Harm Derksen

Northeastern University | NEU · Department of Mathematics

Drug discovery often involves targeting specific members within a family of similar proteins. For example, pyruvate dehydrogenase kinase (PDHK) exists as four isozymes, which exhibit varying expression patterns across multiple tissues. Different PDHK isozymes have been implicated in conditions such as cancer, heart failure, and diabetes, suggesting...

The quick Sequential Organ Failure Assessment (qSOFA) system identifies an individual’s risk to progress to poor sepsis-related outcomes using minimal variables. We used Support Vector Machine, Learning Using Concave and Convex Kernels, and Random Forest to predict an increase in qSOFA score using electronic health record (EHR) data, electrocardiog...

The Anantharam-Jog-Nair inequality [AJN22] in Information Theory provides a unifying approach to the information-theoretic form of the Brascamp-Lieb inequality [CCE09] and the Entropy Power inequality [ZF93]. In this paper, we use methods from Quiver Invariant Theory [CD21] to study Anantharam-Jog-Nair inequalities with integral exponents. For such...

Over the past decades, there has been an increase of attention to adapting machine learning methods to fully exploit the higher order structure of tensorial data. One problem of great interest is tensor classification, and in particular the extension of linear discriminant analysis to the multilinear setting. We propose a novel method for multiline...

Tensors in the form of multilinear arrays are ubiquitous in data science applications. Captured real-world data, including video, hyperspectral images, and discretized physical systems, naturally occur as tensors and often come with attendant noise. Under the additive noise model and with the assumption that the underlying clean tensor has low rank...

We develop a representation theoretic technique for detecting closed orbits that is applicable in all characteristics. Our technique is based on Kempf’s theory of optimal subgroups and we make some improvements and simplify the theory from a computational perspective. We exhibit our technique in many examples and in particular, give an algorithm to...

Images from cameras are a common source of navigation information for a variety of vehicles. Such navigation often requires the matching of observed objects (e.g., landmarks, beacons, stars) in an image to a catalog (or map) of known objects. In many cases, this matching problem is made easier through the use of invariants. However, if the objects...

A model's interpretability is essential to many practical applications such as clinical decision support systems. In this paper, a novel interpretable machine learning method is presented, which can model the relationship between input variables and responses in humanly understandable rules. The method is built by applying tropical geometry to fuzz...

In this paper, we study sample size thresholds for maximum likelihood estimation for tensor normal models. Given the model parameters and the number of samples, we determine whether, almost surely, (1) the likelihood function is bounded from above, (2) maximum likelihood estimates (MLEs) exist, and (3) MLEs exist uniquely. We obtain a complete answ...

Since the seminal works of Strassen and Valiant it has been a central theme in algebraic complexity theory to understand the relative complexity of algebraic problems, that is, to understand which algebraic problems (be it bilinear maps like matrix multiplication in Strassen's work, or the determinant and permanent polynomials in Valiant's) can be...

A model's interpretability is essential to many practical applications such as clinical decision support systems. In this paper, a novel interpretable machine learning method is presented, which can model the relationship between input variables and responses in humanly understandable rules. The method is built by applying tropical geometry to fuzz...

Let V be an n-dimensional algebraic representation over an algebraically closed field K of a group G. For m > 0, we study the invariant rings K[Vm]G for the diagonal action of G on Vm. In characteristic zero, a theorem of Weyl tells us that we can obtain all the invariants in K[Vm]G by the process of polarization and restitution from K[Vⁿ]G. In par...

It is often necessary to identify a pattern of observed craters in a single image of the lunar surface and without any prior knowledge of the camera’s location. This so-called “lost-in-space” crater identification problem is common in both crater-based terrain relative navigation (TRN) and in automatic registration of scientific imagery. Past work...

This paper solves the two-sided version and provides a counterexample to the general version of the 2003 conjecture by Hadwin and Larson. Consider evaluations of linear matrix pencils $L=T_0+x_1T_1+\cdots+x_mT_m$ on matrix tuples as $L(X_1,\dots,X_m)=I\otimes T_0+X_1\otimes T_1+\cdots+X_m\otimes T_m$. It is shown that ranks of linear matrix pencils...

We develop a representation theoretic technique for detecting closed orbits that is applicable in all characteristics. Our technique is based on Kempf's theory of optimal subgroups and we make some improvements and simplify the theory from a computational perspective. We exhibit our technique in many examples and in particular, give an algorithm to...

Let $Q$ be a bipartite quiver, $V$ a real representation of $Q$, and $\sigma $ an integral weight of $Q$ orthogonal to the dimension vector of $V$. Guided by quiver invariant theoretic considerations, we introduce the Brascamp–Lieb (BL) operator $T_{V,\sigma }$ associated to $(V,\sigma )$ and study its capacity, denoted by $\textbf{D}_Q(V, \sigma )...

Acute respiratory distress syndrome (ARDS) is a life-threatening lung injury with global prevalence and high mortality. Chest x-rays (CXR) are critical in the early diagnosis and treatment of ARDS. However, imaging findings may not result in proper identification of ARDS due to a number of reasons, including nonspecific appearance of radiological f...

Patients recovering from cardiovascular surgeries may develop life-threatening complications such as hemodynamic decompensation, making the monitoring of patients for such complications an essential component of postoperative care. However, this need has given rise to an inexorable increase in the number and modalities of data points collected, mak...

This paper introduces a novel method for classifying and predicting cardiac arrhythmia events via a special type of deterministic probabilistic finite-state automata (DPFA). The proposed method constructs the underlying state space and transition probabilities of the DPFA model directly from the input data. The algorithm was employed in the predict...

In this paper, we study sample size thresholds for maximum likelihood estimation for tensor normal models. Given the model parameters and the number of samples, we determine whether, almost surely, (1) the likelihood function is bounded from above, (2) maximum likelihood estimates (MLEs) exist, and (3) MLEs exist uniquely. We obtain a complete answ...

Background This study outlines an image processing algorithm for accurate and consistent lung segmentation in chest radiographs of critically ill adults and children typically obscured by medical equipment. In particular, this work focuses on applications in analysis of acute respiratory distress syndrome – a critical illness with a mortality rate...

It is often necessary to identify a pattern of observed craters in a single image of the lunar surface and without any prior knowledge of the camera's location. This so-called "lost-in-space" crater identification problem is common in both crater-based terrain relative navigation (TRN) and in automatic registration of scientific imagery. Past work...

In this paper, we study the log-likelihood function and Maximum Likelihood Estimate (MLE) for the matrix normal model for both real and complex models. We describe the exact number of samples needed to achieve (almost surely) three conditions, namely a bounded log-likelihood function, existence of MLEs, and uniqueness of MLEs. As a consequence, we...

Using the Grosshans Principle, we develop a method for proving lower bounds for the maximal degree of a system of generators of an invariant ring. This method also gives lower bounds for the maximal degree of a set of invariants that define Hilbert's null cone. We consider two actions: The first is the action of SL(V) on S3(V)⊕4, the space of 4-tup...

We develop algebraic methods for computations with tensor data. We give 3 applications: extracting features that are invariant under the orthogonal symmetries in each of the modes, approximation of the tensor spectral norm, and amplification of low rank tensor structure. We introduce colored Brauer diagrams, which are used for algebraic computation...

Predicting the interactions between drugs and targets plays an important role in the process of new drug discovery, drug repurposing (also known as drug repositioning). There is a need to develop novel and efficient prediction approaches in order to avoid the costly and laborious process of determining drug-target interactions (DTIs) based on exper...

We introduce the $G$-stable rank of a higher order tensors over perfect fields. The $G$-stable rank is related to the Hilbert-Mumford criterion for stability in Geometric Invariant Theory. We will relate the $G$-stable rank to the tensor rank and slice rank. For numerical applications, we express the $G$-stable rank as a solution to an optimization...

The regularity lemma is a stringent condition of the possible ranks of tensor blow-ups of linear subspaces of matrices. It was proved by Ivanyos, Qiao and Subrahmanyam in [5 Ivanyos, G., Qiao, Y., Subrahmanyam, K. V. (2017). Non-commutative Edmonds’ problem and matrix semi-invariants. Comput. Complex. Complexity 26(3):717–763. DOI: 10.1007/s00037-0...

We study the category of wheeled PROPs using tools from Invariant Theory. A typical example of a wheeled PROP is the mixed tensor algebra ${\mathcal V}=T(V)\otimes T(V^\star)$, where $T(V)$ is the tensor algebra on an $n$-dimensional vector space over a field of $K$ of characteristic 0. First we classify all the ideals of the initial object ${\math...

The advent of portable cardiac monitoring devices has enabled real-time analysis of cardiac signals. These devices can be used to develop algorithms for real-time detection of dangerous heart rhythms such as ventricular arrhythmias. This paper presents a Markov model based algorithm for real-time detection of ventricular tachycardia, ventricular fl...

Tensor methods for the analysis of multi-dimensional, multi-modal biomedical signals.

Let $Q$ be a bipartite quiver, $V$ a real representation of $Q$, and $\sigma$ an integral weight of $Q$ orthogonal to the dimension vector of $V$. In this paper, we introduce the Brascamp-Lieb operator $T_{V,\sigma}$ associated to $(V,\sigma)$ and study its capacity, denoted by $\mathbf{D}_Q(V,\sigma)$. Using methods and ideas from quiver invariant...

Fibromyalgia is a medical condition characterized by widespread muscle pain and tenderness and is often accompanied by fatigue and alteration in sleep, mood, and memory. Poor sleep quality and fatigue, as prominent characteristics of fibromyalgia, have a direct impact on patient behavior and quality of life. As such, the detection of extreme cases...

Abstract: With the recent advances in information technology, high-dimensional datasets are commonplace. When working in higher-dimensional spaces a set of new challenges arise, known as the "curse of dimensionality". The curse, not only dictates that computational complexity explodes in memory and time, but also that the problem of noise and missi...

Using the Grosshans Principle, we develop a method for proving lower bounds for the maximal degree of a system of generators of an invariant ring. This method also gives lower bounds for the maximal degree of a set of invariants that define Hilbert's null cone. We consider two actions: The first is the action of ${\rm SL}(V)$ on ${\rm Sym}^3(V)^{\o...

The regularity lemma is a stringent condition of the possible ranks of tensor blow-ups of linear subspaces of matrices. It was proved by Ivanyos, Qiao and Subrahmanyam when the underlying field is sufficiently large. We show that if the field size is too small, the regularity lemma is false.

Introduction: Cardiovascular magnetic resonance (CMR) imaging is currently the gold-standard to analyze cardiac morphology and evaluate global and regional left ventricle (LV) function. Quantitative parameters such as ejection fraction (EF) and LV mass are important indicators for diagnosis and associated with morbidity and mortality. In practice,...

Atrial Fibrillation (AFib) is by itself a strong risk factor for many life-threatening heart diseases. An estimated 2.7 to 6.1 million people in the United States have AFib. With the aging of the U.S. population, this number is expected to increase. In this preliminary study, a heart rate-duration criteria region is proposed to automatically label...

For tensors in , Landsberg provides non-trivial equations for tensors of border rank 2d−3 for d even and 2d−5 for d odd in Landsberg [Non-triviality of equations and explicit tensors in of border rank at least 2m−2. J Pure Appl Algebra. 2015;219(8):3677–3684]. In Derksen and Makam [On non-commutative rank and tensor rank, Linear Multilinear Algebra...

A geometric measure for the entanglement of a tensor $T \in (\mathbb{C}^n)^{\otimes k}$ is given by $- 2 \log_2 ||T||_\sigma$, where $||.||_\sigma$ denotes the spectral norm. A simple induction gives an upper bound of $(k-1) \log_2(n)$ for the entanglement. We show the existence of tensors with entanglement larger than $k \log_2(n) - \log_2(k) - o(...

Consider the ring of invariants $R(n,m)$ for the left-right action of ${\rm SL}_n \times {\rm SL}_n$ on $m$-tuples of $n \times n$ matrices. It was recently proved that the ring $R(n,m)$ is generated by invariants of degrees $\leq n^6$ in characteristic $0$. It is known that an upper bound on the degree of generators that is independent of $m$ cann...

We consider two group actions on $m$-tuples of $n \times n$ matrices. The first is simultaneous conjugation by $\operatorname{GL}_n$ and the second is the left-right action of $\operatorname{SL}_n \times \operatorname{SL}_n$. We give efficient algorithms to decide if the orbit closures of two points intersect. We also improve the known bounds for t...

For tensors in $\mathbb{C}^d \otimes \mathbb{C}^d \otimes \mathbb{C}^d$, Landsberg provides non-trivial equations for tensors of border rank $2d-3$ for $d$ even and $2d-5$ for $d$ odd were found by Landsberg. In previous work, we observe that Landsberg's method can be interpreted in the language of tensor blow-ups of matrix spaces, and using concav...

We study the Pareto frontier for two competing norms $\|\cdot\|_X$ and $\|\cdot\|_Y$ on a vector space. For a given vector $c$, the pareto frontier describes the possible values of $(\|a\|_X,\|b\|_Y)$ for a decomposition $c=a+b$. The singular value decomposition of a matrix is closely related to the Pareto frontier for the spectral and nuclear norm...

We show that the two notions of entanglement: the maximum of the geometric measure of entanglement and the maximum of the nuclear norm is attained for the same states. We affirm the conjecture of Higuchi-Sudberry on the maximum entangled state of four qubits. We introduce the notion of d-density tensor for mixed d-partite states. We show that d-den...

Overweight and obesity are highly prevalent in the population of the United States, affecting roughly 2/3 of Americans. These diseases, along with their associated conditions, are a major burden on the healthcare industry in terms of both dollars spent and effort expended. Volitional weight loss is attempted by many, but weight regain is common. Th...

Given an algebraic $\mathbf{Z}^{d}$ -action corresponding to a prime ideal of a Laurent ring of polynomials in several variables, we show how to find the smallest order $n+1$ of non-mixing. It is known that this is determined by the non-mixing sets of size $n+1$ , and we show how to find these in an effective way. When the underlying characteristic...

We study the left-right action of SL_n×SL_n on m-tuples of n×n matrices with entries in an infinite field K. We show that invariants of degree n^2−n define the null cone. Consequently, invariants of degree ≤n^6 generate the ring of invariants if char(K)=0. We also prove that for m≫0, invariants of degree at least n[sqrt(n+1)] are required to define...

Let $P_3(\mathbf{C}^{\infty})$ be the space of complex cubic polynomials in infinitely many variables. We show that this space is $\mathbf{GL}_{\infty}$-noetherian, meaning that any $\mathbf{GL}_{\infty}$-stable Zariski closed subset is cut out by finitely many orbits of equations. Our method relies on a careful analysis of an invariant of cubics i...

It is well known that the ring of polynomial invariants of a reductive group is finitely generated. However, it is difficult to give strong upper bounds on the degrees of the generators, especially over fields of positive characteristic. In this paper, we make use of the theory of good filtrations along with recent results on the null cone to provi...

We study the relationship between the commutative and the non-commutative rank of a linear matrix. We give examples that show that the ratio of the two ranks comes arbitrarily close to 2. Such examples can be used for giving lower bounds for the border rank of a given tensor. Landsberg used such techniques to give nontrivial equations for the tenso...

Abstract: Due to the ill-posed nature of image denoising problem, good image priors are of great importance for an effective restoration. Non-local self-similarity and sparsity are two popular and widely used image priors which have led to several state-of-the-art methods in natural image denoising. In this paper, we take advantage of these priors...

We use recent results on matrix semi-invariants to give degree bounds on generators for the ring of semi-invariants for quivers with no oriented cycles.

Advanced hemodynamic monitoring is a critical component of treatment in clinical situations where aggressive yet guided hemodynamic interventions are required in order to stabilize the patient and optimize outcomes. While there are many tools at a physician's disposal to monitor patients in a hospital setting, the reality is that none of these tool...

We study the left-right action of $\operatorname{SL}_n \times \operatorname{SL}_n$ on $m$-tuples of $n \times n$ matrices with entries in an infinite field $K$. We show that invariants of degree $n^2- n$ define the null cone. Consequently, invariants of degree $\leq n^6$ generate the ring of invariants if $\operatorname{char}(K)=0$. We also prove t...

Introduction: Hemorrhage (Hem) is one of the leading causes of mortality from trauma, and continuous monitoring for Hem using techniques such as traditional BP, especially in field conditions, poses significant challenges. We have built a portable low-powered noninvasive sensor for continuous vascular tone monitoring to detect significant Hem as an...

Coronary artery disease (CAD) is one of the major causes of death worldwide. Today X-ray angiography is a standard method for CAD diagnosis. Usually, the quality of these images is not good enough. Noise, camera and heart motions, non-uniform illumination and even the presence of catheter are sources of quality degradation. The existence of cathete...

For few decades digital X-ray imaging has been one of the most important tools for medical diagnosis. With the advent of distance medicine and the use of big data in this respect, the need for efficient storage and online transmission of these images is becoming an essential feature. Limited storage space and limited transmission bandwidth are the...

We first propose what we call the Gaussian Moments Conjecture. We then show that the Jacobian Conjecture follows from the Gaussian Moments Conjecture. Note that the the Gaussian Moments Conjecture is a special case of [11, Conjecture 3.2]. The latter conjecture was referred to as the Moment Vanishing Conjecture in [7, Conjecture A] and the Integral...

We give a lower bound for the Waring rank and cactus rank of forms that are invariant under an action of a connected algebraic group. We use this to improve the Ranestad--Schreyer--Shafiei lower bounds for the Waring ranks and cactus ranks of determinants of generic matrices, Pfaffians of generic skew-symmetric matrices, and determinants of generic...

Let Λ be a finite dimensional algebra over an algebraically closed field. We exhibit slices of the representation theory of Λ that are always classifiable in stringent geometric terms. Namely, we prove that, for any semisimple object T ∈ Λ-mod, the class of those Λ-modules with fixed dimension vector (say d) and top T which do not permit any proper...

The Rank Minimization Problem asks to find a matrix of lowest rank inside a linear variety of the space of n x n matrices. The Low Rank Matrix Completion problem asks to complete a partially filled matrix such that the resulting matrix has smallest possible rank. The Tensor Rank Problem asks to determine the rank of a tensor. We show that these thr...

Finding the rank of a tensor is a problem that has many applications. Unfortunately it is often very difficult to determine the rank of a given tensor. Inspired by the heuristics of convex relaxation, we consider the nuclear norm instead of the rank of a tensor. We determine the nuclear norm of various tensors of interest. Along the way, we also do...

In this paper, we show that the low rank matrix completion problem can be reduced to the problem of finding the rank of a certain tensor.

Kruskal proved that a tensor in V1⊗V2⊗⋯⊗Vm of rank r has a unique decomposition as a sum of r pure tensors if a certain inequality is satisfied. We will show the uniqueness fails if the inequality is weakened. We give two different constructions for counterexamples.

http://deepblue.lib.umich.edu/bitstream/2027.42/135647/1/plms1045.pdf

It is unknown whether two graphs can be tested for isomorphism in polynomial time. A classical approach to the Graph Isomorphism Problem is the d-dimensional Weisfeiler-Lehman algorithm. The d-dimensional WL-algorithm can distinguish many pairs of graphs, but the pairs of non-isomorphic graphs constructed by Cai, Furer and Immerman it cannot distin...

Many important invariants for matroids and polymatroids, such as the Tutte polynomial, the Billera–Jia–Reiner quasi-symmetric function, and the invariant G introduced by the first author, are valuative. In this paper we construct the Z-modules of all Z-valued valuative functions for labeled matroids and polymatroids on a fixed ground set, and their...

Let u1, . . ., um be linear recurrences with values in a field K of positive characteristic p. We show that the set of integer vectors (k1, . . ., km) such that u1(k1)+⋯+um(km)=0 is p-normal in a natural sense generalizing that of the first author, who proved the result for m=1. Furthermore the set is effectively computable if K is. We illustrate t...

We continue the study of quivers with potentials and their representations initiated in the first paper of the series. Here we develop some applications of this theory to cluster algebras. As shown in the ``Cluster algebras IV'' paper, the cluster algebra structure is to a large extent controlled by a family of integer vectors called {g} -vectors,...

Algebraic actions of unipotent groups $U$ actions on affine $k-$varieties $X$ ($k$ an algebraically closed field of characteristic 0) for which the algebraic quotient $X//U$ has small dimension are considered$.$ In case $X$ is factorial, $O(X)^{\ast}=k^{\ast},$ and $X//U$ is one-dimensional, it is shown that $O(X)^{U}$=$k[f]$, and if some point in...

International audience Many important invariants for matroids and polymatroids, such as the Tutte polynomial, the Billera-Jia-Reiner quasi-symmetric function, and the invariant $\mathcal{G}$ introduced by the first author, are valuative. In this paper we construct the $\mathbb{Z}$-modules of all $\mathbb{Z}$-valued valuative functions for labelled...

For any finite dimensional basic associative algebra, we study the presentation spaces and their relation to the representation spaces. We prove two propositions about a general presentation, one on its subrepresentations and the other on its canonical decomposition. As a special case, we consider rigid presentations. We show how to complete a rigi...

Recently many scientific and engineering applications have involved the challenging task of analyzing large amounts of unsorted high-dimensional data that have very compli-cated structures. From both geometric and statistical points of view, such unsorted data are considered mixed as different parts of the data have significantly different structur...

Let $A$ be the polynomial ring over $k$ (a field of characteristic zero) in $n+1$ variables. The commuting derivations conjecture states that $n$ commuting locally nilpotent derivations on $A$, linearly independent over $A$, must satisfy $A^{D_1,...,D_m}=k[f]$ where $f$ is a coordinate. The conjecture can be formulated as stating that a $(G_m)^n$-a...

To a directed graph without loops and 2-cycles, we can associate a skew-symmetric matrix with integer entries. Mutations of such skew-symmetric matrices, and more generally skew-symmetrizable matrices, have been defined in the context of cluster algebras by Fomin and Zelevinsky. The mutation class of a graph G is the set of all isomorphism classes...

Let G be an affine algebraic group acting on an affine variety X. We present an algorithm for computing generators of the invariant ring KG[X] in the case where G is reductive. Furthermore, we address the case where G is connected and unipotent, so the invariant ring need not be finitely generated. For this case, we develop an algorithm which compu...

To every subspace arrangement X we will associate symmetric functions ℘[X] and ℋ[X]. These symmetric functions encode the Hilbert series and the minimal projective resolution of the product ideal associated to the subspace arrangement. They can be defined for discrete polymatroids as well. The invariant ℋ[X] specializes to the Tutte polynomial \({\...

We give counterexamples to Okounkov’s log-concavity conjecture for Littlewood–Richardson coefficients.

In this paper, based on ideas from lossy data coding and compression, we present a simple but effective technique for segmenting multivariate mixed data that are drawn from a mixture of Gaussian distributions, which are allowed to be almost degenerate. The goal is to find the optimal segmentation that minimizes the overall coding length of the segm...

We study quivers with relations given by non-commutative analogs of Jacobian ideals in the complete path algebra. This framework allows us to give a representation-theoretic interpretation of quiver mutations at arbitrary vertices. This gives a far-reaching generalization of Bernstein-Gelfand-Ponomarev reflection functors. The motivations for this...

The vanishing ideal I of a subspace arrangement V1∪V2∪⋯∪Vm⊆V is an intersection I1∩I2∩⋯∩Im of linear ideals. We give a formula for the Hilbert polynomial of I if the subspaces meet transversally. We also give a formula for the Hilbert series of the product ideal J=I1I2⋯Im without any assumptions about the subspace arrangement. It turns out that the...

In this paper, based on ideas from lossy data coding and compression, we present a simple but surprisingly effective technique for segmenting multivariate mixed data that are drawn from a mixture of Gaussian distributions or linear subspaces. The goal is to find the optimal segmentation that minimizes the overall coding length of the segmented data...

We consider the problem of simultaneously segmenting data samples drawn from multiple linear subspaces and estimating model parameters for those subspaces. This "subspace segmentation" problem naturally arises in many computer vision applications such as motion and video segmentation, and in the recognition of human faces, textures, and range data....

We give counterexamples to Okounkov's log-concavity conjecture for Littlewood-Richardson coefficients.

We give a description of faces of all codimensions for the cones of weights of rings of semi-invariants of quivers. For a triple flag quiver and faces of codimension 1 this reduces to the result of Knutson-Tao-Woodward on the facets of the Klyachko cone. We give new applications to Littlewood-Richardson coefficients, including a product formula for...

Lech proved in 1953 that the set of zeroes of a linear recurrence sequence in a field of characteristic 0 is the union of a finite set and finitely many infinite arithmetic progressions. This result is known as the Skolem-Mahler-Lech theorem. Lech gave a counterexample to a similar statement in positive characteristic. We will present some more pat...

It is well known that the intersection multiplicities of Schubert classes in the Grassmannian are Littlewood–Richardson coefficients. We generalize this statement in the context of quiver representations. Here the intersection multiplicity of Schubert classes is replaced by the number of subrepresentations of a general quiver representation, and th...