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$$\begin{array}{@{}rcl@{}} f^{\prime}(x, y, t) = \sum_{(i,j,k) \in \Gamma_{f}^{(p)}}f_{{ijk}}^{\prime}x^{i}y^{j}t^{k}, \\ s^{\prime}(x,y,t) = \sum_{(i,j,k) \in \Gamma_{s_{1}}^{(p)}}s_{{ijk}}^{\prime}x^{i}y^{j}t^{k}, \\ r^{\prime}(x,y,t) = \sum_{(i,j,k) \in \Gamma_{r_{1}}^{(p)}}r_{{ijk}}^{\prime}x^{i}y^{j}t^{k}, \\ m^{\prime}(x,y,t) = \sum_{(i,j,k) \in \Gamma_{m}^{(p)}}m_{{ijk}}^{\prime}x^{i}y^{j}t^{k}, \end{array} $$
$$F_{1} - F_{2} = (s_{1} - s_{2})f + (r_{1}-r_{2})X = s^{\prime}f^{\prime} + r^{\prime}X, $$

Since X is made public, one can try to divide F 1 F 2 by X to find f in the remainder. But f does not appear in the remainder because of ( 2 ). For this attack, see also [ 31 ].

$$g(x,y,t) = \sum_{(i,j,k) \in \Gamma_{g}^{(p)}}g_{{ijk}}x^{i}y^{j}t^{k}, $$
$$ g(x_{\ell}, y_{\ell}, t_{\ell}) = F(x_{\ell}, y_{\ell}, t_{\ell}) (\ell = 1,\ldots,L). $$

Then one can find f by factorization and get m as in section Womens EmbellishedHeel Metallic Leather Sandals Miu Miu JLnXQ
. However, one cannot determine f and m uniquely. If \(g_{0}(x,y,t) \in \mathbb {F}_{p}[x,y,t]\) satisfies ( Sneakers for Women White Leather 2017 45 5 65 7 85 Santoni Sneakers for Women VObTJ
), then g 0 + r X also satisfies ( 3 ) and has the same form as g for any polynomial \(r(x,y,t) \in \mathbb {F}_{p}[x,y,t]\) having the same form as f . In [ 1 ], it is pointed out that if \(p^{\# \Gamma _{r}^{(p)}} = p^{\# \Gamma _{f}^{(p)}} > 2^{100}\) , then we may avoid this attack.

$$(F_{1}+z, F_{2}+z, X) + I_{1} = (m+z, f, X). $$

Choose a constant and an integer ≈ deg()· log/. Choose irreducible polynomials ,…, of degree ≈/ log such that \(\sum _{1 \leq i \leq n}\deg P_{i} > \deg _{t}m\) . Set =1.

Let \(K_{i} := \mathbb {F}_{p}[t]/(P_{i})\) .

Let \(F_{k}^{(P_{i})} := F_{k}\phantom {\dot {i}\!}\) (mod ) and \(X^{(P_{i})} := X\phantom {\dot {i}\!}\) (mod ). Compute \(Q(y) := \text {Res}_{x}(F_{1}^{(P_{i})} - F_{2}^{(P_{i})}, X^{(P_{i})}) \in K_{i}[y]\phantom {\dot {i}\!}\) , the resultant of \(F_{1}^{(P_{i})} - F_{2}^{(P_{i})}\phantom {\dot {i}\!}\) and \(X^{(P_{i})}\phantom {\dot {i}\!}\) with respect to .

Factor () and let () be an irreducible factor of highest degree.

Compute a Gröbner basis of the ideal \(J := (F_{1}^{(P_{i})} + z, F_{2}^{(P_{i})} + z, X^{(P_{i})}, Q_{0}) \subset K_{i}[x,y,z]\phantom {\dot {i}\!}\) with respect to the graded reverse lexicographical ordering.

$$NF_{J}(m^{\prime} + z) = 0, $$

If <, then replace by +1 and go back to step 2.

Recover from \(m^{(P_{i})}\phantom {\dot {i}\!}\) by using the Chinese Remainder Theorem.

$$\begin{array}{@{}rcl@{}} \Lambda_{f} := \{\underline{i} \in (\mathbb{Z}_{\geq 0})^{n} \mid f_{\underline{i}} \neq 0\}, \\ \Gamma_{f} := \{(\underline{i}, b_{\underline{i}}) \in \Lambda_{f} \times \mathbb{Z}_{>0} \mid 2^{b_{\underline{i}} - 1} \leq |f_{\underline{i}}| < 2^{b_{\underline{i}}}\}. \end{array} $$
$$\begin{array}{@{}rcl@{}} \Lambda_{f_{1}} := \{ (4,2,1), (2,1,0), (0,0,1), (0,0,0) \}, \\ \Gamma_{f_{1}} := \{ (4,2,1,3), (2,1,0,4), (0,0,1,3), (0,0,0,2) \}, \\ \Lambda_{f_{2}} := \{ (2,2,1), (1,2,0), (0,0,1), (0,0,0) \}, \\ \Gamma_{f_{2}} := \{ (2,2,1,4), (1,2,0,4), (0,0,1,3), (0,0,0,4) \}. \end{array} $$
$$\begin{array}{@{}rcl@{}} H(f) := \max\{|f_{\underline{i}}| \mid \underline{i} \in \Lambda_{f}\}. \end{array} $$

For a vector \(\underline {v} := (v_{1},\ldots,v_{n}) \in \mathbb {Q}^{n}\) , we denote by \(f({\underline {v}})\) the value of f at \(\underline {v}\) . For an integer d , we denote by \(\underline {v}/d\) the vector \(\left (\frac {v_{1}}{d},\ldots,\frac {v_{n}}{d}\right)\) . For each ideal \(J \subset \mathbb {Q}[\underline {x}]\) , each polynomial \(f \in \mathbb {Q}[\underline {x}]\) and each monomial ordering <, we denote by N F J ( f ) a normal form of f with respect to J and <. For a polynomial \(f \in \mathbb {Z}[\underline {x}]\) and an integer m , we denote by \(\overline {f}^{(m)}\) the polynomial f (mod m ) \(\in (\mathbb {Z}/m\mathbb {Z})[\underline {x}]\) .

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Cultivating connections between your program and its stakeholders.

Partners play an important role in sustainability in several ways: connecting you to greater resources or expertise,providing services if your program has to cut back, and advocating on behalf of your cause. Partners can also help rally the community around your program and its goals. They can range from business leaders and media representatives to organizations addressing similar issues and community members. When your program is threatened either politically or financially, your partners can be some of your greatest champions. Building awareness and capacity for sustainability requires a strategic approach and partnerships across sectors, including alliances between private and public organizations.

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Having the internal support and resources needed to effectively manage your program.

Organizational capacity encompasses a wide range of capabilities, knowledge, and resources. For example, having enough staff and strong leadership can make a big difference in accomplishing your program goals. Cultivating and strengthening your program’s internal support can also increase your program’s likelihood of long-term success.

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Assessing your program to inform planning and document results.

Evaluation helps keep your program on track with its goals and outcomes. If evaluation data shows that an activity or strategy isn’t working, you can correct your program’s course to become more effective.

Moreover, collecting data about your program’s successes and impact is a powerful tool for gaining support and funding. If your evaluation data shows that your program is making an important (or irreplaceable) impact, you can make a strong case for why your program needs to continue. Even in times of decreased funding, evaluation and monitoring data are key for the pursuit of new funding sources.

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Taking actions that adapt your program to ensure its ongoing effectiveness.

Circumstances change and sometimes your program needs to as well. The goal is not necessarily to sustain all of a program’s components over time, but rather to sustain the most effective components and their benefits to your target group. This requires flexibility, adaptation to changing conditions, and quality improvement within your program. By using your evaluation data and current evidence-base, you can ensure that your program effectively uses resources and continues having an impact.

So what did investors in 1940 rely on when making investment decisions? Mostly narratives and emotions. Many of them remembered the Kayano sneakers Grey Asics ET7An9xq
of the 1930s as highlighted in the incredible book The Great Depression: A Diary by Benjamin Roth. Examining the plot of U.S. stock drawdowns since 1871 illustrates why:

I bring up this point to reiterate that we shouldn’t take market history for granted, because most investors never had such a detailed history to rely on . This is why grabbing data from 50 years ago and making statements like, “If you had bought X…”, is all market fantasy. The investment world has changed so much in the last half century that many historical comparisons are useless.

I bring up this point to reiterate that we shouldn’t take market history for granted, because most investors never had such a detailed history to rely on

Even in the last few decades our research abilities have improved immensely. Jim O’Shaughnessy once told me that it took him a year working full time in the mid 1990s (using a custom built computer with 2 CD ROM drives) to finish all of the data crunching for What Works on Wall Street . With today’s computational resources it could be done in a few months or less. Even something as obvious now as the importance of global equity returns may not have been as apparent until Triumph of the Optimists was first published in 2002. The data is getting better and we are getting better at analyzing all of it.

The fact is, before modern technology, people simply didn’t have the information to make the same investment decisions as you and I. Our investment ancestors had to be more emotional in their approach because what other choice did they have? At least today we can say that we use data to make investment decisions. Whether or not we actuallyfollow the data and stick to our plan is another story altogether.

The Curse of Knowledge (It Will Never Be Easy)

You might be thinking, why does any of this matter to me? Because though our investment knowledge is a privilege, it can also be a curse. Remember, investing is a game that is based on the preferences and information of other participants. If the other participants have learned things from market history such as “The market goes up in the long run” or “Buy the dip”, then your job as an investor didn’t get any easier though you have more knowledge. If everyone knows to buy the dip, then they may use that knowledge to stay invested longer, possibly irrationally propping up markets. Obviously, this is conjecture on my part, but just because you know more doesn’t mean your investment journey is going to be easier.

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