more likely to have elevated blood pressure values. This suggests treatment resistance and/or medication nonadherence. However, our findings demonstrated no racial differences in self-reported medication adherence. These findings are in contrast to those of Bosworth et al. (20), who found that African Americans in the Veterans Administration system were 81% more likely than European Americans to be nonadherent with medications for blood pressure control. However, the findings of Bosworth et al. were not limited to diabetic subjects. The fact that a smaller proportion of male than female diabetic hypertensive subjects had a measured blood pressure that was less than published target values may contribute to sex differences in premature mortality in this high-risk population.

Most significant, however, was the improvement between 2003 and 2007 in the proportion of diabetic hypertensive enrollees in both races who met target blood pressure control values. These gains were substantial and clinically important for both African Americans and European Americans. This trend has important national public health implications with respect to reducing cardiovascular risk in this high-risk population. The reason for this trend of improved blood pressure control is not clear but may relate to improved dissemination or adoption of more stringent blood pressure control target values, improved patient-specific self-management and/or lifestyle choices, improved treatment intensification, improved access to medical care, and/or improved access to antihypertensive medications as a result of the national implementation of Medicare Part D coverage and generic reduced out-of-pocket cost (WalMart-type) formularies (21). These latter factors may be particularly important, considering that the largest improvement in the proportion of subjects with adequate blood pressure control occurred between 2006 and 2007 following the implementation of both Medicare Part D insurance and WalMart's reduced-cost formulary, both in 2006.

The present study has limitations, including its cross-sectional design, the use of self-reported disease and medication adherence measures, and the lack of data on proteinuria. Specifically, it is possible that for reasons we do not understand, hypertensive diabetic individuals recruited to the study during the latter years were more likely to have improved blood pressure control. The REGARDS study did adjust its sampling rates to meet race and sex recruitment goals; however, these analyses were stratified by race and controlled for sex, and, as such, race would not have played a role in any results and sex would not have played a role in the multivariable results. In addition, we attempted to adjust for other factors that could have changed over the study period (adherence, access to health insurance, etc.), and these factors failed to mediate the observed temporal improvement in control. Nevertheless, it is possible that factors that we did not account for in the analysis could have changed over time and confounded our results. However, these potential weaknesses are offset by the large and representative nature of this national population-based study and by the measurement of blood pressure in the home by a trained professional using a standardized protocol.

These data suggest substantial national improvements in blood pressure control in individuals with concurrent hypertension and diabetes, similar to that reported by McWilliams et al. (22). The etiology of these improvements is unknown, but they may reflect the broad dissemination of tighter blood pressure control targets and improving medication access. However, despite temporal trends suggesting improvement, significant racial disparities in successful blood pressure control persist among African Americans and may contribute to the increased risk for premature cardiovascular disease and stroke in this population. Additional interventions to facilitate the achievement of goal blood pressure values in all hypertensive diabetic subjects appear warranted.

The costs of publication of this article were defrayed in part by the payment of page charges. This article must therefore be hereby marked “advertisement” in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

The REGARDS research project is supported by a cooperative agreement (U01 NS041588) from the National Institute of Neurological Disorders and Stroke, National Institutes of Health, Department of Health and Human Services.

Parts of this article were presented in abstract form at the American Diabetes Association 69th Annual Scientific Sessions, New Orleans, LA, 6 June 2009 (citation: Cummings DM, Howard G, Howard V, Safford M, Prince V, Kissela B, Sands C, Lackland DT, Doherty L. Temporal progress yet persistent racial disparities in blood pressure control in diabetes: national results from the REasons for Geographic And Racial Differences in Stroke [REGARDS] study. Diabetes 2009;58[Suppl. 1]:A27–A28).

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Roccella
EJ
the National Heart, Lung, and Blood Institute Joint National Committee on Prevention, Detection, Evaluation, and Treatment of High Blood Pressure
:
The seventh report of the Joint National Committee on Prevention, Detection, Evaluation, and Treatment of High Blood Pressure: the JNC 7 report
.
JAMA
2003
;
289
:
2560
72
2.
Du
X
,
Cruickshank
K
,
McNamee
R
,
Saraee
M
,
Sourbutts
J
,
Summers
A
,
Roberts
N
,
Walton
E
,
Holmes
S
.
Case-control study of stroke and the quality of hypertension control in northwest England
.
BMJ
1997
;
314
:
272
276
3.
Lazarus
JM
,
Bourgoignie
JJ
,
Buckalew
VM
,
Greene
T
,
Levey
AS
,
Milas
NC
,
Paranandi
L
,
Peterson
JC
,
Porush
JG
,
Rauch
S
,
Soucie
JM
,
Stollar
C
.
Achievement and safety of a low blood pressure goal in chronic renal disease: the Modification of Diet in Renal Disease Study Group
.
Hypertension
1997
;
29
:
641
650
4.
UKPDS Study Group.
Tight blood pressure control and risk of macrovascular and microvascular complications in type 2 diabetes: UKPDS 38
.
BMJ
1998
;
317
:
703
713
5.
Zanchetti
A
,
Hansson
L
,
Ménard
J
,
Leonetti
G
,
Rahn
K
,
Warnold
I
,
Wedel
H
the HOT Study Group
:
Risk assessment and treatment benefit in intensively treated hypertensive patients of the Hypertension Optimal Treatment (HOT) study
.
J Hyperten
2001
;
19
:
819
825
6.
Effects of a fixed combination of perindopril and indapamide on macrovascular and microvascular outcomes in patients with type 2 diabetes mellitus: a randomised controlled trial
.
Lancet
2007
;
370
(
9590
):
829
40
7.
Howard
BV
,
Roman
MJ
,
Devereux
RB
,
Fleg
JL
,
Galloway
JM
,
Henderson
JA
,
Howard
WJ
,
Lee
ET
,
Mete
M
,
Poolaw
B
,
Ratner
RE
,
Russell
M
,
Silverman
A
,
Stylianou
M
,
Umans
JG
,
Wang
W
,
Weir
MR
,
Weissman
NJ
,
Wilson
C
,
Yeh
F
,
Zhu
J
.
Effect of lower targets for blood pressure and LDL cholesterol on atherosclerosis in diabetes: the SANDS randomized trial
.
JAMA
2008
;
299
:
1678
89
8.
ACCORD Study Group
Buse
JB
,
Bigger
JT
,
Byington
RP
,
Cooper
LS
,
Cushman
WC
,
Friedewald
WT
,
Genuth
S
,
Gerstein
HC
,
Ginsberg
HN
,
Goff
DC
Jr
,
Grimm
RH
Jr
,
Margolis
KL
,
Probstfield
JL
,
Simons-Morton
DG
,
Sullivan
MD
.
Action to Control Cardiovascular Risk in Diabetes (ACCORD) trial: design and methods
.
Am J Cardiol
2007
;
99
:
21i
33i
9.
The Action to Control Cardiovascular Risk in Diabetes Study Group.
Effects of intensive glucose lowering in type 2 diabetes
.
N Engl J Med
2008
;
358
:
2545
59
10.
Wang
YR
.
Lack of effect of guideline changes on hypertension control for patients with diabetes in the U.S., 1995–2005
.
Diabetes Care
2007
;
30
:
49
52
11.
Ong
KL
,
Cheung
BMY
,
Wong
LYF
,
Wat
NMS
,
Tan
KCB
,
Lam
KSL
.
Prevalence, treatment, and control of diagnosed diabetes in the U.S. National Health and Nutrition Examination Survey 1999–2004
.
Ann Epidemiol
2008
;
18
:
222
229
12.
Howard
VJ
,
Cushman
M
,
Pulley
L
,
Gomez
CR
,
Go
RC
,
Prineas
RJ
,
Graham
A
,
Moy
CS
,
Howard
G
.
The REasons for Geographic And Racial Differences in Stroke study: objectives and design
.
Neuroepidemiology
2005
;
25
:
135
43
13.
National Cholesterol Education Program Expert Panel
.
Third Report of the National Cholesterol Education Program (NCEP) Expert Panel: Detection, Evaluation, and Treatment of High Blood Cholesterol in Adults (Adult Treatment Panel III), Final Report
.
Bethesda, MD
,
National Cholesterol Education Program, National Heart, Lung, and Blood Institute, National Institutes of Health
,
September
2002
(
NIH publ. no. 02-5215
)
14.
Morisky
DE
,
Green
LW
,
Levine
DM
.
Concurrent and predictive validity of a self-reported measure of medication adherence
.
Med Care
1986
;
24
:
67
74
15.
The Diabetes Control and Complications Trial Research Group
.
The effect of intensive treatment of diabetes on the development and progression of long-term complications in insulin-dependent diabetes mellitus
.
N Engl J Med
1993
;
329
:
977
986
16.
Haffner
SM
,
Lehto
S
,
Ronnemaa
T
,
Pyorala
K
,
Laakso
M
.
Mortality from coronary heart disease in subjects with type 2 diabetes and in nondiabetic subjects with and without prior myocardial infarction
.
N Engl J Med
1998
;
339
:
229
234
17.
Buse
JB
,
Ginsberg
HN
,
Barkis
GL
,
Clark
NG
,
Costa
F
,
Eckel
R
,
Fonseca
V
,
Gerstein
HC
,
Grundy
S
,
Nesto
RW
,
Pignone
MP
,
Plutzky
J
,
Porte
D
,
Redberg
R
,
Stitzel
KF
,
Stone
NJ
.
Primary prevention of cardiovascular diseases in people with diabetes mellitus: a scientific statement from the American Heart Association and the American Diabetes Association
.
Diabetes Care
2007
;
30
:
162
172
18.
Bosworth
HB
,
Powers
B
,
Grubber
JM
,
Thorpe
CT
,
Olsen
MK
,
Orr
M
,
Oddone
EZ
.
Racial differences in blood pressure control: potential explanatory factors
.
J Gen Intern Med
2008
;
23
:
692
8
19.
Safford
MM
,
Halanych
JH
,
Lewis
CE
,
Levine
D
,
Houser
S
,
Howard
G
.
Understanding racial disparities in hypertension control: intensity of hypertension medication treatment in the REGARDS study
.
Ethnicity and Disease
2007
;
17
:
421
6
20.
Bosworth
HB
,
Dudley
T
,
Olsen
MK
,
Voils
CI
,
Powers
B
,
Goldstein
MK
,
Oddone
EZ
.
Racial differences in blood pressure control: potential explanatory factors
.
Am J Med
2006
;
119
:
70.e9
e15
21.
Wagner
TH
,
Heisler
M
,
Piette
JD
.
Prescription drug co-payments and cost-related medication underuse
.
Health Econ Policy Law
2008
;
3
:
51
67
22.
McWilliams
JM
,
Meara
E
,
Zaslavsky
AM
,
Ayanian
JZ
.
Differences in control of cardiovascular disease and diabetes by race, ethnicity, and education: U.S. trends from 1999 to 2006 and effects of Medicare coverage
.
Ann Intern Med
2009
;
150
:
505
515


text:
# User friendly software for math drawings

I am a math grad student and these days I am writing my thesis using $$\LaTeX$$. I have to use few 2D graphs that contain curves and shaded regions with few labeling. Can anybody please recommend me a user friendly free software that I can use to draw my figures?

• It shouldn't be hard with pstricks or tikz-pgf. You'll have a better integration to your document than with an external software. – Bernard Jun 25 at 21:39
• I always used Mathematica with the Save As... functionality. Save to pdf, then include the pdf in your document. – Adrian Keister Jun 25 at 21:41
• Geogebra has an intuitive GUI and it can export to tikz. – mlainz Jun 25 at 21:44
• Another vote for Mathematica, which I have used exclusively for several books, including the book I'm writing now. – David G. Stork Jun 25 at 21:48
• along with the other recommendations, take a quick look at draw.io, might be of some help. – ExtremeRaider Jun 26 at 3:41

• @EthanBolker: from my point of view, pstricks has the advantage to use LaTeX syntax, so the learning curve is not so steep. Furthermore, it is well documented, with many examples, and the documentation has a sensible size. – Bernard Jun 25 at 22:06


text:
# The 35th International Symposium on Lattice Field Theory

18-24 June 2017
Palacio de Congresos
Home > Timetable > Session details > Contribution details

Andalucía II

# More on heavy tetraquarks in lattice QCD at almost physical pion mass

## Speakers

• Dr. Anthony FRANCIS

## Content

We report on our progress with the ongoing study of exotic, heavy tetraquark states, $qq’ \bar Q\bar Q’$. With regards to the previously reported $\bar Q\bar Q’=\bar b\bar b$ case, we have significantly increased our statistics by computing and saving more propagators on publicly available dynamical $n_f =2+1$ Wilson-Clover gauge configurations, generated by the PACS-CS collaboration, with pion masses $\simeq$164, 299 and 415 MeV. We extend our previous analysis to heavy quark components containing heavier than physical bottom quarks $\bar Q\bar Q’=\bar b’\bar b’$, charm and bottom quarks $\bar c\bar b$ and also only charm quarks $\bar c\bar c$. Throughout we employ NRQCD and Fermilab heavy quarks for the heavier than bottom, bottom and charm quarks. Using our previously established diquark-antidiquark and meson-meson operator basis we comment in particular on the dependence of the binding energy on the mass of the heavy quark component $\bar Q\bar Q$, with heavy quarks ranging from $m_Q=0.7…6.3\cdot m_b$. In the heavy flavor non-degenerate case, $\bar Q\bar Q’$, and especially for the tetraquark channel $ud\bar c\bar b$, we extend our work to utilize a $3\times 3$ GEVP to study the ground and threshold states, thereby enabling an identification also of finite volume effects in this channel. Finally, we present some initial work on the $\bar Q\bar Q’=\bar c\bar c$ system where a much larger operator basis is available in comparison to flavor combinations with NRQCD quarks.


text:
## Category Archives: miscellaneous

### The personality of Leonhard Euler

The portrait of Euler that emerges from his publications and letters is that of a genial man of simple tastes and conventional religious faith. He was even wealthy, at least in the second half of his life, but ostentation was not part of his lifestyle. His memory was prodigious, and contemporary accounts have emphasized this. He would delight relatives, friends, and acquaintances with a literal recitation of any song from Virgil’s Aenesis, and he would remember minutes of Academy meetings years after they were held. He was not given to envy, and when someone made an advance on his work his happiness was genuine. For example, when he learnt of Lagrange’s improvements on his work on elliptic integrals, he wrote to him that his admiration knew no bounds and then proceeded to improve upon Lagrange!

But, what is most characteristic of his work is its clarity and openness. He never tries to hide the difficulties from the reader. This is in stark contrast to Newton, who was prone to hide his methods in obscure anagrams, and even from his successor, Gauss, who very often erased his steps to present a monolithic proof that was seldom illuminating. In Euler’s writings there are no comments on how profound his results are, and in his papers one can follow his ideas step by step with the greatest of ease. Nor was he chary of giving credit to others; his willingness to share his summation formula with Maclaurin, his proper citations to Fuguano when he started his work on algebraic integrals, his open admiration for Lagrange when the latter improved on his work in calculus of variations are all instances of his serene outlook. One can only contrast this with Gauss’s reaction to Bolyai’s discovery of non-Euclidean postulates. Euler was secure in his knowledge of what he had achieved but never insisted that he should be the only one on top of the mountain.

Perhaps, the most astounding aspect of his scientific opus is its universality. He worked on everything that had any bearing on mathematics. For instance, his early training under Johann Bernoulli did not include number theory; nevertheless, within a couple of years after reaching St. Petersburg he was deeply immersed in it, recreating the entire corpus of Fermat’s work in that area and then moving well beyond him. His founding of graph theory as a separate discipline, his excursions in what we call combinatorial topology, his intuition that suggested to him the idea of exploring multizeta values are all examples of a mind that did not have any artificial boundaries. He had no preferences about which branch of mathematics was dear to him. To him, they were all filled with splendour, or Herrlichkeit, to use his own favourite word.

Hilbert and Poincare were perhaps last of the universalists of modern era. Already von Neumann had remarked that it would be difficult even to have a general understanding of more than a third of the mathematicians of his time. With the explosive growth of mathematics in the twentieth century we may never see again the great universalists. But who is to say what is and is not possible for the human mind?

It is impossible to read Euler and not fall under his spell. He is to mathematics what Shakespeare is to literature and Mozart to music: universal and sui generis.

Reference:

Euler Through Time: A New Look at Old Themes by V S Varadarajan:

Hindustan Book Agency;

https://www.amazon.in/Euler-Through-Time-Look-Themes/dp/9380250592/ref=sr_1_1?keywords=Euler+Through+Time&qid=1568316624&s=books&sr=1-1

### Stretching is a good exercise but…

Stretching is a good exercise, but stretching the mind through math is even better !!

Ha…ha…ha…LOL 🙂

Nalin Pithwa

PS: An object does not change topologically if it is “stretched”…Ha, ha, ha…LOL 🙂

### You and your research ( You and your studies) : By Richard Hamming, AT and T, Bell Labs mathematician;

Although the title is grand (and quite aptly so)…the reality is that it can be applied to serious studies for IITJEE entrance, CMI entrance, highly competitive math olympiads, and also competitive coding contests…in fact, to various aspects of student life and various professional lifes…

https://www.cs.virginia.edu/~robins/YouAndYourResearch.html

### What motivated Einstein?

The most beautiful thing that we can experience is the mysterious. It is the source of all true art and sciences.

— Albert Einstein, in What I believe, 1930.

### E. T. Bell’s Men of Mathematics, John Nash, Jr., genius mathematician, Nobel Laureate and Abel Laureate; and Albert Einstein

(From A Beautiful Mind by Sylvia Nasar)

The first bite of mathematical apple probably occurred when Nash at around age thirteen or fourteen read E. T. Bell’s extra ordinary book Men of Mathematics — an experience he alludes to in his autobiographical essay (of Nobel Prize, Economics) Bell’s book, which was published in 1937, would have given Nash the first glimpse of real mathematics, a heady realm of symbols and mysteries entirely unconnected to the seemingly arbitrary and dull rules of arithmetic and geometry taught in school or even in the entertaining but ultimately trivial calculations that Nash carried out in the course of chemistry and electrical experiments.

Men of Mathematics consists of lively — and, as it turns out, not entirely accurate — biographical sketches. Its flamboyant author, a professor of mathematics at California Institute of Technology, declared himself disgusted with “the ludicrous untruth of the traditional portrait of the mathematician” as a “slovenly dreamer totally devoid of common sense.” He assured his readers that the great mathematicians of history were an exceptionally virile and even adventuresome breed. He sought to prove his point with vivid accounts of infant precocity, monstrously insensitive educational authorities, crushing poverty, jealous rivals, love affairs, royal patronage, and many varieties of early death, including some resulting from duels. He even went so far in defending mathematicians as to answer the question : “How many of the great mathematicians have been perverts?” None, was his answer. ‘Some lived celibate lives, usually on account of economic disabilities, but the majority were happily married…The only mathematician discussed here whose life might offer something of interest to a Freudian is Pascal.’ The book became a bestseller as soon as it appeared.

What makes Bell’s account not merely charming, but intellectually seductive, are his lively descriptions of mathematical problems that inspired his subjects when they were young, and his breezy assurance that there were still deep and beautiful problems that could be solved by amateurs, boys of fourteen, to be specific. It was Bell’s essay on Fermat, one of the greatest mathematicians of all time, but a perfectly conventional seventeenth century French magistrate, whose life was “quiet, laborious and uneventful,” that caught Nash’s eye. The main interest of Fermat, who shares the credit for inventing calculus with Newton and analytic geometry with Descartes, was number theory — “the higher arithmetic.” Number theory, investigates the natural relationships of those common whole numbers 1, 2, 3, 4, 5…which we utter almost as soon as we learn to talk.

For Nash, proving a theorem known as Fermat’s (Little) Theorem about prime numbers, those mysterious integers that have no divisors besides themselves and one produced an epiphany of sorts. Often mathematical geniuses, Albert Einstein and Bertrand Russell among them recount similar revelatory experiences in early adolescence. Einstein recalled the “wonder” of his first encounter with Euclid at age twelve:

“Here were assertions, as for example the intersection of three altitudes of a triangle at one point which — though by no means evident — could nevertheless be proved with such certainty that any doubt appeared to be out of the question. This lucidity and certainty made an indescribable impression on me.”

Nash does not describe his feelings when he succeeded in devising a proof for Fermat’s assertion that if n is any whole number and p any prime number, then n multiplied by itself p times minus p is divisible by p. But, he notes the fact in his autobiographical essay, and his emphasis on this concrete result of his initial encounter with Fermat suggests that the thrill of discovering and exercising his own intellectual powers — as much as any sense of wonder inspired by hitherto unsuspected patterns and meanings — was what made this moment such a memorable one. That thrill has been decisive for many a future mathematician. Bell describes how success in solving a problem posed by Fermat led Carl Friedrich Gauss, the renowned German mathematician, to choose between two careers for which he was similarly talented. ‘It was this discovery …which induced the young man to choose mathematics instead of philology as his life work.”…

For those readers who are interested:

1. Who wants to be a mathematician:

http://www.ams.org/publicoutreach/students/wwtbam/wwtbam

2. Resonance Journal (India):

https://www.ias.ac.in/Journals/Resonance_%E2%80%93_Journal_of_Science_Education/

3. Ramanujan School of Mathematics; Super30 of Prof Anand Kumar:

http://www.super30.org/rsm.html

Cheers,

Nalin Pithwa

### From Passive to Active Learning: India Today: Jamshed Bharucha: Aug 19 2019

(By Jamshed Bharucha; Vice Chancellor, SRM Amravati University)

https://www.indiatoday.in/magazine/education/story/20190819-from-passive-to-active-learning-1578655-2019-08-09

### Some practical uses of maths

1. MPEG 4, audio/video/speech recognition/speaker identification/face recognition/HDTV and mp3 —- all these use logarithms and trigonometry. The special jargon is — Fourier Series, and Fourier Transforms.
2. In finance, compound interest is calculated by using a power function; the inverse problem of finding the duration of deposit is calculated using logarithms.
3. All (digital) phones are touch phones and they use DTMF (Dual Tone Multi Frequency) standard — implemented using sines and cosines.
4. Quadratic equations are used to design/model/develop certain kind of electronic amplifiers.
5. Probability theory is used in computer networks, routing of telephone calls, and also in Wall Street — stock market !!
6. There are ways to compute the numerical value of the irrational number $\pi$ up to a million digits and these ways are used to test the efficiency and efficacy of supercomputers.
7. Quadratic equations are used to study projectile motion (or to put it playfully, suppose we throw a pebble at a certain angle from horizontal ground, (angle less than 90 degrees (which would mean vertically up)) — the projectile is subject only to the force of gravity of the earth — the path or curve or trajectory of the projectile is a parabole, which is characterized by a quadratic equation. This can be easily proved using laws of straight line motion in two dimensional using resolution of vectors.

More later,

Nalin Pithwa.

### The importance of lines and slopes

1. Light travels along straight lines. In fact, the shortest distance between any two points is the path taken by a light wave to travel from the initial point to the final point. In other words, it is a straight line. (A slight detour: using this elementary fact, can you prove the triangle inequality?)
2. Bodies falling from rest in a planet’s gravitational field do so in a straight line.
3. Bodies coasting under their own momentum (like a hockey puck gliding across the ice) do so in a straight line. (Think of Newton’s First Law of Motion).

So we often use the equations of lines (called linear equations) to study such motions.

Many important quantities are related by linear equations. Once we know that a relationship between two variables is linear, we can find it from any two pairs of corresponding values just as we find the equation of a line from the coordinates of two points.

Slope is important because it gives us a way to say how steep something is (roadbeds, roofs, stairs, banking of railway tracks). The notion of a slope also enables us to describe how rapidly things are changing. (To philosophize, everything in the observable universe is changing). For this reason, slope plays an important role in calculus.

More later,

Nalin Pithwa.

PS: Ref: Calculus and Analytic Geometry by G B Thomas and Finney; or any other book on calculus.

PS: I strongly recommend the Thomas and Finney book : You can get it from Amazon India or Flipkart:

https://www.amazon.in/Thomas-Calculus-George-B/dp/9353060419/ref=sr_1_1?crid=36S3685TG7OYF&keywords=thomas+calculus&qid=1561503390&s=books&sprefix=Thomas+%2Caps%2C259&sr=1-1

or Flipkart:

https://www.flipkart.com/thomas-calculus-1/p/itmebug5kzrnttfj?pid=9789332547278&lid=LSTBOK9789332547278CHN4GH&marketplace=FLIPKART&srno=s_1_23&otracker=AS_Query_OrganicAutoSuggest_2_9&otracker1=AS_Query_OrganicAutoSuggest_2_9&fm=SEARCH&iid=fdc8327b-756c-4f6d-aa10-b45acc900e12.9789332547278.SEARCH&ppt=sp&ppn=sp&ssid=uz7zckp71c0000001561503474614&qH=2488f76736a10369


text:
# Algorithm for building the extended universal Grubner Base

Authors:
Abstract:

The paper is devoted to the description of an algorithm for building the extended universal Grubner base (EUGB) of a polynomial ideal. This algoritm is based on the relationship between multi-dimensional Young diagrams and Grubner bases of polynomial ideals. The algorithm starts with construction of the maximal linearly- independent monomial sets (MLIMS).


text:
This is a walkthrough of training CLIP by OpenAI. CLIP was designed to put both images and text into a new projected space such that they can map to each other by simply looking at dot products.

Traditionally training sets like imagenet only allowed you to map images to a single class (and hence one word). This method allows you to map text to images, but can also be used to map images to text if the need arises.

This particular blog however is specifically how we managed to train this on colab GPUs using huggingface transformers and pytorch lightning.

A Working version of this code can be found on kaggle..

## Acknowledgement

Kudos to the following CLIP tutorial in the keras documentation.

The important thing to notice about the constants is the embedding dim. We will project the output of a resnet and transformers into 512 dimensional space.

EMBED_DIM = 512
TRANSFORMER_EMBED_DIM = 768
MAX_LEN = 32 # Maximum length of text
TEXT_MODEL = "distilbert-base-multilingual-cased"

EPOCHS = 5
BATCH_SIZE = 64

## Data

We download the coco dataset which contains 5 captions per image and has roughly 82k images. We take 20% of it to be our validation set.

Considering that the image backbone is trained using imagenet, we normalise it using the imagenet stats as shown in the transforms normalize step. We also resize the image to 128x128 to make sure it trains in reasonable time.

class Tokenizer:
def __init__(self, tokenizer: BertTokenizer) -> None:
self.tokenizer = tokenizer

def __call__(self, x: str) -> AutoTokenizer:
return self.tokenizer(
)

def decode(self, x: Dict[str, torch.LongTensor]):
return [self.tokenizer.decode(sentence[:sentence_len]) for sentence, sentence_len in

tokenizer = Tokenizer(AutoTokenizer.from_pretrained(TEXT_MODEL))

img = inv_tfm(img)
plt.imshow(np.rot90(img.transpose(0, 2), 3))
plt.title(tokenizer.decode(target)[0])
plt.show()

train_len = int(0.8*len(cap))
train_data, valid_data = random_split(cap, [train_len, len(cap) - train_len])
train_dl = DataLoader(train_data, BATCH_SIZE, pin_memory=True, shuffle=True, num_workers=4, drop_last=True)
valid_dl = DataLoader(valid_data, BATCH_SIZE, pin_memory=True, shuffle=False, num_workers=4, drop_last=False)

/usr/local/lib/python3.7/dist-packages/torch/utils/data/dataloader.py:478: UserWarning: This DataLoader will create 4 worker processes in total. Our suggested max number of worker in current system is 2, which is smaller than what this DataLoader is going to create. Please be aware that excessive worker creation might get DataLoader running slow or even freeze, lower the worker number to avoid potential slowness/freeze if necessary.
cpuset_checked))

## Model

There are two main models, the VisionEncoder and the TextEncoder which have resnet18 and distilbert as backbones. In order to make it multi-lingual, we simply choose the distilbert-multilingual model and that's it! No need to specifically train on non-english words as you will soon see.

The Projection module, takes the embeddings from vision and text encoders and projects them into 512 dimensional space.

Two things to note:

1. We have frozen both the text and vision encoder backbones and do not retrain their weights at all.
2. For both encoders the final output is normalised to be of unit length.
class Projection(nn.Module):
def __init__(self, d_in: int, d_out: int, p: float=0.5) -> None:
super().__init__()
self.linear1 = nn.Linear(d_in, d_out, bias=False)
self.linear2 = nn.Linear(d_out, d_out, bias=False)
self.layer_norm = nn.LayerNorm(d_out)
self.drop = nn.Dropout(p)

def forward(self, x: torch.Tensor) -> torch.Tensor:
embed1 = self.linear1(x)
embed2 = self.drop(self.linear2(F.gelu(embed1)))
embeds = self.layer_norm(embed1 + embed2)
return embeds

class VisionEncoder(nn.Module):
def __init__(self, d_out: int) -> None:
super().__init__()
base = models.resnet34(pretrained=True)
d_in = base.fc.in_features
base.fc = nn.Identity()
self.base = base
self.projection = Projection(d_in, d_out)
for p in self.base.parameters():

def forward(self, x):
projected_vec = self.projection(self.base(x))
projection_len = torch.norm(projected_vec, dim=-1, keepdim=True)
return projected_vec / projection_len

class TextEncoder(nn.Module):
def __init__(self, d_out: int) -> None:
super().__init__()
self.base = AutoModel.from_pretrained(TEXT_MODEL)
self.projection = Projection(TRANSFORMER_EMBED_DIM, d_out)
for p in self.base.parameters():

def forward(self, x):
out = self.base(**x)[0]
out = out[:, 0, :]  # get CLS token output
projected_vec = self.projection(out)
projection_len = torch.norm(projected_vec, dim=-1, keepdim=True)
return projected_vec / projection_len

## CLIP loss function

For someone like me who hasn't played around with contrastive loss, this was the most interesting part.

We know that we want the vectors of the corresponding image and the text to line up. Which means that the dot product has to be as close to one as possible. For everything else we need to push it towards 0.

Therfore for a given caption, we take the softmax of the dot products across all images, and then take cross entropy loss. Similarly for a given image, we repeat the process across all captions. We average these two losses.

In terms of which element is the true positive within a batch, remember that we are sending image, caption pairs already lined up. Therefore we want all the diagonal elements to line up while all off-diagonal elements we want to push towards zero.

def contrastive_loss(logits, dim):
neg_ce = torch.diag(F.log_softmax(logits, dim=dim))
return -neg_ce.mean()

def clip_loss(similarity: torch.Tensor) -> torch.Tensor:
caption_loss = contrastive_loss(similarity, dim=0)
image_loss = contrastive_loss(similarity, dim=1)
return (caption_loss + image_loss) / 2.0

def metrics(similarity: torch.Tensor) -> Tuple[torch.Tensor, torch.Tensor]:
y = torch.arange(len(similarity)).to(similarity.device)
img2cap_match_idx = similarity.argmax(dim=1)
cap2img_match_idx = similarity.argmax(dim=0)

img_acc = (img2cap_match_idx == y).float().mean()
cap_acc = (cap2img_match_idx == y).float().mean()

return img_acc, cap_acc

## Model

If you haven't used pytorch lightning before, the benefit is that you do not need to stress about which device to put it in, remembering to zero the optimizer etc. All of that is taken care of. Just simply specify the training and validation steps, along with the optimizer and you are good to go.

The other benefit that I really like is logging. You just need to write self.log("name", metric_to_track) and it will log to tensorboard by default, or any other kind of logger for that matter.

class Model(pl.LightningModule):
def __init__(self,
lr: float = 1e-3
) -> None:
super().__init__()
self.vision_encoder = VisionEncoder(EMBED_DIM)
self.caption_encoder = TextEncoder(EMBED_DIM)
self.tokenizer = Tokenizer(AutoTokenizer.from_pretrained(TEXT_MODEL))
self.lr = lr

def common_step(self, batch: Tuple[torch.Tensor, List[str]]) -> torch.Tensor:
images, text = batch
text_dev = {k: v.to(self.device) for k, v in self.tokenizer(text).items()}

image_embed = self.vision_encoder(images)
caption_embed = self.caption_encoder(text)
similarity = caption_embed @ image_embed.T

loss = clip_loss(similarity)
img_acc, cap_acc = metrics(similarity)
return loss, img_acc, cap_acc

def training_step(
self, batch: Tuple[torch.Tensor, List[str]], *args: list
) -> torch.Tensor:
loss, img_acc, cap_acc = self.common_step(batch)
self.log("training_loss", loss, on_step=True)
self.log("training_img_acc", img_acc, on_step=True, prog_bar=True)
self.log("training_cap_acc", cap_acc, on_step=True, prog_bar=True)
return loss

def validation_step(
self, batch: Tuple[torch.Tensor, List[str]], *args: list
) -> torch.Tensor:
loss, img_acc, cap_acc = self.common_step(batch)
self.log("validation_loss", loss, on_step=True)
self.log("validation_img_acc", img_acc, on_step=True, prog_bar=True)
self.log("validation_cap_acc", cap_acc, on_step=True, prog_bar=True)
return loss

def configure_optimizers(self) -> torch.optim.Optimizer:
vision_params = {"params": self.vision_encoder.projection.parameters(), "lr": self.lr}
caption_params = {"params": self.caption_encoder.projection.parameters() , "lr": self.lr}

## Train

Training is straight forward as show in the five lines below. Using 16 bit precision almost halved the training time from 16 minutes to 9 minutes per epoch. Notice how easy it was to add half precision training and gradient clipping.

Also one thing to note is that I could not get this working on TPUs so if anyone knows what I need to adjust, please let me know. Setting tpu_cores=8 just did not work.

model = Model(1e-3)
trainer = pl.Trainer(
max_epochs= 1,
gpus=torch.cuda.device_count(),
# tpu_cores=1,
precision=16
)
trainer.fit(model, train_dl, valid_dl) #

Run the following cell if you wish to see the logs in tensorboard. But here's a screenshot I took:

%reload_ext tensorboard
%tensorboard --logdir ./lightning_logs/

## Results

I will compare the text embeddings of the first batch (in the validation set) to all the images of the validation set by taking the dot product between them.

similarity = caption_embed @ image_embed.T
val, closest = similarity.topk(5, dim=-1)
similarity.shape

torch.Size([64, 16557])

draw_result(i, similarity_matrix) is a convenience function that takes the i-th caption and the similarity matrix, and plots the five closest images, along with the true image. The similarity between the caption and the image is shown in the title. The caption is printed first.

The histogram show the similarity of the caption to all images as a histogram.

draw_result(2, similarity)

A baseball player in the outfield with his hands up, standing next to a team mascot.

draw_result(1, similarity)

A watch and clock repair shop window with clocks on display.

draw_result(10, similarity)

A person on a skateboard on the ground.

Below is the google translted version of one of the captions.

English caption: "A zebra standing up with it's head down and eating grass on the dirt ground.", tranlated into Spanish:

text = "Una cebra de pie con la cabeza gacha y comiendo hierba en el suelo de tierra."
text_dev = {k: v.to(device) for k, v in tokenizer(text).items()}
caption_embed_text = caption_encoder(text_dev)

similarity_text = caption_embed_text @ image_embed.T

draw_result_single_query(10, similarity_text)

Skateboarder conducting a trick with bicycles in the background.

Again a translated version, this time to french. English caption: "A laptop is displayed on a small wooden platform."

text = "Un ordinateur portable est affiché sur une petite plate-forme en bois."
text_dev = {k: v.to(device) for k, v in tokenizer(text).items()}
caption_embed_text = caption_encoder(text_dev)

similarity_text = caption_embed_text @ image_embed.T

draw_result_single_query(3, similarity_text)

Laptop computer on a small table on the side of a bed

The russian translation below is doing terrible though, so its clearly not bullet proof. Or perhaps I need to train for a bit longer. English caption: "A shop filled with different kinds of clocks.

text = "Магазин с разными часами"
text_dev = {k: v.to(device) for k, v in tokenizer(text).items()}