text:
## Ledoux, James

Compute Distance To:
 Author ID: ledoux.james Published as: Ledoux, James
 Documents Indexed: 31 Publications since 1993, including 1 Book Co-Authors: 13 Co-Authors with 22 Joint Publications 172 Co-Co-Authors
all top 5

### Co-Authors

 9 single-authored 12 Hervé, Loïc 3 Truffet, Laurent 2 Ferré, Déborah 2 Rubino, Gerardo 1 Ahmane, Mourad 1 Brushe, Gary D. 1 Gaudoin, Olivier 1 Gravereaux, Jean-Bernard 1 Gurvits, Leonid 1 Leguesdron, Patrice 1 Patilea, Valentin 1 Sericola, Bruno 1 White, Langford B.
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### Serials

 7 Journal of Applied Probability 6 Statistics & Probability Letters 5 Advances in Applied Probability 2 Communications in Statistics. Theory and Methods 2 Linear Algebra and its Applications 1 Operations Research Letters 1 Communications in Statistics. Stochastic Models 1 Stochastic Processes and their Applications 1 Annales de l’Institut Henri Poincaré. Probabilités et Statistiques 1 Electronic Communications in Probability 1 Bernoulli 1 Methodology and Computing in Applied Probability 1 Collection Méthodes Stochastiques Appliquées
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### Fields

 28 Probability theory and stochastic processes (60-XX) 7 Systems theory; control (93-XX) 5 Operator theory (47-XX) 3 Linear and multilinear algebra; matrix theory (15-XX) 3 Statistics (62-XX) 3 Operations research, mathematical programming (90-XX) 2 Computer science (68-XX) 1 Associative rings and algebras (16-XX) 1 Dynamical systems and ergodic theory (37-XX) 1 Numerical analysis (65-XX)

### Citations contained in zbMATH Open

23 Publications have been cited 83 times in 63 Documents Cited by Year
Markov property for a function of a Markov chain: A linear algebra approach. Zbl 1077.15017
Gurvits, Leonid; Ledoux, James
2005
Approximating Markov chains and $$V$$-geometric ergodicity via weak perturbation theory. Zbl 1305.60059
Hervé, Loïc; Ledoux, James
2014
Limit theorems for stationary Markov processes with $$L^{2}$$-spectral gap. Zbl 1245.60068
Ferré, Déborah; Hervé, Loïc; Ledoux, James
2012
On weak lumpability of denumerable Markov chains. Zbl 0843.60060
Ledoux, James
1995
Regular perturbation of $$V$$-geometrically ergodic Markov chains. Zbl 1276.60075
Ferré, Déborah; Hervé, Loïc; Ledoux, James
2013
A uniform Berry-Esseen theorem on $$M$$-estimators for geometrically ergodic Markov chains. Zbl 1279.60089
Hervé, Loïc; Ledoux, James; Patilea, Valentin
2012
A geometric invariant in weak lumpability of finite Markov chains. Zbl 0899.60061
Ledoux, James
1997
Poisson approximation for some point processes in reliability. Zbl 1052.60038
Gravereaux, Jean-Bernard; Ledoux, James
2004
Linear dynamics for the state vector of Markov chain functions. Zbl 1076.60058
Ledoux, James
2004
Recursive filters for partially observable finite Markov chains. Zbl 1086.60046
Ledoux, James
2005
Simple formulae for counting processes in reliability models. Zbl 0904.60057
Ledoux, James; Rubino, Gerardo
1997
Stochastic modeling in software reliability. (Modélisation aléatoire en fiabilité des logiciels.) Zbl 1147.68014
Gaudoin, Olivier; Ledoux, James
2007
Markovian bounds on functions of finite Markov chains. Zbl 0995.60069
Ledoux, James; Truffet, Laurent
2001
Exact aggregation of absorbing Markov processes using the quasi- stationary distribution. Zbl 0812.60055
Ledoux, James; Rubino, Gerardo; Sericola, Bruno
1994
Spectral analysis of Markov kernels and application to the convergence rate of discrete random walks. Zbl 1305.60060
Hervé, Loïc; Ledoux, James
2014
Computable bounds of an $$\ell^2$$-spectral gap for discrete Markov chains with band transition matrices. Zbl 1351.60092
Hervé, Loïc; Ledoux, James
2016
A local limit theorem for densities of the additive component of a finite Markov additive process. Zbl 1296.60056
Hervé, Loïc; Ledoux, James
2013
On the relationships between lumpability and filtering of finite stochastic systems. Zbl 1147.93412
Ledoux, James; White, Langford B.; Brushe, Gary D.
2008
Weak lumpability and pseudo-stationarity of finite Markov chains. Zbl 0958.60065
Ledoux, James; Leguesdron, Patrice
2000
Filtering and the EM-algorithm for the Markovian arrival process. Zbl 1130.93054
Ledoux, James
2007
A necessary condition for weak lumpability in finite Markov processes. Zbl 0780.90108
Ledoux, James
1993
Comparison and aggregation of max-plus linear systems. Zbl 1052.15014
Ledoux, James; Truffet, Laurent
2004
Positive invariance of polyhedrons and comparison of Markov reward models with different state spaces. Zbl 1132.93333
Ahmane, Mourad; Ledoux, James; Truffet, Laurent
2006
Computable bounds of an $$\ell^2$$-spectral gap for discrete Markov chains with band transition matrices. Zbl 1351.60092
Hervé, Loïc; Ledoux, James
2016
Approximating Markov chains and $$V$$-geometric ergodicity via weak perturbation theory. Zbl 1305.60059
Hervé, Loïc; Ledoux, James
2014
Spectral analysis of Markov kernels and application to the convergence rate of discrete random walks. Zbl 1305.60060
Hervé, Loïc; Ledoux, James
2014
Regular perturbation of $$V$$-geometrically ergodic Markov chains. Zbl 1276.60075
Ferré, Déborah; Hervé, Loïc; Ledoux, James
2013
A local limit theorem for densities of the additive component of a finite Markov additive process. Zbl 1296.60056
Hervé, Loïc; Ledoux, James
2013
Limit theorems for stationary Markov processes with $$L^{2}$$-spectral gap. Zbl 1245.60068
Ferré, Déborah; Hervé, Loïc; Ledoux, James
2012
A uniform Berry-Esseen theorem on $$M$$-estimators for geometrically ergodic Markov chains. Zbl 1279.60089
Hervé, Loïc; Ledoux, James; Patilea, Valentin
2012
On the relationships between lumpability and filtering of finite stochastic systems. Zbl 1147.93412
Ledoux, James; White, Langford B.; Brushe, Gary D.
2008
Stochastic modeling in software reliability. (Modélisation aléatoire en fiabilité des logiciels.) Zbl 1147.68014
Gaudoin, Olivier; Ledoux, James
2007
Filtering and the EM-algorithm for the Markovian arrival process. Zbl 1130.93054
Ledoux, James
2007
Positive invariance of polyhedrons and comparison of Markov reward models with different state spaces. Zbl 1132.93333
Ahmane, Mourad; Ledoux, James; Truffet, Laurent
2006
Markov property for a function of a Markov chain: A linear algebra approach. Zbl 1077.15017
Gurvits, Leonid; Ledoux, James
2005
Recursive filters for partially observable finite Markov chains. Zbl 1086.60046
Ledoux, James
2005
Poisson approximation for some point processes in reliability. Zbl 1052.60038
Gravereaux, Jean-Bernard; Ledoux, James
2004
Linear dynamics for the state vector of Markov chain functions. Zbl 1076.60058
Ledoux, James
2004
Comparison and aggregation of max-plus linear systems. Zbl 1052.15014
Ledoux, James; Truffet, Laurent
2004
Markovian bounds on functions of finite Markov chains. Zbl 0995.60069
Ledoux, James; Truffet, Laurent
2001
Weak lumpability and pseudo-stationarity of finite Markov chains. Zbl 0958.60065
Ledoux, James; Leguesdron, Patrice
2000
A geometric invariant in weak lumpability of finite Markov chains. Zbl 0899.60061
Ledoux, James
1997
Simple formulae for counting processes in reliability models. Zbl 0904.60057
Ledoux, James; Rubino, Gerardo
1997
On weak lumpability of denumerable Markov chains. Zbl 0843.60060
Ledoux, James
1995
Exact aggregation of absorbing Markov processes using the quasi- stationary distribution. Zbl 0812.60055
Ledoux, James; Rubino, Gerardo; Sericola, Bruno
1994
A necessary condition for weak lumpability in finite Markov processes. Zbl 0780.90108
Ledoux, James
1993
all top 5


text:
# Tag Info

18

dat = {0.71, 0.685, 0.16, 0.82, 0.73, 0.44, 0.89, 0.02, 0.47, 0.65}; Module[{t = 0}, Split[dat, (t += #) <= 1 || (t = 0) &] ] {{0.71, 0.685}, {0.16, 0.82, 0.73}, {0.44, 0.89}, {0.02, 0.47, 0.65}} Credit to Simon Woods for getting me to think about using Or in applications like this. Performance I decided to make an attempt at a higher ...

6

The following two expressions are equivalent. Table[RandomReal[], {10^8}]; // AbsoluteTiming {7.99593, Null} RandomReal[1., 10^8]; // AbsoluteTiming {1.20604, Null} The second expression shows the advantage of RandomReal over Random. Edit Another consideration is the generator used. For example, when the Mersenne twister is specified, there is not ...

5

f[x_, y_] := Module[{new}, If[Total[new = Append[x, y]] >= 1, Sow[new]; {}, new]] Reap[Fold[f, {}, dat]][[2, 1]]

5

(1) Convolution with Plus and Times can be done via FFT. (2) The overall speed complexity cannot be less than the size-of-result complexity. Point (1) might help to explain why the standard ListConvolve is fast under most circumstances. Point (2) on the examples in this post should help to explain why LC2 is likely to be slow. To make this clear we can ...

5

InternalPartitionRagged uses Accumulate internally to generate a list of positions from the sub-list lengths, then MapThread and Take to extract the corresponding elements from the array. You can check the internal definition with Needs["GeneralUtilities"]; PrintDefinitions[InternalPartitionRagged] The reason for pointing this out is that answers which ...

4

Here's my take at making a function as fast as possible. main = Module[{idxs = sub[Accumulate@#]}, InternalPartitionRagged[#, idxs]] &; sub = Compile[{{list, _Real, 1}}, Block[{i, l = Length[list], ref = 1., bag = InternalBag[{0}]}, For[i = 1, i <= l, i++, If[list[[i]] >= ref || i == l, InternalStuffBag[bag, i]; ref = ...

2

Borrowing heavily from other answers here, but I wanted to do as much as I could inside Compile. On my machine, this is a bit faster than LLIAMnYP's main: runsComp = Compile[{{list, _Real, 1}}, Block[{ans = ConstantArray[{0, 0}, Length@list + 1], t = 0., j = 1, len = Length[list]}, Do[(t += list[[i]]) <= 1 || (ans[[j + 1]] = {ans[[j, 2]] + 1, i}; ...

Only top voted, non community-wiki answers of a minimum length are eligible

format: plain


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# 数学代写|常微分方程代写ordinary differential equation代考|MATH3331

## 数学代写|常微分方程代写ordinary differential equation代考|Linear systems with singularities

Now we want to extend the results from the previous section to linear systems
$$w^{\prime}=A(z) w, \quad w\left(z_0\right)=w_0, \quad z, z_0 \in \Omega \subseteq \mathbb{C},$$
where $A(z)$ is a matrix whose coefficients are analytic in $\Omega$.

As in the real case one can show that one can always extend solutions. However, extensions along different paths might give different solutions in general, as we have seen in example (4.8). These problems do not arise if $\Omega$ is simply connected.

Theorem 4.5. Suppose $w^{\prime}=A(z) w+b(z)$ is linear, where $A: \Omega \rightarrow \mathbb{C}^{n \times n}$ and $b: \Omega \rightarrow \mathbb{C}^n$ are analytic in a simply connected domain $\Omega \subseteq \mathbb{C}$. Then for every $z_0 \in \Omega$ the corresponding initial value problem has a unique solution defined on all of $\Omega$.

In particular, the power series for every solution will converge in the largest disc centered at $z_0$ and contained in $\Omega$.

Proof. If $\Omega$ is a disc centered at $z_0$ the result follows as in Corollary $2.6$. For general $\Omega$, pick $z \in \Omega$ and let $\gamma:[0,1] \rightarrow \Omega$ be a path from $z_0$ to $z$. Around each point $\gamma(t)$ we have a solution in a ball with radius independent of the initial condition and of $t \in[0,1]$. So we can define the value of $w(z)$ by analytic continuation along the path $\gamma$. Since $\Omega$ is simply connected, this value is uniquely defined by the monodromy theorem.

This result has the important consequence that a solution of a linear equation can have singularities (poles, essential singularities, or branch points) only at the points where the coefficients have isolated singularities. That is, the singularities are fixed and do not depend on the initial condition. On the other hand, nonlinear equations will in general have movable singularities, as the simple example
$$w^{\prime}=-w^2$$
whose general solution is
$$w(z)=\frac{1}{z-z_0},$$
shows.

## 数学代写|常微分方程代写ordinary differential equation代考|The Frobenius method

In this section we pursue our investigation of simple singularities. Without loss of generality we will set $z_0=0$. Since we know how a fundamental system looks like from Theorem 4.7, we can make the ansatz
$$W(z)=U(z) z^M, \quad U(z)=\sum_{j=0}^{\infty} U_j z^j, \quad U_0 \neq 0 .$$
Using
$$A(z)=\frac{1}{z} \sum_{j=0}^{\infty} A_j z^j$$
and plugging everything into our differential equation yields the recurrence relation
$$U_j(j+M)=\sum_{k=0}^j A_k U_{j-k}$$
for the coefficients $U_j$. However, since we don’t know $M$, this does not help us much. By (4.77) you could suspect that we just have $M=A_0$ and $U_0=\mathbb{I}$. Indeed, if we assume $\operatorname{det}\left(U_0\right) \neq 0$, we obtain $U_0 M=A_0 U_0$ for $j=0$ and hence $W(z) U_0^{-1}=U(z) U_0^{-1} z^{A_0}$ is of the anticipaled form. Unforimalely, we don’t know that $\operatorname{det}\left(U_0\right) \neq 0$ and, even worse, this is wrong in general (examples will follow).

So let us be less ambitious and look for a single solution first. If $\mu$ is an eigenvalue with corresponding eigenvector $u_0$ of $M$, then
$$w_0(z)=W(z) u_0=z^\mu U(z) u_0$$
is a solution of the form
$$w_0(z)=z^\alpha u_0(z), \quad u_0(z)=\sum_{j=0}^{\infty} u_{0, j} z^j, \quad u_{0,0} \neq 0, \alpha=\mu+m .$$
Here $m \in \mathbb{N}0$ is chosen such that $u_0(0)=u{0,0} \neq 0$. Inserting this ansatz into our differential equation we obtain
$$(\alpha+j) u_{0, j}=\sum_{k=0}^j A_k u_{0, j-k}$$
respectively
$$\left(A_0-\alpha-j\right) u_{0, j}+\sum_{k=1}^j A_k u_{0, j-k}=0$$

# 常微分方程代考

## 数学代写|常微分方程代写ordinary differential equation代考|Linear systems with singularities

$$w^{\prime}=A(z) w, \quad w\left(z_0\right)=w_0, \quad z, z_0 \in \Omega \subseteq \mathbb{C},$$

$$w^{\prime}=-w^2$$

$$w(z)=\frac{1}{z-z_0},$$

## 数学代写|常微分方程代写ordinary differential equation代考|The Frobenius method

$$W(z)=U(z) z^M, \quad U(z)=\sum_{j=0}^{\infty} U_j z^j, \quad U_0 \neq 0 .$$

$$A(z)=\frac{1}{z} \sum_{j=0}^{\infty} A_j z^j$$

$$U_j(j+M)=\sum_{k=0}^j A_k U_{j-k}$$

$W(z) U_0^{-1}=U(z) U_0^{-1} z^{A_0}$ 是预期的形式。我们不知道det $\left(U_0\right) \neq 0$ 而且，更糟糕的是，这通常是错䢔的 (示例如下)。

$$w_0(z)=W(z) u_0=z^\mu U(z) u_0$$

$$w_0(z)=z^\alpha u_0(z), \quad u_0(z)=\sum_{j=0}^{\infty} u_{0, j} z^j, \quad u_{0,0} \neq 0, \alpha=\mu+m .$$

$$(\alpha+j) u_{0, j}=\sum_{k=0}^j A_k u_{0, j-k}$$

$$\left(A_0-\alpha-j\right) u_{0, j}+\sum_{k=1}^j A_k u_{0, j-k}=0$$

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text:
## CryptoDB

### Ron Steinfeld

#### Publications

Year
Venue
Title
2019
CRYPTO
We devise new techniques for design and analysis of efficient lattice-based zero-knowledge proofs (ZKP). First, we introduce one-shot proof techniques for non-linear polynomial relations of degree $k\ge 2$, where the protocol achieves a negligible soundness error in a single execution, and thus performs significantly better in both computation and communication compared to prior protocols requiring multiple repetitions. Such proofs with degree $k\ge 2$ have been crucial ingredients for important privacy-preserving protocols in the discrete logarithm setting, such as Bulletproofs (IEEE S&P ’18) and arithmetic circuit arguments (EUROCRYPT ’16). In contrast, one-shot proofs in lattice-based cryptography have previously only been shown for the linear case ($k=1$) and a very specific quadratic case ($k=2$), which are obtained as a special case of our technique.Moreover, we introduce two speedup techniques for lattice-based ZKPs: a CRT-packing technique supporting “inter-slot” operations, and “NTT-friendly” tools that permit the use of fully-splitting rings. The former technique comes at almost no cost to the proof length, and the latter one barely increases it, which can be compensated for by tweaking the rejection sampling parameters while still having faster computation overall.To illustrate the utility of our techniques, we show how to use them to build efficient relaxed proofs for important relations, namely proof of commitment to bits, one-out-of-many proof, range proof and set membership proof. Despite their relaxed nature, we further show how our proof systems can be used as building blocks for advanced cryptographic tools such as ring signatures.Our ring signature achieves a dramatic improvement in length over all the existing proposals from lattices at the same security level. The computational evaluation also shows that our construction is highly likely to outperform all the relevant works in running times. Being efficient in both aspects, our ring signature is particularly suitable for both small-scale and large-scale applications such as cryptocurrencies and e-voting systems. No trusted setup is required for any of our proposals.
2018
JOFC
2017
CRYPTO
2017
CRYPTO
2015
EPRINT
2015
EPRINT
2015
FSE
2015
ASIACRYPT
2014
CRYPTO
2014
EUROCRYPT
2014
EPRINT
2014
EPRINT
2014
EPRINT
2012
PKC
2012
JOFC
We study the natural problem of secure n-party computation (in the computationally unbounded attack model) of circuits over an arbitrary finite non-Abelian group (G,⋅), which we call G-circuits. Besides its intrinsic interest, this problem is also motivating by a completeness result of Barrington, stating that such protocols can be applied for general secure computation of arbitrary functions. For flexibility, we are interested in protocols which only require black-box access to the group G (i.e. the only computations performed by players in the protocol are a group operation, a group inverse, or sampling a uniformly random group element). Our investigations focus on the passive adversarial model, where up to t of the n participating parties are corrupted.Our results are as follows. We initiate a novel approach for the construction of black-box protocols for G-circuits based on k-of-k threshold secret-sharing schemes, which are efficiently implementable over any black-box (non-Abelian) group G. We reduce the problem of constructing such protocols to a combinatorial coloring problem in planar graphs. We then give three constructions for such colorings. Our first approach leads to a protocol with optimal resilience t<n/2, but it requires exponential communication complexity $O({\binom{2 t+1}{t}}^{2} \cdot N_{g})$ group elements and round complexity $O(\binom{2 t + 1}{t} \cdot N_{g})$, for a G-circuit of size Ng. Nonetheless, using this coloring recursively, we obtain another protocol to t-privately compute G-circuits with communication complexity $\mathcal{P}\mathit{oly}(n)\cdot N_{g}$ for any t∈O(n1−ϵ) where ϵ is any positive constant. For our third protocol, there is a probability δ (which can be made arbitrarily small) for the coloring to be flawed in term of security, in contrast to the first two techniques, where the colorings are always secure (we call this protocol probabilistic, and those earlier protocols deterministic). This third protocol achieves optimal resilience t<n/2. It has communication complexity O(n5.056(n+log δ−1)2⋅Ng) and the number of rounds is O(n2.528⋅(n+log δ−1)⋅Ng).
2011
EUROCRYPT
2010
EPRINT
We describe two improvements to Gentry's fully homomorphic scheme based on ideal lattices and its analysis: we provide a refined analysis of one of the hardness assumptions (the one related to the Sparse Subset Sum Problem) and we introduce a probabilistic decryption algorithm that can be implemented with an algebraic circuit of low multiplicative degree. Combined together, these improvements lead to a faster fully homomorphic scheme, with a~$\softO(\lambda^{3})$ bit complexity per elementary binary add/mult gate, where~$\lambda$ is the security parameter. These improvements also apply to the fully homomorphic schemes of Smart and Vercauteren [PKC'2010] and van Dijk et al. [Eurocrypt'2010].
2010
ASIACRYPT
2009
ASIACRYPT
2008
FSE
2008
EPRINT
LSBS-RSA denotes an RSA system with modulus primes, p and q, sharing a large number of least significant bits. In ISC 2007, Zhao and Qi analyzed the security of short exponent LSBS-RSA. They claimed that short exponent LSBS-RSA is much more vulnerable to the lattice attack than the standard RSA. In this paper, we point out that there exist some errors in the calculation of Zhao & Qi's attack. After re-calculating, the result shows that their attack is unable for attacking RSA with primes sharing bits. Consequently, we give a revised version to make their attack feasible. We also propose a new method to further extend the security boundary, compared with the revised version. The proposed attack also supports the result of analogue Fermat factoring on LSBS-RSA, which claims that p and q cannot share more than (n/4) least significant bits, where n is the bit-length of pq. In conclusion, it is a trade-off between the number of sharing bits and the security level in LSBS-RSA. One should be more careful when using LSBS-RSA with short exponents.
2007
CRYPTO
2007
JOFC
2007
EPRINT
We show that the LASH-$x$ hash function is vulnerable to attacks that trade time for memory, including collision attacks as fast as $2^{\frac{4}{11}x}$ and preimage attacks as fast as $2^{\frac47x}$. Moreover, we describe heuristic lattice based collision attacks that use small memory but require very long messages. Based upon experiments, the lattice attacks are expected to find collisions much faster than $2^{x/2}$. All of these attacks exploit the designers' choice of an all zero IV. We then consider whether LASH can be patched simply by changing the IV. In this case, we show that LASH is vulnerable to a $2^{\frac78x}$ preimage attack. We also show that LASH is trivially not a PRF when any subset of input bytes is used as a secret key. None of our attacks depend upon the particular contents of the LASH matrix -- we only assume that the distribution of elements is more or less uniform. Additionally, we show a generalized birthday attack on the final compression of LASH which requires $O\left(x2^{\frac{x}{2(1+\frac{107}{105})}}\right) \approx O(x2^{x/4})$ time and memory. Our method extends the Wagner algorithm to truncated sums, as is done in the final transform in LASH.
2006
ASIACRYPT
2006
EUROCRYPT
2006
PKC
2006
EPRINT
Pseudorandom Generators (PRGs) based on the RSA inversion (one-wayness) problem have been extensively studied in the literature over the last 25 years. These generators have the attractive feature of provable pseudorandomness security assuming the hardness of the RSA inversion problem. However, despite extensive study, the most efficient provably secure RSA-based generators output asymptotically only at most $O(\log n)$ bits per multiply modulo an RSA modulus of bitlength $n$, and hence are too slow to be used in many practical applications. To bring theory closer to practice, we present a simple modification to the proof of security by Fischlin and Schnorr of an RSA-based PRG, which shows that one can obtain an RSA-based PRG which outputs $\Omega(n)$ bits per multiply and has provable pseudorandomness security assuming the hardness of a well-studied variant of the RSA inversion problem, where a constant fraction of the plaintext bits are given. Our result gives a positive answer to an open question posed by Gennaro (J. of Cryptology, 2005) regarding finding a PRG beating the rate $O(\log n)$ bits per multiply at the cost of a reasonable assumption on RSA inversion.
2005
PKC
2005
EPRINT
We introduce VSH, {\em very smooth hash}, a new $S$-bit hash function that is provably collision-resistant assuming the hardness of finding nontrivial modular square roots of very smooth numbers modulo an $S$-bit composite. By very smooth, we mean that the smoothness bound is some fixed polynomial function of~$S$. We argue that finding collisions for VSH has the same asymptotic complexity as factoring using the Number Field Sieve factoring algorithm, i.e., subexponential in~$S$. %We show how our asymptotic argument can be turned into a practical method to %select parameters so that VSH meets a desired security level. VSH is theoretically pleasing because it requires just a single multiplication modulo the~$S$-bit composite per $\Omega(S)$ message-bits (as opposed to $O(\log S)$ message-bits for previous provably secure hashes). It is relatively practical. A preliminary implementation on a 1GHz Pentium III processor that achieves collision resistance at least equivalent to the difficulty of factoring a 1024-bit RSA modulus, runs at 1.1 MegaByte per second, with a moderate slowdown to 0.7MB/s for 2048-bit RSA security. VSH can be used to build a fast, provably secure randomised trapdoor hash function, which can be applied to speed up provably secure signature schemes (such as Cramer-Shoup) and designated-verifier signatures.
2004
ASIACRYPT
2004
PKC
2003
ASIACRYPT
2003
EPRINT
Motivated by privacy issues associated with dissemination of signed digital certificates, we define a new type of signature scheme called a ?Universal Designated-Verifier Signature? (UDVS). A UDVS scheme can function as a standard publicly-verifiable digital signature but has additional functionality which allows any holder of a signature (not necessarily the signer) to designate the signature to any desired designated-verifier (using the verifier?s public key). Given the designated-signature, the designated-verifier can verify that the message was signed by the signer, but is unable to convince anyone else of this fact. We propose an efficient deterministic UDVS scheme constructed using any bilinear group-pair. Our UDVS scheme functions as a standard Boneh-Lynn-Shacham (BLS) signature when no verifier-designation is performed, and is therefore compatible with the key-generation, signing and verifying algorithms of the BLS scheme. We prove that our UDVS scheme is secure in the sense of our unforgeability and privacy notions for UDVS schemes, under the Bilinear Diffie-Hellman (BDH) assumption for the underlying group-pair, in the random-oracle model. We also demonstrate a general constructive equivalence between a class of unforgeable and unconditionally-private UDVS schemes having unique signatures (which includes the deterministic UDVS schemes) and a class of ID-Based Encryption (IBE) schemes which contains the Boneh-Franklin IBE scheme but not the Cocks IBE scheme.
2003
EPRINT
Universal Designated-Verifier Signature (UDVS) schemes are digital signature schemes with additional functionality which allows any holder of a signature to designate the signature to any desired designated-verifier such that the designated-verifier can verify that the message was signed by the signer, but is unable to convince anyone else of this fact. Since UDVS schemes reduce to standard signatures when no verifier designation is performed, it is natural to ask how to extend the classical Schnorr or RSA signature schemes into UDVS schemes, so that the existing key generation and signing implementation infrastructure for these schemes can be used without modification. We show how this can be efficiently achieved, and provide proofs of security for our schemes in the random oracle model.
2002
PKC
2002
EPRINT
Motivated by emerging needs in online interactions, we define a new type of digital signature called a Content Extraction Signature' (CES). A CES allows the owner, Bob, of a document signed by Alice, to produce an extracted signature' on selected extracted portions of the original document, which can be verified to originate from Alice by any third party Cathy, while hiding the unextracted (removed) document portions. The new signature therefore achieves verifiable content extraction with minimal multi-party interaction. We specify desirable functional and security requirements for a CES (including an efficiency requirement: a CES should be more efficient in either computation or communication than the simple multiple signature solution). We propose and analyze four CES constructions which are provably secure with respect to known cryptographic assumptions and compare their performance characteristics.

Asiacrypt 2019
Asiacrypt 2017
Asiacrypt 2016
Crypto 2016
Eurocrypt 2016
Asiacrypt 2014
Eurocrypt 2014
PKC 2012
Asiacrypt 2012
Asiacrypt 2010
Eurocrypt 2010
Asiacrypt 2008
PKC 2006


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# How do you use synthetic division to divide (2x^3-x^2+5x-12) /(2x - 3) ?

Mar 28, 2018

Quotient is ${x}^{2} + x + 4$ and remainder is $0$.

#### Explanation:

$2 {x}^{3} - {x}^{2} + 5 x - 12$

=$2 {x}^{3} - 3 {x}^{2} + 2 {x}^{2} - 3 x + 8 x - 12$

=${x}^{2} \cdot \left(2 x - 3\right) + x \cdot \left(2 x - 3\right) + 4 \cdot \left(2 x - 3\right)$

=$\left(2 x - 3\right) \cdot \left({x}^{2} + x + 4\right)$

Hence quotient is ${x}^{2} + x + 4$ and remainder is $0$.


text:
## If N and M/N are finitely generated as R-modules, then so is M

Let $R$ be a ring with 1, and suppose $M$ is a left $R$-module with submodule $N$. Suppose further that $N$ and $M/N$ are finitely generated. Prove that $M$ is finitely generated.

Say $M/N = (a_i + N \ |\ 1 \leq i \leq m)$ and $N = (b_j \ |\ 1 \leq j \leq n)$. We claim that $M = (a_i,b_j \ |\ 1 \leq i \leq m, 1 \leq j \leq n)$. Certainly $(\supseteq)$ holds. Now suppose $m \in M$. Note that $m + N = \sum_i r_i \cdot (a_i + N)$ $= (\sum_i r_i \cdot a_i) + N$. Now $m - \sum_i r_i \cdot a_i \in N$, so that $m - \sum_i r_i \cdot a_i = \sum_j s_j b_j$. Then $m = \sum_i r_i \cdot a_i + \sum_j s_j \cdot b_j \in (a_i, b_j \ |\ 1 \leq i \leq m, 1 \leq j \leq n)$. In particular, $M$ is finitely generated.


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# zbMATH — the first resource for mathematics

Stochastic allocation and scheduling for conditional task graphs in multi-processor systems-on-chip. (English) Zbl 1232.68017
Summary: Embedded systems designers are turning to multicore architectures to satisfy the ever-growing computational needs of applications within a reasonable power envelope. One of the most daunting challenges for multiprocessor system-on-chip (MPSoC) platforms is the development of tools for efficient mapping multi-task applications onto hardware platforms. Software mapping can be formulated as an optimal allocation and scheduling problem, where the application is modeled as a task graph, the target hardware is modeled as a set of heterogeneous resources, and the objective function represents a design goal $$\alpha$$ (e.g. minimum execution time, minimum usage of communication resources, etc.). Conditional task graphs, where inter-task edges represent data as well as control dependencies, are a well-known computational model to describe complex real-life applications where alternative execution paths, guarded by conditionals, can be specified. Each condition has a probability associated with each possible outcome.
Mapping conditional task graphs is significantly more challenging than mapping pure data-flow graphs (where edges only represent data dependencies). Approaches based on general-purpose complete solvers (e.g. integer linear programming solvers) are limited both by computational blowup and by the fact that the objective is a stochastic functional. The main contribution of our work is an efficient and complete approach to allocation and scheduling of conditional task graphs, based on (i) an exact analytic formulation of the stochastic objective function exploiting task graph analysis and (ii) an extension of the timetable constraint for conditional activities. Moreover, our solver is integrated in a complete application development environment which produces executable code for target multicore platforms. This integrated framework allows us to validate modeling assumptions and to assess constraint satisfaction and objective function optimization. Extensive validation results demonstrate not only that our approach can handle non-trivial instances efficiently, but also that our models are accurate and lead to optimal and highly predictable execution.

##### MSC:
 68M07 Mathematical problems of computer architecture 68M20 Performance evaluation, queueing, and scheduling in the context of computer systems 05C90 Applications of graph theory
JaCoP
Full Text:


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#### [SOLVED] Question regarding the solution of Schrödinger equation for finite potential well and quantum barrier

When solving the Schrödinger equation for finite potential well, the solution outside of the well is $$\psi _{1}=Fe^{{-\alpha x}}+Ge^{{\alpha x}}\,\!$$ and $$\psi _{3}=He^{{-\alpha x}}+Ie^{{\alpha x}}\,\!$$ However, when solving the Schrödinger equation for the quantum barrier, the solution of the regions are \begin{align} \psi _{L}(x)&=A_{r}e^{{ik_{0}x}}+A_{l}e^{{-ik_{0}x}}\quad x<0\\ \psi _{C}(x)&=B_{r}e^{{ik_{1}x}}+B_{l}e^{{-ik_{1}x}}\quad 0<x<a\\ \psi_{R}(x)&=C_{r}e^{{ik_{2}x}}+C_{l}e^{{-ik_{2}x}}\quad x>a \end{align}

The solutions for quantum barrier have the imaginary $i$ on the exponent of $e$ for example $A_{l}e^{{-ik_{0}x}}$. But the solution for the finite potential well does not have the imaginary $i$ on the exponent $e$ for example $Fe^{{-\alpha x}}$.

So why is there an imaginary $i$ for quantum barrier problem while there isn't an imaginary $i$ for potential well problem when both solution of the problem is derived by solving the Schrödinger equation in the form of $(\left[{\frac {d^{2}}{dx^{2}}}\psi (x)\right]=b\psi (x))$ ?

Thank you for the reply, and I understand the quantum barrier and potential well a bit more. However, I have a follow up question: Why is the region with potential $V=0$ in the finite potential well have a wavefunction of the form $$\psi _{2}=Asinkx+Bsinkx\,\!$$ But for the quantum barrier, the regions with potential $V=0$ have a wavefunction of $$\psi _{L}(x)=A_{r}e^{{ik_{0}x}}+A_{l}e^{{-ik_{0}x}}\quad x<0\\$$ So why does the Schrodinger equation produce different result for the region of $V=0$?


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# 勉強日記

## (x'Ax)^2<=(x'AA'x)(x'x)

Let $$A$$ be any real square matrix (not necessarily symmetric).  Prove that: $$(x'A x)^2 \leq (x'A A'x)(x'x)$$

The key point in proving this inequality is to recognize that $$x'A A'x$$ can be expressed as vector norm of $$A'x$$.

Proof:

If $$x=0$$, then the inequality is trival.

Suppose $$x \neq 0$$.

$$\frac{x'A x}{x'x} = \frac{(A'x)'x}{\| x \|^2} = (A'\frac{x}{\| x \|})'\frac{x}{\| x \|}$$

Because $$\frac{x}{\| x \|}$$ is a unit vector, $$A'\frac{x}{\| x \|}$$ can be considered as scale and rotation of $$\frac{x}{\| x \|}$$ by $$A'$$.  Thus, the resulting vector norm of $$A'\frac{x}{\| x \|}$$ is $$\alpha$$ for some $$\alpha > 0$$.  And $$(A'\frac{x}{\| x \|})'\frac{x}{\| x \|}=\alpha \, cos(\beta)$$ for some $$-\pi \leq \beta \leq \pi$$, which is the angle between before and after premulitplying $$A'$$.

Now:

$$( \frac{x'A x}{x'x} )^2$$

$$= ( (A'\frac{x}{\| x \|})'\frac{x}{\| x \|} )^2$$

$$=\alpha^2 \, cos(\beta)^2$$

$$\leq \alpha^2$$

$$= (A'\frac{x}{\| x \|})'A'\frac{x}{\| x \|}$$

$$= \frac{(A'x)'A'x}{\| x \|^2}$$

$$= \frac{x'A A'x}{x'x}$$

Finally, multiplying both sides by $$(x'x)^2$$ completes the proof.

## Ab=0 iff A^+b=0, when A is symmetric.

Let $$A'$$ be transpose of $$A$$, $$A^{+}$$ be MP-inverse of $$A$$.

Prove that if $$A=A'$$, then $$A b=0$$ if and only if $$A^+b=0$$.

Proof:

$$A b=0$$ implies that $$b=(I-A^+A)q$$ for some $$q$$.  Then,

$$A^+ b \\ =A^+(I-A^+A)q \\ =A^+(I-A'^+A')q \\ =A^+(I-(AA^+)')q \\ =A^+(I-AA^+)q \\ =A^+q-A^+AA^+q \\ =A^+q-A^+q \\ =0$$

Because $$A^{++}=A$$, the proof is completed.

## 連続冪等行列関数の階数は定数である

Fは冪等であるため、$$tr(F)=rank(F)$$。Fは連続ですから、$$tr(F)$$も連続です。しかし、$$rank(F)$$の変化は常に1の倍数で飛びます。両方合わせば $$tr(F)=rank(F)$$ は定数でなければならないことが分かります。


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• #### Inverse problems in image processing

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Inverse problems involve estimating parameters or data from inadequate observations; the observations are often noisy and contain incomplete information about the target parameter or data due to physical limitations of the ...
• #### Inversion Formula for Continuous Multifractals

(1997-01-20)
In a previous paper the authors introduced the inverse measure Âµâ   of a probability measure Âµ on [0,1]. It was argued that the respective multifractal spectra are linked by the 'inversion formula' ...
• #### Investigating irregularly patterned deep brain stimulation signal design using biophysical models

(2015-06)
Parkinson's disease (PD) is a neurodegenerative disorder which follows from cell loss of dopaminergic neurons in the substantia nigra pars compacta (SNc), a nucleus in the basal ganglia (BG). Deep brain stimulation (DBS) ...
• #### Investigating Multiple Candidate Genes and Nutrients in the Folate Metabolism Pathway to Detect Genetic and Nutritional Risk Factors for Lung Cancer

(2013)
Purpose: Folate metabolism, with its importance to DNA repair, provides a promising region for genetic investigation of lung cancer risk. This project investigates genes (MTHFR, MTR, MTRR, CBS, SHMT1, TYMS), folate metabolism ...
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(2006)
Scanning tunneling microscopy (STM) is an ideal tool for probing the chemistry and physics of these types of nanostructures. Building upon on our previous carbon nanotube research, we have investigated thiol and thiophene ...
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(2010)
Carbon allotropes such as nanotubes, graphene, and buckyball all preserve the same lattice structure. However, their electronic properties are very different depending on their dimensionality. These nanostructures can be ...
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(1986)
Thin film LiNbO(,3) was deposited on silicon substrates by using r.f. sputtering techniques under different conditions. A set of films was deposited with different substrate temperatures and with different gas mixtures ...
• #### Investigation of Reliability of Hydraulic Robots for Hazardous Environments Using Analytic Redundancy

(1999-01-01)
The Rosie mobile worksystem is a robot that is on the cutting edge of hazardous environment robotics. It is a heavy-duty hydraulic robot designed for nuclear reactor decontamination and dismantlement. The robot consists ...

(1972)
• #### Investigations in improving image visualization and quality in positron emission tomography/computed tomography (PET/CT) imaging

(2010)
Positron Emission Tomography/Computed Tomography (PET/CT) is a widely used imaging modality for managing patients with cancer. The combination of PET and CT can provide both functional and anatomic information of disease ...
• #### IO-lite: A copy-free UNIX I/O system

(1997)
Memory copy speed is known to be a significant barrier to high-speed communication. We perform an analysis of the requirements for a copy-free buffer system, develop an implementation-independent applications programming ...
• #### Ionization of Rydberg atoms by half-cycle pulses: Effect of pulse shape and rise time

(1998)
The ionization of potassium np Rydberg atoms with $n\sim388$ by pulsed unidirectional electric fields, termed half-cycle pulses (HCPs), with various well-characterized shapes (rectangular, triangular, and sawtooth) and ...
• #### Isolating the Performance Impacts of Network Interface Cards through Microbenchmarks

(2004-06-01)
This paper studies the impact of network interface cards (NICs) on network server performance, testing six different Gigabit Ethernet NICs. Even with all other hardware and software configurations unchanged, a network ...
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(2015)
Ternary Mn-Zn ferrite (MnxZn1-xFe2O4) nanoparticles (NPs) have been prepared by the thermal decomposition of an oleate complex, sodium dodecylbenzenesulfonate (SDBS) mediated hydrazine decomposition of the chloride salts, ...
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(1995)
This dissertation introduces a novel method of teleoperation of complex anthropomorphic robotic hands: converting the myoelectric signal generated by an operator's muscles during movement into robot commands replicating ...
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Find books Any comments are appreciated. Nationality. The rising sea. refer to here), so $q\in Supp\frac{m}{s}$ too. The most relevant reference for the course is the book of Hartshorne and the notes of Vakil. Compra Snowbird Lectures in Algebraic Geometry. Mathematical Society I also strongly recommend Foundations of Algebraic Geometry by Ravi Vakil. That is, $p$ contains the annihilator of $m$, $p\supset Ann_A(m)$. The Rising Sea: Foundations of Algebraic Geometry (November 18, 2017 edition) by Ravi Vakil Algebraic Geometry by Robin Hartshorne Fourier-Mukai Transforms in Algebraic Geometry by Daniel Huybrechts Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. And he stated the important property (rather than defining it) first: (A) The associated primes/points of $M$ are precisely the generic points of irreducible \begin{align} a National Science Foundation CAREER grant, the presidential award PECASE, Asking for help, clarification, or responding to other answers. The set of zeros of a set of polynomial equations in finitely many variables over a field is called an affine variety and it is equipped with a particular topology called Zariski topology, whose closed sets are subvarieties. \end{align} The course will begin with a quick review of the theory of varieties as presented in the Part II algebraic geometry â¦ Then there must be a prime ideal $q\supset Ann_A(m)$ and being disjoint from $S$ (e.g. Ravi Vakil, The rising sea: Foundations of algebraic geoemtry (available online). Department of Mathematics Building 380, Stanford, California 94305 Phone: (650) 725-6284 Email The article In fall 2017, I helped Soundararajan run the Polya Seminar. On the other side, now suppose $p\in Supp \frac{m}{s}$ is an associated point of $S^{-1}M$. So before I take you on as a new student, you should be comfortable with the foundations of the subject, which means having done the majority of the exercises in Hartshorne or my course notes, and being able to explain them on demand. I am the Mathematical Association of America's, Undergraduate Introduction. based on this lecture has won the Lester R. Ford Award in 2012 and the Chauvenet Prize in 2014. Overview. components of the support of some element of $M$ (on Spec $A$). Other valuable online sources: Mel Hochster's commutative algebra notes. Because the field is a synthesis of ideas from many different parts of mathematics, it usually â¦ Ravi Vakil T he moduli space of curves has provenitself a central object in geometry. 1.2.A. Mathematical Society, and rm\ne 0, \forall r\notin p. - 18, online Friday 14 c.t. A celebration of algebraic geometry : a conference in honor of Joe Harris' 60th birthday, Harvard University, Cambridge, Massachusetts, August 25-28, 2011 by Algebraic geometry conference ( Book ) 5 editions published in 2013 in English and held by 61 WorldCat member libraries worldwide rev 2020.12.18.38240, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us, \begin{align*} Note to reader: the index and formatting have yet to be properly dealt with. The core topics include affine and projective varieties, morphisms between them, basic properties of such (irreducibility, dimension, finiteness, degree, ...), sheaves, schemes, and the functor of points perspective, some key geometric notions and constructions (birational maps, blowups, projections, divisors, ...), some key exaâ¦ for Gifted Youth (EPGY), page above for people thinking of doing a Ph.D. with me, In fall 2017, I helped Soundararajan run the Polya Seminar. SARS-COV2) survive in food? Now since $p$ is disjoint from $S$, for any $t\in S$ we have $rt\notin p$, hence $rtm\ne 0$. \end{align}, Vakil 5.5.F Associated primes/points behaves well with respect to localizing, Exercise 5.5.F. The most relevant reference for the course is the book of Hartshorne and the notes of Vakil. In what way would invoking martial law help Trump overturn the election? I was the 2009 Earle Raymond Hedrick Lecturer at Mathfest, and Stanford Math Circle (for high school students). Exam on March 18 canceled !!! Degenerations in algebraic geometry August 17 to August 21, 2015 at the American Institute of Mathematics, San Jose, California organized by Brian Osserman and Ravi Vakil This workshop, sponsored by AIM and the NSF, will be devoted to the study of degeneration techniques in algebraic geometry, in several differing but interrelated contexts. In this webinar, Professors Ravi Vakil and Bernd Sturmfels discuss the history and applications of algebraic geometry, the branch of mathematics that studies zeros of polynomials and solves geometrical problems about these sets of zeros. Did the Allies try to "bribe" Franco to join them in World War II? which is nothing but the condition for $p\in Supp\frac{m}{s}$ (for any $s\in S$). Faculty in algebraic geometry study a diverse set of topics including the cohomology and geometry of the moduli space of curves, the foundations of Gromov-Witten theory, the geometry of algebraic cycles, and problems of enumerative geometry Was wood used in the construction of the TU-144? The year before, I taught a year-long course: From 2001 until 2007, I coordinated the William Lowell Putnam competition In 2013, I was a Simons Fellow in Mathematics. How long can a virus (e.g. I have received the Coxeter-James Prize from the Canadian Fiducial marks: Do they need to be a pad or is it okay if I use the top silk layer? In this class, you will be introduced to some of the central ideas in algebraic geometry. \begin{align*} The first lecture will take place on Friday, October 30. I am the Mathematical Association of America's Pólya Lecturer 2012-2014. Snowbird Lectures in Algebraic Geometry: Vakil, Ravi, AMS-IMS-SIAM JOINT SUMMER RESEARCH CONFE: Amazon.sg: Books Monday 16 c.t. In other words a weekly Pólya problem-solving seminar for talented undergraduates, as well Robert K. Packard University Fellow Ravi Vakil in 2008. on Ravi Vakil's Notes related to associated points, Hypersurfaces have no embedded points (Vakil 5.5.I), Question about associated points in Vakil's notes, If $A$ is reduced, Spec $A$ has no embedded points (How to use the hints), Proper flat morphism with geometrically connected and smooth generic fiber. The most relevant reference for the course is the book of Hartshorne and the notes of Vakil. algebraic geometry (Math 145), Topics in Algebraic Geometry (M24) Mark Gross This will be a basic course introducing the tools of modern algebraic geometry. Commutative algebra notes variations ) in TikZ/PGF, Capital gains tax when proceeds were immediately used another. Of America 's Pólya Lecturer 2012-2014 this has being asked here ( is! Your answer ”, you will be introduced to some of the TU-144, 2017 draft âc by. To Mathematics Stack Exchange the 2009 Earle Raymond Hedrick Lecturer at Mathfest, and André-Aisenstadt from!  bribe '' Franco to join them in World War II and cookie policy { m {! Polynomial equations and generalizations attack examples that use the top silk layer object is a question and site... The Polya Seminar with references or personal experience the hash collision I also strongly recommend foundations of geometry... Atmospheric layer T he moduli space of curves has provenitself a central object in geometry did the try... Them up with references or personal experience plural in the name for PCA silk layer for a year-old! Equations and generalizations $Ann_A ( m )$ and $p\cap S=\emptyset$ 2009!, see our tips on writing great answers and formatting have yet to be properly with! Idea of delivering a letter from past to the situation with the Coronavirus the. ) \$ ) sea: foundations of algebraic geometry Winter 2017 rigid body states they are not deformable technology public... 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# Java Tetris - Using rotation matrix math to rotate piece

I'm working on building tetris now in Java and am at the point of rotations...

I originally hardcoded all of the rotations, but found that linear algebra (matrix rotations) was the better way to go.

I'm trying to use a rotation matrix to rotate my pieces, and found I need a good understanding of trigonometry.

I don't understand how R(90 degrees) equals a rotation matrix of R(-theta) = $$\begin{bmatrix}cos0 &sin0 \\-sin0 & cos0\end{bmatrix}$$, aka $$\begin{bmatrix} 0 & 1 \\ -1& 0 \end{bmatrix}$$ (btw, if you know how to make a matrix on stackoverflow, please let me know). How does that equal a 90 degree rotation? Where do those zeros and ones come from? Moreover, how would that translate into code? I'm not looking for someone to code it for me, I'm looking for a concept with maybe a snippet of pseudocode.

I'm trying to visualize it with this drawing by putting the grid and tiles on a graph and drawing out the angles... but still don't understand it.

Can anyone help?

Thanks!

Transposing

-
For some basic information about writing math at this site see e.g. here, here, here and here. Particularly the first answer to the question in the first link will help you typeset matrices. –  Lord_Farin Apr 16 '13 at 18:09
Very helpful! Thanks –  Growler Apr 16 '13 at 18:17
Your rotation matrix is a vector space rotation matrix and you want an affine transformation, not a linear one. –  xavierm02 Apr 16 '13 at 18:23
Since you're looking as much for coding help as for mathematics help, I'd suggest that you start at gamedev.SE; unless you're physically rotating pieces for display (e.g., you expect to have a piece rotated by $45^\circ$ at some point), then hard-coding is almost certainly better than rotation matrices, but that's more a gamedev discussion than a mathematical one. –  Steven Stadnicki Apr 16 '13 at 19:06
@StevenStadnicki Well my goal is to 1) Learn some useful math, 2) I want to code a function that can handle the rotation of any shape... so I want to stay away from hardcoding rotations for 7 shapes. –  Growler Apr 16 '13 at 19:15

The operations you are wanting to perform can be easily implemented without the need of rotation matrices since the transforms you desire are limited to right angle rotations and mirroring.

Assuming $B$ is a two dimensional array representing a block and $B[i,j]$ is the $i^{\text{th}}$ row and $j^{\text{th}}$ column of the block taking on values of 1 for a single block segment and 0 for an empty space.

The first operation you need is a transpose which is effectively a 90 degree clockwise rotation:

B T(B b)
B b' <- new B[b.columns, b.rows]

for i = 0 to b.rows - 1
for j = 0 to b.columns -1
b'[j,i] <- b[i, j]

return b'

The second operation is mirroring a block about the y-axis:

B Y(B b)
B b' <- new B[b.rows, B.columns]

for i = 0 to b.rows - 1
for j = 0 to b.columns - 1
b'[i,j] <- b[i, b.columns - 1 - j]

return b'

And finally, mirroring a block about the x-axis:

B X(B b)
B b' <- new B[b.rows, B.columns]

for i = 0 to b.rows - 1
for j = 0 to b.columns - 1
b'[i,j] <- b[b.rows - 1 - i, j]

return b'

Given a block $B$ in a neutral position, a 90 degree clockwise rotation is $T(B)$, 180 is $T(X(T(B)))$, 270 is $Y(T(B))$ and 360 is simply $B$. counter clockwise rotations are just 360 minus the complementing clockwise rotations.

Also, as @StevenStadnicki pointed out in the comments, you can hard code much of this and you will find that if you are replicating the original Tetris, you'll encounter more that requires a hard coded solution than not.

-
Okay, but that's how I started off doing it (see here: stackoverflow.com/questions/15884051/java-tetris-rotations), but it seemed pretty hairy to have to calculate each rotation for each block for either left or right rotations... unless I'm doing it wrong? –  Growler Apr 18 '13 at 14:23
ahh... I finally understand why you were mirroring it... Once you transpose it... you have your 0 degree and 90 degree rotation... then you can mirror both of those to get your missing 2 rotations (180 and 270 degree). Clever... haha –  Growler Apr 18 '13 at 15:32
Is my graph above the correct depiction of what you've said? –  Growler Apr 18 '13 at 17:37
Yes, your graph is correct. I was mistaken however when I said the transpose would rotate clockwise, as implemented, it will rotate counter clockwise. Also note that $T(X(T(B)))$ can be simplified to just $X(B)$. So the final clockwise rotation of 0, 90, 180, 270 degree is $B$, $Y(T(B))$, $X(T)$, $T(B)$ respectively. –  lewellen Apr 19 '13 at 13:52
Okay I'll give that a try. Thanks! –  Growler Apr 19 '13 at 13:54


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# What can we say about the function if its derivative is Strictly increasing.

Let we have a function $$f:R\rightarrow R$$ such that $$f'(x)$$ is Strictly increasing. Let $$a$$ and $$b$$ denotes the Minimum and maximum on the intervals $$[2,3]$$. Then, is $$b=f(3)$$ true? This can only happen when the function is Strictly increasing in the given Interval. But, can we say the Strictly increasing nature of derivative can be extended to say that function is also Strictly increasing?

I am not able to understand the relationship between the function and the monotonicity of its derivative. One thing is sure that it's second derivative is greater that zero. Any help would be beneficial for me. Thanks

• Consider the function $f(x) = (x-4)^2$ on the interval $[2,3]$. Is its derivative strictly increasing? Where does it take a maximum value on that interval? Feb 5 '18 at 17:37
• However, what you can conclude is that the function is convex, so it must have its maximum at the ends of the interval. It is going to be either $f(3)$ or $f(2)$.
– user491874
Feb 5 '18 at 17:47
• I suspect all you can conclude is that $f$ is concave up everywhere
– MPW
Feb 5 '18 at 17:57

Hint: What if the derivative is negative on the entire interval but increasing? For instance, say $f'(x)=x-5$.
Bonus: Does that mean that $f(a)$ and $f(b)$ are always going to be local extrema?
Well , Rahul what you can comment about$\ f(x)$ is it's concavity . What you claim about$\ b$ is right only if in the interval$\ [2,3]$ ,$\ f'(x) \ge0$. Otherwise not .
Hint: To understand the relation between functions and monotonicity of its derivative , take few coordinates of your choice and draw arbitrary graphs of$\ f(x)$ satisfying monotonicity of$\ f'(x)$ .
I don't think it's true. Take for example the function $f(x)=e^{-x}$ then $f''(x)=e^{-x} >0,$ thus $f'(x)$ is strictly increasing. Whereas $f(x)$ is strictly decreasing. And $b= f(3)$ does not hold true. But if you take $f(x)=e^{x}$, your claim holds true. So you can't conclude monotonicity of the function from the monotonicity of its derivative.


text:
pythagorean triple diophantine equation problem

• August 29th 2008, 11:00 AM
Proggy
pythagorean triple diophantine equation problem
Find all solutions to the equation $x^2 + 2y^2 = w^2$ with x>0, y>0, w>0.

I have several problems similar to this one and I do not understand how to do them. The chapter focuses on pythagorean tripes of $a^2 + b^2 = c^2$ but I can not figure out how to apply that when there are additional parts to the equation.
• August 30th 2008, 05:44 PM
Proggy
so, I mostly figured it out, but hopefully somebody can clarify something for me...

$x^2 + 2y^2 = w^2$
$2y^2$ is always even due to the 2
so, when x is even, $x^2$ is even and $x^2 + 2y^2$ will be an even plus an even, so $w^2$ is even which means that w is even.
when x is odd, $x^2$ is odd and $x^2 + 2y^2$ will be an odd plus an even, so $w^2$ is odd which means that w is odd.
So x and w are both even or x and w are both odd, so w - x and w + x will always be even.

The equation becomes $2y^2 = w^2 - x^2 = (w - x)(w + x)$
So I divide through by two, apparently just because I can since they are all even (is there another reason?) which gives me
$2(y/2)^2 = [(w - x)/2][(w + x)/2]$
$(y/2)^2 = [(w - x)/4][(w + x)/2]$

Using an earlier theorem, i can say that $(w - x)/4 = u^2 and (w + x)/2 = v^2$

doing the math, that gives me $x = v^2 - 2u^2$ and $w = v^2 + 2u^2$ and this leads to the further result that $y = 2uv$

This result is correct according to the back of the book except for two things...
first, apparently since it wants all solutions, I multiply all my answers by $d$.
second, the book puts absolute value signs around the entire x answer. The actual answers are:
$x = |d(v^2 - 2u^2)|$
$w = d(v^2 + 2u^2)$
$y = 2duv$

The second problem I did, $x^2 + 3y^2 = w^2$, I did in much the same way except it does matter what y is in this case, so I end up not dividing through by 2. I end up with the equations $(w - x)/3 = u^2$ and $(w + x) = v^2$ I come up with the correct answer, but the answer for x is again enclosed with absolute value signs. The same answers result for the third equation of $x^2 + py^2 = w^2 for p is any odd prime$.

So where are the absolute value signs coming from? I barely understand the problems as it is and the book of course does not mention absolute values in the section.
• August 31st 2008, 12:50 AM
Moo
Hello,
Quote:

Originally Posted by Proggy
The equation becomes $2y^2 = w^2 - x^2 = (w - x)(w + x)$
So I divide through by two, apparently just because I can since they are all even (is there another reason?)

You have to prove that y is even (which is actually easy since the RHS is at least divisible by 4)

Quote:

which gives me
$2(y/2)^2 = [(w - x)/2][(w + x)/2]$
$(y/2)^2 = [(w - x)/4][(w + x)/2]$

Using an earlier theorem, i can say that $(w - x)/4 = u^2 and (w + x)/2 = v^2$
Okay, I'm just curious (Blush) what is this theorem ? How can one know that w-x is divisible by 4 ?

Quote:

doing the math, that gives me $x = v^2 - 2u^2$ and $w = v^2 + 2u^2$ and this leads to the further result that $y = 2uv$

This result is correct according to the back of the book except for two things...
first, apparently since it wants all solutions, I multiply all my answers by $d$.
Yup.

Imagine you have d which divides x,y and w, hence x=dx', y=dy' and w=dw' :
$(dx')^2+2(dy')^2=(dw')^2$
Divide the equation by d² and you're back to the same equation.
Conversely, you get the same thing (that is if you multiply x,y and w by d, you'll get the same equation too)

Quote:

second, the book puts absolute value signs around the entire x answer. The actual answers are:
$x = |d(v^2 - 2u^2)|$
$w = d(v^2 + 2u^2)$
$y = 2duv$
Because you can't tell whether u or v is the greatest and the solutions have to be > 0 ?
• August 31st 2008, 04:58 PM
Proggy
Quote:

Originally Posted by Moo
Hello,

You have to prove that y is even (which is actually easy since the RHS is at least divisible by 4)

so, since both of the factors on the right are even and thus divisible by 2, then the left side is divisible by 2x2 or 4. So that makes y squared divisible by 2 and if it is divisible by 2, then it is divisible by 4 and y itself is divisible by 2 and even. That kinds of makes sense now. But my question then becomes why do I bother to divide both sides by 4? It seems like I come out with the exact same answer if I do not do the division and work out the math.

Quote:

Originally Posted by Moo
Okay, I'm just curious (Blush) what is this theorem ? How can one know that w-x is divisible by 4 ?

heh, well, the theorem i am referring to is the fact that a square is the product of two squares. The division by 4 comes from dividing both sides by the 2 in front of the y squared. So I do not really konw that w-x is divisible by 4, but one of them has to be disibile by 4 and the equations work out to be the same whichever one I divide by the extra 2. I am only figuring some of this out due to already knowing what the answers were for these particular problems. I can not seem to grasp all the concepts behind them.

Quote:

Originally Posted by Moo
Yup.

Imagine you have d which divides x,y and w, hence x=dx', y=dy' and w=dw' :
$(dx')^2+2(dy')^2=(dw')^2$
Divide the equation by d² and you're back to the same equation.
Conversely, you get the same thing (that is if you multiply x,y and w by d, you'll get the same equation too)

I did try the math with the thought that (x,y,z)=d and then plugged in x'/d and so forth and proved to myself that the answer was the same as simply multiplying by d at the end.

Quote:

Originally Posted by Moo
Because you can't tell whether u or v is the greatest and the solutions have to be > 0 ?

I know what you are saying here, but I do not understand it. Eliminating the negative sign simply because you want a positive number does not seem right. I would have expected it more to lead to saying that if it resulted in a negative number, then that simply was not a valid answer. But I do not see why removing the negative sign because it is inconvenient works. I mean I know it does, i just do not understand why it does.

I was really good at math before I got to Number Theory. It is a rather discouraging class, but maybe that is because I am trying it online. Abstract Algebra and Probablility and Statistics (not intro, but an advanced one) are my next two, and last math classes, I need for my grade 7-12 integrated math licensure. I seem to be struggling alot, even though I got 100 percent on my midterm because it was basically nowhere near as hard as the homework. Anyway, just rambling. Thanks for helping.


text:
Accelerating the pace of engineering and science

# analyze

Analyze circuit object in frequency domain

## Syntax

analyze(h,freq)
analyze(h,freq,zl,zs,zo,aperture)
analyze(h,freq,'condition1',value1,...,'conditionm',valuem)

## Description

analyze(h,freq) calculates the following circuit data at the specified frequency values:

• Circuit network parameters

• Noise figure

• Output third-order intercept point

• Power data

• Phase noise

• Voltage standing-wave ratio

• Power gain

• Group delay

• Reflection coefficients

• Stability data

• Transfer function

h is the handle of the circuit object to be analyzed. freq is a vector of frequencies, specified in hertz, at which to analyze the circuit. OIP3 is always infinite for passive circuits.

analyze(h,freq,zl,zs,zo,aperture) calculates the circuit data at the specified frequency values. The arguments zl, zs, zo, and aperture are optional. zl, zs, and zo represent the circuit load, circuit source, and reference impedances of the S-parameters, respectively. The default value of all these arguments is 50 ohms.

 Note:   When you specify impedance values, the analyze method changes the object's values to match your specification.

The aperture argument determines the two frequency points that the analyze method uses to compute the group delay for each frequency in freq. aperture can be a positive scalar or a vector of the same length of as freq.

 Note:   For rfckt.datafile, rfckt.passive, rfckt.amplifier, and rfckt.mixer objects that contain measured S-parameter data, the analyze method uses the two nearest measurement points to compute the group delay, regardless of the value of aperture.

Group delay τg at each frequency point f is the negative slope of the phase angle of S21 with respect to f:

${\tau }_{g}\left(f\right)=-\frac{\Delta \varphi }{\Delta \omega }=-\frac{\mathrm{arg}\left({S}_{21}\left({f}_{+}\right)\right)-\mathrm{arg}\left({S}_{21}\left({f}_{-}\right)\right)}{2\pi \left({f}_{+}-{f}_{-}\right)}$

where:

• f+ is:

• f(1 + aperture/2) for aperture < 1.

• f + aperture/2 for aperture ≥ 1.

If f is the maximum value of freq, then f+ = f.

• f is:

• f (1 – aperture/2) for aperture < 1.

• f – aperture/2 for aperture ≥ 1.

If f is the minimum value of freq, then f = f.

By default, analyze calculates the group delay in nanoseconds.

The value of aperture affects the accuracy of the computed group delay. If aperture is too large, the slope estimate may be not accurate. If aperture is too small, the computer numerical error may affect the accuracy of the group delay result.

analyze(h,freq,'condition1',value1,...,'conditionm',valuem) calculates the circuit data at the specified frequency values and operating conditions for the object h. The inputs 'condition1',value1,...,'conditionm',valuem are the condition/value pairs at which to analyze the object. Use this syntax for rfckt.amplifier, rfckt.mixer, and rfdata.data objects where the condition/value pairs are operating conditions from a .p2d or .s2d file.

 Note:   When you specify condition/value pairs, the analyze method changes the object's values to match your specification.

When you analyze a network that contains several objects, RF Toolbox™ software does not issue an error or warning if the specified conditions cannot be applied to all objects. For some networks, because there is no error or warning, you can call the analyze method once to apply the same set of operating conditions to any objects where operating conditions are applicable. However, you may want to analyze a network that contains one or more of the following:

• Several objects with different sets of operating conditions.

• Several objects with the same set of operating conditions that are configured differently.

To analyze such a network, you should use the setop method to configure the operating conditions of each individual object before analyzing the network.

### Analysis of Circuit Objects

For most circuit objects, the AnalyzedResult property is empty until the analyze method is applied to the circuit object. However, the following four circuit objects are the exception to this rule:

• rfckt.datafile — By default, the AnalyzedResult property of rfckt.datafile objects contains the S-parameter, noise figure, and group delay values that are calculated over the network parameter frequencies in the passive.s2p data file. OIP3 is ∞ by default because the data in passive.s2p is passive.

• rfckt.passive — By default, the AnalyzedResult property of rfckt.passive objects contains the S-parameter, noise figure, and group delay values that are the result of analyzing the values stored in the passive.s2p file at the frequencies stored in this file. These frequency values are also stored in the NetworkData property. OIP3 is always ∞ for rfckt.passive objects because the data is passive.

• rfckt.amplifier — By default, the AnalyzedResult property of rfckt.amplifier objects contains the S-parameter, noise figure, OIP3, and group delay values that result from analyzing the values stored in the default.amp file at the frequencies stored in this file. These frequency values are also stored in the NetworkData property.

• rfckt.mixer — By default, the AnalyzedResult property of rfckt.mixer objects contains the S-parameter, noise figure, OIP3, and group delay values that result from analyzing the values stored in the default.s2p file at the frequencies stored in this file. These frequency values are also stored in the NetworkData property.

For a detailed explanation of how the analyze method calculates the network parameters, noise figure values, and OIP3 values for a particular object, see the AnalyzedResult property on the reference page for that object.


text:
### Derivatives And Integrals Of Vector Functions

(requires JavaScript)

1. The figure shows a curve $C$ given by a vector function $\mathbf{r}\left(t\right)$.

1. Draw the vectors $\mathbf{r}\left(4.5\right)-\mathbf{r}\left(4\right)$, $\mathbf{r}\left(4.2\right)-\mathbf{r}\left(4\right)$, $\frac{\mathbf{r}\left(4.5\right)-\mathbf{r}\left(4\right)}{0.5}$, and $\frac{\mathbf{r}\left(4.2\right)-\mathbf{r}\left(4\right)}{0.2}$.
2. Write expressions for $\mathbf{r}\prime \left(4\right)$ and the unit tangent vector $\mathbf{T}\left(4\right)$. Draw the vector $\mathbf{T}\left(4\right)$.

2. Make a large sketch of the curve described by the vector function $\mathbf{r}\left(t\right)=⟨{t}^{2},t⟩,\phantom{\rule{0.5em}{0ex}}0\le t\le 2$, and draw the vectors $\mathbf{r}\left(1\right)$, $\mathbf{r}\left(1.1\right)$, and $\mathbf{r}\left(1.1\right)-\mathbf{r}\left(1\right)$. Moreover, draw the vector $\mathbf{r}\prime \left(1\right)$ starting at $\left(1,1\right)$ and compare it with the vector

$\frac{\mathbf{r}\left(1.1\right)-\mathbf{r}\left(1\right)}{0.1}$

Explain why these vectors are similar.

3. Let $\mathbf{r}\left(t\right)=\left(1+\mathrm{cos}\left(t\right)\right)\mathbf{i}+\left(2+\mathrm{sin}\left(t\right)\right)\mathbf{j}$ and $t=\frac{\pi }{6}$, sketch the plane curve with the given vector equation, find $\mathbf{r}\prime \left(t\right)$, and sketch the position vector $\mathbf{r}\left(t\right)$ and the tangent vector $\mathbf{r}\prime \left(t\right)$ for the given value of $t$.
4. Find the derivative of the vector function

$\mathbf{r}\left(t\right)={e}^{{t}^{2}}\mathbf{i}-\mathbf{j}+\mathrm{ln}\left(1+3t\right)\mathbf{k}$.

$\mathbf{r}\prime \left(t\right)=2t{e}^{{t}^{2}}\mathbf{i}+\frac{3}{1+3t}\mathbf{k}$
5. Find the unit tangent vector $\mathbf{T}\left(t\right)$ at the point where $t=0$ for the curve given by the equation

$\mathbf{r}\left(t\right)=\mathrm{cos}\left(t\right)\mathbf{i}+3t\mathbf{j}+2\mathrm{sin}\left(2t\right)\mathbf{k}$.

$\frac{3}{5}\mathbf{j}+\frac{4}{5}\mathbf{k}$
6. Find parametric equations for the tangent line to the curve

$x=\mathrm{ln}\left(t\right),\phantom{\rule{0.5em}{0ex}}y=2\sqrt{t},\phantom{\rule{0.5em}{0ex}}z={t}^{2}$

at the point $\left(0,2,1\right)$.

$x=t$, $y=2+t$, $z=1+2t$
7. At what point do the curves

${\mathbf{r}}_{1}\left(t\right)=⟨t,1-t,3+{t}^{2}⟩$

and

${\mathbf{r}}_{2}\left(s\right)=⟨3-s,s-2,{s}^{2}⟩$

intersect? Find the angle of intersection to the nearest degree.

$⟨1,0,4⟩$, the angle is approx. $54.74°$
8. Evaluate the integral

${\int }_{0}^{\pi /2}\left(3{\mathrm{sin}}^{2}\left(t\right)\mathrm{cos}\left(t\right)\mathbf{i}+3\mathrm{sin}\left(t\right){\mathrm{cos}}^{2}\left(t\right)\mathbf{j}+2\mathrm{sin}\left(t\right)\mathrm{cos}\left(t\right)\mathbf{k}\right)\phantom{\rule{0.2em}{0ex}}dt$

$\mathbf{i}+\mathbf{j}+\mathbf{k}$
9. Find $\mathbf{r}\left(t\right)$ if $\mathbf{r}\prime \left(t\right)=t\mathbf{i}+{e}^{t}\mathbf{j}+t{e}^{t}\mathbf{k}$ and $\mathbf{r}\left(0\right)=\mathbf{i}+\mathbf{j}+\mathbf{k}$.
$\mathbf{r}\left(t\right)=⟨\frac{{t}^{2}}{2}+1,{e}^{t},t{e}^{t}-{e}^{t}+2⟩$


text:
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# If m and n are positive integers, is root(m)^n an integer?

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If m and n are positive integers, is root(m)^n an integer? [#permalink]

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09 Jul 2012, 04:56
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If m and n are positive integers, is $$(\sqrt{m})^n$$ an integer?

(1) $$\sqrt{m}$$ is an integer.
(2) $$\sqrt{n}$$ is an integer.

Diagnostic Test
Question: 44
Page: 26
Difficulty: 600
[Reveal] Spoiler: OA

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Re: If m and n are positive integers, is root(m)^n an integer? [#permalink]

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SOLUTION

If m and n are positive integers, is $$(\sqrt{m})^n$$ an integer?

(1) $$\sqrt{m}$$ is an integer --> $$(\sqrt{m})^n=(integer)^{positive \ integer}=integer$$. Sufficient.

(2) $$\sqrt{n}$$ is an integer. If $$m=n=1$$, then the answer is YES but if $$m=2$$ and $$n=1$$, then $$(\sqrt{m})^n=\sqrt{2}\neq{integer}$$, so in this case the answer is NO. Not sufficient.

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Re: If m and n are positive integers, is root(m)^n an integer? [#permalink]

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09 Jul 2012, 05:27
Bunuel wrote:
If m and n are positive integers, is $$(\sqrt{m})^n$$ an integer?

(1) $$\sqrt{m}$$ is an integer.
(2) $$\sqrt{n}$$ is an integer.

Hi,

Difficulty level: 600

We have to find whether $$(\sqrt{m})^n$$ is an integer,
or $$m^{\frac n2}$$ is an integer.

Given, m & n are integers,

Using (1),
$$\sqrt{m}$$ is an integer
then, $$(integer)^{(integer)}$$ is Integer. Sufficient.

Using (2),
$$\sqrt{n}$$ is an integer, or n is an integer.
lets say, m=3, n=9,
$$(\sqrt{m})^n$$=$$(\sqrt{3})^9$$ which is not an integer. Insufficient.

Regards,
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Re: If m and n are positive integers, is root(m)^n an integer? [#permalink]

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13 Jul 2012, 04:09
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SOLUTION

If m and n are positive integers, is $$(\sqrt{m})^n$$ an integer?

(1) $$\sqrt{m}$$ is an integer --> $$(\sqrt{m})^n=(integer)^{positive \ integer}=integer$$. Sufficient.

(2) $$\sqrt{n}$$ is an integer. If $$m=n=1$$, then the answer is YES but if $$m=2$$ and $$n=1$$, then $$(\sqrt{m})^n=\sqrt{2}\neq{integer}$$, so in this case the answer is NO. Not sufficient.

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text:
## University Calculus: Early Transcendentals (3rd Edition)

The sum of a geometric series can be found as: $S=\dfrac{a}{1-r}$ Here, $a=1, r=-3$, and $|-3| \gt 1$; so the series diverges.


text:
Distribution (or transformation) for data with heavy lower tail and light upper tail?

The data is maximum yield in pulse amplitude modulated fluorometry $$Y(\text{II})$$ or $$\phi_{\text{II}}$$, so the support is $$x\in[0,1]$$. Below are some qqnorm plots for the data and some ML distributions fit with fitdistrplus. I choose Beta and Gamma distributions based on the output of descdist. Gamma seems interesting given the tail behavior is similar (low tail lighter than normal , upper tail heavier), but still far from being compatible. After reading about Lambert W transformation and the LambertW package, I was able to 'gaussianize' this data using MLE_LambertW with type = 's'. My question is whether there are any other distributions/transformations better suited for this data? The maximum yield is one variable in a multivariate GLMM using the MCMCglmm package. While this transformation seems sound (and reduces my analysis to simple MANOVA if I end up disregarding random effects), I'm curious if there is anyway to model this distribution directly?

EDIT: Better is arbitrary of course, but for the case of most model fitting I know, a transformation is either to induce better compatibility with a normality assumption or to stabilize mean-variance relationship. Maybe I'm wrong, but it seems inappropriate to use data that is clearly not normal (and not even trivially so) for a classical MANOVA. A generalized linear model would allow for the specification of the residual distribution for each response. So my question is how might you model such a distribution?

• Can you expand on what criteria you are uisng to decide on whether one transformation is better than another and for what purpose you are doing this? Nov 30 '16 at 16:59
• If your data is in $[0,1]$ then why not take a logit transformation and go from there ( using Lambert W x Gaussian trafo on transformed data if you need to really transform it)? Since you are already in a glm setting thenwhy not model your outcome via logistic link function in a logistic or beta regression ? Dec 24 '16 at 14:44
• That was the answer I was potentially looking for, though wasn't sure if that is a common practice. The data is not particularly compatible with a beta distribution from my tests with fitdistrplus, but logit might be promising. How robust are GLMs in general to discrepancy in the assumed distribution? Thanks. Dec 24 '16 at 20:34
• Is this variable the response or one of the predictor variables for sthg else? If the response variable, then a marginal fit on it does not have to be Beta, just conditionally on X matrix. If a predictor variable, then throw in all transformations you can think of and do variable selection ( LASSO, etc). Dec 28 '16 at 1:44
• It's a response variable, and the kurtosis is similarly represented in the conditionals at the level of two factor interaction (the main exposure variables), but there are ultimately 3 to 4 total factors. But at that point n = 5-10, so maybe it doesn't matter anyway at such small sample size. I'll check again, do some fitting, etc. Beta does seem more statisitically sound than binomial, since this isn't count data but normalized change in fluorescent flux under two modes of irradiation. Dec 28 '16 at 2:37